# Emergent centrality in rank-based supplanting process

Kenji Shimomura1, Yasuhiro Ishitsuka2, and Hiroki Ohta3 1Center for
Gravitational Physics and Quantum Information,
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606–8502,
Japan
2Institute of Mathematics for Industry, Kyushu University, Fukuoka, 819–0395,
Japan
3Department of Human Sciences, Obihiro University of Agriculture and
Veterinary Medicine, Hokkaido, 080-8555, Japan

###### Abstract

We propose a stochastic process of interacting many agents, which is inspired
by rank-based supplanting dynamics commonly observed in a group of Japanese
macaques. In order to characterize the breaking of permutation symmetry with
respect to agents’ rank in the stochastic process, we introduce a rank-
dependent quantity, overlap centrality, which quantifies how often a given
agent overlaps with the other agents. We give a sufficient condition in a wide
class of the models such that overlap centrality shows perfect correlation in
terms of the agents’ rank in zero-supplanting limit. We also discuss a
singularity of the correlation in the case of interaction induced by a Potts
energy.

##  I Introduction

One of promising candidates for going a step further in studying a many-body
system is to construct a lattice model of interacting elements or agents which
describes a many-body system. This strategy is not only applied to equilibrium
systems Baxter  but also nonequilibrium systems Odor . One of such attentions
has been put on nonequilibrium lattice models such as driven lattice gas DLG ,
ASEP ASEP , ABC model ABC , zero range process ZRP1 ; ZRP2 , etc., where
emergent macroscopic property such as phase transition is one of the topics to
be elucidated. Recent development on active matter has focused on
experimentally realizable systems such as colloidal or biological systems
showing various phase transition such as flocking transition, lane formation,
or motility-induced phase separation Vicsek ; Lane1 ; Bacteria . In the
framework of statistical physics, it is of interest to look for an
analytically tractable and minimal model for such phenomena V-Solon ; Lane2 ;
Hydro .

Apart from such model-based studies, in the context of network theory, the
concept of centrality plays one of important roles in studying a given network
induced by a many-body system consisting of inhomogeneous agents. Centrality
has been particularly used in the literature of social network analysis to
characterize which element on a network is the most influential. Depending on
the purpose of network analysis, various measures of centrality such as degree
centrality, closeness centrality, PageRank, eigenvector centrality, etc., have
been proposed and found to be useful to characterize network structures
Freeman ; Bonacich ; Barrat ; Newman .

As an example of inhomogeneous agents, primate species often live through
interacting with members in a group primatebook . It has been reported in a
primate species that the individuals, which we call agents, with high rank in
social dominance tend to have high rank also in the eigenvector centrality of
adjacency matrix for a graph composed from agents positions pecentrality1 ;
pecentrality2 . In particular, Japanese macaques form a group living together
and each agent in a group has its rank along the linear social dominance in
the group, leading to rank-dependent repulsion between two agents supplanting
; Nakagawa . This is so-called supplanting phenomenon which we mainly focus
on.

In this paper, we focus on one of the intersections between lattice models and
the network theory from the viewpoint of inhomogeneous agents. We propose a
new type of nonequilibrium lattice model, which is inspired by a supplanting
phenomenon occurring between two agents in a group of Japanese macaques. The
main objective of this paper is to show that a new type of macroscopic
correlation appears when supplanting process, which is a rank-based
interaction with broken detailed balance, is added to an equilibrium system.
It turns out that this problem can be mapped to computing a type of
centrality, which we call overlap centrality, for a complete graph derived
naturally from the correlations of agents’ positions.

This paper consists of five sections. In Section II, we introduce a class of
models breaking permutation symmetry that we study in this paper. In Section
III, focusing on the case of the model where the interaction induced by the
Potts energy is assumed, we provide a brief review of the equilibrium
properties, introduce overlap centrality, and compute it by exact
diagonalization of transition matrix. In Section IV, we provide the proof for
the main result that overlap centrality characterizing how often a given agent
overlaps with the other agent shows perfect correlation with respect to the
ranking of the given agent in zero-supplanting limit. This result holds rather
generally, which is not limited to the case where the Potts energy is assumed
as the source of an equilibrium interaction. Further, a conjecture about the
existence of a singularity of the correlation is discussed for the case of the
Potts energy. In Section V as concluding remarks, we summarize the results and
some subjects of future considerations.

##  II Model

Let $N\geq 2$ be the number of agents, and $L\geq 3$ be the length of the one-
dimensional lattice
$X\coloneqq{\mathbb{Z}}/L{\mathbb{Z}}=\left\\{{0,1,\ldots,L-1}\right\\}$. Let
us denote by $i\in\\{1,2,\cdots,N\\}$ an agent, and by integer $x_{i}\in X$
the position of agent $i$. We also regard the number $i$ identifying an agent
as rank of that agent. We say that rank $i$ is higher (lower) than $j$ if
$i<j$ ($i>j$) ; for example, rank $i$ is higher than rank $i+1$. Let us also
write the collection of elements $a_{i}$ labelled by $1\leq i\leq n$ as
$(a_{i})_{i=1}^{n}$ or the bold symbol ${\bm{a}}$. In particular we write a
set of positions of agents as $\bm{x}=(x_{i})_{i=1}^{N}$. Hereafter, we call
${\bm{x}}=(x_{i})_{i=1}^{N}$ a configuration. We consider a hopping map
$f_{i}^{\pm}$ such that
$f_{i}^{\pm}\bm{x}\coloneqq(x_{j}\pm\delta(i,j))_{j=1}^{N}$, where
$\delta(i,j)$ is the Kronecker delta. Note that the periodic boundary
condition in terms of positions is automatically assumed by definition of $X$.

###  II.1 Equilibrium dynamics

We consider a general class of energy function
$E(\bm{x})=E(x_{1},x_{2},\dots,x_{N})$ which is permutation symmetric in the
following sense:

$E(x_{1},x_{2},\dots,x_{N})=E(x_{\sigma(1)},x_{\sigma(2)},\dots,x_{\sigma(N)}),$
(1)

for any permutation $\sigma\in\mathfrak{S}_{N}$ of $N$ elements, where
$\mathfrak{S}_{N}$ is the symmetric group of order $N$. In addition, let
$\beta$ be a parameter determining the magnitude of the energy including the
sign $\pm$.

As an example of the models belonging to the above class, one can consider the
following $L$-state Potts energy $E({\bm{x}})$ on the complete graph where
each agent connects with all agents Potts :

$\displaystyle
E({\bm{x}})=-\frac{2(L-1)\log(L-1)}{L-2}\dfrac{1}{2N}\sum_{i=1}^{N}\sum_{j=1}^{N}\delta(x_{i},x_{j}).$
(2)

This case means that an agent interacts with the other agents only if they
have overlaps. In this sense, this Potts model on the complete graph is
equivalent to the agents with an on-site interaction in one dimension, which
may be a simple model to describe interacting agents. The coefficient of (2)
is adjusted so that the phase transition point $\beta_{c}$ in the equilibrium
state is equal to 1, which we will discuss in more detail in Section III.1.

Let us consider a Markov process with discrete time $t$, where during one time
step between $t$ and $t+1$, only one of the following possible transitions may
occur. The transition probability $T_{0}({\bm{x}}\to f_{i}^{\pm}{\bm{x}})$
from each configuration ${\bm{x}}$ to the configuration $f_{i}^{\pm}{\bm{x}}$
for any agent $i$ is

$\displaystyle T_{0}({\bm{x}}\to
f_{i}^{\pm}{\bm{x}})=\dfrac{1}{2N}\dfrac{1}{1+\exp\left(\beta\mathcal{D}_{i}^{\pm}E({\bm{x}})\right)},$
(3)

where

$\displaystyle\mathcal{D}_{i}^{\pm}E({\bm{x}}):=E(f_{i}^{\pm}{\bm{x}})-E({\bm{x}}).$
(4)

This leads to that the joint probability $P_{t}({\bm{x}})$ of configuration
${\bm{x}}$ at time $t$ satisfies the following master equation:

$\displaystyle P_{t+1}({\bm{x}})=$
$\displaystyle\sum_{{\bm{x}}^{\prime}\neq{\bm{x}}}P_{t}({\bm{x}}^{\prime})T_{0}({\bm{x}}^{\prime}\to{\bm{x}})$
$\displaystyle+P_{t}({\bm{x}})\Big{(}1-\sum_{{\bm{x}}^{\prime}\neq{\bm{x}}}T_{0}({\bm{x}}\to{\bm{x}}^{\prime})\Big{)},$
(5)

where the summation over ${\bm{x}^{\prime}}$ is done for all of the possible
configurations such that $T_{0}({\bm{x}}\to{\bm{x}}^{\prime})$ and
$T_{0}({\bm{x}}^{\prime}\to{\bm{x}})$ are defined above.

The Gibbs distribution

$\displaystyle
P_{\mathrm{can}}({\bm{x}}):=\dfrac{1}{Z_{N}(\beta)}\exp\left(-\beta
E({\bm{x}})\right),$ (6)

where $Z_{N}(\beta):=\sum_{{\bm{x}}}\exp(-\beta E({\bm{x}}))$, is the
stationary solution $P_{\mathrm{st}}({\bm{x}})$ of the master equation (5),
satisfying

$\displaystyle\sum_{{\bm{x}}^{\prime}}P_{\mathrm{st}}({\bm{x}}^{\prime})T_{0}({\bm{x}}^{\prime}\to{\bm{x}})=P_{\mathrm{st}}({\bm{x}})\sum_{{\bm{x}}^{\prime}\neq{\bm{x}}}T_{0}({\bm{x}}\to{\bm{x}}^{\prime}),$
(7)

because the Gibbs distribution satisfies the detailed balance condition:

$\displaystyle
P_{\mathrm{can}}({\bm{x}})T_{0}({\bm{x}}\to{\bm{x}^{\prime}})=P_{\mathrm{can}}({\bm{x}^{\prime}})T_{0}({\bm{x}^{\prime}}\to{\bm{x}}),$
(8)

for any pair ${\bm{x}},{\bm{x}^{\prime}}$ realized by the above dynamics.

Figure 1: (Color online) Schematic illustration of a transition step from a
configuration described by (a) to a configuration described by (c) in the
model. From (a) to (b), agent $2$ hops to the right site, and from (b) to (c),
agent $3$ among two possibly supplanted agents, which are $3$ and $5$, is
supplanted by agent $2$ and hops to the left site. Four arrows in (b) means
that, in the above transition, two agents $3$ and $5$ in $S(\bm{x},2,+)$ could
be supplanted, and the direction of supplanting process could be either to the
left or to the right.

###  II.2 Broken detailed balance by supplanting

Next, we consider to add a supplanting process to the equilibrium dynamics
introduced above, which breaks the detailed balance condition.

Let us imagine a process in which an agent $i$ hops to a position
$y_{i}=x_{i}\pm 1$ from position $x_{i}$ in accordance with the equilibrium
transition probability $T_{0}$, and then an agent at $y_{i}$, which has a rank
$j$ such that $i<j$, is stochastically forced to hop to the position $y_{i}+1$
for $d=+$, or $y_{i}-1$ for $d=-$. In this process, an agent with a higher
rank $i$ supplants another agent with a lower rank $j$. This is why we call
such a process the supplanting process. Note that the configuration $\bm{x}$
turns to be the configuration $f_{j}^{d}f_{i}^{\pm}\bm{x}$ through the whole
supplanting process. See Fig. 1 for a graphical reference of the process. For
convenience, let us introduce the following set

$S(\bm{x},i,\pm)\coloneqq\\{i<j\leq N\mid x_{j}=x_{i}\pm 1\\},$ (9)

which is a set of every agent whose rank is lower than $i$ at position
$x_{i}\pm 1$. That is, those agents could be supplanted by the agent $i$ when
the agent $i$ hops to position $x_{i}\pm 1$.

Suppose that every agent in $S(\bm{x},i,\pm)$ has the same chance to be chosen
as the supplanted one, and that the direction $d$ of hopping by supplanting is
determined with equal probability $1/2$. Explicitly, we define the transition
probability from ${\bm{x}}$ to $f_{j}^{d}f_{i}^{\pm}{\bm{x}}$ for
$d\in\\{+,-\\}$ and $j\in S(\bm{x},i,\pm)$:

$\displaystyle T({\bm{x}}\to
f_{j}^{d}f_{i}^{\pm}{\bm{x}})=\dfrac{p}{2}\frac{1}{1+p\\#S(\bm{x},i,\pm)}T_{0}({\bm{x}}\to
f_{i}^{\pm}{\bm{x}}),$ (10)

where $p\in\mathbb{R}_{+}\coloneqq[0,\infty)$ is the parameter of supplanting
rate and $\\#S$ is the number of all the elements of a set $S$. When $p\to 0$,
supplanting rarely occurs, and when $p\to\infty$, supplanting almost always
occurs. Note that at most one agent is supplanted in a single transition
regardless of the value of $p$.

On the other hand, for the case of $j\notin S(\bm{x},i,\pm)$, it holds that

$\displaystyle T({\bm{x}}\to f_{j}^{d}f_{i}^{\pm}{\bm{x}})=0.$ (11)

Further, the probability of transition $T({\bm{x}}\to f_{i}^{\pm}{\bm{x}})$ is
modified from $T_{0}$ as

$\displaystyle T({\bm{x}}\to
f_{i}^{\pm}{\bm{x}})=\frac{1}{1+p\\#S(\bm{x},i,\pm)}T_{0}({\bm{x}}\to
f_{i}^{\pm}{\bm{x}}).$ (12)

Totally, the following holds:

$\displaystyle T_{0}({\bm{x}}\to f_{i}^{\pm}{\bm{x}})$
$\displaystyle=T({\bm{x}}\to
f_{i}^{\pm}{\bm{x}})+\sum_{\begin{subarray}{c}j\in S(\bm{x},i,\pm)\\\
d=\pm\end{subarray}}T({\bm{x}}\to f_{j}^{d}f_{i}^{\pm}{\bm{x}}).$ (13)

Then, the master equation for the joint probability $P_{t}({\bm{x}})$
governing the above stochastic process is as follows:

$\displaystyle P_{t+1}({\bm{x}})=$
$\displaystyle\sum_{{\bm{x}}^{\prime}}P_{t}({\bm{x}}^{\prime})T({\bm{x}}^{\prime}\to{\bm{x}})$
$\displaystyle+P_{t}({\bm{x}})(1-\sum_{{\bm{x}}^{\prime}\neq{\bm{x}}}T({\bm{x}}\to{\bm{x}}^{\prime})),$
(14)

where the summation over ${\bm{x}^{\prime}}$ is done for all of the possible
configurations such that $T({\bm{x}}^{\prime}\to{\bm{x}})$ and
$T({\bm{x}}\to{\bm{x}}^{\prime})$ are defined. Since $T(\bm{x}\to
f_{i}^{\pm}\bm{x})$ is positive for any $\bm{x}$ and $i$ with finite
$\beta,p,N,L$, and $E$, any state can reach any state in this stochastic
process; that is, the stochastic process defined above is an irreducible
Markov process.

In the presence of $p>0$, the supplanting process does not hold the detailed
balance condition because of the asymmetric property in terms of agents rank;
when a supplanting occurs in one step, i.e., an agent supplants another agent,
the reverse process never occurs in any single step. Thus, obviously the
stationary solution $P_{\mathrm{st}}({\bm{x}})$ is no longer the Gibbs
distribution of a given energy function. Note that in the limit of $p\to 0$,
the detailed balance condition in terms of a given energy function is
recovered. Thus, we can also regard $p$ as the strength of violation of
detailed balance condition.

##  III The case of the Potts energy

In this section, we focus on the case of the Potts energy defined by (2). We
briefly review the known equilibrium properties and compute nonequilibrium
stationary distribution by exact diagonalization of transition matrix
corresponding to the master equation (II.2). Further, we introduce overlap
centrality and its correlation with agents’ rank, which are calculated using
the computed stationary distribution.

###  III.1 Computation of partition function at equilibrium

As a preliminary, we consider the equilibrium case with $p=0$. The equilibrium
ferromagnetic Potts model on the complete graph has two phases; one is the
ordered phase for stronger interaction, and another is the disordered phase
for weaker interaction, which are separated by a first-order transition point
if $L\geq 3$ Potts . In the context of this paper, the ordered state can be
regarded as the condensate state of agents, that is, the state where all
agents are located at the same position.

Let us look into more detail for the computation of the above results. When
$p=0$ and interaction strength $\beta$ is positive ($\beta>0$), corresponding
to the case of attractive interaction, the partition function
$Z_{N}(\beta):=\sum_{{\bm{x}}}\exp\big{(}{-}\beta E({\bm{x}})\big{)}$ for
$N\to\infty$ can be explicitly expressed, by which the equilibrium transition
point is computed exactly. Concretely, by performing the Stratonovich–Hubbard
transformation mean-field_Potts2013 ; mean-field_Potts2020  with

$\displaystyle\exp\left[{\frac{K\beta}{2N}\sum_{x\in
X}\left({\sum_{i=1}^{N}\delta(x_{i},x)}\right)^{2}}\right]$
$\displaystyle=\prod_{x\in
X}\sqrt{\frac{NK\beta}{2\pi}}\int_{\mathbb{R}}\mathrm{d}q\exp\left[{-\frac{NK\beta}{2}q^{2}+K\beta
q\sum_{i=1}^{N}\delta(x_{i},x)}\right],$ (15)

we obtain

$\displaystyle
Z_{N}(\beta)=\left({\frac{NK\beta}{2\pi}}\right)^{L/2}\int_{\mathbb{R}^{L}}\mathrm{d}^{L}q\exp\left(-N\beta
f_{\beta}({\bm{q}})\right),$ (16) $\displaystyle
f_{\beta}({\bm{q}})=\frac{K}{2}\sum_{x\in
X}q_{x}^{2}-\beta^{-1}\log\left({\sum_{x\in X}\exp\left({K\beta
q_{x}}\right)}\right),$ (17)

where

$\displaystyle K=\frac{2(L-1)\log(L-1)}{L-2}.$ (18)

For $N\gg 1$, the minimal value of $f_{\beta}(\bm{q})$ as a function of the
order parameter $\bm{q}$ behaves effectively as the free energy density of the
Potts model as follows:

$\displaystyle\lim_{N\to\infty}\frac{-\beta^{-1}\log
Z_{N}(\beta)}{N}=\min_{\bm{q}\in{\mathbb{R}}^{L}}f_{\beta}(\bm{q}).$ (19)

Taking $\frac{\partial f_{\beta}}{\partial q_{x}}(\bm{q})=0$ to minimize
$f_{\beta}(\bm{q})$, we obtain the stationary condition

$\displaystyle q_{y}\exp(-K\beta q_{y})=\Big{(}\sum_{x\in X}\exp(K\beta
q_{x})\Big{)}^{-1}$ (20)

for each $y\in X$. From (20), we see that

$\displaystyle q_{x}\exp(-K\beta q_{x})=q_{y}\exp(-K\beta q_{y}),$ (21)

for any $x,y\in X$. This indicates that there is a constant $c$ such that
$q_{x}\exp(-K\beta q_{x})=c$, which has at most two real solutions $a,b$ as an
equation for $q_{x}$. Then, we have $q_{x}\in\\{a,b\\}$ for each $x\in X$.
From (20), we also have

$\displaystyle\sum_{x\in X}q_{x}=1.$ (22)

Thus, a necessary condition for the order parameter $\bm{q}$ to minimize
$f_{\beta}(\bm{q})$ is described below. Keeping with (21) and (22), one of the
following conditions (i) and (ii) is satisfied:

  1. (i)

It holds that $\bm{q}=\tilde{\bm{q}}^{(0)}\coloneqq\dfrac{1}{L}(1,1,\dots,1).$

  2. (ii)

There exist an integer $n\in\left\\{{1,\dots,L-1}\right\\}$ and two distinct
real numbers $a_{n}=a_{n}(\beta),b_{n}=b_{n}(\beta)$ satisfying

$\displaystyle a_{n}\exp(-K\beta a_{n})=b_{n}\exp(-K\beta b_{n}),$ (23)
$\displaystyle na_{n}+(L-n)b_{n}=1.$ (24)

Moreover, $n$ components of $\bm{q}$ are $a_{n}$ and remaining $(L-n)$
components of $\bm{q}$ are $b_{n}$.

For example, if $n=1$, the solutions are described by
$\bm{q}=\tilde{\bm{q}}^{(\alpha)}$ for $1\leq\alpha\leq L$, where

$\displaystyle\tilde{q}^{(\alpha)}_{x}\coloneqq\begin{dcases}a_{1}(\beta)&(\text{if
$x\equiv\alpha\mod L$})\\\ b_{1}(\beta)&(\text{if $x\not\equiv\alpha\mod
L$})\end{dcases}.$ (25)

In Ref. Ellis , it has been shown that the set of the global minimum points
$\bm{q}$ of $f_{\beta}(\bm{q})$ corresponds to the case of
$\bm{q}=\tilde{\bm{q}}^{(0)}$ or $n=1$, depending on $\beta$. Concretely, the
set is described as

$\displaystyle\begin{dcases}\left\\{{\tilde{\bm{q}}^{(1)}(\beta),\tilde{\bm{q}}^{(2)}(\beta),\ldots,\tilde{\bm{q}}^{(L)}(\beta)}\right\\}&(\text{if}\
0<\beta<1)\\\
\left\\{{\tilde{\bm{q}}^{(0)},\tilde{\bm{q}}^{(1)}(1),\ldots,\tilde{\bm{q}}^{(L)}(1)}\right\\}&(\text{if}\
\beta=1)\\\ \left\\{{\tilde{\bm{q}}^{(0)}}\right\\}&(\text{if}\ \beta>1).\\\
\end{dcases}$ (26)

Note that the equations (23) and (24) with $n=1$ determine the value
$a_{1}\neq b_{1}$ uniquely, and the resulting functions
$a_{1}(\beta),b_{1}(\beta)$ are differentiable in the region $0<\beta<1$.

The expectation value of energy density is also expressed as

$\displaystyle\lim_{N\to\infty}\frac{\braket{E}_{\text{can}}}{N}=\begin{dcases}\frac{\partial}{\partial\beta}\beta
f_{\beta}(\tilde{\bm{q}}^{(1)}(\beta))&(\text{if}\ 0<\beta<1)\\\
\frac{\partial}{\partial\beta}\beta
f_{\beta}(\tilde{\bm{q}}^{(0)})&(\text{if}\ \beta>1).\end{dcases}$ (27)

Thus, one can show that the energy density (27) exhibits a discontinuous jump
at $\beta=1$, which is the phase transition point of the Potts model.

###  III.2 State vector description

Let us move onto the model with general $p\geq 0$. In this case, we need to
explicitly consider the dynamics in order to compute the stationary
distribution of the model. We would like to describe the stochastic process by
transition matrices with some basic linear operators. For more detailed
description and derivation, see Appendix A.2.

Let $H_{X}$ be the one-agent state space, which is of dimension $L$. It is
considered as a complex vector space with inner product
$\braket{\cdot}{\cdot}$, and has an orthonormal basis $\\{|x\rangle\mid x\in
X\\}$ over the set of complex numbers $\mathbb{C}$. Then the $N$-times self-
tensored space $H_{X}^{\otimes N}$ can be identified to the $N$-agent state
spaces, which is of dimension $L^{N}$. For a configuration of agents
$\bm{x}=(x_{1},x_{2},\dots,x_{N})\in X^{N}$, the corresponding state vector is
$|\bm{x}\rangle=|x_{1}\rangle\otimes|x_{2}\rangle\otimes\dots\otimes|x_{N}\rangle$.
The space $H_{X}^{\otimes N}$ has a natural inner product induced by
$\langle\cdot|\cdot\rangle$, and the set of the state vectors
$\\{|\bm{x}\rangle\\}_{\bm{x}\in X^{N}}$ is an orthonormal basis. We use the
same symbol $\langle\cdot|\cdot\rangle$ to write the inner product on
$H_{X}^{\otimes N}$.

We associate the probability $P(\bm{x})$ for agents’ configuration $\bm{x}$
with a state $\ket{P}\in H_{X}^{\otimes N}$ as follows:

$\displaystyle\begin{split}\braket{\bm{x}}{P}=P(\bm{x}),\end{split}$ (28)

or

$\displaystyle\begin{split}\ket{P}=\sum_{\bm{x}\in
X^{N}}P(\bm{x})\ket{\bm{x}}.\end{split}$ (29)

For a given state $\ket{P_{t}}$ at time $t$, the time evolution of the state
is described as follows:

$\displaystyle\ket{P_{t+1}}=\widehat{T}\ket{P_{t}},$ (30)

where $\widehat{T}$ is a transition matrix on $H_{X}^{\otimes N}$ such that
(30) is equivalent to the master equation (II.2) for the joint probability.
Similarly, $\widehat{T}_{0}$ is the transition matrix $\widehat{T}$ when
$p=0$.

We introduce some basic operators. A hopping map $f_{i}^{\pm}$ to the right
(resp. the left) corresponds to the operator defined by
$\widehat{\Delta}_{i}^{\pm}$; for $\bm{x}=(x_{1},x_{2},\dots,x_{N})\in X^{N}$,

$\displaystyle\widehat{\Delta}_{i}^{+}|\bm{x}\rangle$
$\displaystyle=|f_{i}^{+}\bm{x}\rangle$
$\displaystyle=\ket{x_{1}}\otimes\ket{x_{2}}\otimes\dots\otimes\ket{x_{i}+1}\otimes\dots\otimes\ket{x_{N}},$
(31) $\displaystyle\widehat{\Delta}_{i}^{-}|\bm{x}\rangle$
$\displaystyle=|f_{i}^{-}\bm{x}\rangle$
$\displaystyle=\ket{x_{1}}\otimes\ket{x_{2}}\otimes\dots\otimes\ket{x_{i}-1}\otimes\dots\otimes\ket{x_{N}}.$
(32)

Next we define projection operators. For a site $y\in X$ and an agent $1\leq
i\leq N$, we define $\widehat{\Xi}_{i}^{y}$ by

$\widehat{\Xi}_{i}^{y}|\bm{x}\rangle=\begin{dcases}|\bm{x}\rangle&\text{(if
$x_{i}=y$)}\\\ 0&\text{(if $x_{i}\neq y$)},\end{dcases}$ (33)

for $\bm{x}=(x_{1},x_{2},\dots,x_{N})\in X^{N}$. Then for a configuration
$\bm{y}\in X^{N}$, we define the operator $\widehat{\Xi}^{\bm{y}}$ as
$\prod_{1\leq i\leq N}\widehat{\Xi}_{i}^{y_{i}}$. Finally, we denote
$\mathop{\widehat{\mathrm{id}}}\nolimits_{H}$ (resp.
$\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes N}$) as the identity
operator on $H_{X}$ (resp. $H_{X}^{\otimes N}$).

Using these notions, we can describe the transition matrices $\widehat{T}_{0}$
and $\widehat{T}$. First, $\widehat{T}_{0}$ is

$\displaystyle\widehat{T}_{0}$ $\displaystyle=\sum_{\begin{subarray}{c}1\leq
i\leq N\\\ d=\pm\end{subarray}}\sum_{\bm{x}\in X^{N}}\Big{(}T_{0}(\bm{x}\to
f_{i}^{d}\bm{x})\widehat{\Delta}_{i}^{d}$
$\displaystyle\qquad+\left(\frac{1}{2N}-T_{0}(\bm{x}\to
f_{i}^{d}\bm{x})\right)\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N}\Big{)}\widehat{\Xi}^{\bm{x}}.$ (34)

Then $\widehat{T}$ is written as

$\displaystyle\widehat{T}$ $\displaystyle=\sum_{\begin{subarray}{c}1\leq i\leq
N\\\ d=\pm\end{subarray}}\sum_{\bm{x}\in
X^{N}}\Bigg{(}\sum_{\begin{subarray}{c}j\in S(\bm{x},i,d)\\\
d^{\prime}=\pm\end{subarray}}T(\bm{x}\to
f_{j}^{d^{\prime}}f_{i}^{d}\bm{x})\widehat{\Delta}_{j}^{d^{\prime}}\widehat{\Delta}_{i}^{d}+T(\bm{x}\to
f_{i}^{d}\bm{x})\widehat{\Delta}_{i}^{d}+\Big{(}\frac{1}{2N}-T_{0}(\bm{x}\to
f_{i}^{d}\bm{x})\Big{)}\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N}\Bigg{)}\widehat{\Xi}^{\bm{x}}$ (35)
$\displaystyle=\widehat{T}_{0}+\sum_{\begin{subarray}{c}1\leq i\leq N\\\
d=\pm\end{subarray}}\sum_{\bm{x}\in X^{N}}\sum_{\begin{subarray}{c}j\in
S(\bm{x},i,d)\\\ d^{\prime}=\pm\end{subarray}}T(\bm{x}\to
f_{j}^{d^{\prime}}f_{i}^{d}\bm{x})(\widehat{\Delta}_{j}^{d^{\prime}}-\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N})\widehat{\Delta}_{i}^{d}\widehat{\Xi}^{\bm{x}},$ (36)

where we used (II.2). For the definition of coefficients, see (3), (10), and
(12).

On this settings, the transition matrix $\widehat{T}=\widehat{T}(\beta,p)$ is
naturally regarded as a linear operator on $H_{X}^{\otimes N}$. Let
$\ket{P(\beta,p)}$ be the unique stationary state of $\widehat{T}(\beta,p)$
satisfying

$\displaystyle\begin{split}\widehat{T}(\beta,p)\ket{P(\beta,p)}=\ket{P(\beta,p)}.\end{split}$
(37)

One can show that, since $\widehat{T}$ is irreducible, $\ket{P(\beta,p)}$
exists and is uniquely determined by Perron–Frobenius theorem.

For the latter discussion, let us consider the symmetry of the transition
matrices $\widehat{T}_{0}$ and $\widehat{T}$. We introduce permutation
operators $\widehat{\Pi}_{\sigma}$ on $H_{X}^{\otimes N}$. For a given element
$\sigma\in\mathfrak{S}_{N}$ of the symmetry group $\mathfrak{S}_{N}$ of
agents, we define

$\widehat{\Pi}_{\sigma}\ket{\bm{x}}\coloneqq\ket{\sigma^{-1}(\bm{x})},$ (38)

where
$\sigma^{-1}(\bm{x})\coloneqq\left({x_{\sigma^{-1}(j)}}\right)_{j=1}^{N}$.
Then, we have

$\displaystyle\widehat{\Pi}_{\sigma}^{\dagger}\widehat{T}_{0}\widehat{\Pi}_{\sigma}=\widehat{T}_{0},$
(39)
$\displaystyle\widehat{\Pi}_{\sigma}^{\dagger}\widehat{T}\widehat{\Pi}_{\sigma}\neq\widehat{T},$
(40)

where $\widehat{\Pi}_{\sigma}^{\dagger}$ is the Hermitian conjugate of
$\widehat{\Pi}_{\sigma}$ (see (162) and (182)). Note that
$\widehat{\Pi}_{\sigma}$ is unitary:
$\widehat{\Pi}_{\sigma}^{\dagger}=\widehat{\Pi}_{\sigma}^{-1}=\widehat{\Pi}_{\sigma^{-1}}$.
In the sense of relation (39), the equilibrium dynamics described by
$\widehat{T}_{0}$ holds permutation symmetry. In contrast, the whole dynamics
by $\widehat{T}$ breaks the permutation symmetry, as described by (40).

###  III.3 Exact diagonalization of transition matrices

Figure 2: (Color online) Probabilities of the configurations determined by
stationary distribution for $\beta=2$ (circles), $\beta=0$ (rectangles), and
$\beta=-1$ (diamonds); $p=0$ (blue or the left at each column) and $p=10$ (red
or the right); $N=L=3$. The state of $(0,0,0)$ means that three agents are
located at the same site. The state of $(0,1,2)$ means that each of three
agents is located at the different position, respectively. The other three
states mean that two agents are located at the same site and another agent is
located at one of the different sites.

We perform exact diagonalization for the transition matrix $\widehat{T}$ to
obtain the eigenvalues and their corresponding eigenvectors. Thus, the
stationary distribution corresponds to the eigenvector with the maximum real
part, which is 1, of the eigenvalue. Note that the number of the states is
$L^{N}$, which gets exponentially large as a function of $N$.

As shown in Fig. 2 with $p=0$, at $\beta=0$, the joint probability of each
configuration shows the same value. As $\beta$ is increased from $0$, the
joint probability of condensate configuration $(0,0,0)$ is much higher than
that of the other configurations. Conversely, as $\beta$ is decreased from
$0$, the joint probability of the configuration $(0,1,2)$, where all the
agents are separated, is much higher than that of the other configurations. On
the other hand, the joint probability of configuration with a pair of two
agents overlapping at the same site and the other located at the different
site such as $(1,0,0),(0,1,0),(0,0,1)$ does not depend on the pair for $p=0$.

When $p=10$, the joint probabilities of configurations $(1,0,0)$, $(0,1,0)$,
$(0,0,1)$ are distinct. Concretely, those probabilities with $\beta=2$ and
$\beta=-1$ increase and decrease, respectively, in the order of $(1,0,0)$,
$(0,1,0)$, and $(0,0,1)$. This means that higher-ranked agents (resp. lower-
ranked agents) tend to overlap more frequently for $\beta=2$ (resp. for
$\beta=-1$). This can be interpreted as a typical consequence of supplanting
process.

In order to discuss how the configuration is condensed, let us introduce the
normalized expectation value of the Potts energy in terms of a probability
distribution $P(\bm{x})$ as follows:

$\displaystyle
M:=\frac{1}{N^{2}}\sum_{{\bm{x}}}\sum_{i=1}^{N}\sum_{j=1}^{N}\delta(x_{i},x_{j})P({\bm{x}}).$
(41)

Note that by definition, $M$ takes $1$ as the maximum value in the case of
$P(\bm{x})=\prod_{k=1}^{N}\delta(x,x_{k})$. In Fig. 3, using the computation
of the stationary distribution by the exact diagonalization, $M$ is shown as a
function of $p$ and $\beta$.

Relating to Section III.1, in the case of the equilibrium distribution
corresponding to the case with $p=0$, $M$ is rewritten by using
$\braket{E}_{\text{can}}$ as

$\displaystyle M=-\frac{2}{NK}\sum_{{\bm{x}}}E(\bm{x})\frac{e^{-\beta
E({\bm{x}})}}{Z_{N}(\beta)}=-\frac{2}{K}\frac{\braket{E}_{\text{can}}}{N},$
(42)

which means that $M$ is also discontinuous at the equilibrium phase transition
point $\beta=1$ in the thermodynamic limit $N\to\infty$.

Figure 3: (Color online) The $\beta$-dependence of the normalized expectation
value of the energy
$M=\dfrac{1}{N^{2}}\sum_{x}\sum_{i,j}\delta(x_{i},x_{j})\braket{\bm{x}}{P(\beta,p)}$
for various values of $p$ with $L=6,N=6$.

###  III.4 Overlap centrality and its correlation coefficient

Figure 4: (Color online) Heatmap of neighbor matrix $\mathcal{R}$ determined
by the stationary distribution with the diagonal components left out. The
color corresponds to $r_{ij}$ for pair of two agents $(i,j)$. Parameters:
(top) $\beta=-1,p=1,L=6,N=6$. (bottom) $\beta=1,p=1,L=6,N=6$.
---
Figure 5: (Color online) The overlap centrality $O_{i}$ determined by the
stationary distribution obtained by the exact diagonalization as a function of
agent $i$ for $L=N=6$. The solid lines are linear regressions for $O_{i}$ and
$i$. The parameters are set as follows; $p=0.1$ (left) and $p=10.0$ (right);
$\beta=1$ (top), $\beta=0$ (center), and $\beta=-1$ (bottom), respectively.
For negative $\beta$, $\phi$ is close to $1$ regardless of the value of $p$.

In order to characterize the correlation among agents, we may consider the
neighbor matrix $\mathcal{R}:=(r_{ij})_{i,j}$ defined as

$\displaystyle r_{ij}:=\sum_{{\bm{x}}}\delta(x_{i},x_{j})P({\bm{x}}).$ (43)

For example, if $P(\bm{x})$ is the uniform distribution then $r_{ij}=1/L$, and
if $P(\bm{x})=\prod_{k=1}^{N}\delta(x,x_{k})$ then $r_{ij}=1$. The latter
gives the maximum value of $r_{ij}$. Note that $r_{jj}=1$ for any $j$.

The entry $r_{ij}$ means how often agent $i$ and $j$ are located at the same
site under the distribution $P(\bm{x})$. In Fig. 4, we show heatmaps of
neighbor matrices computed from the stationary distribution. It demonstrates
that at $\beta=1$, the agents with higher rank have more overlaps with the
other agents, and conversely at $\beta=-1$, the agents with lower rank have
more overlaps with the other agents.

In order to quantify how often a given agent overlaps with the other agents in
total, we introduce the overlap centrality as a function of rank $i$ using the
entries of the neighbor matrix:

$\displaystyle O_{i}:=\sum_{\begin{subarray}{c}1\leq j\leq N\\\ j\neq
i\end{subarray}}r_{ij}.$ (44)

That is, we regard the agent $i$ having larger value of the overlap centrality
as more influential one compared to the other agents having lower values of
the overlap centrality. When the probability distribution $P(\bm{x})$ is
permutation symmetric, i.e., $P(\sigma(\bm{x}))=P(\bm{x})$ for any
$\sigma\in\mathfrak{S}_{N}$, the overlap centrality does not depend on rank
$i$. Note that

$\displaystyle
M=\displaystyle\dfrac{1}{N^{2}}\sum_{i=1}^{N}O_{i}+\dfrac{1}{N}$ (45)

holds by the definition. As shown in Fig. 5, we compute the overlap centrality
$O_{i}$ computed from the stationary distribution, showing that the overlap
centrality has a plus slope at attractive interaction of $\beta=1$ and has a
minus slope at repulsive interaction of $\beta=-1$.

In order to quantify the class of the overlap centrality in terms of the
slope, we measure the correlation coefficient $\phi$ of the overlap centrality
with respect to agents’ rank. This is defined as

$\displaystyle\phi:=\frac{1}{N}\sum_{i=1}^{N}\dfrac{(O_{i}-\frac{1}{N}\sum_{j=1}^{N}O_{j})(i-\frac{1}{N}\sum_{j=1}^{N}j)}{s_{O}s_{I}},$
(46)

where we define

$\displaystyle s_{O}^{2}$
$\displaystyle:=\frac{1}{N}\sum_{i}(O_{i}-\frac{1}{N}\sum_{j=1}^{N}O_{j})^{2},$
(47) $\displaystyle s_{I}^{2}$
$\displaystyle:=\frac{1}{N}\sum_{i}(i-\frac{1}{N}\sum_{j=1}^{N}j)^{2}=\frac{(N-1)(N+1)}{12}.$
(48)

By definition, when $|\phi|=1$, $O_{i}$ is a linear function of $i$. We check
the condition when this quantity $\phi$ is not defined. Since we set $N\geq
2$, the denominator of $\phi$ is zero exactly when $s_{O}^{2}=0$. This
corresponds to the case when $O_{i}$ is constant as a function of $i$. In this
case, we say that the quantity $\phi$ is singular.

In Fig. 6, we show the $\beta$-dependence of the correlation coefficient
$\phi$. In the weak supplanting condition with $p\ll 1$ such as $p=0.1$ or
$p=0.01$, $\phi$ is close to $+1$ for negative $\beta$. As $\beta$ increases,
$\phi$ sharply changes its sign around $\beta=0$, and turns out to be $-1$ for
positive $\beta$. We will discuss this behavior in a more general setting in
Section IV.

Figure 6: (Color online) Correlation coefficient $\phi$ as a function of
$\beta$ for $L=N=6$.

##  IV Analytic results for overlap centrality

Let us discuss the weak-supplanting limit of $p\ll 1$, where general results
in terms of overlap centrality are available. Although we have focused on only
the Potts energy as $E(\bm{x})$ in Section III, hereafter, we consider all of
the models which belong to the general class of energy functions satisfying
the permutation symmetry condition (1). The neighbor matrix $\mathcal{R}$
(43), the overlap centrality $O_{i}\;(1\leq i\leq N)$ (44), and the
correlation coefficient $\phi$ of the overlap centrality with agents’ rank
(46) can also be defined for the general cases in the same manner. These are
indeed the main subject in this section and also this paper. Hereafter, we use
the state vector description, fix parameters $\beta,L,N$ as arbitrary values,
and consider $p$-dependence of the dynamics unless otherwise specified.

The main goal in this section is to derive perfect correlation, which means
that the correlation coefficient (46) satisfies $\phi=\pm 1$, in the weak-
supplanting limit of $p\to 0$ as long as $\phi$ is not singular in the sense
mentioned after the definition of $\phi$. In order to take a step forward, we
start with introducing auxiliary stochastic processes (49), by which the
transition matrix of the model can be completely reconstructed. Then, using
this decomposition property (50), we construct another decomposition form (53)
of the transition matrix, which we call beta decomposition, where the
asymptotic behaviors in terms of $p\to 0$ can be rigorously estimated. Note
that the assumption of (1) is essential in the derivation of key properties
(IV.2) and (IV.2).

###  IV.1 Decompositions of transition matrix

In this subsection, we would like to introduce beta decomposition (53) of the
operator $\widehat{T}$. This decomposition enables us to investigate an
asymptotic behavior of $\widehat{T}$ for $p\ll 1$ because of an asymptotic
property (55). To describe it, first we introduce another decomposition,
called supplanting decomposition. This is relatively easy to describe, and
simplifies the description of beta decomposition. For details, see Appendix
A.3 and A.4.

Let us define the partial sum in (35). For an integer $1\leq n\leq N-1$, we
define

$\displaystyle\widehat{T}_{n}\coloneqq$
$\displaystyle\sum_{\begin{subarray}{c}1\leq i\leq N\\\
d=\pm\end{subarray}}\sum_{\begin{subarray}{c}\bm{x}\in
X^{N}\end{subarray}}\delta(\\#S(\bm{x},i,d),n)$ (49)
$\displaystyle\Bigg{[}\sum_{\begin{subarray}{c}j\in S(\bm{x},i,d)\\\
d^{\prime}=\pm\end{subarray}}T(\bm{x}\to
f_{j}^{d^{\prime}}f_{i}^{d}\bm{x})(\widehat{\Delta}_{j}^{d^{\prime}}-\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N})\Bigg{]}\widehat{\Delta}_{i}^{d}\widehat{\Xi}^{\bm{x}}.$

This operator $\widehat{T}_{n}$ is the second term in (36) restricting indices
$\bm{x},i,d$ to those with $\\#S(\bm{x},i,d)=n$. For any subset
$\mathcal{S}\subseteq\\{1,2,\dots,N-1\\}$, the matrix
$\widehat{T}_{0}+\sum_{i\in\mathcal{S}}\widehat{T}_{i}$ is also a stochastic
matrix. For example, the matrix $\widehat{T}_{0}+\widehat{T}_{n}$ represents
the supplanting process only when $\\#S(\bm{x},i,d)=n$. By definition, we have

$\widehat{T}=\widehat{T}_{0}+\widehat{T}_{1}+\widehat{T}_{2}+\dots+\widehat{T}_{N-1}.$
(50)

This is a decomposition of the operator $\widehat{T}$, which we call the
supplanting decomposition. Note that the coefficients
$\langle\bm{y}|\widehat{T}_{n}|\bm{x}\rangle$ of $n$-th term $\widehat{T}_{n}$
for $n\geq 1$ are estimated as

$|\langle\bm{y}|\widehat{T}_{n}|\bm{x}\rangle|\leq\frac{p}{2}.$ (51)

For details, see Appendix A.3.2. In particular, the coefficients of
$\widehat{T}_{n}$ are estimated by $\mathcal{O}(p)$ when $p\to+0$. Moreover,
since at most one of $\langle\bm{y}|\widehat{T}_{n}|\bm{x}\rangle\;(1\leq
n\leq N-1)$ is non-zero for fixed $\bm{x}$ and $\bm{y}$, we also have

$|\langle\bm{y}|(\widehat{T}-\widehat{T}_{0})|\bm{x}\rangle|\leq\frac{p}{2}.$
(52)

Though we explicitly estimate the coefficients of $\widehat{T}_{n}$ in (51),
the supplanting decomposition does not give effective truncation of
$\widehat{T}$ in terms of small $p$. Thus, we look for another decomposition
of $\widehat{T}$

$\widehat{T}=\widehat{T}_{0}+\widehat{U}_{1}+\dots+\widehat{U}_{N-1}$ (53)

satisfying

$\langle\bm{y}|\widehat{U}_{m}|\bm{x}\rangle=\mathcal{O}(p^{m})\text{ as
}p\to+0.$ (54)

If we find such an expansion, we have

$\widehat{T}=\widehat{T}_{0}+\widehat{U}_{1}+\dots+\widehat{U}_{m}+\mathcal{O}(p^{m+1})$
(55)

for $1\leq m\leq N-1$.

It would not be straightforward to practically find such an expansion.
Nevertheless, in this case, through a rather tricky procedure as shown in
Appendix A.4.2, one can prove that (53) and (54) are satisfied by the
following definition of $\widehat{U}_{m}$:

$\displaystyle\widehat{U}_{m}\coloneqq$
$\displaystyle\frac{(-1)^{m+1}B(m,1+1/p)}{p}$ $\displaystyle\quad\sum_{m\leq
n\leq N}\binom{n-1}{m-1}(1+np)\widehat{T}_{n},$ (56)

for $1\leq m\leq N-1$. Here, $B(a,b)$ is the beta function and
$\displaystyle\binom{n-1}{m-1}$ is the binomial coefficient. See (204) and
(221) of Appendix A for the detail of the derivation; see also Remark A.2 for
the motivation of this decomposition. We call this expansion beta
decomposition. Note that, for an integer $m\geq 1$,

$B\left(m,1+\frac{1}{p}\right)=\frac{(m-1)!p^{m}}{(1+p)(1+2p)\dots(1+mp)}.$
(57)

By substituting (49) and (57) into (IV.1) with certain sets of transformations
in Appendix A.3.3 and A.4.4, we reach another representation of
$\widehat{U}_{m}$ as in (195) and (231):

$\displaystyle\widehat{U}_{m}=$
$\displaystyle\frac{(-1)^{m-1}(m-1)!p^{m}}{2(1+p)(1+2p)\dots(1+mp)}$ (58)
$\displaystyle\times\sum_{\begin{subarray}{c}1\leq i<i_{1}<\dots<i_{m}\leq
N\end{subarray}}\left[\sum_{1\leq k\leq
m}(\widehat{\Delta}_{i_{k}}^{+}+\widehat{\Delta}_{i_{k}}^{-}-2\mathop{\widehat{\mathrm{id}}}\nolimits^{\otimes
N}_{H})\right]$
$\displaystyle\qquad\times\widehat{\Xi}_{i,i_{1},\dots,i_{m}}\widehat{T}_{0,\text{move}}^{i},$

where we define

$\displaystyle\widehat{T}_{0,\text{move}}^{i}$
$\displaystyle\coloneqq\sum_{d=\pm}\sum_{\bm{x}\in X^{N}}T_{0}(\bm{x}\to
f_{i}^{d}\bm{x})\widehat{\Delta}_{i}^{d}\widehat{\Xi}^{\bm{x}},$ (59)
$\displaystyle\widehat{\Xi}_{i,i_{1},\dots,i_{m}}$
$\displaystyle\coloneqq\sum_{x\in
X}\widehat{\Xi}_{i}^{x}\widehat{\Xi}_{i_{1}}^{x}\dots\widehat{\Xi}_{i_{m}}^{x}.$
(60)

This representation of $\widehat{U}_{m}$ is suitable for the further
calculation related to permutation symmetry in Section IV.2. In the following
sections, $\widehat{U}_{1}$ is the one we mainly consider in the weak
supplanting limit $p\to+0$.

Recall that $\widehat{\Pi}_{\sigma}$ defined in (38) denotes the permutation
operator corresponding to a permutation $\sigma\in\mathfrak{S}_{N}$. In
Appendix A.2.5 and A.2.7, we obtain commutation relations between permutation
operators and other operators:

$\displaystyle\widehat{\Pi}_{\sigma}\widehat{\Delta}_{i_{1}}^{\pm}$
$\displaystyle=\widehat{\Delta}_{\sigma(i_{1})}^{\pm}\widehat{\Pi}_{\sigma},$
(61) $\displaystyle\widehat{\Pi}_{\sigma}\widehat{\Xi}_{i_{0},i_{1}}$
$\displaystyle=\widehat{\Xi}_{\sigma(i_{0}),\sigma(i_{1})}\widehat{\Pi}_{\sigma},$
(62) $\displaystyle\widehat{\Pi}_{\sigma}\widehat{T}_{0,\text{move}}^{i_{0}}$
$\displaystyle=\widehat{T}_{0,\text{move}}^{\sigma(i_{0})}\widehat{\Pi}_{\sigma}.$
(63)

These relations will be used in (IV.2).

###  IV.2 Existence of perfect correlation

By using beta decomposition of the transition matrix obtained above, we are
going to show that the correlation coefficient $\phi$ exhibits perfect
correlation $|\phi|=1$ in the limit of $p\to 0$.

One can use the Brillouin–Wigner type perturbation theory Brilloin-Wigner-type
perturbation theory  to rewrite the stationary state $\ket{P(\beta,p)}$ of
$\widehat{T}(\beta,p)$ as a perturbation expansion from the stationary state
of $\widehat{T}_{0}$. For that purpose, we introduce some symbols. Let
$\ket{P_{\mathrm{can}}(\beta)}$ be the stationary state of
$\widehat{T}_{0}(\beta)$, i.e., the state corresponding to the Gibbs
distribution (6):

$\displaystyle\begin{split}\ket{P_{\mathrm{can}}(\beta)}\coloneqq\sum_{\bm{x}}\frac{\exp(-\beta
E(\bm{x}))}{Z(\beta)}\ket{\bm{x}},\end{split}$ (64)

which satisfies

$\displaystyle\begin{split}\widehat{T}_{0}(\beta)\ket{P_{\mathrm{can}}(\beta)}=\ket{P_{\mathrm{can}}(\beta)},\end{split}$
(65)

and

$\displaystyle\sum_{\bm{x}}\braket{\bm{x}}{P_{\mathrm{can}}(\beta)}=1.$ (66)

Note that $\ket{P_{\mathrm{can}}(\beta)}$ is invariant under a permutation,
that is, it holds that

$\displaystyle\widehat{\Pi}_{\sigma}\ket{P_{\mathrm{can}}(\beta)}=\ket{P_{\mathrm{can}}(\beta)}$
(67)

for any $\sigma\in\mathfrak{S}_{N}$, because of the permutation symmetry
condition (1) for energy function. Let $\widehat{\mathrm{pr}}(\beta)$ be a
projection operator on $H_{X}^{\otimes N}$ to the orthogonal complement of the
subspace ${\mathbb{C}}\ket{P_{\mathrm{can}}(\beta)}$:

$\displaystyle\begin{split}\widehat{\mathrm{pr}}(\beta)\coloneqq\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N}-\frac{\ket{P_{\mathrm{can}}(\beta)}\bra{P_{\mathrm{can}}(\beta)}}{\braket{P_{\mathrm{can}}(\beta)}{P_{\mathrm{can}}(\beta)}},\end{split}$
(68)

and $\widehat{G}_{0}(\beta)$ be a linear operator from $H_{X}^{\otimes N}$ to
itself:

$\displaystyle\begin{split}\widehat{G}_{0}(\beta)\coloneqq\left({\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N}-\widehat{T}_{0}(\beta)}\right)^{-1}\widehat{\mathrm{pr}}(\beta).\end{split}$
(69)

Here, we need to set the coefficient of the term
$\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes N}$ in
$\widehat{G}_{0}(\beta)$ as the eigenvalue $1$ corresponding to the
eigenvector $\ket{P(\beta,p)}$ of $\widehat{T}(\beta,p)$ (see (37)). With the
above notations, $\ket{P(\beta,p)}$ can be written as follows:

$\displaystyle\begin{split}&\ket{P(\beta,p)}\\\
&=C(\beta,p)\left[{\ket{P_{\mathrm{can}}(\beta)}+\sum_{n=1}^{\infty}\left({\widehat{G}_{0}(\widehat{T}-\widehat{T}_{0})}\right)^{n}\ket{P_{\mathrm{can}}(\beta)}}\right],\end{split}$
(70)

where $C(\beta,p)$ is the positive normalization factor of $\ket{P(\beta,p)}$
such that $\sum_{\bm{x}\in X^{N}}\braket{\bm{x}}{P(\beta,p)}=1$. Since
$\widehat{T}-\widehat{T}_{0}=\mathcal{O}(p)$, one can estimate that
$C(\beta,p)=1+\mathcal{O}(p)$.

As obtained in (53) and (58), within the asymptotic regime of small $p$,
$\widehat{T}-\widehat{T}_{0}$ is can be expanded with powers of $p$, and then
we have

$\displaystyle\begin{split}\widehat{T}-\widehat{T}_{0}=\widehat{U}_{1}+\mathcal{O}(p^{2}),\end{split}$
(71)

where

$\displaystyle\begin{split}\widehat{U}_{1}=\frac{p/2}{1+p}\sum_{1\leq
i_{0}<i_{1}\leq
N}\left({\widehat{\Delta}_{i_{1}}^{+}+\widehat{\Delta}_{i_{1}}^{-}-2\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N}}\right)\widehat{\Xi}_{i_{0},i_{1}}\widehat{T}_{0,\text{move}}^{i_{0}}.\end{split}$
(72)

By substituting (72) into (LABEL:ptheory), we have

$\displaystyle\begin{split}&\ket{P(\beta,p)}\\\
&=C(\beta,p)\left[{\ket{P_{\mathrm{can}}(\beta)}+\widehat{G}_{0}\widehat{U}_{1}\ket{P_{\mathrm{can}}(\beta)}+\mathcal{O}(p^{2})}\right]\\\
&=C(\beta,p)\left[{\ket{P_{\mathrm{can}}(\beta)}+\frac{p}{1+p}\sum_{1\leq
i_{0}<i_{1}\leq
N}\widehat{B}_{i_{0},i_{1}}\ket{P_{\mathrm{can}}(\beta)}}\right]\\\ &\hskip
199.16928pt+\mathcal{O}(p^{2}),\end{split}$ (73)

where we define:

$\displaystyle\begin{split}\widehat{B}_{i_{0},i_{1}}\coloneqq\frac{1}{2}\widehat{G}_{0}\left({\widehat{\Delta}_{i_{1}}^{+}+\widehat{\Delta}_{i_{1}}^{-}-2\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N}}\right)\widehat{\Xi}_{i_{0},i_{1}}\widehat{T}_{0,\text{move}}^{i_{0}}.\end{split}$
(74)

This operator $\widehat{B}_{i_{0},i_{1}}$ is dependent on $\beta$ but
independent of $p$. Thus, using $r_{ii}=1$ for any $i$, the overlap centrality
can be written as follows:

$\displaystyle\begin{split}&O_{i}=\sum_{1\leq j\leq N}\sum_{\bm{x}\in
X^{N}}\delta\left({x_{i},x_{j}}\right)\braket{\bm{x}}{P(\beta,p)}-1\\\
&=C(\beta,p)\left[{\mathcal{A}_{0}(i)+\frac{p}{1+p}\sum_{1\leq i_{0}<i_{1}\leq
N}\mathcal{A}_{1}(i,i_{0},i_{1})}\right]-1\\\ &\hskip
199.16928pt+\mathcal{O}(p^{2}),\end{split}$ (75)

where

$\displaystyle\mathcal{A}_{0}(i)=\sum_{1\leq j\leq N}\sum_{\bm{x}\in
X^{N}}\delta\left({x_{i},x_{j}}\right)\braket{\bm{x}}{P_{\mathrm{can}}(\beta)},$
(76) $\displaystyle\mathcal{A}_{1}(i,i_{0},i_{1})=\sum_{1\leq j\leq
N}\sum_{\bm{x}\in
X^{N}}\delta\left({x_{i},x_{j}}\right)\bra{\bm{x}}\widehat{B}_{i_{0},i_{1}}\ket{P_{\mathrm{can}}(\beta)}.$
(77)

Here $\mathcal{A}_{0}(i)$ and $\mathcal{A}_{1}(i,i_{0},i_{1})$ are constant as
a function of $p$.

Recall that $\widehat{\Pi}_{\sigma}$ is the permutation operator corresponding
to a permutation $\sigma\in\mathfrak{S}_{N}$ (see (38) for definition). By
using (67), we have

$\displaystyle\mathcal{A}_{0}(i)$ $\displaystyle=\sum_{1\leq j\leq
N}\sum_{\bm{x}\in
X^{N}}\delta\left({x_{i},x_{j}}\right)\braket{\bm{x}}{\widehat{\Pi}_{\sigma}}{P_{\mathrm{can}}(\beta)}$
$\displaystyle=\sum_{1\leq j\leq N}\sum_{\bm{x}\in
X^{N}}\delta\left({x_{\sigma(i)},x_{\sigma(j)}}\right)\braket{\bm{x}}{P_{\mathrm{can}}(\beta)}$
$\displaystyle=\sum_{1\leq j\leq N}\sum_{\bm{x}\in
X^{N}}\delta\left({x_{\sigma(i)},x_{j}}\right)\braket{\bm{x}}{P_{\mathrm{can}}(\beta)}$
$\displaystyle=\mathcal{A}_{0}(\sigma(i)).$ (78)

This means that $\mathcal{A}_{0}(i)$ does not depend on $i$:

$\mathcal{A}_{0}(i)=\mathcal{A}_{0}(1)\eqqcolon\mathcal{B}_{0}.$ (79)

Moreover, by using (61), (62), (63), and (67), we have
$\widehat{\Pi}_{\sigma}\widehat{B}_{i_{0},i_{1}}=\widehat{B}_{\sigma(i_{0}),\sigma(i_{1})}\widehat{\Pi}_{\sigma}$.
Therefore, it follows that

$\displaystyle\mathcal{A}_{1}(i,i_{0},i_{1})$ $\displaystyle=\sum_{1\leq j\leq
N}\sum_{\bm{x}\in
X^{N}}\delta\left({x_{i},x_{j}}\right)\bra{\bm{x}}\widehat{\Pi}_{\sigma}^{-1}\widehat{B}_{\sigma(i_{0}),\sigma(i_{1})}\widehat{\Pi}_{\sigma}\ket{P_{\mathrm{can}}(\beta)}$
$\displaystyle=\sum_{1\leq j\leq N}\sum_{\bm{x}\in
X^{N}}\delta\left({x_{\sigma(i)},x_{\sigma(j)}}\right)\bra{\bm{x}}\widehat{B}_{\sigma(i_{0}),\sigma(i_{1})}\ket{P_{\mathrm{can}}(\beta)}$
$\displaystyle=\sum_{1\leq j\leq N}\sum_{\bm{x}\in
X^{N}}\delta\left({x_{\sigma(i)},x_{j}}\right)\bra{\bm{x}}\widehat{B}_{\sigma(i_{0}),\sigma(i_{1})}\ket{P_{\mathrm{can}}(\beta)}$
$\displaystyle=\mathcal{A}_{1}(\sigma(i),\sigma(i_{0}),\sigma(i_{1}))$ (80)

for any $\sigma\in\mathfrak{S}_{N}$. In this sense, $\mathcal{A}_{1}$
preserves permutation symmetry in spite of the permutation-symmetry breaking
of $\widehat{T}$. By this equation, we obtain

$\displaystyle\mathcal{A}_{1}(i,i_{0},i_{1})=\begin{dcases}\mathcal{A}_{1}(1,1,2)\eqqcolon\mathcal{B}_{1}&(\text{if}\
i=i_{0})\\\ \mathcal{A}_{1}(2,1,2)\eqqcolon\mathcal{B}_{2}&(\text{if}\
i=i_{1})\\\ \mathcal{A}_{1}(3,1,2)\eqqcolon\mathcal{B}_{3}&(\text{if}\ i\neq
i_{0},i_{1})\end{dcases}$ (81)

for any $i$ and any pair $i_{0}<i_{1}$. These quantities
$\mathcal{B}_{1},\mathcal{B}_{2}$, and $\mathcal{B}_{3}$ are independent of
$p$.

Note that for a given agent $i$, the number of pairs $(i_{0},i_{1})$ which
satisfies each of the above conditions corresponding with $\mathcal{B}_{1}$,
$\mathcal{B}_{2}$, and $\mathcal{B}_{3}$ is $N-i$, $i-1$, and $(N-1)(N-2)/2$,
respectively. Therefore, we obtain

$\displaystyle\begin{split}&\sum_{1\leq i_{0}<i_{1}\leq
N}\mathcal{A}_{1}(i,i_{0},i_{1})\\\
&=(N-i)\mathcal{B}_{1}+(i-1)\mathcal{B}_{2}+\frac{(N-1)(N-2)}{2}\mathcal{B}_{3}.\end{split}$
(82)

Substituting (LABEL:eq:sum_of_A1) into
(LABEL:eq:expression_overlap_centrality_by_A), we find that

$\displaystyle
O_{i}=\frac{p}{1+p}C(\beta,p)(\mathcal{B}_{2}-\mathcal{B}_{1})i+c_{0}+\mathcal{O}(p^{2}),$
(83)

where $c_{0}$ is a real number independent of rank $i$:

$\displaystyle c_{0}$
$\displaystyle=\frac{p}{1+p}C(\beta,p)\left({N\mathcal{B}_{1}-\mathcal{B}_{2}+\frac{(N-1)(N-2)}{2}\mathcal{B}_{3}}\right)$
$\displaystyle\qquad+C(\beta,p)\mathcal{B}_{0}-1.$ (84)

With the estimation of $C(\beta,p)=1+\mathcal{O}(p)$, we can rewrite (83) as

$\displaystyle
O_{i}=\frac{p}{1+p}(\mathcal{B}_{2}-\mathcal{B}_{1})i+c_{0}+\mathcal{O}(p^{2}).$
(85)

Note that, in (LABEL:eq:expression_overlap_centrality_by_A), (83), and (85),
the terms in $\mathcal{O}(p^{2})$ could depend on $i$. As shown in Fig. 7, we
have confirmed that $p$-dependence in the first term of (85) is consistent to
that computed by the exact diagonalization. Moreover, as shown in Fig. 8, we
can see good agreement of the value $\mathcal{B}_{2}-\mathcal{B}_{1}$ by two
distinct methods: one method is by definition (81). The other is by comparing
$O_{i}$ computed by the exact diagonalization with rank-dependence in (85).

---
Figure 7: (Color online) The overlap centrality $O_{j}$ computed by the exact
diagonalization for $L=N=6$ is expressed by $O_{j}\simeq a(p)j+b(p)$, where
coefficient $a(p)$ and $b(p)$ are determined by linear regression. These two
figures plot the regression coefficient $a(p)$ versus $p/(1+p)$ for
$p=0.001,0.002,\dots,0.01$; $\beta=-1$ (left) and $\beta=1$ (right). The
correlation coefficient between $a(p)$ and $p/(1+p)$ is $0.9999998...$ for
$\beta=-1$ and $-0.99997...$ for $\beta=1$. Note that $p$-dependence of the
first term in (85) corresponds to that the correlation coefficient is equal to
$1$ for $\beta=-1$ and $-1$ for $\beta=1$, respectively. Figure 8: (Color
online) $\mathcal{B}_{2}-\mathcal{B}_{1}$ as a function of $\beta$ for
$L=N=3$. The line corresponds to the computation through its definition (81).
The blue crosses correspond to the estimation through (85) where ${O}_{i}$ is
calculated by the exact diagonalization. Practically, the estimated value is
calculated as $\tilde{a}(\beta)$ where
$a(p)\simeq\tilde{a}(\beta)\frac{p}{1+p}+\tilde{b}(\beta)$ through linear
regression.

We write the remaining term in (85) as

$\varepsilon_{i}(\beta,p)\coloneqq
O_{i}-\frac{p}{1+p}(\mathcal{B}_{2}-\mathcal{B}_{1})i-c_{0}=\mathcal{O}(p^{2}).$
(86)

Let us suppose $\mathcal{B}_{2}\neq\mathcal{B}_{1}$. The remainder term
$\varepsilon_{i}$ can be ignored when $p\ll 1$ compared to the term
$\dfrac{p}{1+p}(\mathcal{B}_{2}-\mathcal{B}_{1})i.$ After ignoring
$\varepsilon_{i}$, the overlap centrality $O_{i}$ is a linear function with
respect to rank $i$. Therefore the perfect correlation holds:

$\displaystyle\phi\to\begin{dcases}+1&(\text{if}\
\mathcal{B}_{1}<\mathcal{B}_{2})\\\ -1&(\text{if}\
\mathcal{B}_{1}>\mathcal{B}_{2})\end{dcases}\quad\text{as}\ p\to+0.$ (87)

This is consistent with the observation in Fig. 6 as long as $p$ is small
because the value of $\phi$ approaches $+1$ or $-1$ very closely.

The above discussion gives another indication. Ignoring $\varepsilon_{i}$, the
linear dependency of the overlap centrality $O_{i}$ with respect to rank $i$
comes from permutation symmetry (IV.2) in $\mathcal{A}_{1}$. Note that, as
mentioned by (40), the permutation symmetry is broken in the transition matrix
if $p\neq 0$, but this symmetry is partially recovered in the quantity
$\mathcal{A}_{1}$. Note that one can also derive

$\displaystyle M$
$\displaystyle=\frac{p}{1+p}\left({\dfrac{N-1}{2N}(\mathcal{B}_{1}+\mathcal{B}_{2})+\dfrac{(N-1)(N-2)}{2N}\mathcal{B}_{3}}\right)$
$\displaystyle\qquad+C(\beta,p)\dfrac{\mathcal{B}_{0}}{N}+\mathcal{O}(p^{2})$
(88)

by using (45).

Evaluating the sign of $\phi$ requires concrete calculation of both
$\mathcal{B}_{1}$ and $\mathcal{B}_{2}$, but it gets complicated to obtain
their analytic expressions for a given energy function such as the Potts
energy (2). Nevertheless, it is still feasible to perform such a calculation
in the case of $\beta=0$ as discussed in the next subsection. It is because at
$\beta=0$, transition matrix $\widehat{T}_{0}(\beta)$ gets independent of the
form of energy function, resulting in getting close to that of the free random
walk process.

###  IV.3 Singularity in $\phi$ at $\beta=0$

One can calculate the overlap centrality at $\beta=0$ for the asymptotic
regime of small $p$ concretely. As a result, we show that at $\beta=0$,
$\mathcal{B}_{1}=\mathcal{B}_{2}$ holds, which means that the correlation
coefficient $\phi$ is singular at $\beta=0$.

From the definitions of $\mathcal{B}_{1}$, $\mathcal{B}_{2}$ in (81) and
$\mathcal{A}_{1}$ in (77), we have

$\displaystyle\mathcal{B}_{1}=\sum_{1\leq j\leq N}\sum_{\bm{x}\in
X^{N}}\delta(x_{1},x_{j})\braket{\bm{x}}{\widehat{B}_{1,2}}{P_{\mathrm{can}}(\beta)},$
(89) $\displaystyle\mathcal{B}_{2}=\sum_{1\leq j\leq N}\sum_{\bm{x}\in
X^{N}}\delta(x_{2},x_{j})\braket{\bm{x}}{\widehat{B}_{1,2}}{P_{\mathrm{can}}(\beta)}.$
(90)

In order to calculate $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ at $\beta=0$, we
need derive an explicit expression of
$\braket{\bm{x}}{\widehat{B}_{1,2}}{P_{\mathrm{can}}(0)}$. First, let us
define

$\displaystyle|k)\coloneqq\sum_{x\in X}\frac{e^{ikx}}{\sqrt{L}}\ket{x}\in
H_{X},$ (91)

for $k=2\pi n/L$ and $n\in\mathbb{Z}/L\mathbb{Z}$. The orthonormal system
$\left\\{{|k)}\right\\}_{k}$ spans the whole space $H_{X}$ over $\mathbb{C}$.
In particular

$\displaystyle\ket{P_{\mathrm{can}}(0)}=\left({\dfrac{|0)}{\sqrt{L}}}\right)^{\otimes
N}$ (92)

holds. Note that, for $\bm{k}=(k_{1},k_{2},\dots,k_{N})$ with $k_{i}=2\pi
n_{i}/L$ and $n_{i}\in\mathbb{Z}/L\mathbb{Z}$ with $1\leq i\leq N$,

$\displaystyle|\bm{k})\coloneqq|k_{1})\otimes\cdots\otimes|k_{N})\in
H_{X}^{\otimes N}$ (93)

is an eigenvector of $\widehat{T}_{0}(\beta=0)$ with an eigenvalue

$\displaystyle\dfrac{1}{N}\left({\cos^{2}\dfrac{k_{1}}{2}+\cdots+\cos^{2}\dfrac{k_{N}}{2}}\right).$
(94)

The vector $|\bm{k})$ is also an eigenvector of $\widehat{G}_{0}(\beta=0)$,
and the eigenvalue is

$\displaystyle
N\left({\sin^{2}\dfrac{k_{1}}{2}+\cdots+\sin^{2}\dfrac{k_{N}}{2}}\right)^{-1}$
(95)

for any $\bm{k}\neq\bm{0}=(0,0,\dots,0)$.

Next, substituting the definition (74) of $\widehat{B}_{1,2}$ into the term
$\braket{\bm{x}}{\widehat{B}_{1,2}}{P_{\mathrm{can}}(0)}$, we obtain

$\displaystyle\braket{\bm{x}}{\widehat{B}_{1,2}}{P_{\mathrm{can}}(0)}$
$\displaystyle=\frac{1}{2}\bra{\bm{x}}\widehat{G}_{0}(0)(\widehat{\Delta}_{2}^{+}+\widehat{\Delta}_{2}^{-}-2\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N})$
$\displaystyle\quad\quad\quad\times\widehat{\Xi}_{1,2}\widehat{T}_{0,\mathrm{move}}^{1}{(\beta=0)}\ket{P_{\mathrm{can}}(0)}.$
(96)

In order to obtain more explicit expression, let us multiply
$\ket{P_{\mathrm{can}}(0)}$ by $\widehat{T}_{0,\mathrm{move}}^{1}$,
$\widehat{\Xi}_{1,2}$,
$(\widehat{\Delta}_{2}^{+}+\widehat{\Delta}_{2}^{-}-2\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N})$, and $\widehat{G}_{0}(0)$ from the left, successively. Reminding of the
definition (59) of $\widehat{T}_{0,\mathrm{move}}^{1}$, we have

$\displaystyle\widehat{T}_{0,\mathrm{move}}^{1}(0)\ket{P_{\mathrm{can}}(0)}=\frac{1}{2N}\ket{P_{\mathrm{can}}(0)},$
(97)

and can show that

$\displaystyle\widehat{\Xi}_{1,2}\ket{P_{\mathrm{can}}(0)}$
$\displaystyle=\sum_{x\in
X}\frac{\ket{x}}{L}\otimes\frac{\ket{x}}{L}\otimes\left({\frac{|0)}{\sqrt{L}}}\right)^{\otimes(N-2)}$
$\displaystyle=\frac{1}{L^{2}}\sum_{k_{1}}|k_{1})\otimes|{-}k_{1})\otimes\left({\frac{|0)}{\sqrt{L}}}\right)^{\otimes(N-2)}.$
(98)

By using (97) and (IV.3), we obtain

$\displaystyle(\widehat{\Delta}_{2}^{+}+\widehat{\Delta}_{2}^{-}-2\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N})\widehat{\Xi}_{1,2}\widehat{T}_{0,\mathrm{move}}^{1}(0)\ket{P_{\mathrm{can}}(0)}$
$\displaystyle=-\frac{1}{2N}\frac{1}{L^{2}}\sum_{k_{1}}4\sin^{2}\frac{k_{1}}{2}|k_{1})\otimes|{-}k_{1})\otimes\left({\frac{|0)}{\sqrt{L}}}\right)^{\otimes(N-2)}.$
(99)

Substituting (IV.3) into (IV.3), and using (95), we find

$\displaystyle\braket{\bm{x}}{\widehat{B}_{1,2}}{P_{\mathrm{can}}(0)}$
$\displaystyle=-\frac{1}{2L^{2}}\sum_{k_{1}\neq
0}\frac{e^{ik_{1}x_{1}}}{\sqrt{L}}\frac{e^{-ik_{1}x_{2}}}{\sqrt{L}}\left({\frac{1}{L}}\right)^{N-2}$
$\displaystyle=-\frac{1}{2L^{N+1}}\left({L\delta(x_{1},x_{2})-1}\right).$
(100)

Recalling (89) and (90) with

$\displaystyle\sum_{\bm{x}\in
X^{N}}\delta(x_{i},x_{j})(L\delta(x_{1},x_{2})-1)$
$\displaystyle=\begin{dcases}L^{N-1}(L-1)&(\text{if}\ (i,j)=(1,2)\text{ or
}(2,1))\\\ 0&(\text{otherwise}),\end{dcases}$ (101)

one can calculate

$\displaystyle\mathcal{B}_{1}=\mathcal{B}_{2}=-\frac{L-1}{2L^{2}}.$ (102)

Thus, it turns out that $\phi$ is singular at $\beta=0$.

One can also calculate $\mathcal{B}_{0}=N/L$, and using (IV.3),
$\mathcal{B}_{3}=0$. By substituting the value of $\mathcal{B}_{\ell}$
($\ell=0,1,2,3$) into (84) and (85), the overlap centrality $O_{i}$ is
$\mathcal{O}(N/L)$. This is consistent with the uniform distribution
corresponding to the case of $\beta=0$. Therefore, it is reasonable that
$\mathcal{B}_{0}=\mathcal{O}(N/L)$ and
$\mathcal{B}_{1}=\mathcal{B}_{2}=\mathcal{O}(L^{-1})$ as functions of $N$ and
$L$. Note that one can also evaluate

$M=\dfrac{1}{L}\left({C(0,p)-\dfrac{p}{2(1+p)}\dfrac{N-1}{N}\dfrac{L-1}{L}}\right)+\mathcal{O}(p^{2}).$
(103)

###  IV.4 Comparison between exact diagonalization and analytic result

In this subsection, we shall illustrate the behavior of the correlation
coefficient $\phi$ computed by exact diagonalization in comparison with
analytic discussion in Section IV.

In order to consider $\beta$-dependence of $\phi$, let us fix $p$ as a small
but non-zero value and change the value of $\beta$. When the parameter $\beta$
varies with satisfying the condition

$\mathcal{B}_{2}(\beta,p)-\mathcal{B}_{1}(\beta,p)=\mathcal{O}(\varepsilon_{i}(\beta,p)),$
(104)

the linear term $(\mathcal{B}_{2}-\mathcal{B}_{1})i$ in $O_{i}$ is no longer
dominant in the rank-dependence. As a result, $\phi$ could change continuously
from $\phi=-1$ to $\phi=1$. In this case, (87) does not necessarily hold.
Exact diagonalization indicates that the range of $\beta$, where $\phi$ takes
a value close to $\pm 1$, is wider as $p$ is smaller.

Let us remind of the observation in the exact diagonalization in the case of
the Potts energy that the value of $\phi$ sharply changes around $\beta=0$ for
small $p$ as shown in Fig. 6. Assuming that this observation is universal for
sufficiently small $p$, by combining the existence of perfect correlation and
the singularity of $\phi$ at $\beta=0$, it may be a reasonable conjecture
that, at least, in the case of the Potts energy, $\phi$ becomes discontinuous
at $\beta=0$ as a function of $\beta$ in the limit of $p\to+0$.

###  IV.5 Correspondence between Overlap centrality and Eigenvector centrality

Let us discuss the relation between the overlap centrality
${\bm{O}}=(O_{i})_{i=1}^{N}$ and the other existing ways to define centrality.

First, the overlap centrality has a connection to another existing centrality
in the following sense. Let us consider a weighted complete graph, where each
agent is regarded as a vertex, and an element $r_{ij}$ of the neighbor matrix
$\mathcal{R}$ for the pair of agents $i,j$ is regarded as the weight of the
edge $(i,j)$. Then, the overlap centrality defined above is equivalent to the
strength centrality of the complete graph constructed above, which has been
introduced in the field of network science Freeman ; Barrat .

Second, one can also define the eigenvector centrality of the complete graph
mentioned above as the eigenvector ${\bm{V}}=(V_{i})_{1\leq i\leq N}$ of
$\mathcal{R}$ with the maximum eigenvalue. Indeed, one may show that the
eigenvector centrality is directly related to the overlap centrality when
$p\ll 1$ in the following manner:

$\displaystyle{\bm{V}}\propto\frac{1}{N^{3/2}c}{\bm{O}}-\gamma\times(1,1,\ldots,1)^{T}+\mathcal{O}(p^{2}),$
(105)

where $c$ and $\gamma$ are constants with $\mathcal{O}(1)$. In particular,
combining with (85), we have

$\displaystyle V_{i}$
$\displaystyle\propto\frac{p}{1+p}\frac{\mathcal{B}_{2}-\mathcal{B}_{1}}{N^{3/2}c}i+\left(\frac{c_{0}}{N^{3/2}c}-\gamma\right)+\mathcal{O}(p^{2})$
(106)

where the coefficient of proportionality is independent of $i$. Thus, the
eigenvector centrality as well as the overlap centrality depends on the rank
$i$ linearly if the term of $\mathcal{O}(p^{2})$ is ignored in (105).
Remarkably, that relation (105) holds for general probability distribution
which breaks the permutation symmetry of agents weakly. See Appendix B for the
explicit two conditions to hold the relation (105) in a more general form. As
shown in Fig. 9, one can see better agreement of both centralities for smaller
$p$.

---
Figure 9: (Color online) The overlap centrality (blue circles) $O_{i}$ and the
normalized eigenvector centrality (black crosses) determined by the stationary
distribution obtained by the exact diagonalization as a function of agent $i$
for $L=N=6$. The parameters are set as follows; $p=0.1$ (left) and $p=10.0$
(right); $\beta=1$ (top), $\beta=0$ (center), and $\beta=-1$ (bottom),
respectively. The eigenvector centrality is normalized so that the two data
points of $i=1$ and $i=2$ in the ranking axis are perfectly matched. The
overlap centrality gets closer to the eigenvector centrality for $p=0.1$
compared to the case of $p=10.0$.

##  V Concluding remarks

In this paper, we have proposed a stochastic process without both detailed
balance and permutation symmetry, which is inspired by the supplanting
phenomenon of Japanese macaques. We have introduced a new type of centrality,
which we call overlap centrality, to characterize a rank-dependent correlation
of agents’ positions. Then we have found that the new model of interacting
ranked agents exhibits the unexpected linearity (85) with respect to agent’s
rank $i$ in overlap centrality $O_{i}$ at small supplanting condition, which
could be regarded as a new type of collective phenomenon. Precisely speaking,
when $\mathcal{B}_{1}\neq\mathcal{B}_{2}$, the perfect correlation
corresponding to the linearity mentioned above appears between $O_{i}$ and
rank $i$ in the regime of weak supplanting limit $p\to+0$. Even for small but
non-zero $p$, concrete analysis by exact diagonalization shows that $\phi$ is
very close to the perfect correlation in the case of the Potts energy if
$\beta$ is far from $0$.

One might ask about the meaning of perfect correlation. So far, we do not have
clear answer this question. However, one might have speculation that the
perfect correlation implies divergence of characteristic scale in ranking.
From this viewpoint, it could be intriguing to quantify such a characteristic
scale as future studies.

Another problem on the singular behaviors of $\phi$ around $\beta=0$ is to
identify the effects which essentially cause those singular behaviors.
Compared to the equilibrium Potts model, the model with supplanting process
does not have permutation symmetry in terms of agents, and also not have
detailed balance. In our derivation of perfect correlation, broken permutation
symmetry is one of essential parts, but we are not aware of the effects from
broken detailed balance. Concerning this point, one can consider an
equilibrium model keeping with detailed balance without permutation symmetry
by, for example, an energy function $\sum_{i,j}J_{i,j}\delta(x_{i},x_{j})$,
where $J_{i,j}$ is a function of agents $i,j$ such as $i\times j$. If one
could prove that there does not exist the singularity of $\phi$ for such
general equilibrium models, one could presumably expect that both permutation
symmetry breaking and broken detailed balance are essential for causing the
singularity. Such a motivation has been raised for broken rotational symmetry
observed in active matter V-Tasaki . Indeed, it has been proven for a general
class of systems having potentials dependent on position and velocity that
rotational symmetry cannot be broken in equilibrium. This implies that the
observed phase transitions associating with broken rotational symmetry in
active matters are caused purely by nonequilibrium effects such as broken
detailed balance. We will need somewhat similar ideas. We remark that the term
permutation symmetry has been used in this paper in various manners depending
on the quantity to which the term is applied. See the list of the various ways
of the term in Table 1.

| Class of Quantity |  | Definition for the quantity
---
to be permutation symmetric
(a) | function $f(i_{1},i_{2},\ldots,i_{n})$ depending on ranks $i_{1},i_{2},\ldots,i_{n}$ | $f(\sigma(i_{1}),\sigma(i_{2}),\ldots,\sigma(i_{n}))=f(i_{1},i_{2},\ldots,i_{n})$ | for any $\sigma\in\mathfrak{S}_{N}$
(b) | function $f(\bm{x})$ depending on agents’ configuration $\bm{x}\in X^{N}$ | $f(\sigma(\bm{x}))=f(\bm{x})$
(c) | state vector $\ket{\psi}\in H_{X}^{\otimes N}$ | $\widehat{\Pi}_{\sigma}\ket{\psi}=\ket{\psi}$
(d) | linear operator $\widehat{A}$ from $H_{X}^{\otimes N}$ to itself | $\widehat{\Pi}_{\sigma}\widehat{A}\widehat{\Pi}_{\sigma}^{-1}=\widehat{A}$
Table 1:  How to use the term permutation symmetry depending on the classes of
quantity. For example of each class, the case (a) is applied to (IV.2), (b) to
(1), (c) to (67), and (d) to (39).

Let us briefly discuss the obtained results in the context of linear response
theory. If one defines a susceptibility of $M$ as
$\chi(p):={\mathrm{d}M}/{\mathrm{d}p}$ in terms of $p$, we can obtain

$\displaystyle\chi(0)$
$\displaystyle=\frac{N-1}{2N}(\mathcal{B}_{1}+\mathcal{B}_{2})+\frac{(N-1)(N-2)}{2N}\mathcal{B}_{3}$
$\displaystyle\qquad+\partial_{p}C(\beta,0)\mathcal{B}_{0}.$ (107)

Note that the coefficient $\partial_{p}C(\beta,0)$ of $\mathcal{B}_{0}$ is
described as

$\displaystyle-\binom{N}{2}\sum_{\bm{x}\in
X^{N}}\bra{\bm{x}}\widehat{B}_{i_{0},i_{1}}\ket{P_{\mathrm{can}}(\beta)}$
(108)

for some $1\leq i_{0}<i_{1}\leq N$. By a similar computation to (IV.2), we can
find that (108) does not depend on $i_{0}$, $i_{1}$, and $p$. So far, it is
not obvious for us to connect those quantities with the other known
quantities, which remains as an open problem.

Whereas our above discussions focus on the case of small $p\ll 1$, we move
onto the case of any $p$. The result of exact diagonalization implies that
perfect correlation with $\phi=1$ would hold for negative $\beta$ sufficiently
far from $0$. This behavior is of interest in that the strong correlation
effect originating from hard core repulsion may stabilize perfect correlation
with $\phi=1$. However, our strategy of the perturbation with respect to $p$
is unavailable for not small $p\not\ll 1$. For this reason, it is not clear
whether the perfect correlation is derived by use of permutation symmetry in a
similar way to the case of $p\ll 1$. In order to tackle this problem, the
situation with restricted values of $L$ and $N$ is to be considered. As an
example, let us take the case satisfying $N=L+1$ with the Potts energy. In
this case, if the repulsion is sufficiently strong, then each site is occupied
by at least an agent, and there is a single site occupied by two agents. Then,
one of the two agents occupying the site can hop in accordance with the
equilibrium dynamics, while all of the other agents cannot hop effectively.
This leads to reduction of the transition of states and could help us to
analyze the overlap centrality of this model. Note that the above discussion
is based on the Potts energy, and is not necessarily applied to the other case
with general energy form. It remains for future work to perform further
numerical calculation for energy functions other than the Potts energy as well
as to explore analytic methods for general $p$.

Let us briefly discuss the possibility of phase transition lines in parameter
space $(\beta,p)$ for non-zero $p$. First, the equilibrium phase transition
point $\beta=\beta_{c}$ with $p=0$ for the case of Potts energy might extend
toward non-zero $p$ as a nonequilibrium phase transition line
$\beta=\beta_{c}(p)$ where $M$ shows singular jump. Second, the point
$\beta=0$ with $p\to+0$ where correlation coefficient shows singularity might
also extend toward non-zero $p$ as another nonequilibrium phase transition
line $\beta=\beta_{0}(p)$ where $\phi$ shows a singular jump. However, since
our analysis is limited to the parameter region close to $p=0$, it is
necessary to perform large scale numerical simulations or develop another
analytical methods in order to capture the true limiting behavior of large
system size for non-zero $p$. This remains to be an intriguing future study.

From a mathematical viewpoint, beta decomposition of the transition matrix
could be useful in more general context. This expression gives the exact
lowest order of transition matrix in terms of $p$. Nevertheless, the concrete
expression of it looks complicated and then it would not be straightforward to
use the expression for a given purpose. Further, it should be noted that the
minor modification of supplanting process prevents us to obtain the beta
decomposition. In this sense, the present version of the supplanting process
can be regarded as a specially tractable case. It remains as an open question
how beta decomposition can be applied to calculation of the other quantities
and whether one can find other tractable cases in this direction.

Let us remark on the direct relation between eigenvector centrality and
strength centrality. Unifying two relations among centralities discussed in
Section IV.5, one can also have another conclusion in the context of network
theory. Consider a complete graph whose edges are weighted by the same value
with small fluctuation. Then the neighbor matrix $\mathcal{R}$ whose entries
are weights of edges has a decomposition
$\tilde{\mathcal{R}}^{(1)}+\tilde{\mathcal{R}}^{(2)}$ similar to (247), and
the graph holds a linear relation similar to (105) between the strength
centrality and the eigenvector centrality.

Lastly, we mention the results in this paper relevant to behaviors of members
in a group of primate species. Ranking has been widely known to be one
characteristic structure which primates species has when they live as a group
primatebook . Indeed, it has been found that ranking affects the spatial
location of the members in a group rank_location1 ; rank_location2 ;
pecentrality1 ; pecentrality2  and the distance between the members
rank_distance1 ; rank_distance2 . The overlap centrality observed in the model
proposed in this paper might shed light on how one could estimate the ranking
structures of such a group and its environmental condition.

###### Acknowledgements.

The authors thank T. Yamaguchi for discussing his field study about Japanese
macaques and telling us the relevant references to our work.

##  Appendix A Derivation of two decompositions of transition matrix

In this section, we would like to explain, in detail, the derivations of two
decompositions of transition matrix $\widehat{T}$, which we call supplanting
decomposition and beta decomposition as mentioned in Section IV.1. To do this,
we give a detailed account of our models in terms of operators. Hereafter, in
order to make this Appendix self-contained, many definitions in the main text
are repeated.

## Notations

For a set $\bm{S}$, the number of elements is denoted as $\\#\bm{S}$. The
Kronecker’s delta $\delta(i,j)$ is defined as

$\delta(i,j)=\begin{dcases}1&(\text{if $i=j$})\\\ 0&(\text{if $i\neq j$}).\\\
\end{dcases}$ (109)

We consider any vector space appearing in this section as complex vector
space. We denote the set of complex numbers as $\mathbb{C}$, and the set of
real numbers as $\mathbb{R}$.

###  A.1 Describing the model

In this subsection, as a preliminary, we introduce basic concepts which are
necessary to explain the model.

####  A.1.1 Agents and states

We consider $N\geq 2$ agents labeled by $1,2,\dots,N$. They have a total
ordering (i.e. linear dominance) meaning that if $1\leq i<j\leq N$, agent $i$
is higher than agent $j$. The set of all agents is written as

$[N]\coloneqq\\{1,2,\dots,N\\}.$ (110)

The agents lie in the lattice
$X={\mathbb{Z}}/L{\mathbb{Z}}=\\{0,1,\dots,L-1\\}$ with $L\geq 3$. For an
agent $i$, the position is written as $x_{i}\in X$. Thus, the configuration
space of agents is described by

$X^{N}=\left\\{\bm{x}=(x_{1},x_{2},\dots,x_{N})\mid x_{i}\in X\right\\}.$
(111)

####  A.1.2 Hopping map

We define a map $f_{i}^{+}\colon X^{N}\to X^{N}$ (resp. $f_{i}^{-}\colon
X^{N}\to X^{N}$) of configurations as the increment (resp. decrement) of
position of agent $i$ under periodic boundary condition. Explicitly, we have

$\displaystyle f_{i}^{+}(x_{1},x_{2},\dots,x_{i},\dots,x_{N})$
$\displaystyle\coloneqq(x_{1},x_{2},\dots,x_{i}+1,\dots,x_{N}),$ (112)
$\displaystyle f_{i}^{-}(x_{1},x_{2},\dots,x_{i},\dots,x_{N})$
$\displaystyle\coloneqq(x_{1},x_{2},\dots,x_{i}-1,\dots,x_{N}).$ (113)

####  A.1.3 The permutation-invariant energy function

We take an energy function $E(\bm{x})=E(x_{1},x_{2},\dots,x_{N})$ which is
symmetric in the following sense:

$E(x_{1},x_{2},\dots,x_{N})=E(x_{\sigma(1)},x_{\sigma(2)},\dots,x_{\sigma(N)})$
(114)

for any permutation $\sigma\in\mathfrak{S}_{N}$ of $N$ elements in $[N]$. The
symmetry will be used in Appendix A.2.5 and A.4.5, and is essential in (IV.2)
and (IV.2). For example, in (2), we take the normalized Potts model

$\displaystyle E(\bm{x})$
$\displaystyle=-\frac{(L-1)\log(L-1)}{N(L-2)}\sum_{1\leq i,j\leq
N}\delta(x_{i},x_{j})$ (115)

as an energy function.

To simplify the notation, we introduce the $(i,d)$-th difference of the energy
function $E$ for $1\leq i\leq N$ and $d\in\\{+,-\\}$:

$\displaystyle\mathcal{D}_{i}^{d}E(\bm{x})$ $\displaystyle\coloneqq
E(f_{i}^{d}\bm{x})-E(\bm{x}).$ (116)

####  A.1.4 Describing the hopping by equilibrium dynamics

First we describe the hopping by equilibrium dynamics as follows.

  1. (i)

Determine an agent $1\leq i\leq N$ with equal probability $1/N$.

  2. (ii)

Choose a direction $d=+$ or $-$ with equal probability $1/2$.

  3. (iii)

Decide whether the agent $i$ stays or hops to the direction $d$.

     * •

The probability that the agent $i$ hops to direction $d$ is

$\frac{1}{1+\exp(\beta\mathcal{D}_{i}^{d}E(\bm{x}))}.$ (117)

     * •

The probability that the agent $i$ stays is

$1-\frac{1}{1+\exp(\beta\mathcal{D}_{i}^{d}E(\bm{x}))}=\frac{1}{1+\exp(-\beta\mathcal{D}_{i}^{d}E(\bm{x}))}.$
(118)

For a graphical explanation, see Fig. 10.

Figure 10: Probability tree of hopping by equilibrium dynamics

####  A.1.5 Describing the hopping by supplanting process

For a given configuration $\bm{x}\in X^{N}$, an agent $i\in[N]$, and a
direction $d\in\\{+,-\\}$, we define the set $S(\bm{x},i,d)\subseteq[N]$ as
follows:

$\displaystyle S(\bm{x},i,\pm)$ $\displaystyle\coloneqq\left\\{i<j\leq N\mid
x_{j}=x_{i}\pm 1\right\\}.$ (119)

In other words, $S(\bm{x},i,d)$ is defined as a set of agents $j\in[N]$
satisfying the two conditions: (1) $j\in S(\bm{x},i,d)$ is lower than $i$ (in
other words $j>i$), and (2) sits on the site $x_{i}+1$ (when $d=+$) or
$x_{i}-1$ (when $d=-$).

Let us fix the supplanting rate $0\leq p<\infty$. After the step (iii) of the
first hopping, we define the supplanting process as described below.

  1. (iv)

One of the following events occurs in the probabilities described below.

     * •

The agent $j\in S(\bm{x},i,d)$ hops with the probability

$\frac{p}{1+p\\#S(\bm{x},i,d)}.$ (120)

     * •

In a probability

$\frac{1}{1+p\\#S(\bm{x},i,d)},$ (121)

no one hops.

  2. (v)

If a hopping occurs in (iv), choose a direction $d^{\prime}=+$ or $-$ of the
hopping of the agent $j$ with uniform probability $1/2$.

Note that if $p=0$ or $S(\bm{x},i,d)$ is empty, then no supplanting occurs.
The following diagram in Fig. 11 describes the probability tree after the
first hopping of the agent $i$.

Figure 11: Probability tree of supplanting process after the first hopping

###  A.2 Transition matrices

In this subsection, we write down the transition matrices as linear operators
on a state space.

####  A.2.1 State vector spaces

The state space $H_{X}$ is the complex vector space with a basis

$|0\rangle,|1\rangle,\dots,|{L-1}\rangle$ (122)

corresponding to sites $0,1,\dots,L-1\in X$, respectively.

Similarly the multi-state space $H_{X^{N}}$ is the complex vector space with a
basis $|\bm{x}\rangle$ corresponding to configurations $\bm{x}\in X^{N}$. It
is identified to the tensor space $H^{\otimes N}_{X}$ with a map

$H_{X^{N}}\to H_{X}^{\otimes N}\coloneqq\underbrace{H_{X}\otimes
H_{X}\otimes\dots\otimes
H_{X}}_{N}\quad;\quad|\bm{x}\rangle\mapsto|x_{1}\rangle\otimes|x_{2}\rangle\otimes\dots\otimes|x_{N}\rangle.$
(123)

With this identification, we freely use tensor notation to write down
operators.

We introduce the inner product $\braket{\cdot}{\cdot}$ on $H_{X}$ such that
the basis $(|x\rangle)_{x\in X}$ is orthonormal. This induces an inner product
on $H_{X}^{\otimes N}$, and the basis $(|\bm{x}\rangle)_{\bm{x}\in X^{N}}$ on
$H_{X}^{\otimes N}$ is orthonormal. We use the same symbol
$\langle\cdot|\cdot\rangle$ to write this induced inner product on
$H_{X}^{\otimes N}$.

####  A.2.2 Coefficients of operators

An operator $\widehat{A}\colon H_{X}^{\otimes N}\to H_{X}^{\otimes N}$ is
described by its coefficients. The $(\bm{x},\bm{y})$-th coefficient of
$\widehat{A}$ is written by $\langle\bm{y}|\widehat{A}|\bm{x}\rangle$. In
other words, we can write

$\widehat{A}|\bm{x}\rangle=\sum_{\bm{y}\in
X^{N}}|\bm{y}\rangle\langle\bm{y}|\widehat{A}|\bm{x}\rangle.$ (124)

####  A.2.3 Shift operators

As pieces of transition matrices, we define notations of shift operators.

The shift operators $\widehat{\Delta}_{i}^{+}$ and $\widehat{\Delta}_{i}^{-}$
on $H_{X}^{\otimes N}$ for $1\leq i\leq N$ are defined as

$\displaystyle\widehat{\Delta}_{i}^{+}|\bm{x}\rangle$
$\displaystyle\coloneqq|f_{i}^{+}\bm{x}\rangle$ (125)
$\displaystyle=|x_{1}\rangle\otimes|x_{2}\rangle\dots\otimes|x_{i}+1\rangle\otimes\dots\otimes|x_{N}\rangle,$
(126) $\displaystyle\widehat{\Delta}_{i}^{-}|\bm{x}\rangle$
$\displaystyle\coloneqq|f_{i}^{-}\bm{x}\rangle$ (127)
$\displaystyle=|x_{1}\rangle\otimes|x_{2}\rangle\dots\otimes|x_{i}-1\rangle\otimes\dots\otimes|x_{N}\rangle,$
(128)

for $\bm{x}=(x_{1},x_{2},\dots,x_{N})\in X^{N}$. In terms of coefficients, the
followings hold:

$\displaystyle\langle\bm{y}|\widehat{\Delta}_{i}^{+}|\bm{x}\rangle$
$\displaystyle=\begin{dcases}1&(\text{if $\bm{y}=f_{i}^{+}\bm{x}$})\\\
0&(\text{otherwise}),\end{dcases}$ (129)
$\displaystyle\langle\bm{y}|\widehat{\Delta}_{i}^{-}|\bm{x}\rangle$
$\displaystyle=\begin{dcases}1&(\text{if $\bm{y}=f_{i}^{-}\bm{x}$})\\\
0&(\text{otherwise}).\end{dcases}$ (130)

####  A.2.4 Projection operators

We denote the identity operator on the state space $H_{X}$ as
$\mathop{\widehat{\mathrm{id}}}\nolimits_{H}$. Then we can write the identity
operator on the multi-state space $H_{X}^{\otimes N}$ as
$\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes N}$.

The projection operator $\widehat{\Xi}_{i}^{y}$ on $H_{X}^{\otimes N}$ for
$1\leq i\leq N$ and $y\in X$ is defined as

$\widehat{\Xi}_{i}^{y}|\bm{x}\rangle\coloneqq\begin{dcases}|\bm{x}\rangle&(\text{if
$x_{i}=y$})\\\ 0&(\text{otherwise}).\end{dcases}$ (131)

This projection operator checks whether $x_{i}=y$ or not; if $x_{i}=y$ return
$|\bm{x}\rangle$, otherwise return the zero vector. For a configuration
$\bm{x}=(x_{1},x_{2},\dots,x_{N})\in X^{N}$, we define

$\widehat{\Xi}^{\bm{x}}\coloneqq\prod_{1\leq j\leq
N}\widehat{\Xi}_{j}^{x_{j}}.$ (132)

It satisfies

$\widehat{\Xi}^{\bm{y}}|\bm{x}\rangle=\begin{dcases}|\bm{x}\rangle&(\text{if
$\bm{x}=\bm{y}$})\\\ 0&(\text{otherwise}).\end{dcases}$ (133)

and, in terms of coefficients,

$\displaystyle\langle\bm{y}|\widehat{\Xi}^{\bm{z}}|\bm{x}\rangle$
$\displaystyle=\begin{dcases}1&(\text{if $\bm{y}=\bm{x}=\bm{z}$})\\\
0&(\text{otherwise}).\end{dcases}$ (134)

The operator $\widehat{\Xi}^{\bm{z}}$ checks whether $\bm{x}=\bm{z}$.

We define some other projection operators: first, for $x\in X$ and
$\emptyset\neq\bm{S}\subseteq[N]$,

$\displaystyle\widehat{\Xi}_{\bm{S}}$
$\displaystyle\coloneqq\begin{dcases}\sum_{x\in
X}\prod_{i\in\bm{S}}\widehat{\Xi}_{i}^{x}&(\text{if $\bm{S}\neq\emptyset$})\\\
\mathop{\widehat{\mathrm{id}}}\nolimits^{\otimes N}_{H}&(\text{if
$\bm{S}=\emptyset$}).\end{dcases}$ (135)

The operator $\widehat{\Xi}_{\bm{S}}$ checks whether all of the agents in
$\bm{S}$ sit on the same site. Explicitly,

$\displaystyle\widehat{\Xi}_{\bm{S}}|\bm{x}\rangle=\begin{dcases}|\bm{x}\rangle&(\text{if
$x_{i}=x_{j}$ for any $i,j\in\bm{S}$})\\\ 0&(\text{otherwise}).\end{dcases}$
(136)

If $\bm{S}$ is empty, it checks nothing; it is the identity operator.

Another projection operator $\widehat{\Upsilon}_{i;\bm{S}}$ for $1\leq i\leq
N$ and $\bm{S}\subseteq\\{i+1,i+2,\dots,N\\}$ checks two conditions: (1)
$x_{j}=x_{i}$ if $j\in\bm{S}$, and (2) $x_{j^{\prime}}\neq x_{i}$ if
$j^{\prime}\not\in\bm{S}$ and $j^{\prime}>i$. Explicitly, it is defined as

$\displaystyle\widehat{\Upsilon}_{i;\bm{S}}|\bm{x}\rangle$
$\displaystyle\coloneqq\begin{dcases}|\bm{x}\rangle&(\text{if $x_{j}=x_{i}$
for $j\in\bm{S}$, and
$x_{i}\not\in\\{x_{j^{\prime}}\\}_{\begin{subarray}{c}j^{\prime}>i\\\
j^{\prime}\not\in\bm{S}\end{subarray}}$})\\\
0&(\text{otherwise}).\end{dcases}$ (137)

In terms of $\widehat{\Xi}_{i}^{x}$ and $\widehat{\Xi}_{\bm{S}}$, this
operator is written as

$\displaystyle\widehat{\Upsilon}_{i;\bm{S}}$
$\displaystyle=\begin{dcases}\sum_{x\in
X}\widehat{\Xi}_{i}^{x}\cdot\prod_{j\in\bm{S}}\widehat{\Xi}_{j}^{x}\cdot\prod_{\begin{subarray}{c}j^{\prime}>i\\\
j^{\prime}\not\in\bm{S}\end{subarray}}(\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N}-\widehat{\Xi}_{j^{\prime}}^{x})&(\text{if $\bm{S}\neq\emptyset$ and
$\bm{S}\neq\\{i+1,i+2,\dots,N\\}$})\\\ \sum_{x\in
X}\widehat{\Xi}_{i}^{x}\cdot\prod_{\begin{subarray}{c}j^{\prime}>i\end{subarray}}(\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N}-\widehat{\Xi}_{j^{\prime}}^{x})&(\text{if $\bm{S}=\emptyset$ and $i\leq
N-1$})\\\ \sum_{x\in
X}\widehat{\Xi}_{i}^{x}\cdot\prod_{j\in\bm{S}}\widehat{\Xi}_{j}^{x}&(\text{if
$\bm{S}\neq\emptyset$ and $\bm{S}=\\{i+1,i+2,\dots,N\\}$})\\\ \sum_{x\in
X}\widehat{\Xi}_{i}^{x}&(\text{if $\bm{S}=\emptyset$ and $i=N$})\end{dcases}$
(138)
$\displaystyle=\sum_{\begin{subarray}{c}\bm{S}\subseteq\bm{S}^{\prime}\subseteq\\{i+1,\dots,N\\}\end{subarray}}(-1)^{\\#\bm{S}^{\prime}-\\#\bm{S}}\widehat{\Xi}_{\bm{S}^{\prime}}.$
(139)

A relation between these operators is

$\displaystyle\widehat{\Xi}_{\\{i\\}\cup\bm{S}}$
$\displaystyle=\sum_{\begin{subarray}{c}\bm{S}\subseteq\bm{S}^{\prime}\subseteq\\{i+1,\dots,N\\}\end{subarray}}\widehat{\Upsilon}_{i;\bm{S}}.$
(140)

####  A.2.5 Permutation operators

For a permutation $\sigma\in\mathfrak{S}_{N}$ of $[N]=\\{1,2,\dots,N\\}$, we
define

$\sigma(\bm{x})\coloneqq(x_{\sigma(1)},x_{\sigma(2)},\dots,x_{\sigma(n)})$
(141)

for $\bm{x}=(x_{1},x_{2},\dots,x_{n})\in X^{N}$. Let $\widehat{\Pi}_{\sigma}$
be the corresponding matrix to $\sigma^{-1}$. Explicitly,

$\displaystyle\widehat{\Pi}_{\sigma}|(x_{1},x_{2},\dots,x_{n})\rangle$
$\displaystyle=|\sigma^{-1}(x_{1},x_{2},\dots,x_{n})\rangle$ (142)
$\displaystyle=|(x_{\sigma^{-1}(1)},x_{\sigma^{-1}(2)},\dots,x_{\sigma^{-1}(N)})\rangle.$
(143)

For an operator $\widehat{A}$ on $H_{X}^{\otimes N}$, the daggered symbol
$\widehat{A}^{\dagger}$ denotes the Hermitian conjugate of $\widehat{A}$. Then
we have
$\widehat{\Pi}_{\sigma}^{\dagger}=\widehat{\Pi}_{\sigma}^{-1}=\widehat{\Pi}_{\sigma^{-1}}$.
We also have some commutation relations as follows:

$\displaystyle\widehat{\Delta}_{j}^{\pm}\widehat{\Pi}_{\sigma}$
$\displaystyle=\widehat{\Pi}_{\sigma}\widehat{\Delta}_{\sigma(j)}^{\pm},$
(144) $\displaystyle\widehat{\Xi}_{j}^{x}\widehat{\Pi}_{\sigma}$
$\displaystyle=\widehat{\Pi}_{\sigma}\widehat{\Xi}_{\sigma(j)}^{x},$ (145)
$\displaystyle\widehat{\Xi}_{\bm{S}}\widehat{\Pi}_{\sigma}$
$\displaystyle=\widehat{\Pi}_{\sigma}\widehat{\Xi}_{\sigma(\bm{S})}.$ (146)

We remark that $\widehat{\Upsilon}_{i;\bm{S}}$ has a factor
$\prod_{\begin{subarray}{c}j>i\\\
j\not\in\bm{S}\end{subarray}}(\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N}-\widehat{\Xi}_{x}^{j})$. Since this factor involves an ordering $j>i$, it
does not satisfy simple relation in terms of permutation operators. A key
procedure in our calculation is to rewrite an operator
$\widehat{\Upsilon}_{i;\bm{S}}$ in terms of
$\widehat{\Xi}_{\\{i\\}\cup\bm{S}}$. This enables us to use the commutation
relations in terms of permutations.

Besides, since the energy function depends only on the numbers of agents on
each site, the energy function does not depend on the labeling of the agents.
In other terms,

$\displaystyle
E(f_{\sigma(i)}^{d}(x_{\sigma^{-1}(1)},x_{\sigma^{-1}(2)},\dots,x_{\sigma^{-1}(N)}))$
$\displaystyle=E(f_{i}^{d}(x_{1},x_{2},\dots,x_{N})).$ (147)

####  A.2.6 Other commutation relations of operators

The shift operators (resp. the projection operators) are commutative at each
other, while a shift operator and a projection operator does not necessarily
commute. Explicitly, it holds that

$\displaystyle\widehat{\Xi}_{i}^{x}\widehat{\Delta}_{j}^{+}$
$\displaystyle=\begin{dcases}\widehat{\Delta}_{j}^{+}\widehat{\Xi}_{i}^{x}&(\text{if
$i\neq j$})\\\ \widehat{\Delta}_{j}^{+}\widehat{\Xi}_{i}^{x-1}&(\text{if
$i=j$}),\end{dcases}$ (148)
$\displaystyle\widehat{\Xi}_{i}^{x}\widehat{\Delta}_{j}^{-}$
$\displaystyle=\begin{dcases}\widehat{\Delta}_{j}^{-}\widehat{\Xi}_{i}^{x}&(\text{if
$i\neq j$})\\\ \widehat{\Delta}_{j}^{-}\widehat{\Xi}_{i}^{x+1}&(\text{if
$i=j$}).\end{dcases}$ (149)

####  A.2.7 Transition matrix for the first hopping

According to the description in Appendix A.1.4, the transition matrix
$\widehat{T}_{0}$ for the first hopping can be described as

$\widehat{T}_{0}\coloneqq\sum_{1\leq i\leq
N}\frac{1}{N}\sum_{d=\pm}\frac{1}{2}\sum_{\bm{x}\in
X^{N}}\Big{(}\frac{\widehat{\Delta}_{i}^{d}}{1+\exp(\beta\mathcal{D}_{i}^{d}E(\bm{x}))}+\frac{\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N}}{1+\exp(-\beta\mathcal{D}_{i}^{d}E(\bm{x}))}\Big{)}\widehat{\Xi}^{\bm{x}}.$
(150)

For the latter use, we introduce some symbols. First, for $\bm{x}\in X^{N}$,
$1\leq i\leq N$, and $d\in\\{+,-\\}$, we write the coefficient of
$\widehat{T}_{0}$ as

$c_{\beta}(\bm{x},i,d)\coloneqq\frac{1}{2N(1+\exp(\beta\mathcal{D}_{i}^{d}E(\bm{x})))}.$
(151)

Then we have

$\frac{1}{2N}-c_{\beta}(\bm{x},i,d)=\frac{1}{2N(1+\exp(-\beta\mathcal{D}_{i}^{d}E(\bm{x})))}.$
(152)

Since $\exp(x)>0$ for any real $x\in\mathbb{R}$, we can estimate

$0<c_{\beta}(\bm{x},i,d)\leq\frac{1}{2N}$ (153)

for any $\beta,E,\bm{x},i,d,N$, and $L$. We define a set of operators: for
$d\in\\{+,-\\}$,

$\displaystyle\widehat{T}_{0,\text{move}}^{i,d}$
$\displaystyle\coloneqq\sum_{\bm{x}\in
X^{N}}c_{\beta}(\bm{x},i,d)\widehat{\Delta}_{i}^{d}\widehat{\Xi}^{\bm{x}},$
(154) $\displaystyle\widehat{T}_{0,\text{stay}}^{i,d}$
$\displaystyle\coloneqq\sum_{\bm{x}\in
X^{N}}\left(\frac{1}{2N}-c_{\beta}(\bm{x},i,d)\right)\widehat{\Xi}^{\bm{x}},$
(155) $\displaystyle\widehat{T}_{0,\text{move}}^{i}$
$\displaystyle\coloneqq\widehat{T}_{0,\text{move}}^{i,+}+\widehat{T}_{0,\text{move}}^{i,-},$
(156) $\displaystyle\widehat{T}_{0,\text{stay}}^{i}$
$\displaystyle\coloneqq\widehat{T}_{0,\text{stay}}^{i,+}+\widehat{T}_{0,\text{stay}}^{i,-}.$
(157)

Then we have

$\displaystyle\widehat{T}_{0}$ $\displaystyle=\sum_{1\leq i\leq
N}(\widehat{T}_{0,\text{move}}^{i}+\widehat{T}_{0,\text{stay}}^{i})$ (158)
$\displaystyle=\sum_{\begin{subarray}{c}1\leq i\leq N\\\
d=\pm\end{subarray}}(\widehat{T}_{0,\text{move}}^{i,d}+\widehat{T}_{0,\text{stay}}^{i,d}).$
(159)

Note that, using relations in Appendix A.2.5, we have

$\displaystyle\widehat{\Pi}_{\sigma}^{\dagger}\widehat{T}_{0,\text{move}}^{i,d}\widehat{\Pi}_{\sigma}$
$\displaystyle=\widehat{T}_{0,\text{move}}^{\sigma(i),d},$ (160)
$\displaystyle\widehat{\Pi}_{\sigma}^{\dagger}\widehat{T}_{0,\text{stay}}^{i,d}\widehat{\Pi}_{\sigma}$
$\displaystyle=\widehat{T}_{0,\text{stay}}^{\sigma(i),d},$ (161)
$\displaystyle\widehat{\Pi}_{\sigma}^{\dagger}\widehat{T}_{0}\widehat{\Pi}_{\sigma}$
$\displaystyle=\widehat{T}_{0}.$ (162)

####  A.2.8 Transition matrix for supplanting process

According to Appendix A.1.5, in order to describe the transition matrix
$\widehat{T}$ for the supplanting process, it suffices to replace
$\widehat{\Delta}_{i}^{d}\widehat{\Xi}^{\bm{x}}$ appearing in
$\widehat{T}_{{0,\text{move}}}^{i}$ by

$\displaystyle\sum_{j\in
S(\bm{x},i,d)}\frac{1}{\\#S(\bm{x},i,d)}\sum_{d^{\prime}=\pm}\frac{1}{2}\Big{(}\frac{p\\#S(\bm{x},i,d)}{1+p\\#S(\bm{x},i,d)}\widehat{\Delta}_{j}^{d^{\prime}}+\frac{1}{1+p\\#S(\bm{x},i,d)}\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N}\Big{)}\widehat{\Delta}_{i}^{d}\widehat{\Xi}^{\bm{x}}$ (163)
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}j\in S(\bm{x},i,d)\\\
d^{\prime}=\pm\end{subarray}}\frac{1}{2\\#S(\bm{x},i,d)}\Big{(}\frac{p\\#S(\bm{x},i,d)}{1+p\\#S(\bm{x},i,d)}\widehat{\Delta}_{j}^{d^{\prime}}+\frac{1}{1+p\\#S(\bm{x},i,d)}\mathop{\widehat{\mathrm{id}}}\nolimits_{H}^{\otimes
N}\Big{)}\widehat{\Delta}_{i}^{d}\widehat{\Xi}^{\bm{x}}$ (164)
$\displaystyle=$
$\displaystyle\Bigg{(}\widehat{\Delta}_{i}^{d}+\sum_{\begin{subarray}{c}j\in
S(\bm{x},i,d)\\\
d^{\prime}=\pm\end{subarray}}\frac{p}{2(1+p\\#S(\bm{x},i,d))}(\widehat{\Delta}_{j}^{d^{\prime}}-\mathop{\widehat{\mathrm{id}}}\nolimits^{\otimes
N}_{H})\widehat{\Delta}_{i}^{d}\Bigg{)}\widehat{\Xi}^{\bm{x}}.$ (165)

Thus, by (159), the transition matrix $\widehat{T}$ is written as

$\displaystyle\widehat{T}$ $\displaystyle=\sum_{\begin{subarray}{c}1\leq i\leq
N\\\ d=\pm\end{subarray}}\sum_{\bm{x}\in
X^{N}}\Big{[}c_{\beta}(\bm{x},i,d)\Big{(}\widehat{\Delta}_{i}^{d}+\sum_{\begin{subarray}{c}j\in
S(\bm{x},i,d)\\\
d^{\prime}=\pm\end{subarray}}\frac{p}{2(1+p\\#S(\bm{x},i,d))}(\widehat{\Delta}_{j}^{d^{\prime}}-\mathop{\widehat{\mathrm{id}}}\nolimits^{\otimes
N}_{H})\widehat{\Delta}_{i}^{d}\Big{)}\widehat{\Xi}^{\bm{x}}+\widehat{T}_{0,\text{stay}}^{i,d}\Big{]}$
(166) $\displaystyle=\widehat{T}_{0}+\sum_{\bm{x}\in
X^{N}}\sum_{\begin{subarray}{c}1\leq i\leq N\\\
d=\pm\end{subarray}}\sum_{\begin{subarray}{c}j\in S(\bm{x},i,d)\\\
d^{\prime}=\pm\end{subarray}}\frac{pc_{\beta}(\bm{x},i,d)}{2(1+p\\#S(\bm{x},i,d))}(\widehat{\Delta}_{j}^{d^{\prime}}-\mathop{\widehat{\mathrm{id}}}\nolimits^{\otimes
N}_{H})\widehat{\Delta}_{i}^{d}\widehat{\Xi}^{\bm{x}}.$ (167)

####  A.2.9 Coefficients of two operators

From the descriptions of operators
$\widehat{\Delta}_{i}^{d},\widehat{\Xi}^{\bm{x}}$ in (129), (130), and (134),
we obtain

$\displaystyle\langle\bm{y}|\widehat{\Delta}_{i}^{d}\widehat{\Xi}^{\bm{z}}|\bm{x}\rangle$
$\displaystyle=\begin{dcases}1&(\text{if $\bm{x}=\bm{z}$ and
$\bm{y}=f_{i}^{d}\bm{z}$})\\\ 0&(\text{otherwise}),\end{dcases}$ (168)
$\displaystyle\langle\bm{y}|\widehat{\Delta}_{j}^{d^{\prime}}\widehat{\Delta}_{i}^{d}\widehat{\Xi}^{\bm{z}}|\bm{x}\rangle$
$\displaystyle=\begin{dcases}1&(\text{if $\bm{x}=\bm{z}$ and
$\bm{y}=f_{j}^{d^{\prime}}f_{i}^{d}\bm{z}$})\\\
0&(\text{otherwise}).\end{dcases}$ (169)

Using this and (159), the coefficients of $\widehat{T}_{0}$ is

$\displaystyle\langle\bm{y}|\widehat{T}_{0}|\bm{x}\rangle$
$\displaystyle=\begin{dcases}c_{\beta}(\bm{x},i,d)&(\text{if
$\bm{y}=f_{i}^{d}\bm{x}$})\\\ 1-\sum_{\begin{subarray}{c}1\leq i\leq N\\\
d=\pm\end{subarray}}c_{\beta}(\bm{x},i,d)&(\text{if $\bm{y}=\bm{x}$})\\\
0&(\text{otherwise}).\end{dcases}$ (170)

Similarly by (167), the coefficients of $\widehat{T}$ is

$\displaystyle\langle\bm{y}|\widehat{T}|\bm{x}\rangle$
$\displaystyle=\begin{dcases}\frac{pc_{\beta}(\bm{x},i,d)}{2(1+np)}&(\text{if
$\bm{y}=f_{j}^{d^{\prime}}f_{i}^{d}\bm{x}$ and $j\in S(\bm{x},i,d)$})\\\
\frac{2pc_{\beta}(\bm{x},i,d)}{2(1+np)}&(\text{if
$\bm{y}=f_{i}^{d}\bm{x}$})\\\ 1-\sum_{\begin{subarray}{c}1\leq i\leq N\\\
d=\pm\end{subarray}}c_{\beta}(\bm{x},i,d)&(\text{if $\bm{y}=\bm{x}$})\\\
0&(\text{otherwise}).\end{dcases}$ (171)

####  A.2.10 The broken permutation symmetry of transition matrix

In this subsection, we explain an example of the broken permutation symmetry
of transition matrix $\widehat{T}$ by using commutative relations. Recall
that, for an operator $\widehat{A}$ on $H_{X}^{\otimes N}$, the daggered
symbol $\widehat{A}^{\dagger}$ denotes the Hermitian conjugate of
$\widehat{A}$.

By (171), for $i\in[N]$ and $j\in S(\bm{x},i,d)$, we have

$\langle
f_{j}^{d^{\prime}}f_{i}^{d}\bm{x}|\widehat{T}|\bm{x}\rangle=\frac{pc_{\beta}(\bm{x},i,d)}{2(1+p\\#S(\bm{x},i,d))}.$
(172)

On the other hand, for a transposition $\sigma=(i\;j)\in\mathfrak{S}_{N}$, one
has

$\displaystyle\langle
f_{j}^{d^{\prime}}f_{i}^{d}\bm{x}|\widehat{\Pi}_{\sigma}^{\dagger}\widehat{T}\widehat{\Pi}_{\sigma}|\bm{x}\rangle$
$\displaystyle=\langle\bm{x}|\widehat{\Delta}_{i}^{d,{\dagger}}\widehat{\Delta}_{j}^{d^{\prime},{\dagger}}\widehat{\Pi}_{\sigma}^{\dagger}\widehat{T}\widehat{\Pi}_{\sigma}|\bm{x}\rangle$
(173)
$\displaystyle=\langle\bm{x}|(\widehat{\Pi}_{\sigma}\widehat{\Delta}_{j}^{d^{\prime}}\widehat{\Delta}_{i}^{d})^{\dagger}\widehat{T}\widehat{\Pi}_{\sigma}|\bm{x}\rangle.$
(174)

Using the commutative relation (144), one can see

$\displaystyle\langle\bm{x}|(\widehat{\Pi}_{\sigma}\widehat{\Delta}_{j}^{d^{\prime}}\widehat{\Delta}_{i}^{d})^{\dagger}\widehat{T}\widehat{\Pi}_{\sigma}|\bm{x}\rangle$
$\displaystyle=\langle\bm{x}|(\widehat{\Delta}_{\sigma(j)}^{d^{\prime}}\widehat{\Delta}_{\sigma(i)}^{d}\widehat{\Pi}_{\sigma})^{\dagger}\widehat{T}\widehat{\Pi}_{\sigma}|\bm{x}\rangle$
(175)
$\displaystyle=\langle\bm{x}|(\widehat{\Delta}_{i}^{d^{\prime}}\widehat{\Delta}_{j}^{d}\widehat{\Pi}_{\sigma})^{\dagger}\widehat{T}\widehat{\Pi}_{\sigma}|\bm{x}\rangle$
(176)
$\displaystyle=\langle\bm{x}|\widehat{\Pi}_{\sigma}^{\dagger}\widehat{\Delta}_{j}^{d,{\dagger}}\widehat{\Delta}_{i}^{d^{\prime},{\dagger}}\widehat{T}\widehat{\Pi}_{\sigma}|\bm{x}\rangle.$
(177)

Proceeding with the transformation using (141), it follows that

$\displaystyle\langle\bm{x}|\widehat{\Pi}_{\sigma}^{\dagger}\widehat{\Delta}_{j}^{d,{\dagger}}\widehat{\Delta}_{i}^{d^{\prime},{\dagger}}\widehat{T}\widehat{\Pi}_{\sigma}|\bm{x}\rangle$
$\displaystyle=\langle\sigma^{-1}(\bm{x})|\widehat{\Delta}_{j}^{d,{\dagger}}\widehat{\Delta}_{i}^{d^{\prime},{\dagger}}\widehat{T}|\sigma^{-1}(\bm{x})\rangle$
(178) $\displaystyle=\langle
f_{i}^{d^{\prime}}f_{j}^{d}\sigma^{-1}(\bm{x})|\widehat{T}|\sigma^{-1}(\bm{x})\rangle.$
(179)

Since $j\in S(\bm{x},i,d)$, we have $j>i$, hence $i\not\in S(\bm{x},j,d)$.
With (171), we obtain

$\langle
f_{i}^{d^{\prime}}f_{j}^{d}\sigma^{-1}(\bm{x})|\widehat{T}|\sigma^{-1}(\bm{x})\rangle=0.$
(180)

Thus, since $c_{\beta}(\bm{x},i,d)\neq 0$, we obtain the broken permutation
symmetry of $\widehat{T}$ with respect to the permutation of components:

$\langle
f_{j}^{d^{\prime}}f_{i}^{d}\bm{x}|\widehat{\Pi}_{\sigma}^{\dagger}\widehat{T}\widehat{\Pi}_{\sigma}|\bm{x}\rangle\neq\langle
f_{j}^{d^{\prime}}f_{i}^{d}\bm{x}|\widehat{T}|\bm{x}\rangle.$ (181)

This shows that, for any energy function $E(\bm{x})$ as in Appendix A.1.3 and
any transposition $\sigma=(i\;j)\in\mathfrak{S}_{N}$, we find that

$\widehat{\Pi}_{\sigma}^{\dagger}\widehat{T}\widehat{\Pi}_{\sigma}\neq\widehat{T}.$
(182)

In fact, for any non-trivial permutation $\sigma\in\mathfrak{S}_{N}$, we can
show (182). Explicitly, for a non-trivial permutation
$\sigma\in\mathfrak{S}_{N}$, we can find the following four data
$(i,j,\bm{x},d)$ satisfying the following two conditions: (1) $i<j$ with
$\sigma(i)>\sigma(j)$ and (2) $\bm{x}\in X^{N}$ and $d=+$ or $-$ such that
$j\in S(\bm{x},i,d)$. Then, in parallel, the above argument works to show
(181) and thus (182) for the permutation $\sigma$.

###  A.3 The supplanting decompotision

From here to the next section, we introduce two decompositions of the operator
$\widehat{T}$. The idea of our first decomposition of $\widehat{T}$ is to
split the sums by the number $\\#S(\bm{x},i,d)$.

####  A.3.1 The $n$-th term

First we note that a coefficient in (167),

$\frac{pc_{\beta}(\bm{x},i,d)}{2(1+p\\#S(\bm{x},i,d))},$ (183)

does not depend on $j\in S(\bm{x},i,d)$ and $d^{\prime}\in\\{+,-\\}$.
Reminding of (167), equivalently,

$\widehat{T}-\widehat{T}_{0}=\sum_{\bm{x}\in
X^{N}}\sum_{\begin{subarray}{c}1\leq i\leq N\\\
d=\pm\end{subarray}}\sum_{\begin{subarray}{c}j\in S(\bm{x},i,d)\\\
d^{\prime}=\pm\end{subarray}}\frac{pc_{\beta}(\bm{x},i,d)}{2(1+p\\#S(\bm{x},i,d))}(\widehat{\Delta}_{j}^{d^{\prime}}-\mathop{\widehat{\mathrm{id}}}\nolimits^{\otimes
N}_{H})\widehat{\Delta}_{i}^{d}\widehat{\Xi}^{\bm{x}},$ (184)

we define the operators

$\widehat{T}_{n}\coloneqq\sum_{\begin{subarray}{c}\bm{x},i,d\end{subarray}}\delta(\\#S(\bm{x},i,d),n)\sum_{\begin{subarray}{c}j\in
S(\bm{x},i,d)\\\
d^{\prime}=\pm\end{subarray}}\frac{pc_{\beta}(\bm{x},i,d)}{2(1+np)}(\widehat{\Delta}_{j}^{d^{\prime}}-\mathop{\widehat{\mathrm{id}}}\nolimits^{\otimes
N}_{H})\widehat{\Delta}_{i}^{d}\widehat{\Xi}^{\bm{x}},$ (185)

where $1\leq n\leq N-1$. Then we obtain

$\widehat{T}=\widehat{T}_{0}+\widehat{T}_{1}+\widehat{T}_{2}+\dots+\widehat{T}_{N-1}.$
(186)

####  A.3.2 Coefficients of $n$-th terms

Next we describe coefficients of $\widehat{T}_{n}$ as in (124). Using (168)
and (169), we have

$\displaystyle\langle\bm{y}|\widehat{T}_{n}|\bm{x}\rangle$
$\displaystyle=\begin{dcases}\frac{pc_{\beta}(\bm{x},i,d)}{2(1+np)}&(\text{if
$\bm{y}=f_{j}^{d^{\prime}}f_{i}^{d}\bm{x}$ and $\\#S(\bm{x},i,d)=n$})\\\
\frac{-2npc_{\beta}(\bm{x},i,d)}{2(1+np)}&(\text{if $\bm{y}=f_{i}^{d}\bm{x}$
and $\\#S(\bm{x},i,d)=n$})\\\ 0&(\text{otherwise}).\end{dcases}$ (187)

By the estimate (153), using $n<N$ and $1+np\geq 1$, we see that

$0\leq|\langle\bm{y}|\widehat{T}_{n}|\bm{x}\rangle|\leq\frac{np}{2N(1+np)}<\frac{p}{2}.$
(188)

This shows that the nonzero coefficients of $\widehat{T}_{n}$ are estimated as
$\mathcal{O}(p)$ as $p\to+0$, independent of $n$. See Table 2 for the
coefficients $\langle\bm{y}|\widehat{T}_{n}|\bm{x}\rangle$ if
$\bm{y}=f_{j}^{d^{\prime}}f_{i}^{d}\bm{x}$.

Table 2: A part of $(\bm{x},\bm{y})$-th coefficients of $\widehat{T}-\widehat{T}_{0}$ and $\widehat{T}_{n}$ with $\bm{y}=f_{j}^{d^{\prime}}f_{i}^{d}\bm{x}$. $\\#S(\bm{x},i,d)$ | $\widehat{T}-\widehat{T}_{0}$ | $\widehat{T}_{1}$ | $\widehat{T}_{2}$ | $\widehat{T}_{3}$ | $\widehat{T}_{4}$ | $\cdots$ | $\widehat{T}_{n}$
---|---|---|---|---|---|---|---
$1$ | $\frac{pc_{\beta}}{2(1+p)}$ | $\frac{pc_{\beta}}{2(1+p)}$ | $0$ | $0$ | $0$ | $\cdots$ | $0$
$2$ | $\frac{pc_{\beta}}{2(1+2p)}$ | $0$ | $\frac{pc_{\beta}}{2(1+2p)}$ | $0$ | $0$ | $\cdots$ | $0$
$3$ | $\frac{pc_{\beta}}{2(1+3p)}$ | $0$ | $0$ | $\frac{pc_{\beta}}{2(1+3p)}$ | $0$ | $\cdots$ | $0$
$4$ | $\frac{pc_{\beta}}{2(1+4p)}$ | $0$ | $0$ | $0$ | $\frac{pc_{\beta}}{2(1+4p)}$ | $\cdots$ | $0$
$n$ | $\frac{pc_{\beta}}{2(1+np)}$ | $0$ | $0$ | $0$ | $0$ | $\cdots$ | $\frac{pc_{\beta}}{2(1+np)}$

####  A.3.3 Another description of $n$-th terms

Here we give another description (196) of $n$-th terms. We change the ordering
of summations in (185) as follows: we first choose $1\leq i_{0}\leq N$, an
$n$-elements set $\bm{S}=\\{i_{1}<\dots<i_{n}\\}\subset[N]$ with
$i_{0}<i_{1}$, and then configurations $\bm{x}=(x_{1},x_{2},\dots,x_{N})$ so
that $S(\bm{x},i_{0},d)=\bm{S}$. Then explicitly we reach

$\displaystyle\widehat{T}_{n}$ $\displaystyle=\sum_{\begin{subarray}{c}1\leq
i_{0}\leq N\end{subarray}}\sum_{\bm{S}=\\{i_{0}<i_{1}<\dots<i_{n}\leq
N\\}}\sum_{\begin{subarray}{c}\bm{x}\\\ S(\bm{x},i_{0},d)=\bm{S}\\\
d=\pm\end{subarray}}\frac{pc_{\beta}(\bm{x},i_{0},d)}{2(1+np)}\sum_{\begin{subarray}{c}j\in{S}(\bm{x},i_{0},d)\\\
d^{\prime}=\pm\end{subarray}}(\widehat{\Delta}_{j}^{d^{\prime}}-\mathop{\widehat{\mathrm{id}}}\nolimits^{\otimes
N}_{H})\widehat{\Delta}_{i_{0}}^{d}\widehat{\Xi}^{\bm{x}}$ (189)
$\displaystyle=\frac{p}{2(1+np)}\sum_{\begin{subarray}{c}1\leq i_{0}\leq
N\end{subarray}}\sum_{\bm{S}}\sum_{\begin{subarray}{c}\bm{x}\\\
S(\bm{x},i_{0},d)=\bm{S}\\\
d=\pm\end{subarray}}c_{\beta}(\bm{x},i_{0},d)\Bigg{[}\sum_{\begin{subarray}{c}j\in\bm{S}\\\
d^{\prime}=\pm\end{subarray}}(\widehat{\Delta}_{j}^{d^{\prime}}-\mathop{\widehat{\mathrm{id}}}\nolimits^{\otimes
N}_{H})\Bigg{]}\widehat{\Delta}_{i_{0}}^{d}\widehat{\Xi}^{\bm{x}}$ (190)
$\displaystyle=\frac{p}{2(1+np)}\sum_{\begin{subarray}{c}1\leq i_{0}\leq
N\end{subarray}}\sum_{\bm{S}}\Bigg{[}\sum_{\begin{subarray}{c}j\in\bm{S}\\\
d^{\prime}=\pm\end{subarray}}(\widehat{\Delta}_{j}^{d^{\prime}}-\mathop{\widehat{\mathrm{id}}}\nolimits^{\otimes
N}_{H})\Bigg{]}\sum_{\begin{subarray}{c}\bm{x}\\\ S(\bm{x},i_{0},d)=\bm{S}\\\
d=\pm\end{subarray}}c_{\beta}(\bm{x},i_{0},d)\widehat{\Delta}_{i_{0}}^{d}\widehat{\Xi}^{\bm{x}}.$
(191)

Using $\widehat{\Upsilon}_{i_{0};\bm{S}}$ defined in (137), we can write

$\displaystyle\sum_{\begin{subarray}{c}\bm{x}\\\ S(\bm{x},i_{0},d)=\bm{S}\\\
d=\pm\end{subarray}}c_{\beta}(\bm{x},i_{0},d)\widehat{\Delta}_{i_{0}}^{d}\widehat{\Xi}^{\bm{x}}$
$\displaystyle=\widehat{\Upsilon}_{i_{0};\bm{S}}\sum_{d=\pm}\sum_{\begin{subarray}{c}\bm{x}\end{subarray}}c_{\beta}(\bm{x},i_{0},d)\widehat{\Delta}_{i_{0}}^{d}\widehat{\Xi}^{\bm{x}}$
(192)
$\displaystyle=\widehat{\Upsilon}_{i_{0};\bm{S}}\widehat{T}_{0,\text{move}}^{{i}_{0}}.$
(193)

By putting $\widehat{P}_{\emptyset}=0$ and

$\displaystyle\widehat{P}_{\bm{S}}$
$\displaystyle\coloneqq\sum_{\begin{subarray}{c}j\in\bm{S}\\\
d^{\prime}=\pm\end{subarray}}(\widehat{\Delta}_{j}^{d^{\prime}}-\mathop{\widehat{\mathrm{id}}}\nolimits^{\otimes
N}_{H})$ (194)
$\displaystyle=\sum_{\begin{subarray}{c}j\in\bm{S}\end{subarray}}\left(\widehat{\Delta}_{j}^{+}+\widehat{\Delta}_{j}^{-}-2\mathop{\widehat{\mathrm{id}}}\nolimits^{\otimes
N}_{H}\right),$ (195)

we can write

$\widehat{T}_{n}=\frac{p}{2(1+np)}\sum_{\begin{subarray}{c}1\leq i_{0}\leq
N\end{subarray}}\sum_{\begin{subarray}{c}\bm{S}\subseteq\\{i_{0}+1,\dots,N\\}\end{subarray}}\delta(\\#\bm{S},n)\widehat{P}_{\bm{S}}\widehat{\Upsilon}_{i_{0};\bm{S}}\widehat{T}_{0,\text{move}}^{{i}_{0}}.$
(196)

###  A.4 The beta decomposition

In this section, we introduce the second decomposition (204) of $\widehat{T}$
which we call the beta decomposition. The coefficients of $m$-th term in the
beta decomposition is estimated as $\mathcal{O}(p^{m})$ (see (221)).

####  A.4.1 Definition of beta terms

Here we give a definition of $m$-th term $\widehat{U}_{m}$ of beta
decomposition by using $\widehat{T}_{n}$ which appear in the supplanting
decomposition (see (185)). For $1\leq m\leq N-1$, we define

$\displaystyle\widehat{U}_{m}$
$\displaystyle\coloneqq\frac{(-1)^{m-1}B(m,1+1/p)}{p}\sum_{m\leq n\leq
N-1}\binom{n-1}{m-1}(1+np)\widehat{T}_{n}.$ (197)

Here $B(a,b)$ is the beta function:

$\displaystyle B(a,b)$
$\displaystyle\coloneqq\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$ (198)
$\displaystyle=\int_{0}^{1}x^{a-1}(1-x)^{b-1}dx.$ (199)

In particular, we have

$\displaystyle B\left(m,1+\frac{1}{p}\right)$
$\displaystyle=\frac{\Gamma(m)\Gamma(1+1/p)}{\Gamma(m+1+1/p)}$ (200)
$\displaystyle=\frac{(m-1)!}{(1+1/p)(2+1/p)\dots(m+1/p)}$ (201)
$\displaystyle=\frac{(m-1)!p^{m}}{(1+p)(1+2p)\dots(1+mp)}.$ (202)

By these descriptions, one can write

$\displaystyle\widehat{U}_{m}$
$\displaystyle=\frac{(-1)^{m-1}(m-1)!p^{m-1}}{(1+p)(1+2p)\dots(1+mp)}\sum_{m\leq
n\leq N-1}\binom{n-1}{m-1}(1+np)\widehat{T}_{n}.$ (203)

####  A.4.2 The beta decomposition

We can prove the following decomposition:

$\widehat{T}=\widehat{T}_{0}+\sum_{1\leq m\leq N-1}\widehat{U}_{m}.$ (204)

In fact,

$\displaystyle\sum_{1\leq m\leq N-1}\widehat{U}_{m}$
$\displaystyle=\sum_{1\leq m\leq N-1}\frac{(-1)^{m-1}B(1+1/p,m)}{p}\sum_{m\leq
n\leq N-1}\binom{n-1}{m-1}(1+np)\widehat{T}_{n}$ (205)
$\displaystyle=\sum_{1\leq n\leq N-1}\left[\sum_{1\leq m\leq
n}(-1)^{m-1}\frac{B(1+1/p,m)}{p}\binom{n-1}{m-1}\right](1+np)\widehat{T}_{n}.$
(206)

Then the following lemma is enough to show the desired decomposition (204).

######  Lemma A.1.

$\sum_{1\leq m\leq
n}\frac{(-1)^{m-1}B(1+1/p,m)}{p}\binom{n-1}{m-1}=\frac{1}{1+np}.$ (207)

###### Sketch of proof.

We only sketch the proof. With two generating series of exponential type

$\displaystyle F_{0}(t)$ $\displaystyle\coloneqq\sum_{m\geq
0}a_{m}\frac{t^{m}}{m!},$ (208) $\displaystyle F_{1}(t)$
$\displaystyle\coloneqq F_{0}(t)e^{t},$ (209)

where the coefficients of $F_{0}(t)$ is

$\displaystyle a_{m}$ $\displaystyle\coloneqq\frac{(-1)^{m}B(1+1/p,m+1)}{p}$
(210) $\displaystyle=\frac{(-1)^{m}m!p^{m}}{(1+p)(1+2p)\dots(1+(m+1)p)},$
(211)

it is enough to prove

$F_{1}(t)=\sum_{r\geq 0}\frac{1}{1+(r+1)p}\frac{t^{r}}{r!}.$ (212)

Since $a_{0}=F_{0}(0)={(1+p)^{-1}}$ and

$\displaystyle(1+(m+2)p)a_{m+1}=(-1)\cdot(m+1)pa_{m}$ (213)
$\displaystyle\iff$
$\displaystyle(m+1)a_{m+1}+(m+1)a_{m}=-\left(1+\frac{1}{p}\right)a_{m+1}$
(214)

for $m\geq 0$, we have a differential equation for $F_{0}(t)$

$\displaystyle t\left(\frac{dF_{0}}{dt}(t)+F_{0}(t)\right)$
$\displaystyle=-\frac{1}{p}({(1+p)}F_{0}(t)-1).$ (215)

Thus $F_{1}(t)=F_{0}(t)e^{t}$ satisfies

$\displaystyle t\frac{dF_{1}}{dt}(t)$
$\displaystyle=-{\left(1+\frac{1}{p}\right)}F_{1}(t)+\frac{1}{p}e^{t}.$ (216)

Now the equation (212) can be proved by comparison of coefficients. ∎

####  A.4.3 Coefficients of beta terms

With our description (187) of coefficients of $\widehat{T}_{n}$, we can write
down the coefficients of $\widehat{U}_{m}$.

For $\bm{y}=f_{j}^{d^{\prime}}f_{i}^{d}\bm{x}$ or $f_{i}^{d}\bm{x}$, the
coefficients $\langle\bm{y}|\widehat{T}_{n}|\bm{x}\rangle$ of
$\widehat{T}_{n}$ is zero when $n\neq\\#S(\bm{x},i,d)$. Hence we have

$\displaystyle\langle\bm{y}|\widehat{U}_{m}|\bm{x}\rangle$
$\displaystyle=\begin{dcases}\binom{n-1}{m-1}\frac{(-1)^{m+1}B(1+1/p,m)}{2}c_{\beta}(\bm{x},i,d)&(\text{if
$\bm{y}=f_{j}^{d^{\prime}}f_{i}^{d}\bm{x}$ and $\\#S(\bm{x},i,d)=n\geq m$})\\\
-2n\binom{n-1}{m-1}\frac{(-1)^{m+1}B(1+1/p,m)}{2}c_{\beta}(\bm{x},i,d)&(\text{if
$\bm{y}=f_{i}^{d}\bm{x}$ and $\\#S(\bm{x},i,d)=n\geq m$})\\\
0&(\text{otherwise}).\end{dcases}$ (217)

Using the description

$B(m,1+1/p)=\frac{(m-1)!p^{m}}{(1+p)(1+2p)\dots(1+mp)},$ (218)

we can write those coefficients in another form:

$\displaystyle\langle\bm{y}|\widehat{U}_{m}|\bm{x}\rangle$
$\displaystyle=\begin{dcases}\binom{n-1}{m-1}\frac{(-1)^{m+1}(m-1)!p^{m}}{2(1+p)(1+2p)\dots(1+mp)}c_{\beta}(\bm{x},i,d)&(\text{if
$\bm{y}=f_{j}^{d^{\prime}}f_{i}^{d}\bm{x}$ and $\\#S(\bm{x},i,d)=n\geq m$})\\\
-2n\binom{n-1}{m-1}\frac{(-1)^{m+1}(m-1)!p^{m}}{2(1+p)(1+2p)\dots(1+mp)}c_{\beta}(\bm{x},i,d)&(\text{if
$\bm{y}=f_{i}^{d}\bm{x}$ and $\\#S(\bm{x},i,d)=n\geq m$})\\\
0&(\text{otherwise}).\end{dcases}$ (219)

This allows us to estimate the coefficients of $\widehat{U}_{m}$. With $p\to
0$ and fixing other parameters $\beta,N,L$, and $E$, we have

$\langle\bm{y}|\widehat{U}_{m}|\bm{x}\rangle=\mathcal{O}(p^{m}).$ (220)

In particular, we have

$\widehat{T}=\widehat{T}_{0}+\widehat{U}_{1}+\dots+\widehat{U}_{m}+\mathcal{O}(p^{m+1})$
(221)

for $1\leq m\leq N-1$. See Table 3 for the coefficients
$\langle\bm{y}|\widehat{U}_{m}|\bm{x}\rangle$ if
$\bm{y}=f_{j}^{d^{\prime}}f_{i}^{d}\bm{x}$.

Table 3: A part of $(\bm{x},\bm{y})$-th coefficients of $\widehat{T}-\widehat{T}_{0}$ and $\widehat{U}_{m}$ with $\bm{y}=f_{j}^{d^{\prime}}f_{i}^{d}\bm{x}$. $\\#S(\bm{x},i,d)$ | $\widehat{T}-\widehat{T}_{0}$ | $\widehat{U}_{1}$ | $\widehat{U}_{2}$ | $\widehat{U}_{3}$ | $\widehat{U}_{4}$ | $\cdots$ | $\widehat{U}_{m}$
---|---|---|---|---|---|---|---
$1$ | $\frac{pc_{\beta}}{2(1+p)}$ | $\binom{0}{0}\frac{pc_{\beta}}{2(1+p)}$ | $0$ | $0$ | $0$ | $\cdots$ | $0$
$2$ | $\frac{pc_{\beta}}{2(1+2p)}$ | $\binom{1}{0}\frac{pc_{\beta}}{2(1+p)}$ | $\binom{1}{1}\frac{(-1)1!p^{2}c_{\beta}}{2(1+p)(1+2p)}$ | $0$ | $0$ | $\cdots$ | $0$
$3$ | $\frac{pc_{\beta}}{2(1+3p)}$ | $\binom{2}{0}\frac{pc_{\beta}}{2(1+p)}$ | $\binom{2}{1}\frac{(-1)1!p^{2}c_{\beta}}{2(1+p)(1+2p)}$ | $\binom{2}{2}\frac{(-1)^{2}2!p^{3}c_{\beta}}{2(1+p)\dots(1+3p)}$ | $0$ | $\cdots$ | $0$
$4$ | $\frac{pc_{\beta}}{2(1+4p)}$ | $\binom{3}{0}\frac{pc_{\beta}}{2(1+p)}$ | $\binom{3}{1}\frac{(-1)1!p^{2}c_{\beta}}{2(1+p)(1+2p)}$ | $\binom{3}{2}\frac{(-1)^{2}2!p^{3}c_{\beta}}{2(1+p)\dots(1+3p)}$ | $\binom{3}{3}\frac{(-1)^{3}3!p^{4}c_{\beta}}{2(1+p)\dots(1+4p)}$ | $\cdots$ | $0$
$m$ | $\frac{pc_{\beta}}{2(1+mp)}$ | $\binom{m-1}{0}\frac{pc_{\beta}}{2(1+p)}$ | $\binom{m-1}{1}\frac{(-1)1!p^{2}c_{\beta}}{2(1+p)(1+2p)}$ | $\binom{m-1}{2}\frac{(-1)^{2}2!p^{3}c_{\beta}}{2(1+p)\dots(1+3p)}$ | $\binom{m-1}{3}\frac{(-1)^{3}3!p^{4}c_{\beta}}{2(1+p)\dots(1+4p)}$ | $\cdots$ | $\binom{m-1}{m-1}\frac{(-1)^{m-1}p^{m}c_{\beta}}{2(1+p)\dots(1+mp)}$

####  A.4.4 Another decomposition of beta terms

Substituting (196) into (203), we can give a combinatorial decomposition of
beta terms: for $m\geq 1$,

$\displaystyle\widehat{U}_{m}$ $\displaystyle=\sum_{m\leq n\leq
N-1}\frac{(-1)^{m+1}(m-1)!p^{m-1}(1+np)}{(1+p)(1+2p)\dots(1+mp)}\binom{n-1}{m-1}\frac{p}{2(1+np)}\sum_{\begin{subarray}{c}1\leq
i_{0}\leq
N\end{subarray}}\sum_{\begin{subarray}{c}\bm{S}\subseteq\\{i_{0}+1,\dots,N\\}\end{subarray}}\delta(\\#\bm{S},n)\widehat{P}_{\bm{S}}\widehat{\Upsilon}_{i_{0};\bm{S}}\widehat{T}_{0,\text{move}}^{{i}_{0}}$
(222)
$\displaystyle=\frac{(-1)^{m+1}(m-1)!p^{m}}{2(1+p)(1+2p)\dots(1+mp)}\sum_{\begin{subarray}{c}1\leq
i_{0}\leq
N\end{subarray}}\bigg{[}\sum_{\begin{subarray}{c}\bm{S}\subseteq\\{i_{0}+1,\dots,N\\}\\\
\\#\bm{S}\geq
m\end{subarray}}\binom{\\#\bm{S}-1}{m-1}\widehat{P}_{\bm{S}}\widehat{\Upsilon}_{i_{0};\bm{S}}\bigg{]}\widehat{T}_{0,\text{move}}^{{i}_{0}}.$
(223)

To obtain a more convenient formula of (223), we use the following
combinatorial equation: for $\bm{S}\subseteq\\{i_{0}+1,\dots,N\\}$ with
$\\#\bm{S}\geq m$,

$\binom{\\#\bm{S}-1}{m-1}\widehat{P}_{\bm{S}}=\sum_{\begin{subarray}{c}\bm{S}^{\prime}\subseteq\bm{S}\end{subarray}}\delta(\\#\bm{S}^{\prime},m)\widehat{P}_{\bm{S}^{\prime}}.$
(224)

Then we can perform the following transformation of the part surrounded by
square brackets in (223):

$\displaystyle\qquad\sum_{\begin{subarray}{c}\bm{S}\subseteq\\{i_{0}+1,\dots,N\\}\\\
\\#\bm{S}\geq
m\end{subarray}}\binom{\\#\bm{S}-1}{m-1}\widehat{P}_{\bm{S}}\widehat{\Upsilon}_{i_{0};\bm{S}}$
(225)
$\displaystyle=\sum_{\begin{subarray}{c}\bm{S}\subseteq\\{i_{0}+1,\dots,N\\}\\\
\\#\bm{S}\geq
m\end{subarray}}\sum_{\begin{subarray}{c}\bm{S}^{\prime}\subseteq\bm{S}\end{subarray}}\delta(\\#\bm{S}^{\prime},m)\widehat{P}_{\bm{S}^{\prime}}\widehat{\Upsilon}_{i_{0};\bm{S}}$
(226)
$\displaystyle=\sum_{\begin{subarray}{c}\bm{S}^{\prime}\subseteq\\{i_{0}+1,\dots,N\\}\end{subarray}}\sum_{\begin{subarray}{c}\bm{S}\subseteq\\{i_{0}+1,\dots,N\\}\\\
\bm{S}\supseteq\bm{S}^{\prime}\end{subarray}}\delta(\\#\bm{S}^{\prime},m)\widehat{P}_{\bm{S}^{\prime}}\widehat{\Upsilon}_{i_{0};\bm{S}}$
(227)
$\displaystyle=\sum_{\begin{subarray}{c}\bm{S}^{\prime}\subseteq\\{i_{0}+1,\dots,N\\}\end{subarray}}\delta(\\#\bm{S}^{\prime},m)\widehat{P}_{\bm{S}^{\prime}}\sum_{\begin{subarray}{c}\bm{S}\subseteq\\{i_{0}+1,\dots,N\\}\\\
\bm{S}\supseteq\bm{S}^{\prime}\end{subarray}}\widehat{\Upsilon}_{i_{0};\bm{S}}$
(228)
$\displaystyle=\sum_{\begin{subarray}{c}\bm{S}^{\prime}\subseteq\\{i_{0}+1,\dots,N\\}\end{subarray}}\delta(\\#\bm{S}^{\prime},m)\widehat{P}_{\bm{S}^{\prime}}\widehat{\Xi}_{\\{i_{0}\\}\cup\bm{S}^{\prime}},$
(229)

where we used (140) in the last equality. Hence, we obtain

$\displaystyle\widehat{U}_{m}$
$\displaystyle=\frac{(-1)^{m+1}(m-1)!p^{m}}{2(1+p)(1+2p)\dots(1+mp)}\sum_{\begin{subarray}{c}1\leq
i_{0}\leq
N\end{subarray}}\bigg{[}\sum_{\begin{subarray}{c}\bm{S}^{\prime}\subseteq\\{i_{0}+1,\dots,N\\}\end{subarray}}\delta(\\#\bm{S}^{\prime},m)\widehat{P}_{\bm{S}^{\prime}}\widehat{\Xi}_{\\{i_{0}\\}\cup\bm{S}^{\prime}}\bigg{]}\widehat{T}_{0,\text{move}}^{{i}_{0}}$
(230)
$\displaystyle=\frac{(-1)^{m+1}(m-1)!p^{m}}{2(1+p)(1+2p)\dots(1+mp)}\sum_{\begin{subarray}{c}1\leq
i_{0}<i_{1}<\dots<i_{m}\leq
N\end{subarray}}\widehat{P}_{\\{i_{1},\dots,i_{m}\\}}\widehat{\Xi}_{\\{i_{0},i_{1},\dots,i_{m}\\}}\widehat{T}_{0,\text{move}}^{{i}_{0}}.$
(231)

For example when $m=1$, we have

$\displaystyle\widehat{U}_{1}$ $\displaystyle=\frac{p}{2(1+p)}\sum_{1\leq
i_{0}<i_{1}\leq
N}\widehat{P}_{\\{i_{1}\\}}\widehat{\Xi}_{\\{i_{0},i_{1}\\}}\widehat{T}_{0,\text{move}}^{i_{0}}$
(232) $\displaystyle=\frac{p}{2(1+p)}\sum_{1\leq i_{0}<i_{1}\leq
N}(\widehat{\Delta}_{i_{1}}^{+}+\widehat{\Delta}_{i_{1}}^{-}-2\mathop{\widehat{\mathrm{id}}}\nolimits^{\otimes
N}_{H})\widehat{\Xi}_{\\{i_{0},i_{1}\\}}\widehat{T}_{0,\text{move}}^{i_{0}}.$
(233)

When $m=N-1$, we have

$\displaystyle\widehat{U}_{N-1}$
$\displaystyle=\frac{(-1)^{N}N!p^{N-1}}{2(1+p)(1+2p)\dots(1+(N-1)p)}\widehat{P}_{\\{2,3,\dots,N\\}}\widehat{\Xi}_{\\{1,2,\dots,N\\}}\widehat{T}_{0,\text{move}}^{1}.$
(234)

For an agent $i_{0}$ and a subset $\bm{S}\subseteq[N]\setminus\\{i_{0}\\}$,
let us define

$\displaystyle\widehat{U}_{i_{0};\bm{S}}\coloneqq\frac{(-1)^{\\#\bm{S}}\\#\bm{S}!p^{\\#\bm{S}-1}}{2(1+p)(1+2p)\dots(1+(\\#\bm{S}-1)p)}\widehat{P}_{\bm{S}}\widehat{\Xi}_{\\{i_{0}\\}\cup\bm{S}}\widehat{T}_{0,\text{move}}^{i_{0}}.$
(235)

Then we can rewrite (231) as

$\displaystyle\widehat{U}_{m}=\sum_{1\leq i_{0}\leq
N}\sum_{\begin{subarray}{c}\bm{S}\subseteq\\{i_{0}+1,\dots,N\\}\end{subarray}}\delta(\\#\bm{S},m)\widehat{U}_{i_{0};\bm{S}}.$
(236)

Note that we can define $\widehat{U}_{i_{0};\bm{S}}$ even if
$\bm{S}\not\subseteq\\{i_{0}+1,\dots,N\\}$.

######  Remark A.2 (The origin of the beta decomposition).

With the operators $\widehat{U}_{i_{0};\bm{S}}$, we can define a variant model
of supplanting process. For a subset $\bm{S}_{0}\subseteq\\{1,\dots,N\\}$, we
define

$\widehat{T}_{\bm{S}_{0}}\coloneqq\sum_{i_{0}\in\bm{S}_{0}}\sum_{\begin{subarray}{c}\bm{S}^{\prime}\subseteq\\{i_{0}+1,\dots,N\\}\cap\bm{S}_{0}\end{subarray}}\widehat{U}_{i_{0};\bm{S}^{\prime}}.$
(237)

This operator $\widehat{T}_{\bm{S}_{0}}$ represents the model where
supplanting process only occurs on pairs of agents $i,j$ with
$i,j\in\bm{S}_{0}$.

Conversely, we can define $\widehat{U}_{i_{0};\bm{S}}$ from these variants
$\widehat{T}_{\bm{S}_{0}}$ by inclusion-exclusion principle:

$\widehat{U}_{i_{0};\bm{S}_{0}}=\sum_{\begin{subarray}{c}\bm{S}^{\prime}\subsetneq\\{i_{0}+1,\dots,N\\}\cap\bm{S}_{0}\end{subarray}}(-1)^{\\#\bm{S}^{\prime}-\\#\bm{S}_{0}}\widehat{T}_{\\{i_{0}\\}\cup\bm{S}^{\prime}}.$
(238)

The beta decomposition was originally derived from this point of view.

####  A.4.5 Commutation relation between beta terms and permutation operators

Though it is not used in the main text, we write down a commutation relation
between beta terms and permutation operators.

Using relations in Section A.2.5, we have the following forms (the third
equation is equivalent to (160)):

$\displaystyle\widehat{P}_{\bm{S}}\widehat{\Pi}_{\sigma}$
$\displaystyle=\widehat{\Pi}_{\sigma}\widehat{P}_{\sigma(\bm{S})},$ (239)
$\displaystyle\widehat{\Xi}_{\\{i_{0}\\}\cup\bm{S}}\widehat{\Pi}_{\sigma}$
$\displaystyle=\widehat{\Pi}_{\sigma}\widehat{\Xi}_{\\{\sigma(i_{0})\\}\cup\sigma(\bm{S})},$
(240) $\displaystyle\widehat{T}_{0,\text{move}}^{i_{0}}\widehat{\Pi}_{\sigma}$
$\displaystyle=\widehat{\Pi}_{\sigma}\widehat{T}_{0,\text{move}}^{\sigma(i_{0})}.$
(241)

With these three relations, we obtain that

$\widehat{U}_{i_{0};\bm{S}}\widehat{\Pi}_{\sigma}=\widehat{\Pi}_{\sigma}\widehat{U}_{\sigma(i_{0});\sigma(\bm{S})}.$
(242)

This enables us to investigate the commutation relation between
$\widehat{U}_{m}$ and permutation operators with using (236) or between
$\widehat{T}_{\bm{S}_{0}}$ and permutation operators with using (237).

##  Appendix B Eigenvector centrality and overlap centrality

In this section we see a relation between the eigenvector centrality and the
overlap centrality of the neighbor matrix. Let us recall the definitions of
their centrality in our context. The eigenvector centrality is defined as a
normalized eigenvector with the maximum eigenvalue of the neighbor matrix
$\mathcal{R}$. Such an eigenvector exists uniquely and its components can be
taken to be real and positive because of the Perron–Frobenius theorem Bonacich
; Newman . The overlap centrality (44) is the expectation value of how many
agents are at the same site with a given agent.

First we describe our settings. We take a state $\ket{P}\in H_{X}^{\otimes N}$
corresponding to a probability distribution $P(\bm{x})$. $\ket{P}$ does not
have to be a stationary state of a certain stochastic process. The neighbor
matrix $\mathcal{R}=(r_{ij})_{1\leq i,j\leq N}$ of $\ket{P}$ is defined
similar to (43), explicitly

$r_{ij}=\sum_{\bm{x}}\delta(x_{i},x_{j})\braket{\bm{x}}{P}.$ (243)

We consider a decomposition of the state $\ket{P}$

$\displaystyle\ket{P}=\ket{P_{1}}+\ket{P_{2}},$ (244)

and define matrices $\mathcal{R}^{(\ell)}=(r^{(\ell)}_{ij})$ as

$\displaystyle r_{ij}^{(\ell)}$
$\displaystyle=\sum_{\bm{x}}\delta(x_{i},x_{j})\braket{\bm{x}}{P_{\ell}}$
(245)

for $\ell=1,2$. By definition, we also have a decomposition of the neighbor
matrix $\mathcal{R}$

$\mathcal{R}={\mathcal{R}}^{(1)}+{\mathcal{R}}^{(2)}.$ (246)

We assume that the decomposition (244) satisfies the following two conditions.

  1. (i)

$\ket{P_{1}}$ is symmetric under permutations: that is,
$\widehat{\Pi}_{\sigma}\ket{P_{1}}=\ket{P_{1}}$ for any
$\sigma\in\mathfrak{S}_{N}$. From this assumption there is a constant $c$ with
$r_{ij}^{(1)}=c$ for any $i\neq j$. We assume that $c\neq 0$.

  2. (ii)

The off-diagonal entries $r_{ij}^{(2)}$ of $\mathcal{R}^{(2)}$ are
sufficiently smaller than $\left|{c}\right|$: that is, $|r_{ij}^{(2)}|\ll|c|$
for any $i\neq j$.

We modify the matrices $\mathcal{R}^{(1)},\mathcal{R}^{(2)}$ to
$\tilde{\mathcal{R}}^{(1)}=(\tilde{r}_{ij}^{(1)}),\tilde{\mathcal{R}}^{(2)}=(\tilde{r}_{ij}^{(2)})$
such that all of the diagonal entries of $\tilde{\mathcal{R}}^{(2)}$ are zero
and the following decomposition of $\mathcal{R}$ holds:

$\mathcal{R}=\tilde{\mathcal{R}}^{(1)}+\tilde{\mathcal{R}}^{(2)}.$ (247)

Considering all of the diagonal entries of $\mathcal{R}$ are one, we can take

$\displaystyle\tilde{r}_{ij}^{(1)}=\begin{dcases}1&\text{(if $i=j$)}\\\
r_{ij}^{(1)}=c&\text{(if $i\neq j$)}\end{dcases}\quad\text{ and
}\quad\tilde{r}_{ij}^{(2)}=\begin{dcases}0&\text{(if $i=j$)}\\\
r_{ij}^{(2)}&\text{(if $i\neq j$)}.\end{dcases}$ (248)

From the assumption (ii), the matrix $\tilde{\mathcal{R}}^{(2)}$ can be
regarded as a perturbative part in $\mathcal{R}$.

The eigenvalue problem of $\tilde{\mathcal{R}}^{(1)}$ can be solved easily:
eigenvalues of $\tilde{\mathcal{R}}^{(1)}$ are $1+(N-1)c$ and $1-c$, and their
corresponding eigenspaces are ${\mathbb{C}}\bm{v}_{0}$ and the orthogonal
complement $({\mathbb{C}}\bm{v}_{0})^{\perp}$, respectively, where
$\bm{v}_{0}\coloneqq\frac{1}{\sqrt{N}}(1,1,\ldots,1)^{T}\in{\mathbb{C}}^{N}$.
In particular, if $\tilde{\mathcal{R}}^{(2)}$ vanishes, then the eigenvector
centrality of $\mathcal{R}=\tilde{\mathcal{R}}^{(1)}$ is the vector
$\bm{v}_{0}$.

Let us use a first-order perturbation theory to calculate the eigenvector
centrality ${\bm{V}}$ of $\mathcal{R}$. Here we introduce an orthonormal
system $(\bm{u}_{i})_{i=1}^{N-1}$ of the vector space
$({\mathbb{C}}\bm{v}_{0})^{\perp}$. According to the Rayleigh–Schrödinger type
perturbation theory, we have

$\displaystyle{\bm{V}}$
$\displaystyle\propto\bm{v}_{0}+\sum_{i=1}^{N-1}\frac{\bm{u}_{i}\bm{u}_{i}^{\dagger}\tilde{\mathcal{R}}^{(2)}\bm{v}_{0}}{1+(N-1)c-(1-c)}+\mathcal{O}(|\tilde{r}^{(2)}/c|^{2})$
(249)
$\displaystyle=\bm{v}_{0}+\frac{(1-\bm{v}_{0}\bm{v}_{0}^{\dagger})\tilde{\mathcal{R}}^{(2)}\bm{v}_{0}}{Nc}+\mathcal{O}(|\tilde{r}^{(2)}/c|^{2})$
(250)
$\displaystyle=\frac{1}{Nc}\mathcal{R}\bm{v}_{0}-\left({\frac{\bm{v}_{0}^{\dagger}\tilde{\mathcal{R}}^{(2)}\bm{v}_{0}}{Nc}+\frac{1-c}{Nc}}\right)\bm{v}_{0}+\mathcal{O}(|\tilde{r}^{(2)}/c|^{2})$
(251)

up to the first order of $|\tilde{r}^{(2)}/c|\coloneqq\max_{1\leq i,j\leq
N}|\tilde{r}_{ij}^{(2)}/c|$. Since
$(\mathcal{R}\bm{v}_{0})_{i}=\frac{1}{\sqrt{N}}\sum_{j=1}^{N}r_{ij}=\frac{1}{\sqrt{N}}(O_{i}-1)$,
we obtain

$\displaystyle{\bm{V}}\propto\frac{1}{N^{3/2}c}{\bm{O}}-\gamma\times(1,1,\ldots,1)^{T}+\mathcal{O}(|\tilde{r}^{(2)}/c|^{2}),$
(252)

where ${\bm{O}}=(O_{i})_{i=1}^{N}$ is the vector consisting of the overlap
centrality, and $\gamma$ is a constant which is expressed by

$\displaystyle\gamma=\frac{1}{\sqrt{N}}\left({\frac{\bm{v}_{0}^{\dagger}\tilde{\mathcal{R}}^{(2)}\bm{v}_{0}}{Nc}+\frac{2-c}{Nc}}\right).$
(253)

Thus we find that the eigenvector centrality and the overlap centrality are
equal up to multiplying by a constant and adding a vector in
${\mathbb{C}}(1,1,\ldots,1)^{T}$. Note that the higher order terms of this
perturbation can have a nontrivial ${\bm{O}}$-dependence, but are ignored in
the approximation.

Let us apply this result to the case of the stationary state
$\ket{P(\beta,p)}$ of the transition matrix $\widehat{T}(\beta,p)$. Suppose
that $p$ is small enough to be able to perform the perturbation expansion
(LABEL:ptheory). Under this assumption, we give the decomposition of
$\ket{P(\beta,p)}$ as follows:

$\displaystyle\ket{P(\beta,p)}$
$\displaystyle=\ket{P_{1}(\beta,p)}+\ket{P_{2}(\beta,p)},$ (254)
$\displaystyle\ket{P_{1}(\beta,p)}$
$\displaystyle=C(\beta,p)\ket{P_{\text{can}}(\beta)},$ (255)
$\displaystyle\ket{P_{2}(\beta,p)}$
$\displaystyle=C(\beta,p)\sum_{n=1}^{\infty}\left({\widehat{G}_{0}(\widehat{T}-\widehat{T}_{0})}\right)^{n}\ket{P_{\mathrm{can}}(\beta)}.$
(256)

Let us check that this decomposition satisfies the above condition (i) and
(ii).

  1. (i)

From (67), $\ket{P_{1}(\beta,p)}$ is symmetric under permutations. Since
$\beta,E(\bm{x})\in{\mathbb{R}}$, we have

$\displaystyle c=\dfrac{C(\beta,p)}{Z_{N}(\beta)}\sum_{\bm{x}\in
X^{N}}\delta(x_{1},x_{2})e^{-\beta E(\bm{x})}\neq 0.$ (257)

  2. (ii)

$\widehat{T}-\widehat{T}_{0}=\mathcal{O}(p)$ leads to
$\tilde{r}_{ij}^{(2)}=\mathcal{O}(p)$, and $c$ is of order of unity in terms
of $p$, which follows that $\left|{\tilde{r}^{(2)}/c}\right|=\mathcal{O}(p)$.

Hence, the relation (252) can be applied in the present case, and we obtain
the expression (105), in which the term of
$\mathcal{O}(\left|{\tilde{r}^{(2)}/c}\right|^{2})$ in (252) is replaced with
$\mathcal{O}(p^{2})$. Note that the manner of such a decomposition (254) is
not unique. In particular, one can choose the coefficient of
$\ket{P_{\mathrm{can}}(\beta)}$ in $\ket{P_{1}(\beta,p)}$ as any form of
$1+\mathcal{O}(p)$, though the values of $c$ and $\gamma$ depend on the
manner.

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