# Constrained Multi-Relational Hyper-Graphons with Maximum Entropy

Juan<EMAIL_ADDRESS>KU Leuven, Department of Computer Science
Jan<EMAIL_ADDRESS>INRIA Lille Yuyi Wang<EMAIL_ADDRESS>ETH
Zurich, Switzerland & CRRC Zhuzhou Institute.

###### Abstract

This work has two contributions. The first one is extending the Large
Deviation Principle for uniform hyper-graphons from Lubetzky and Zhao [19] to
the multi-relational setting where each hyper-graphon can have different
arities. This extension enables the formulation of the most typical possible
world in Relational Probabilistic Logic with symmetric relational symbols in
terms of entropy maximization subjected to constraints of quantum sub-
hypergraph densities.

The second contribution is to prove the most typical constrained multi-
relational hyper-graphons (the most typical possible worlds) are computable by
proving the solutions of the maximum entropy subjected by quantum sub-
hypergraph densities in the space of multi-relational hyper-graphons are step
functions except for in a zero measure set of combinations of quantum hyper-
graphs densities with multiple relations. This result proves in a very general
context the conjecture formulated by Radin et al. [21] that states the
constrained graphons with maximum entropy are step functions.

keywords: Graphon Theory and Principle of Maximum Entropy and Large Random
Graph and Exponential Random Graph Model and Constrained optimization and
Differential Geometry and Statistical Relational Learning

##  1 Introduction

In the space of graphons, the problem of finding the most typical random
graphs satisfying subgraph density constraints is formulated in terms of
entropy maximization which is described as follows [22]. Let $\mathcal{F}$ be
an ordered set of simple graphs and $u\in(0,1)^{|\mathcal{F}|}$ be a vector of
prescribed subgraph densities.

$W^{*}(\mathcal{F},u)=\operatorname*{arg\,max}_{W\in S(\mathcal{F},u)}-I(W)$

where

$S(\mathcal{F},u)=\\{W\in\mathcal{W}\,|\,t(\mathcal{F}_{i},W)=u_{i}\mbox{ for
all }i=1,\cdots,|\mathcal{F}||\\}$

and $-I$ is the Shannon entropy on graphons and $\mathcal{W}$ the space of
labeled graphons444This is formulation originally was established in the space
of unlabeled graphons, but for the sake of simplicity, we formulate in terms
of labeled graphons.. Here, we are interested in extending the formulation to
the most typical multi-relational hyper-graphs regarding entropy maximization.

Lubetzky and Zhao [19] extend the maximum entropy principle of graphons to
uniform hyper-graphons and prove the counting lemma holds for linear hyper-
graphs. Here, we further extend this principle to multi-relational hyper-
graphons (Section 3) following the same ideas of our previous work [1] and the
contraction principle of Large Deviation Principles [5]. There are two
motivations for establishing this extension.

  * •

To establish a micro-canonical semantics of possible worlds of probabilistic
logics whose signatures are only relational symbols describing symmetric
relations, the symbols $\wedge$ and $\implies$ and universal quantifiers. In
this probabilistic logic, each axiom is associated with a probability number.

  * •

To prove the problem of finding the most typical worlds (the micro-canonical
distribution) satisfying the probabilistic axioms can be computable by proving
the solutions of the maximum entropy problem in the space of multi-relational
uniform hyper-graphons are step functions. This result proves a general
version of the conjecture raised by Radin et al. [21], which states
$W^{*}(\mathcal{F},u)$ is a step function.

Since the formulation of this conjecture, it has been proved for several
special cases. The conjecture was rigorously confirmed by Kenyon et al. [12]
when $\mathcal{F}$ contains the edge and a $k$-stars graph. Moreover Kenyon et
al. in [11] confirm the conjecture is true when $\mathcal{F}=[\leavevmode\hbox
to3.25pt{\vbox to8.94pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},F]$ and
$u=[\rho,\tau]$ when $\tau$ is slightly higher than $\rho^{|F|}$, where $F$ is
a simple graph and $|F|$ is the number of edges of $F$. Moreover, Kenyon et
al. in [13] show that the number of steps that $W^{*}(\mathcal{F},u)$ has may
not be bounded when the vector $u$ gets arbitrarily close to an extremal
sufficient statistics vector. in the case of edge and triangle constraints.

Roughly speaking, our approach to proving the conjecture is based on the
observation any step function can be seen as a function and a finitely
dimensional vector of parameters that describe the step function. We study the
solutions $W^{*}(\mathcal{F},u)$ restricted to the space of step functions of
size $m$, these solutions are denoted by $W^{*(m)}(\mathcal{F},u)$. Using
elementary results of differential geometry and constrained optimization, we
prove when the solutions $W^{*(m)}(\mathcal{F},u)$ are embedded in a higher
dimensional vector space whose vectors parameterize step functions of size
$m+1$ then $W^{*(m)}(\mathcal{F},u)$ are also solutions in the extended space.
In other words, we prove $W^{*(m)}(\mathcal{F},u)=W^{*(m+1)}(\mathcal{F},u)$.

###  1.1 Outline

Thus, this paper is organized as follows. Section 2 provides results from
differential geometry and constrained optimization to support the proofs.
Section 3 develops the notion of multi-relational hyper-graphons. Section 4
studies the properties of step functions in the space of multi-relational
hyper-graphons providing formulas to compute partial derivatives of subgraph
densities when the step functions are finitely parameterized and defining the
split map on step functions which embeds any stepfuncion of lower dimensional
space into a higher one. Section 5 extends key results of graphon theory as
Szemeredi Regularity Lemma, Compactness, Counting Lemma, and Large Deviation
Principle to multi-relational uniform hyper-graphons where each relation has
the same arity. Section 6 extends the results developed in Section 5, we
obtain the large deviation principle for Erdős-Renyi version of multi-
relational random hyper-graphs and prove the most typical random hyper-graphs
constrained by quantum subgraph densities are solution of a maximum entropy
problem in the space of multi-relational hyper-graphs. Section 7 provides an
example of modeling possible worlds in a probabilistic logic using the most
typical multi-relational random hyper-graphs. Section 8 proves that the most
typical multi-relational random hyper-graphs are step functions. Finally,
Section 9 provides concluding remarks and some open problems. The Appendix
contains the proofs of theorems and lemmas.

###  1.2 Notations

Let us start with some common notations. We denote by $\mathbbm{1}_{X}$ the
indicator function, i.e., for any $x$, if $x\in X$ then $\mathbbm{1}_{X}(x)=1$
else $\mathbbm{1}_{X}(x)=0$. For any positive integer $n$, we denote by $[n]$
the set of all positive integers smaller than or equal to $n$. Let $S$ be a
set. Then any sequence in $S$ is denoted by $(a_{i})$ where $a_{i}\in S$.

We denote by $\nu$ the Lebesgue measure on reals. We denote by $\Sigma$ all
bijective measure-preserving maps and by $\Sigma_{m}$ the symmetric group on
$[m]$. We use the standard notation of $A^{\circ}$, $\overline{A}$ and
$\partial A$ for the topological interior, closure, and boundary of the set
$A$ and by $|A|$ the cardinality of a finite set $A$. The set of finite,
simple, and non-isomorphic graphs is denoted by $\mathcal{G}$. A multi-
relational graph $F$ is defined by $(V,E_{1},\cdots,E_{r})$ where $V$ is the
set of vertices and $(E_{1},\cdots,E_{r})$ are the set of relations. The
number of vertices is denoted $|F|$.

Let $F:\mathbb{R}^{n}\to\mathbb{R}^{m}$ be a smooth function then $J_{x}(F)$
denotes the Jacobian matrix of $F$ at $x$. Let $f:\mathbb{R}^{n}\to\mathbb{R}$
be a smooth function, $H_{x}f$ denotes the Hessian matrix of $f$ at $x$.

Symbol | Meaning
---|---
$\mathcal{W}_{\mathbb{R}}^{(r,d)}$ | All labeled graphon with real-valued
$\widetilde{\mathcal{W}}_{\mathbb{R}}^{(r,d)}$ | All unlabeled graphons the quotient of $\mathcal{W}_{\mathbb{R}}^{(r,d)}$ by elements of $\Sigma$
$\mathcal{W}^{(r,d)}$ | All labeled graphon with $[0,1]$-valued
$\widetilde{\mathcal{W}}^{(r,d)}$ | The quotient of $\mathcal{W}^{(r,d)}$ by elements of $\Sigma$
$(A,\pi)$ | A step function with partition $\pi$(Sec.4)
$\mathcal{W}^{({m,r,d})}_{\mathbb{R}}$ | All $m$-step functions with real-valued (Sec. 4)
${\mathcal{W}}^{({m},r,d)}_{\mathbb{R}}(\pi)$ | All $m$-step functions with real-valued with a fixed partition vector $\pi$
$S^{(r,d)}_{\mathbb{R}}(\mathcal{F},u)$ | All unlabeled graphons with real-valued satisfying constraints (6.2)
${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$ | All $m$-step functions with real-valued satisfying constraints.
${S_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u,\pi)}$ | All $m$-step functions with real-valued satisfying constraints and $\pi$ fixed.
$t(F,W)$ | Subgraph density of $F$ from the graphon $W$ (3.1)
$t_{x_{1}\cdots x_{k}}(F,W)$ | Partial subgraph density (4.1)
Table 1:  Overview of important notations in this paper

##  2 Mathematical Background

This section reviews results from Differential Geometry and Optimization on
constrained sets.

###  2.1 Riemannian Geometry

Riemannian Geometry endows a smooth manifold $M$ with a geometric structure by
defining a positive definite bilinear form $g_{p}:T_{p}M\otimes
T_{p}M\to\mathbb{R}$ at each point $p\in M$, such that the mapping $p\to
g_{p}(X;Y)$ is smooth. This structure allows us to measure distances and
angles on the manifold.

By utilizing the Riemannian metric $g_{p}(\cdot,\cdot)$, we can define the
length of a curve $\gamma:[a,b]\to M$ in the following manner: Let $g_{p;ij}$
denote the components of the Riemannian metric, and let $y^{(i)}$ represent
the coordinates of the curve $\gamma$. Then, the path length $L(\gamma)$ is
given by the integral

$L(\gamma)=\int_{a}^{b}\sqrt{\sum_{ij}g_{\gamma(s);ij}\frac{d\gamma^{(i)}}{ds}\frac{d\gamma^{(j)}}{ds}}ds.$

This formulation quantifies the length of a curve on the manifold $M$ based on
the underlying Riemannian metric. It enables us to compute distances and study
the geometry of $M$ through the lens of Riemannian Geometry.

####  2.1.1 Exponential Map on Riemannian Manifolds.

In the context of Riemannian manifolds, geodesic curves serve as paths with
minimal distance between points. The notion of a geodesic curve is formalized
as follows:

######  Definition 2.1 (Definition $1.4.2$ in [9]).

Let $M$ be a Riemannian manifold. A curve $\gamma:[0,a]\to M$ is said to be a
geodesic if it satisfies the second-order differential equation:

$\frac{d^{2}\gamma^{(i)}}{ds^{2}}+\Gamma^{i}{j,k}\frac{d\gamma^{(j)}}{ds}\frac{d\gamma^{(k)}}{ds}=0,\quad
i,j,k=1,\ldots,m,$ (2.1)

where $\gamma^{(i)}$ denotes the $i$-th component of $\gamma$ with respect to
a local coordinate system, and $\Gamma^{i}{j,k}$ are the Christoffel symbols
associated with the Levi-Civita connection.

A fundamental tool in studying Riemannian manifolds is the _Exponential Map_ ,
denoted as $Exp_{p}:U\to M$, which provides a correspondence between tangent
vectors and points on the manifold. Specifically, for each point $p\in M$,
there exists an open neighborhood $U$ of $0\in T_{p}M$ such that $Exp_{p}$ is
defined as follows:

$Exp_{p}(v)=\gamma_{p,v}(1),$ (2.2)

where $\gamma_{p,v}(1)$ represents the unique geodesic curve that satisfies
the initial conditions $\gamma_{p,v}(0)=p$ and $\gamma^{\prime}_{p,v}(0)=v$.

In other words, the Exponential Map takes a tangent vector $v\in T_{p}M$ and
”exponentiates” it to a point $q=Exp_{p}(v)\in M$ along the geodesic that
starts at $p$ with the velocity $v$.

A crucial property, as established by Proposition 20.8 in [15], is that
$Exp_{p}$ is a local diffeomorphism when restricted to an open neighborhood of
$0\in T_{p}M$. This property ensures the existence of a well-defined inverse
map, which is essential for various applications in Riemannian Geometry.

###  2.2 Theory of Constrained Optimization

Here, we assume $h:\mathbb{R}^{n}\to\mathbb{R}^{k}$ is differentiable. Then,
we define the constrained set

$M_{h}=\\{x\in\mathbb{R}^{n}\,|\,h(x)=0\\}.$ (2.3)

We review the Karush-Kuhn-Tucker (KKT) conditions for which any $x^{*}\in
M_{h}$ is a local solution of $\min_{x\in M_{h}}f(x)$. The contents of this
section are from [8].

######  Definition 2.2 (Lagrangian of a function).

Let $f:\mathbb{R}^{n}\to\mathbb{R}$ be a differentiable function and let
$\beta\in\mathbb{R}^{k}$. Then the Lagrangian function of $f$ on $x\in M_{h}$
the constraints $h=0$,

$\mathcal{L}(f)(x,\beta)=f(x)-\beta^{\top}h(x).$

######  Theorem 2.3 (First-Order Necessary Conditions).

If $x^{*}\in M_{h}$ is a local minimum of $\min_{x\in M_{h}}f(x)$ then there
is a Lagrange multiplier vector $\beta^{*}\in\mathbb{R}^{k}$ such that

$\nabla_{x^{*}}\mathcal{L}(f)(x,\beta^{*})=0,$

thus

$\nabla_{x^{*}}f=\beta^{*\top}\nabla_{x^{*}}h.$

Hence $x^{*}\in M_{h}$ must be a critical point of $f:M_{h}\to\mathbb{R}$.

The sufficient condition for the optimality of $x^{*}$ comes from second-order
conditions; hence, we need feasible directions.

######  Definition 2.4 (The set of linearized feasible directions).

The set of linearized feasible directions on $M_{g}h$ at $x\in M_{h}$
corresponds to vectors orthogonal to the gradient of the constraints of
$M_{h}$, i.e.

$C_{x}M_{h}=\\{v|v^{\top}J_{x}h=0\\}.$

It is clear when the columns $J_{x}h$ of the constraints of $M_{h}$ are
linearly independent, the tangent space $T_{x}M_{h}$ is defined.

First, we have a necessary second-order condition for the optimality. Let
$H_{x}^{g}f$ be the geodesic hessian of $f$ on $M_{h}$ defined by
$H_{x}^{g}f=H_{x}\mathcal{L}(f)$.

######  Theorem 2.5 (Second-order necessary conditions).

Let $x^{*}$ be a critical point of $f:M_{h}\to\mathbb{R}$. Then if $x^{*}$ is
a local minimum $f$ then $v^{\top}\left(H_{x}^{g}f\right)v\geq 0$ for all
$v\in C_{x^{*}}M_{h}$.

and a sufficient condition,

######  Theorem 2.6 (Second-order sufficient conditions).

Let $x^{*}$ be a critical point of $f:M_{h}\to\mathbb{R}$. Then if
$v^{\top}H_{x^{*}}^{g}fv>0$ for all $v\in C_{x^{*}}M_{h}\setminus\\{0\\}$,
then $x^{*}$ is a strict local minimum of $f$ on $M_{h}$.

###  2.3 Preservation of critical Points by Diffeomorphisms

Here, we review that diffeomorphisms preserve critical points, the
positive/negative definiteness, and the full rankness of Hessian matrices at
any critical points of $f$ on $M$.

Let $M$ be a smooth manifold with dimension $m$. Since $M$ and
$\mathbb{R}^{m}$ are locally diffeomorphic. Let $f:M\to\mathbb{R}$ be a smooth
function, let $u^{*}\in\mathbb{R}^{m}$ be a critical point of $f$ and let
$\phi:\mathbb{R}^{m}\to M$ be a local diffeomorphism at $u$ and
$x^{*}=\phi(u^{*})$. By taking the first and second derivatives of
$f\circ\phi$, we have

$\displaystyle\nabla_{u}f(x(u))=\nabla_{x}f(x(u))J_{u}(x(u)),$

and

$H_{u}f(x(u))=J_{u}(x(u))^{\top}H_{x}f(x(u))J_{u}(x(u))+\nabla_{x}f(x(u))H_{u}x(u),$

where $J_{u}(x(u))$ is a Jacobian matrix of $\phi$. Since
$\phi:\mathbb{R}^{m}\to M$ is a diffeomorphism, $J_{u}(x(u))$ is invertible
and if $u^{*}$ is a critical point then $\nabla_{u^{*}}f(x(u))=0$. Thus
$x^{*}=\phi(u^{*})$ is a critical point since $\nabla_{x}f(x(u^{*}))=0$.
Moreover if $u^{*}$ is critical point then $H_{u^{*}}f\circ\phi$ is
positive/negative (semi)definite iff $H_{x(u^{*})}f$ is positive/negative
(semi)definite. Hence, we have the following lemma,

######  Lemma 2.1.

Let $M$ be a smooth manifold with dimension $m$ and let $f:M\to\mathbb{R}$ be
a smooth function. Let $\phi:\mathbb{R}^{m}\to M$ be a diffeomorphism and let
$H_{x}f$ be the Hessian matrix of $f$ at $x\in M$. Let $u^{*}$ be a critical
point of $f$ and $u^{*}=\phi(x^{*})$ then $H_{u^{*}}f\circ\phi$ is
positive/negative (semi)definite iff $H_{x^{*}}f$ is positive/negative
(semi)definite.

##  3 Hypergraphons

A multi-relational hyper-graph $G=(V,E_{1},\cdots,E_{r})$ is a tuple of sets
where $V$ is the set of vertices, and $(V,E_{i})$ is a uniform hyper-graph,
and $E_{i}$ is a relation on $V$ with arity $d_{i}$. A hyper-graph is uniform
when all edges have the same arity. A uniform hyper-graph is linear when each
pair of vertices has at most one hyperedge connecting them.

Let $r$ be a positive integer and $d$ be a vector of positive integers of $r$
dimensions. Then, a $(r,d)$-hyper-graph is the tuple $(V,E_{1},\cdots,E_{r})$
where $V$ is the set of vertices and each $E_{i}$ is a set of uniform hyper-
edges with arity $d_{i}$. $(r,d)$-hyper-graphs are intended to be possible
worlds (semantic models) of first-order languages with a signature of $r$
symmetric relational symbols where each relational symbol $E_{i}$ has an arity
of $d_{i}$.

A multi-relational hypergraph with $r$ relations is the tuple
$(V,E_{1},\cdots,E_{r})$ where $V$ is the set of vertices and each $E_{i}$ is
a set of $d_{i}$-hyper-edges. We denote Let
$W:\prod_{k=1}^{r}([0,1]^{d_{k}}\mapsto\mathbb{R})$ be the cartesian product
of $r$ functions $[0,1]^{d_{k}}\mapsto\mathbb{R}$. Hence each coordinate is
$W_{k}:[0,1]^{d_{k}}\to\mathbb{R}$.

The $(r,d)$-hyper-graphs intend to model the limit of a growing sequence of
hyper-graphs $(r,d)$. The space of $(r,d)$-hyper-graphons is denoted by
$\mathcal{W}^{(r,d)}$ and defined by

$\displaystyle\mathcal{W}^{(r,d)}=\\{W:\prod_{k=1}^{r}([0,1]^{d_{k}}\mapsto[0,1])\mid
W_{k}(x_{1},\cdots,x_{d_{k}})$
$\displaystyle=W_{k}(x_{\psi(1)},\cdots,x_{\psi(d_{k})})\mbox{ for
}\psi\in\Sigma_{d},k\in[r]\\}.$

The elements of $\mathcal{W}^{(r,d)}$ can be considered tuples of non-
parametric stochastic block models of random hypergraphs.

Now, it is convenient to define the space of $d$-hypergraphons with $r$
relations on reals is

$\displaystyle\mathcal{W}_{\mathbb{R}}^{(r,d)}=\\{W:\prod_{k=1}^{r}([0,1]^{d_{k}}\mapsto\mathbb{R})\mid
W_{k}(x_{1},\cdots,x_{d_{k}})$
$\displaystyle=W_{k}(x_{\psi(1)},\cdots,x_{\psi(d_{k})})\mbox{ for
}\psi\in\Sigma_{d},k\in[r]\\}$

In general, if $A\subseteq\mathbb{R}$ then

$\displaystyle\mathcal{W}^{(r,d)}_{A}=\\{W:\prod_{k=1}^{r}([0,1]^{d_{k}}\mapsto
A)\mid W_{k}(x_{1},\cdots,x_{d_{k}})$
$\displaystyle=W_{k}(x_{\psi(1)},\cdots,x_{\psi(d_{k})})\mbox{ for
}\psi\in\Sigma_{d},k\in[r]\\}$

We call elements of $\mathcal{W}^{(r,d)}_{(0,1)}$ and
$\mathcal{W}^{(r,d)}_{\mathbb{R}}$ respectively purely random graphons and
real-valued graphons and when $A$ is omitted then $A=[0,1]$.

Let $\sim$ be the equivalence relation on $\mathcal{W}_{\mathbb{R}}^{(r,d)}$
defined by $W\sim V$ iff it exists $\sigma\in\Sigma$ such that for all
$k\in[r]$ and $(x_{1},\cdots,x_{d_{k}})\in\mathbb{R}^{d_{k}}$ we have
$V_{k}^{\sigma}(x_{1},\cdots,x_{d_{k}})=V(\sigma(x_{1}),\cdots,\sigma(x_{d}{d_{k}})$
which is abbreviated by $W=V^{\sigma}$. The quotient spaces of
$\mathcal{W}^{(r,d)}$ and $\mathcal{W}_{\mathbb{R}}^{(r,d)}$ by $\sim$ are the
spaces unlabeled d-hypergraphons with $r$ relations i.e.
$\widetilde{\mathcal{W}}^{(r,d)}=\mathcal{W}^{(r,d)}/\sim$ and
$\widetilde{\mathcal{W}}_{\mathbb{R}}^{(r,d)}=\mathcal{W}_{\mathbb{R}}^{(r,d)}/\sim$.

We endow $\mathcal{W}_{\mathbb{R}}^{(r,d)}$ a topology by the cut-norm

$\|W\|_{\Box}=\sum_{k=1}^{r}\sup_{S_{1},\cdots
S_{d_{k}}\subset[0,1]}\left|\int_{S_{1}\times\cdots
S_{d_{k}}}W_{k}(x_{1},\cdots x_{d_{k}})dx_{1}\cdots dx_{d_{k}}\right|$

and for $\widetilde{\mathcal{W}}_{\mathbb{R}}^{(r,d)}$ the cut-distance

$\delta_{\Box}(W,V)=\inf_{\sigma\in\Sigma}\|W-V^{\sigma}\|_{\Box}$

###  3.1 Subgraph density

Let $F$ be a $(r,d)$-hypergraph and let $W$ a $(r,d)$-hypergraphon the
subgraph density of $F$ in $W$ is computed by the formula

$t(F,W)=\int_{[0,1]^{|V(F)|}}\prod_{k=1}^{r}\prod_{(i_{1},\cdots,i_{d_{k}})\in
E_{k}(F)}W(x_{i_{1}},\cdots,x_{i_{d_{k}}})\prod_{i\in V(F)}dx_{i}$

###  3.2  Quantum graphs

######  Definition 3.1 (Quantum graphs).

A quantum graph $F$ is the linear combination of a finite number of linear
$(r,d)$-hypergraphs $F_{i}$ with real coefficients, more precisely

$F=\sum_{i}\alpha_{i}F_{i}\quad\mbox{ and }\alpha_{i}\in\mathbb{R}.$

The $(r,d)$-hypergraphs $F_{i}$ are the constituents of the quantum graph.
Hence the definition of $t(F,W)$ extends to quantum graphs linearly, i.e.
$t(F,W)=\sum_{i}\alpha_{i}t(F_{i},W)$.

##  4 Step functions in $\mathcal{W}_{\mathbb{R}}^{(r,d)}$

A $(r,d)$-step function in $\mathcal{W}_{\mathbb{R}}^{(r,d)}$ is a function
which is constant in each box
$(\frac{i_{1}-1}{m},\frac{i_{1}}{m}]\times\cdots\times(\frac{i_{d}-1}{m},\frac{i_{d_{k}}}{m}]$
for each coordinate $k\in[r]$. A step function has a natural parameterization
as follows:

Let $\nu$ be the Lebesgue measure on $\mathbb{R}$. Let
$S=\\{S_{1},\cdots,S_{m}\\}$ be a partition of $[0,1]$ where every $S_{i}$ is
an interval, let $\pi\in P^{({m})}$ such that $\nu(S_{i})=\pi_{i}$. Let
$\mathbb{R}_{S}^{(m,r,d}$ be the set of $r$ dimensional vectors where each
coordinate is a symmetric $d$-dimensional array with $m^{d}$ steps. Let
$P^{({m})}_{q}$ be the $k$-simplex of probability vectors of length $m$

$P^{({m})}=\\{\pi\in[0,1]^{m}\mid\sum_{i}\pi_{i}=1\\}.$

We call the elements $\pi\in P^{({m})}$ partition vectors of the step
functions. Let $A\in\mathbb{R}_{S}^{(m,r,d}$ and $\pi\in P^{({m})}$. Then a
$m$-step function
$(A,\pi):\prod_{k=1}^{r}([0,1]^{d_{k}}\mapsto\mathbb{R}^{r})$ is a function
that locally constant on $S_{i_{1}}\times\cdots\times S_{i_{d}}$ for all
$i_{1},\cdots,i_{d}\in[m]^{d}$ and in each coordinate $k\in[r]$, more
precisely

$(A,\pi)_{k}=\sum_{i_{1},\cdots,i_{d_{k}}=1}^{m}A_{i_{1},\cdots,i_{d},k}\mathbbm{1}_{S_{i_{1}}\times\cdots\times
S_{i_{d_{k}}}}.$

We denote the space of $m$-step functions by
$\mathcal{W}^{({m,r,d})}_{\mathbb{R}}$. Thus
$\mathcal{W}^{({m,r,d})}_{\mathbb{R}}$ is identified by
$\mathbb{R}_{S}^{(m,r,d)}\times P^{(m)}$. It is convenient to define the class
of step functions when $\pi$ is fixed. Hence
${\mathcal{W}}^{({m},r,d)}_{\mathbb{R}}(\pi)=\mathbb{R}_{S}^{(m,r,d}\times\\{\pi\\}$.
Let

$n(m,r,d)=\sum_{k=1}^{r}\binom{m+d_{k}-1}{d_{k}}$

then the number of parameters to define a $m$-step function in
$\mathcal{W}^{({m,r,d})}_{\mathbb{R}}$ and
${\mathcal{W}}^{({m},r,d)}_{\mathbb{R}}(\pi)$ are $n(m,r,d)+m-1$ and
$n(m,r,d)$. In ${\mathcal{W}}^{({m,r,d})}$, $t(F,W)$ is reduced to

$t(F,(A,\pi))=\sum_{\tiny
x_{1},\cdots,x_{|V(F)|}=1}^{m}\prod_{k=1}^{r}\prod_{(i_{1},\cdots,i_{d_{k}})\in
E_{k}(F)}A_{i_{1},\cdots,i_{d_{k}},k}\prod_{i\in V(F)}\pi_{i}$

Let $E(F)=\cup_{k}E_{k}(F)$. It sometimes convenient to write $t(F,(A,\pi))$
as follows,

$t(F,(A,\pi))=\sum_{\tiny
x_{1},\cdots,x_{|V(F)|}=1}^{m}\prod_{(i_{1},\cdots,i_{d_{k}})\in
E(F)}A_{i_{1},\cdots,i_{d},k}\prod_{i\in V(F)}\pi_{i}$

###  4.1 Partial Derivatives of $t(F,(A,\pi))$

In this section, we develop formulas for the partial derivative of the
polynomials $t(F,(A,\pi))$. First, we define the notion of the labeled
hypergraph and partial subgraph density.

####  4.1.1 Labeled Graphs and Partial Subgraph Densities

######  Definition 4.1 (Labeled Graphs).

A k-labeled hypergraph is a $(r,d)-$hyper-graph $F$ to which we associate a
vector of its vertices $(a_{1}\cdots a_{k})\in(V(F))^{k}$ (for some
$k\in\mathbb{N}$). We denote this $k$-labeled graph by $F^{\bullet a_{1}\cdots
a_{k}}$. For a $k$-labeled graph, its $i$-th label is $a_{i}$ for $i\in[k]$.

We define the partial subgraph density of a $k$-labeled graph $F^{\bullet
a_{1}\cdots a_{k}}$ as a $k$-dimensional function
$t_{x_{1},\cdots,x_{k}}(F^{\bullet a_{1}\cdots a_{k}},W):[0,1]^{k}\to[0,1]$
where

$t_{x_{1},\cdots,x_{k}}(F^{\bullet a_{1}\cdots
a_{k}},(A,\pi))=\int_{[0,1]^{|F|-k}}\prod_{j=1}^{r}\prod_{(i_{1},\cdots,i_{d_{j}})\in
E_{j}(F)}A_{i_{1},\cdots,i_{d_{j}},j}\prod_{i\in V\setminus[k]}dx_{i}.$ (4.1)

By definition, if $k=0$ then $t_{x_{1},\cdots,x_{k}}(F^{\bullet a_{1}\cdots
a_{k}},(A,\pi))$ is $t(F,(A,\pi))$. When considering step functions $(A,\pi)$,
we will sometimes use $t_{\pi:i_{1}\cdots i_{k}}(F^{\bullet a_{1}\cdot
a_{k}},(A,\pi))$ as a shorthand for $t_{x_{1}\cdots x_{k}}(F^{\bullet
a_{1}\cdot a_{k}},(A,\pi))$ where for $i\in[k]$ the value $x_{i}$ is in the
$i$-th partition class, e.g., $x_{i}=\pi_{i}/2+\sum_{j=1}^{i-1}\pi_{j}$.

Moreover, a notation for the linear combination of partial subgraph densities
is also necessary.

######  Definition 4.2 (Linear combination of partial subgraph densities.).

Let $F$ be a $(r,d)$-hypergraph and $k\in[r]$. We define a notation for a
linear combination of partial subgraph densities

$\partial_{k}^{(\bullet\cdots\bullet)}F=\sum_{(a_{1}\cdots a_{d_{k}})\in
E_{k}(F)}F^{\bullet(a_{1}\cdots a_{d_{k}})}$

and

$\partial^{\bullet}F=\sum_{a\in V(F)}F^{\bullet a}.$

where $F^{\bullet(a_{1}\cdots a_{d_{k}})}$ denotes the $d_{k}$-labeled
hypergraph obtained from $F$ by deleting the hyperedge $(a_{1}\cdots
a_{d_{k}})$ and labeling the vertices $\\{a_{1},\cdots,a_{d_{k}}\\}$ and
$F^{\bullet a}$ denotes the $1$-labeled graph obtained from $F$ by labeling
the node $a$, The partial subgraph densities of these labeled hypergraphs are

$\displaystyle t_{i_{1}\cdots i_{d_{k}}}(F^{\bullet(a_{1}\cdots
a_{d_{k}})},(A,\pi))=\sum_{x_{1},\cdots,x_{|F|-d_{k}}}\prod_{(s_{1}\cdots
s_{d_{j}})\in E(F)\setminus\\{a_{1}\cdots q_{d_{k}}\\}}A_{x_{s_{1}},\cdots
x_{d_{j}},j}$ $\displaystyle\prod_{k\in
V(F)\setminus\\{a_{1},\cdots,a_{d_{k}}\\}}\pi_{x_{k}}$ (4.2)

and

$t_{i}(F^{\bullet a},(A,\pi))=\sum_{x_{1},\cdots,x_{|F|-1}}\prod_{(s_{1}\cdots
s_{d_{k}})\in E_{k}(F)}A_{x_{s_{1}},\cdots x_{d_{k}}}\prod_{k\in
V(F)\setminus\\{a\\}}\pi_{x_{k}}$

Note that $F^{\bullet(ab)}$ and $F^{\bullet(ba)}$ are not always equivalent in
the computation of the partial subgraph density. Consider $F^{\bullet(ab)}$
and $F^{\bullet(ba)}$ as shown in Figure 1, then

$t_{uv}(F^{\bullet(ab)},W)=\int_{[0,1]}W(x,v)dx\quad\mbox{ and }\quad
t_{uv}(F^{\bullet(ba)},W)=\int_{[0,1]}W(u,y)dy.$

Figure 1: $F^{\bullet(ab)}$ and $F^{\bullet(ba)}$ are not equivalent.

####  4.1.2 Partial Derivatives of $t(F,)$ in
$\mathcal{W}^{({m,r,d})}_{\mathbb{R}}$

We identify uniquely the partial derivative operator $\frac{\partial}{\partial
A_{i_{1}\cdots i_{d_{k}},k}}$ by the non-increasing sequence
$i_{1}\leq\cdots\leq i_{d_{k}}$. $\frac{\partial{A}_{x_{1}\cdots
x_{d_{j}},j}}{\partial{A}_{i_{1}\cdots i_{d_{k}},k}}$ is non-zero when $j=k$
and $\\{x_{1}\cdots x_{d_{j}}\\}=\\{i_{1}\cdots i_{d_{k}}\\}$ thus

$\frac{\partial{A}_{x_{1}\cdots x_{d_{j}},j}}{\partial{A}_{i_{1}\cdots
i_{d_{k}},k}}=\delta_{j,k}\delta_{\\{x_{1}\cdots x_{d_{j}}\\},\\{i_{1}\cdots
i_{d_{k}}\\}}$

where $\delta_{i,j}$ is Kronecker’s delta, and we overload Kronecker’s delta
for finite sets, i.e.

$\delta_{A,B}=\begin{cases}1&A=B\\\ 0&A\neq B\end{cases}$

where $A$ and $B$ are finite sets.

######  Theorem 4.3.

Let $(A,\pi)\in\mathcal{W}^{({m,r,d})}_{\mathbb{R}}$ and let
$orb((i_{1},i_{2},\cdots i_{d_{k}}))$ by the orbit of
$A_{i_{1},\cdots,i_{d_{k}}}$ the actions of $\Sigma_{d_{k}}$. Then the first
partial derivative of $t(F,(A,\pi))$ are

$\displaystyle\frac{\partial t(F)}{\partial A_{i_{1},\cdots,i_{d_{k}},k}}$
$\displaystyle=$
$\displaystyle\pi_{i_{1}}\cdots\pi_{i_{d_{k}}}\sum_{(j_{1},\cdots,j_{d_{k}})\in
orb((i_{1}\cdots i_{d_{k}}))}t_{j_{1}\cdots
j_{d_{k}}}(\partial_{k}^{\bullet(\bullet\cdots\bullet)}F,(A,\pi))$ (4.3)

and

$\frac{\partial
t(F,(A,\pi))}{\partial\pi_{i}}=t_{i}(\partial^{\bullet}F,(A,\pi)).$ (4.4)

######  Example 4.4.

Let $F=\leavevmode\hbox to8.94pt{\vbox
to8.94pt{\pgfpicture\makeatletter\hbox{\hskip 1.62271pt\lower-1.62271pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{} }
{}{}{{{}} {}{}{}{}{}{}{}{} } {}{}{{{}} {}{}{}{}{}{}{}{} }
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ and let
$(A,\pi)\in\mathcal{W}^{(m,1,2)}_{\mathbb{R}}$. Then $t(\leavevmode\hbox
to8.94pt{\vbox to8.94pt{\pgfpicture\makeatletter\hbox{\hskip
1.62271pt\lower-1.62271pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{} }
{}{}{{{}} {}{}{}{}{}{}{}{} } {}{}{{{}} {}{}{}{}{}{}{}{} }
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},(A,\pi))$ is computed
by the formula

$t(\leavevmode\hbox to8.94pt{\vbox
to8.94pt{\pgfpicture\makeatletter\hbox{\hskip 1.62271pt\lower-1.62271pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{} }
{}{}{{{}} {}{}{}{}{}{}{}{} } {}{}{{{}} {}{}{}{}{}{}{}{} }
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},(A,\pi)=\sum_{i,j,k=1}^{m}A_{ij}A_{jk}A_{ki}\pi_{i}\pi_{j}\pi_{k}.$

Then $F^{\bullet(a,b)}=\leavevmode\hbox to8.74pt{\vbox
to8.74pt{\pgfpicture\makeatletter\hbox{\hskip 1.42271pt\lower-1.62271pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{{{}}
{}{}{}{}{}{}{}{} } {}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ hence
$\partial\leavevmode\hbox to8.94pt{\vbox
to8.94pt{\pgfpicture\makeatletter\hbox{\hskip 1.62271pt\lower-1.62271pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{} }
{}{}{{{}} {}{}{}{}{}{}{}{} } {}{}{{{}} {}{}{}{}{}{}{}{} }
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\leavevmode\hbox
to8.74pt{\vbox to8.74pt{\pgfpicture\makeatletter\hbox{\hskip
1.42271pt\lower-1.62271pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{{{}}
{}{}{}{}{}{}{}{} } {}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to8.74pt{\vbox to8.74pt{\pgfpicture\makeatletter\hbox{\hskip
1.62271pt\lower-1.62271pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{{{}}
{}{}{}{}{}{}{}{} } {}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{2.84544pt}{5.69046pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to8.54pt{\vbox to8.74pt{\pgfpicture\makeatletter\hbox{\hskip
1.42271pt\lower-1.42271pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{{{}}
{}{}{}{}{}{}{}{} } {}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ and
$t_{ij}(\leavevmode\hbox to8.74pt{\vbox
to8.74pt{\pgfpicture\makeatletter\hbox{\hskip 1.42271pt\lower-1.62271pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{{{}}
{}{}{}{}{}{}{}{} } {}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},(A,\pi))=\sum_{k=1}^{m}A_{ik}A_{jk}\pi_{k}$.
Hence the partial derivative $\frac{\partial t(\leavevmode\hbox to8.94pt{\vbox
to8.94pt{\pgfpicture\makeatletter\hbox{\hskip 1.62271pt\lower-1.62271pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{} }
{}{}{{{}} {}{}{}{}{}{}{}{} } {}{}{{{}} {}{}{}{}{}{}{}{} }
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},(A,\pi))}{\partial
A_{ij}}$ is computed by

$\displaystyle\frac{\partial t(\leavevmode\hbox to8.94pt{\vbox
to8.94pt{\pgfpicture\makeatletter\hbox{\hskip 1.62271pt\lower-1.62271pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}{}{}{}{}{} } {}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{2.84544pt}{5.69046pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to8.54pt{\vbox to8.74pt{\pgfpicture\makeatletter\hbox{\hskip
1.42271pt\lower-1.42271pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{{{}}
{}{}{}{}{}{}{}{} } {}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},(A,\pi))\right)$
$\displaystyle=3\pi_{i}\pi_{j}\left(t_{ij}(\leavevmode\hbox to8.74pt{\vbox
to8.74pt{\pgfpicture\makeatletter\hbox{\hskip 1.42271pt\lower-1.62271pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{{{}}
{}{}{}{}{}{}{}{} } {}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},(A,\pi))+(1-\delta_{ij})t_{ji}(\leavevmode\hbox
to8.74pt{\vbox to8.74pt{\pgfpicture\makeatletter\hbox{\hskip
1.42271pt\lower-1.62271pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{{{}}
{}{}{}{}{}{}{}{} } {}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},(A,\pi))\right)$

By the symmetry of triangle, we have $t_{ij}(\leavevmode\hbox to8.74pt{\vbox
to8.74pt{\pgfpicture\makeatletter\hbox{\hskip 1.42271pt\lower-1.62271pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{{{}}
{}{}{}{}{}{}{}{} } {}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},(A,\pi))=t_{ij}(\leavevmode\hbox
to8.74pt{\vbox to8.74pt{\pgfpicture\makeatletter\hbox{\hskip
1.62271pt\lower-1.62271pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{{{}}
{}{}{}{}{}{}{}{} } {}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{2.84544pt}{5.69046pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},(A,\pi))=t_{ij}(\leavevmode\hbox
to8.54pt{\vbox to8.74pt{\pgfpicture\makeatletter\hbox{\hskip
1.42271pt\lower-1.42271pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{{{}}
{}{}{}{}{}{}{}{} } {}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},(A,\pi))$. then we
have

$\displaystyle\frac{\partial t(\leavevmode\hbox to8.94pt{\vbox
to8.94pt{\pgfpicture\makeatletter\hbox{\hskip 1.62271pt\lower-1.62271pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{} }
{}{}{{{}} {}{}{}{}{}{}{}{} } {}{}{{{}} {}{}{}{}{}{}{}{} }
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},(A,\pi))}{\partial
A_{ij}}$ $\displaystyle=3(2-\delta_{ij})\pi_{i}\pi_{j}t_{ij}(\leavevmode\hbox
to8.74pt{\vbox to8.74pt{\pgfpicture\makeatletter\hbox{\hskip
1.42271pt\lower-1.62271pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.42271pt}{0.0pt}\pgfsys@curveto{1.42271pt}{0.78575pt}{0.78575pt}{1.42271pt}{0.0pt}{1.42271pt}\pgfsys@curveto{-0.78575pt}{1.42271pt}{-1.42271pt}{0.78575pt}{-1.42271pt}{0.0pt}\pgfsys@curveto{-1.42271pt}{-0.78575pt}{-0.78575pt}{-1.42271pt}{0.0pt}{-1.42271pt}\pgfsys@curveto{0.78575pt}{-1.42271pt}{1.42271pt}{-0.78575pt}{1.42271pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{{}}
{}{}{}{}{}{}{}{} }\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@moveto{4.26816pt}{5.69046pt}\pgfsys@curveto{4.26816pt}{6.47621pt}{3.6312pt}{7.11317pt}{2.84544pt}{7.11317pt}\pgfsys@curveto{2.0597pt}{7.11317pt}{1.42273pt}{6.47621pt}{1.42273pt}{5.69046pt}\pgfsys@curveto{1.42273pt}{4.90471pt}{2.0597pt}{4.26775pt}{2.84544pt}{4.26775pt}\pgfsys@curveto{3.6312pt}{4.26775pt}{4.26816pt}{4.90471pt}{4.26816pt}{5.69046pt}\pgfsys@closepath\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{}{{{}}
{}{}{}{}{}{}{}{} } {}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{2.84544pt}{5.69046pt}\pgfsys@lineto{5.69046pt}{0.0pt}\pgfsys@moveto{7.11317pt}{0.0pt}\pgfsys@curveto{7.11317pt}{0.78575pt}{6.47621pt}{1.42271pt}{5.69046pt}{1.42271pt}\pgfsys@curveto{4.90471pt}{1.42271pt}{4.26775pt}{0.78575pt}{4.26775pt}{0.0pt}\pgfsys@curveto{4.26775pt}{-0.78575pt}{4.90471pt}{-1.42271pt}{5.69046pt}{-1.42271pt}\pgfsys@curveto{6.47621pt}{-1.42271pt}{7.11317pt}{-0.78575pt}{7.11317pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{5.69046pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},(A,\pi))$
$\displaystyle=3(2-\delta_{ij})\pi_{i}\pi_{j}\sum_{k=1}^{m}A_{ik}A_{jk}\pi_{k}$

###  4.2 The split map

On the space of step functions, we define the split of step functions,

######  Definition 4.5 (Split of step functions).

Let $\lambda\in(0,1)$ and let $k\in[m]$. we denote by
$\theta(\cdot,\lambda,k):\mathcal{W}^{({m,r,d})}_{\mathbb{R}}\to\mathcal{W}^{({m+1,r,d})}_{\mathbb{R}}$
the map defined by
$\theta((A,\pi),\lambda,k)=(\theta(A,\lambda,k),\theta(\pi,\lambda,k))$ where
$\theta(A,\lambda,k)\in\mathbb{R}_{S}^{(m+1,r,d)}$ and
$\theta(\pi,\lambda,k)\in P^{({m+1})}$ such that

$\theta_{i}(\pi,\lambda,k)=\left\\{\begin{array}[]{c c}\pi_{i}&i<k\\\
\lambda\pi_{k}&i=k\\\ (1-\lambda)\pi_{k}&i=k+1\\\ \pi_{i-1}&k+1<i\leq
m+1\end{array}\right.$

and

$\theta_{i_{1},\cdots,i_{d_{k}},k}(A,\lambda,m)=\left\\{\begin{array}[]{c
c}A_{i_{1},\cdots,i_{d_{k}},k}&\forall_{n\in[d_{k}]}i_{n}\in[k]\\\
A_{i_{1},\cdots,i_{d-s}m\cdots
m,k}&\exists_{s\in[d_{k}]}i_{d_{k}-s+1}=i_{d_{k}-s+2}=\cdots=i_{d}=m+1\end{array}\right.$

and $\theta_{i_{1},\cdots,i_{d_{k}}}(A,\lambda,k)$ is obtained from
$\theta_{i_{1},\cdots,i_{d_{k}}}(A,\lambda,m)$ by a suitable permutation
$\sigma\in\Sigma_{m}$ i.e.
$\theta_{i_{1},\cdots,i_{d_{k}}}(A,\lambda,k)=\theta_{i_{1},\cdots,i_{d_{k}}}(A,\lambda,m)^{\sigma}$.

Note that the action of $\theta(\cdot,\lambda,k)$ on $(A,\pi)$ is the split
the $k$-th row/column of $(A,\pi)$ in two rows/columns whose weights are
$\lambda\pi_{k}$ and $(1-\lambda)\pi_{k}$.

Figure 2: A split map transforms the representation of a $3$ step function
into a $4$ step function

The split map does not change $(A,\pi):[0,1]^{d}\to\mathbb{R}^{r}$ as function
only increases the dimensions to represent $(A,\pi)$. Figure 2 shows how a
split map transforms a bidimensional $3$ step function into a $4$-step
function, which is the same step function.

##  5 Uniform multi-relational hypergraphons

Here we extend key results in Graphon Theory as Szemeredi’s regularity lemma,
Compactness, Counting Lemma, and Large Deviation principle from the space of
one-relation hypergraphons, i.e., $\widetilde{\mathcal{W}}^{(1,d)}$ to
$\widetilde{\mathcal{W}}^{(r,d)}$. To do that, we extend these key results for
the case when $d$ is a vector of constant value. Hence, $\mathcal{W}^{(r,d)}$
is the space of vectors of uniform multi-relational hypergraphons and
$\widetilde{\mathcal{W}}^{(r,d)}=\mathcal{W}^{(r,d)}/\sim$. When $r=1$,
Lubetzky and Zhao [19], extend these key results. Here, we extend the results
when $r\geq 1$.

###  5.1 Key results in graphons

######  Theorem 5.1 (Szemeredi’s regularity lemma).

When $r=1$. For every $\epsilon>0$ there exists some $M(\epsilon)>0$ such that
for every $W\in\mathcal{W}^{(r,d)}$ there exist some $m\leq M(\epsilon)$ and
some $g\in\mathcal{W}^{(r,d)}$ with $\|f-g\|_{\Box}\leq\epsilon$, and such
that $g$ is constant in each box
$(\frac{i_{1}-1}{m},\frac{i_{1}}{m}]\times\cdots\times(\frac{i_{d}-1}{m},\frac{i_{d}}{m}]$.

Hence, we have the Szemeredi Regularity Lemma for $\mathcal{W}^{(r,d)}$.

######  Theorem 5.2.

For every $\epsilon>0$ there exists some $M(\epsilon)>0$ such that for every
$W\in\mathcal{W}^{(r,d)}$ there exist some $m\leq M(\epsilon)$ and some
$g\in\mathcal{W}^{(r,d)}$ with $\|f-g\|_{\Box}\leq\epsilon$, and such that g
is constant in each box
$(\frac{i_{1}-1}{m},\frac{i_{1}}{m}]\times\cdots\times(\frac{i_{d}-1}{m},\frac{i_{d}}{m}]$.

###### Proof.

The proof is direct from Lemma $4.1$ in [17]. ∎

Using Szemeredi’s regularity lemma, we prove the compactness of
$\widetilde{\mathcal{W}}^{(r,d)}$.

######  Theorem 5.3.

($\widetilde{\mathcal{W}}^{(r,d)}$, $\delta_{\Box}$) is compact.

###### Proof.

Based on Theorem 5.2, the proof is identical to the compactness proof for
classical graphons $\widetilde{\mathcal{W}}^{(1,2)}$ given in [16]. ∎

Another key result is the Counting Lemma.

######  Lemma 5.1 (Counting Lemma ).

Let $F$ be a linear $(r,d)$-hypergraph then for all
$V,W\in\widetilde{\mathcal{W}}^{(r,d)}$ we have

$|t(F,W)-t(F,V)|\leq\delta_{\Box}(W,V)$

###### Proof.

The proof is direct for the graph case given in [3]. ∎

###  5.2 Large Deviation Principle for random hypergraphons

To state properly the large deviation principle, we need some definition.

######  Definition 5.4 (Graphon representation of a $(r,d)$-hypergraph ).

Let $G$ be a $(r,d)$-hyper-graph. Then $f^{G}$ is the graphon representation
of $G$ whose $k$ coordinate is defined by,

$f_{k}^{G}(x_{1},\cdots,x_{d})=\begin{cases}1&\mbox{ if }(v_{\lceil
nx_{1}\rceil},\cdots,v_{\lceil nx_{d}\rceil})\in E_{k}(G)\\\ 0&\mbox{
otherwise}\end{cases}$

where $n=|V(G)|$.

######  Definition 5.5 (Random hypergraph $G^{(r,d)}(n,p)$).

Let $n$ be a positive integer and $p\in[0,1]$. The Erdős-Renyi random
$(r,d)$-hyper-graph is obtained by sampling each $k$ coordinate by taking
every $d$-tuple of vertices from $[n]$ and connecting them with a $p$
probability.

Let $\mathbb{P}^{(r,d)}_{n,p}$ be the probability distribution induced by
$f^{G}$ where $G$ are random $(r,d)$-hyper-graphs.

######  Definition 5.6 (Rate function for $\mathbb{P}^{(r,d)}_{n,p}$).

Let $I^{(r,d)}:\widetilde{\mathcal{W}}^{(r,d)}\to[0,\infty)$ be the function
defined by,

$I^{(r,d)}_{p}(W)=\sum_{k}\frac{1}{d_{k}!}\int_{[0,1]^{d_{k}}}h_{p}(W_{k}(x_{1},\cdots,x_{d_{k}})dx_{1}\cdots
dx_{d_{k}}$

where

$h_{p}(x)=x\log(\frac{x}{p})+(1-x)\log(\frac{1-x}{1-p})$

######  Theorem 5.7.

(Theorem $5.4$ in [19]) When $r=1$. For each fixed $p\in(0,1)$, the sequence
$\mathbb{P}_{n,p}$ obeys a large deviation principle in the space
$\widetilde{\mathcal{W}}^{(r,d)}$ with rate function hp. Explicitly, for any
closed set $F\subseteq\widetilde{\mathcal{W}}^{(r,d)}$

$\limsup_{n\to\infty}\frac{1}{(n)_{d_{1}}}\log\mathbb{P}^{(r,d)}_{n,p}(F)\leq-\inf_{W\in
F}I^{(r,d)}_{p}(W)$

and for any open set $U\subseteq\widetilde{\mathcal{W}}^{(r,d)}$

$\liminf_{n\to\infty}\frac{1}{(n)_{d_{1}}}\mathbb{P}^{(r,d)}_{n,p}(U)\geq-\inf_{W\in
U}I^{(r,d)}_{p}(W)$

where $(n)_{x}=n(n-1)\cdots(n-x+1)$.

To extend LDP for random $(r,d)$-hypergraphons when $d$ is uniform, we need
the following Lemma

######  Lemma 5.2.

When $r=1$. Let $x\in\widetilde{\mathcal{W}}^{(r,d)}$ and let
$B(x,\eta)=\\{y\mid\delta_{\Box}(x,y)\leq\eta\\}$. Then for any
$x\in\widetilde{\mathcal{W}}^{(r,d)}$

$\lim_{\eta\to
0}\limsup_{n\to\infty}\frac{1}{(n)_{d_{1}}}\log\mathbb{P}_{n,p}^{(r,d)}(B(x,\eta))\leq
I^{(r,d)}_{p}(x)$

and for any arbitrary $\eta>0$

$\liminf_{n\to\infty}\frac{1}{(n)_{d_{1}}}\log\mathbb{P}_{n,p}^{(r,d)}(B(x,\eta))\geq
I^{(r,d)}_{p}(x)$

###### Proof.

The proof is identical to the proof Theorem $2.3$ in [4]. ∎

Hence, we extend Theorem 5.7 for the general case.

######  Theorem 5.8.

For each fixed $p\in(0,1)$, the sequence $\mathbb{P}_{n,p}$ obeys a large
deviation principle in the space $\widetilde{\mathcal{W}}^{(r,d)}$ with rate
function hp. Explicitly, for any closed set
$F\subseteq\widetilde{\mathcal{W}}^{(r,d)}$

$\limsup_{n\to\infty}\frac{r}{(n)}_{d_{1}}\log\mathbb{P}^{(r,d)}_{n,p}(F)\leq-\inf_{W\in
F}I^{(r,d)}_{p}(W)$

and for any open set $U\subseteq\widetilde{\mathcal{W}}^{(r,d)}$

$\liminf_{n\to\infty}\frac{r}{(n)_{d_{1}}}\mathbb{P}^{(r,d)}_{n,p}(U)\geq-\inf_{W\in
U}I^{(r,d)}_{p}(W)$

###### Proof.

Using Lemma 5.2, the proof is identical to Theorem $5$ in [1] ∎

##  6 Non-uniform multi-relational hyper-graphons

Here, we extend the results of the previous section to
$\widetilde{\mathcal{W}}^{(r,d)}$ when $d$ is a non-uniform vector.

Let $d$ be a vector of arities and $\max(d)$ the maximum arity. Note that
$(\widetilde{\mathcal{W}}^{(r,d)},\delta)$ is a closed space of
$(\widetilde{\mathcal{W}}^{(r,\max(d))},\delta_{\Box})$. Hence, compactness
and counting lemma hold in $(\widetilde{\mathcal{W}}^{(r,d)},\delta)$.

To extend LDP, we first define the projection map $h$ and apply the
contraction principle of Large Deviation Theory [5]. Let
$h:\widetilde{\mathcal{W}}^{(r,\max(d))}\to\widetilde{\mathcal{W}}^{(r,d)}$ be
the map defined by

$h(W_{k}(x_{1},\cdots,x_{\max(d)}))=W_{k}(x_{1},\cdots,x_{d_{k}},x_{d_{k}},\cdots,x_{d_{k}})$
(6.1)

for all $k\in[r]$. This map forgets for each $k$ hypergraphon the last
$max(d)-d_{k}$ coordinates. In this setting, a $(r,1)$-hypergraphon is the
limit of growing hypergraphs with isolated vertices and self-loops.

######  Lemma 6.1.

The map
$h:\widetilde{\mathcal{W}}^{(r,\max(d))}\to\widetilde{\mathcal{W}}^{(r,d)}$ is
continuous in $(\widetilde{\mathcal{W}}^{(r,\max(d))},\delta_{\Box})$.

###### Proof.

It is sufficient to prove that $\|h(W)\|_{\Box}\leq\|W\|_{\Box}$. Indeed, it
is clear that the inequality holds

$\displaystyle\sup_{S_{1},\cdots
S_{d_{k}}\subseteq[0,1]}\left|\int_{S_{1}\times\cdots
S_{d_{k}}}W(x_{1},\cdots,x_{d_{k}})dx_{1}\cdots dx_{d_{k}}\right|\leq$
$\displaystyle\sup_{S_{1},\cdots
S_{\max(d)}\subseteq[0,1]}\left|\int_{S_{1}\times\cdots
S_{\max(d)}}W(x_{1},\cdots,x_{\max(d)})dx_{1}\cdots dx_{\max(d)}\right|$

∎

###  6.1 Large deviation principle for random $(r,d)$-hypergraphs

We prove the LDP principle for $\widetilde{\mathcal{W}}^{(r,d)}$. To do that,
we apply the contraction principle in Large Deviation Theory.

Let $\mathbb{P}^{(r,d)}_{k,n,p}$ be the probability distribution induced by
$f_{k}^{G}$ where $G$ are random $(r,d)$-hypergraaphs.

######  Theorem 6.1 (Contraction Principle Theorem $4.2.1$ in [5]).

Let $\mathcal{X}$ and $\mathcal{Y}$ be Hausdorff topological spaces and
$f:\mathcal{X}\to\mathcal{Y}$ a continuous function. Consider a good rate555A
good rate function is a rate function for which all the level sets
$I([0,\alpha])$ are compact subsets of $\mathcal{X}$. function
$I:\mathcal{X}\to[0,\infty]$.

  1. 1.

For each $y\in\mathcal{Y}$, define

$I^{\prime}(y)=\inf\\{I(x)\mid x\in\mathcal{X},y=f(x)\\}$

Then $I^{\prime}$ is a good rate function on $\mathcal{Y}$, where as usual the
infimum over the empty set is taken as $\infty$.

  2. 2.

If $I$ controls the LDP associated with a family of probability measures
$\mathbb{P}$ on $\mathcal{X}$, then $I^{\prime}$ controls the LDP associated
with the family of probability measures $\mathbb{P}\circ f^{-1}$ on
$\mathcal{Y}$.

A consequence of the above lemma is $\widetilde{\mathcal{W}}^{(r,d)}$ is
compact when $d$ is a vector of arities. Now, we prove LDP for random multi-
relational hypergraphs with variable arities.

######  Theorem 6.2.

Let $d$ be a $r$-dimensional vector of arities. For each fixed $p\in(0,1)$,
the sequence $\mathbb{P}_{n,p}$ obeys a large deviation principle in the space
$\widetilde{\mathcal{W}}^{(r,d)}$ with rate function hp. Explicitly, for any
closed set $F\subseteq\widetilde{\mathcal{W}}^{(r,d)}$

$\limsup_{n\to\infty}\sum_{k}\frac{1}{(n)_{d_{k}}}\log\mathbb{P}^{(r,d)}_{k,n,p}(F)\leq-\inf_{W\in
F}I^{(r,d)}_{p}(W)$

and for any open set $U\subseteq\widetilde{\mathcal{W}}^{(r,d)}$

$\liminf_{n\to\infty}\sum_{k}\frac{1}{(n)_{d_{k}}}\mathbb{P}^{(r,d)}_{k,n,p}(U)\geq-\inf_{W\in
U}I^{(r,d)}_{p}(W)$

where $\mathbb{P}^{(r,d)}_{k,n,p}$ is the probability distribution induced by
$G_{k}^{(r,d_{k})}(n,p)$.

###### Proof.

The proof is direct from Theorem 6.1 and Lemma 6.1. ∎

######  Definition 6.3 (Constrained region defined by quantum graphs).

Let $\mathcal{F}$ be an ordered set of quantum $(r,d)$-graphs
$\mathcal{F}=[F_{1},\cdots,F_{k}]$ and let $u\in\mathbb{R}^{k}$ be a vector of
multi-relational subgraph densities. Then we define the constrained region
defined by $(\mathcal{F},u)$

$\widetilde{S}^{(r,d)}(\mathcal{F},u)=\\{W\in\widetilde{\mathcal{W}}^{(r,d)}\mid\wedge_{i}t(F_{i},W)=u_{i}\\{.$
(6.2)

Hence, we prove the most typical random $(r,d)$-hyper-graphs of
$\widetilde{S}^{(r,d)}(\mathcal{F},u)$.

######  Theorem 6.4.

Let $\mathcal{F}$ be an ordered set of quantum $(r,d)$-graphs
$\mathcal{F}=[F_{1},\cdots,F_{k}]$ and let $u\in\mathbb{R}^{k}$ be a vector of
multi-relational subgraph densities. Then the limits $W^{*}(\mathcal{F},u)$ of
the sequence $(G^{(r,d)}(n,1/2))_{n=1}^{\infty}$ of growing random
$(r,d)$-hypergraphs which are uniformly sampled such that
$\lim_{n\rightarrow\infty}G^{[r]}(n,1/2)\in\widetilde{S}^{(r,d)}(\mathcal{F},u)$
satisfy

$W^{*}(\mathcal{F},u)=\operatorname*{arg\,min}_{W\in\widetilde{S}^{(r,d)}(\mathcal{F},u)}I_{1/2}^{(r,d)}(W)$
(6.3)

###### Proof.

The proof is identical to Theorem $3.1$ in [4]. ∎

Hereafter we write $I_{1/2}^{(r,d)}$ as $I$ and $h_{1/2}$ as $I_{0}$.

##  7 Modeling Possible Worlds in Probabilistic Logic

Let $L=(\forall,\wedge,\Rightarrow,R_{1},\cdots,R_{r})$ be a signature of a
first-order language with only symmetric and relational symbols. If we have an
ordered set of logical closed formulas $\mathcal{F}$ from $L$ and vector
$u\in(0,1)^{|\mathcal{F}|}$ then we can form a probabilistic relational theory
$(\mathcal{F}),u$ by attaching each logical formula $\mathcal{F}_{i}$ with
$u_{i}$.

Let $L=(\forall,\wedge,\Rightarrow,Friends;Sm)$ be a relational signature with
the predicates Friends and Smoker(Sm). From this signature, we build the
following probabilistic theory

$\displaystyle u_{1}:\forall_{x}Sm(x)$ $\displaystyle
u_{2}:\forall_{x,y}Friends(x,y)$ $\displaystyle
u_{3}:\forall_{x,y,z}Friends(x,y)\wedge Friends(y,z)\wedge Friends(z,x)$
$\displaystyle u_{4}:\forall_{x,y}Friends(x,y)\Rightarrow(Sm(x)\Leftrightarrow
Sm(y))$

From this theory, we can make probabilistic logical inferences. For example
What is the probability of the following formula

$\displaystyle\forall_{x,y,z}Friends(x,y)\wedge Friends(y,z)\wedge
Friends(z,x)$ $\displaystyle\Rightarrow(Sm(x))\wedge Sm(y)\wedge Sm(z))$

is true?

From Theorem 6.4, The most typical random $(2,[2,1])$-hyper-graphs
$W^{*(r,d)}(\mathcal{F},u)$ are the solution of the following optimization
problem.

$\min_{[Friends,Sm]\in\widetilde{\mathcal{W}}^{(r,d)}}\frac{1}{2}\int_{[0,1]^{2}}I_{0}(Friends(x,y))dxdy+\int_{[0,1]}I_{0}(Sm(x))dx$
(7.1)

subjected to

$\displaystyle u_{1}=\int_{[0,1]}Sm(x)dx$ $\displaystyle
u_{2}=\int_{[0,1]^{2}}Friends(x,y)dxdy$ $\displaystyle
u_{3}=\int_{[0,1]^{3}}Friends(x,y)Friends(y,z)Friends(z,x)dxdydz$

For the fourth constraint, we transform the formula to the equivalent one

$\forall_{x,y}Friends(x,y)\Rightarrow((Sm(x)\Rightarrow
S(y))\wedge(Sm(y)\Rightarrow S(x))$ (7.2)

and using the formula $p\Rightarrow q\equiv 1-(p\wedge\neg q$)

$\displaystyle 1-u_{4}=\int_{[0,1]^{2}}Friends(x,y)$
$\displaystyle(1-(1-Sm(x)(1-Sm(y)))(1-Sm(y)(1-Sm(x)))dxdy.$

The last constraint can be reduced to a constraint of a quantum graph.

Let $W^{*}(\mathcal{F},u)$ be the set of solutions of (7.1). Then, the
probability of (7.2) is computed by

$1-\frac{1}{|W^{*}(\mathcal{F},u)|}\sum_{[Friend,Sz]\in
W^{*}(\mathcal{F},u)}\int_{[0,1]^{3}}Friend(x,y)Friend(y,z)Friend(z,x)\\\
(1-Sz(x))(1-Sz(y))(1-Sz(z))dxdydz.$

##  8 Computability of $W^{*(r,d)}(\mathcal{F},u)$

$W^{*(r,d)}(\mathcal{F},u)$ is defined as the solution of an optimization
problem in an infinity dimensional space. Hence, it is not clear whether
$W^{*(r,d)}(\mathcal{F},u)$ is computable. Here we show
$W^{*(r,d)}(\mathcal{F},u)$ is computable in almost all cases when
$(\mathcal{F},u)$ is feasible.

###  8.1 Marginal Polytope

######  Definition 8.1 (Marginal map).

Let $\mathcal{F}$ be an ordered vector of quantum graphs. Then the marginal
map
$t(\mathcal{F},\cdot):\mathcal{W}_{\mathbb{R}}^{(r,d)}\to\mathbb{R}^{|\mathcal{F}|}$
is defined by

$t(\mathcal{F},W)=(t(\mathcal{F}_{1},W),\cdots,t(\mathcal{F}_{k},W)).$

We denote by $J_{x}(\mathcal{F})$ the Jacobian of $t(\mathcal{F},\cdot)$.

######  Definition 8.2 (Marginal Polytope).

The marginal polytope $T^{(m)}(\mathcal{F})$ of $\mathcal{F}$ is the image of
$t(\mathcal{F},\cdot)$ on ${\mathcal{W}}^{({m,r,d})}$, i.e.,
$T^{(m)}(\mathcal{F})=t(\mathcal{F},{\mathcal{W}}^{({m,r,d})})$. We also
define the total marginal polytopes of
$T(\mathcal{F})=t(\mathcal{F},\mathcal{W}^{(r,d)})$ and
$T(\mathcal{F},\pi)=t(\mathcal{F},{\mathcal{W}}^{({m,r,d})}(\pi))$.

######  Definition 8.3 (Constrained regions).

If $u\in T(\mathcal{F})$, the level sets of $t(\mathcal{F},u)$ define regions
constrained by subgraph densities. Hence

$S^{(r,d)}_{\mathbb{R}}(\mathcal{F},u)=t^{-1}(\mathcal{F},u)$

and

${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}=S^{(r,d)}_{\mathbb{R}}(\mathcal{F},u)\cap\mathcal{W}^{({m,r,d})}_{\mathbb{R}}$

and if we fix $\pi\in P^{({m})}$

${S_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u,\pi)}={{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}\cap{\mathcal{W}}^{({m},r,d)}_{\mathbb{R}}(\pi)$

####  8.1.1 Non-empty interior of $T^{(m)}(\mathcal{F})$

To guarantee that ${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$ are analytical
manifolds for almost all $u\in T^{(m)}(\mathcal{F})$ by applying Theorem 8.10
(Sard’s Theorem), we need a sufficient condition for which
$T^{(m)}(\mathcal{F})$ has a non-empty interior. Hence, it is sufficient that
$\mathcal{F}$ is a set of independent quantum graphs.

######  Definition 8.4 (Independent graphs).

Let $\mathcal{F}$ be an ordered and finite set of quantum graphs. We say that
$\mathcal{F}$ is a set of independent graphs if for all $m$ such that
$n(m,r,d)\geq|\mathcal{F}|$ there is
$(A,\pi)\in\mathcal{W}^{({m,r,d})}_{\mathbb{R}}$ such that
$\pi\in(P^{({m})})^{\circ}$ and $J_{(A,\pi)}(\mathcal{F})$ is full rank666This
criterion is equivalent to say the polynomials $t(\mathcal{F}_{i},X)$
$i\in[|\mathcal{F}|]$ are algebraically independent..

The following lemma shows that $T^{(m)}(\mathcal{F})$ and
$T^{(m)}(\mathcal{F},\pi)$ have a non-empty interior when $\mathcal{F}$ is a
set of independent graphs.

######  Lemma 8.1.

The following statements are equivalent.

  1. 1.

$\mathcal{F}$ is a set of independent graphs and $n(m,r,d)\geq|\mathcal{F}|$.

  2. 2.

$T^{(m)}(\mathcal{F},\pi)$ has a non-empty interior for all
$\pi\in(P^{({m})})^{\circ}$.

  3. 3.

$t(\mathcal{F},U)$ has a non-empty interior for any non-empty open set
$U\subset{\mathcal{W}}^{({m,r,d})}$.

######  Definition 8.5 (Density function).

Let $f_{0}:\mathbb{R}\to\mathbb{R}$ be an analytic function. Then
$f_{s}:\mathcal{W}_{\mathbb{R}}^{(r,d)}\to\mathbb{R}$ is defined by

$f_{s}(W)=\sum_{k=1}^{r}\int_{[0,1]^{d_{k}}}f_{0}(W_{k}(x_{1},\cdots,x_{d_{k}}))dx_{1}\cdots
dx_{d_{k}}$

On $\mathcal{W}^{({m,r,d})}_{\mathbb{R}}$ the density function $f_{s}$ has the
form

$f_{s}((A,\pi))=\sum_{k=1}^{r}\sum_{x_{1}\cdots
x_{k}=1}^{m}f_{0}(A_{x_{1},\cdots
x_{d_{k}},k})\pi_{x_{1}}\cdots.\pi_{x_{d_{k}}}$ (8.1)

Hence, we need to prove the solutions

$W^{*(r,d)}(\mathcal{F},u)=\operatorname*{arg\,min}_{W\in\widetilde{S}^{(r,d)}(\mathcal{F},u)}I(W)$

are step functions. To prove that, we prove the solutions

$W^{*}(\mathcal{F},u,f_{s})=\operatorname*{arg\,min}_{W\in
S^{(r,d)}_{\mathbb{R}}(\mathcal{F},u)}f_{s}(W)$

are step functions for any density function $f_{s}$. Note that the solutions
of the above optimization problems are labeled graphons. However, the orbits
of these solutions are unlabeled graphons since the subgraph density
constraints and the density functions are invariant under the measure-
preserving map. The statement that
$W_{\mathbb{R}}^{*(r,d)}(\mathcal{F},u,f_{s})$ are step functions here is
called the Principle of Optimization of Density functions $(POD)$. The
computability of $W^{*(r,d)}(\mathcal{F},u)$ is a special case of this
principle.

###  8.2 The principle of optimization of density functions

This principle is defined in the following theorem

######  Definition 8.6.

Let $\mathcal{F}$ be a set of quantum graphs and let $u\in T(\mathcal{F})$.
Then, the base size of step functions is,

$m_{0}(\mathcal{F},u)=\min\\{m|n(m,r,d)>|\mathcal{F}|\mbox{ and }u\in
T(\mathcal{F})\\}$

######  Theorem 8.7.

Let $\mathcal{F}$ be a set of independent graphs and let $m\geq
m_{0}(\mathcal{F},u)$. For almost all cases $u\in
T^{(m_{0})}(\mathcal{F})\cap\Omega(\mathcal{F})$ the minima of $f_{s}$ on
$S^{(r,d)}_{\mathbb{R}}(\mathcal{F},u)$ lies in
${{S}_{\mathbb{R}}^{(m_{0},r,d)}(\mathcal{F},u)}$.

The approach to prove POD is to approximate
$W_{\mathbb{R}}^{*(r,d)}(\mathcal{F},u,f_{s})$ by a sequence of solutions from
the step function spaces $W_{\mathbb{R}}^{*(m,r,d)}(\mathcal{F},u,f_{s})$ for
increasing $m$, defined by

$W_{\mathbb{R}}^{*(m,r,d)}(\mathcal{F},u,f_{s})=\operatorname*{arg\,min}_{W\in{{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}}f_{s}(W),$

The above optimization is on finite-dimensional space, and the idea is to
prove the split $W_{\mathbb{R}}^{*(m,r,d)}(\mathcal{F},u,f_{s})$ is in
$W_{\mathbb{R}}^{*(m+1,r,d)}(\mathcal{F},u,f_{s})$ and there is no new
solution in

$W_{\mathbb{R}}^{*(m+1,r,d)}(\mathcal{F},u,f_{s})\setminus\theta(W_{\mathbb{R}}^{*(m,r,d)}(\mathcal{F},u,f_{s}),\lambda,k)$

for all $k\in[m]$. To prove Theorem 8.7, we need develop results for
${S_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u,\pi)}$ and
${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$,

Figure 3 shows the theorem dependencies to prove the computability of
$W^{*}(\mathcal{F},u,f_{s})$. There are three cornerstones to prove POD. The
first one is Theorem 8.15, which shows the number of connected components of
${{S}_{\mathbb{R}}^{(m+1,r,d)}(\mathcal{F},u)}$ is not higher than the number
of connected components of ${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$. The
second cornerstone is Theorem 8.19 that ensures the split of local minima of
$f_{s}$ in ${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$ is again a local
minimum if ${{S}_{\mathbb{R}}^{(m+1,r,d)}(\mathcal{F},u)}$. The third
cornerstone is Theorem 8.17, which states the split of critical points of
$f_{s}$ on ${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$ does not bifurcate
into other critical points for almost all cases $u\in T(\mathcal{F})$. With
these three theorems, we prove in Theorem 8.20 that any local minimum of
$f_{s}$ in ${{S}_{\mathbb{R}}^{(m+1,r,d)}(\mathcal{F},u)}$ is the split of a
local minimum of $f_{s}$ in ${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$.
Theorem 8.7 proves POD by the closure of Theorem 8.20. Lemma 8.7 shows that if
$u$ is not an extremal value, then $W^{*(m)}(\mathcal{F},u,I)$ has values only
in $(0,1)$. Finally, Theorem 8.21 proves the computability of
$W^{*(r,d)}(\mathcal{F},u)$.

$\begin{array}[]{c}\mbox{No new connected component}\\\ \mbox{Theorem
}\ref{thm:numbercomponents}\end{array}$$\begin{array}[]{c}\mbox{No bifurcation
of critical points}\\\ \mbox{Theorem
}\ref{thm:almostsmoothnessCr}\end{array}$$\begin{array}[]{c}\mbox{Split of
local minima}\\\ \mbox{Theorem
}\ref{thm:splitminimum}\end{array}$$\begin{array}[]{c}\mbox{Inductive local
minima}\\\ \mbox{Theorem
}\ref{thm:localminimumsplit}\end{array}$$\begin{array}[]{c}\mbox{POD}\\\
\mbox{Theorem }\ref{thm:POD}\end{array}$$\begin{array}[]{c}\mbox{Non Extremal
Statistics }\\\ \mbox{ Lemma
}\ref{lem:randomsolutions}\end{array}$$\begin{array}[]{c}\mbox{Computability
of $W^{*(r,d)}(\mathcal{F},u)$ proof}\\\ \mbox{Theorem
}\ref{thm:radin}\end{array}$ Figure 3: Theorem dependencies to prove the
computability of $W^{*(r,d)}(\mathcal{F},u)$

###  8.3 Smoothness of ${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$

Here, we develop sufficient conditions to ensure
${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$ is an analytical manifold.

######  Definition 8.8.

Let $F:M\to N$ be a smooth function. A point $c\in N$ is said to be a regular
value of $F$ if every point of the level set $F^{-1}(c)$ is a regular point;
otherwise, it is a critical value.

######  Definition 8.9 (Regular and Critical Values of
$t(\mathcal{F},\cdot)$).

Let $Cri^{(m)}(\mathcal{F})$ be the set of critical values of the map
$t(\mathcal{F},\cdot):\mathcal{W}^{({m,r,d})}_{\mathbb{R}}\to\mathbb{R}^{|\mathcal{F}|}$.
Then, the set of regular values is defined by

$\hbox{Reg}(\mathcal{F})=\cup_{m>0}T^{(m)}(\mathcal{F})\setminus\cup_{m>0}Cri^{(m)}(\mathcal{F})$
(8.2)

and

$\hbox{Reg}^{(m)}(\mathcal{F})=T^{(m)}(\mathcal{F})\cap\hbox{Reg}(\mathcal{F}).$

It is convenient to define the set of non-extremal values of $\mathcal{F}$
defined by

$\Omega(\mathcal{F})=\hbox{Reg}(\mathcal{F})\setminus\cup_{m>0}\partial
T^{(m)}(\mathcal{F})$

Note that $\cup_{m>0}Cri^{(m)}(\mathcal{F})$ has zero measure since it is the
countable union of zero measure sets. Thus if $T^{(m)}(\mathcal{F})$ has a
non-empty interior then $Reg^{(m)}(\mathcal{F})$ and $Reg(\mathcal{F})$ are
dense in $T^{(m)}(\mathcal{F})$ and $T(\mathcal{F})$. If $u\in
Reg(\mathcal{F})$ then ${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}\subset
t^{-1}(\mathcal{F},u)\subset\mathcal{W}^{({m,r,d})}_{\mathbb{R}}$ has only
regular points for hence ${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$ is an
analytical manifold of $n(m,,r,d)+m-1-|\mathcal{F}|$ dimensions since the
marginal map is polynomial.

To ensure that $u\in T^{(m)}(\mathcal{F})$ is a regular value for almost all
cases, we apply Sard’s theorem in [15]

######  Theorem 8.10 (Theorem $6.10$ in [15] (Sard’s Theorem)).

Suppose $M$ and N are smooth manifolds with or without boundary, and $F:M\to
N$ is a smooth map. Then, the set of critical values of $F$ has zero measure
in $N$.

Hence if $T^{(m)}(\mathcal{F})$ has a non-empty interior then from Sard’s
theorem $u\in T^{(m)}(\mathcal{F})$ is a regular value for almost everywhere
in $T^{(m)}(\mathcal{F})$. In other words, if $\mathcal{F}$ is a set of
independent graphs and we pick $u\in Reg(\mathcal{F})$ at random, then the
probability that ${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$ is a smooth
manifold is $1$. Hereafter, we assume that $\mathcal{F}$ is a set of
independent graphs.

###  8.4 The smoothness of ${S_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u,\pi)}$
almost everywhere

${S_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u,\pi)}$ is a real algebraic
variety777A real algebraic variety is a subset of $\mathbb{R}^{n}$ defined by
polynomial equations., and every real algebraic variety $V$ can be decomposed
on smooth manifolds using the so-called Whitney Stratification [10]. The
difference between real algebraic varieties and smooth manifolds is subtle.
Real algebraic varieties can have non-uniform dimensions; when a real
algebraic variety has a uniform dimension, it is a smooth manifold. However,
we are interested in seeing ${S_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u,\pi)}$ as
a smooth manifold and investigating when it is smooth or smooth almost
everywhere. Here, we use the same ideas of Whitney Stratification to prove
that ${S_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u,\pi)}$ is almost everywhere but
not necessarily with a uniform dimension. First we prove that
${{S}}^{({m,r,d})}_{\mathbb{R}}({[F]},{[u]},{\pi})$ is a smooth manifold.

######  Lemma 8.2.

Let $F$ be a simple graph. Then

$\langle(A,\pi),\nabla t(F,(A,\pi)\rangle=|E(F)|t(F,(A,\pi))$

any $(A,\pi)$ and $|E(F)|$ is the number of edges in $F$.

######  Lemma 8.3.

Let $F=\sum_{k}a_{k}F_{k}$ be a quantum graph. If $u\neq 0$ and
$\pi\in(P^{({m})})^{\circ}$ then
${{S}}^{({m,r,d})}_{\mathbb{R}}({[\mathcal{F}},{[u]},{\pi})$ is smooth almost
everywhere.

Hence ${{S}}^{({m,r,d})}_{\mathbb{R}}({[F]},{[u]},{\pi})$ is an analytical
manifold almost everywhere for any quantum graph $F$, $u\neq 0$ and $\pi\in
P^{({m})}$. Hence, we prove that
${S_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u,\pi)}$ is analytical almost
everywhere.

######  Theorem 8.11.

Let $\mathcal{F}$ be a set of independent graphs. If $u\in
T^{(m)}(\mathcal{F})^{\circ}$ then
${S_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u,\pi)}$ is analytical almost
everywhere for all $\pi\in(P^{({m})})^{\circ}$.

###### Proof.

The proof is based on the induction of the number of constraints. The base
step is ensured by Lemma 8.3.

The induction step.
${{S}}^{({m,r,d})}_{\mathbb{R}}({[\mathcal{F}_{1},\cdots\mathcal{F}_{n}]},{[u_{1},\cdots,u_{n}]},{\pi})$
has regular points almost everywhere and
${{S}}^{({m,r,d})}_{\mathbb{R}}({[\mathcal{F}_{1},\cdots\mathcal{F}_{n}]},{[u_{1},\cdots,u_{n}]},{\pi})$
can have different connected components and for each connected component $N$
of $d_{N}$ dimensions, we have two possibilities

  1. 1.

The critical points of $t(\mathcal{F}_{n+1},\cdot)$ on
${{S}}^{({m,r,d})}_{\mathbb{R}}({[\mathcal{F}_{1},\cdots\mathcal{F}_{n}]},{[u_{1},\cdots,u_{n}]},{\pi})$
have a non-empty interior. Hence $t(\mathcal{F}_{n+1},\cdot)$ has critical
points to the entire $N$. In this case, $t(\mathcal{F}_{n+1},\cdot)$ has
dependency with $[\mathcal{F}_{1},\cdots\mathcal{F}_{n}]$, thus the constraint
$t(\mathcal{F}_{n+1},(A,\pi))=u_{n+1}$ is redundant. It follows,
$N_{1}=N\cap{{S}}^{({m,r,d})}_{\mathbb{R}}({[\mathcal{F}_{n+1}]},{[u_{n+1}]},{\pi})$
is analytical almost everywhere and $N_{1}$ has $d_{N}$ dimensions.

  2. 2.

The critical points of $t(\mathcal{F}_{n+1},\cdot)$ on
${{S}}^{({m,r,d})}_{\mathbb{R}}({[\mathcal{F}_{1},\cdots\mathcal{F}_{n}]},{[u_{1},\cdots,u_{n}]},{\pi})$
have an empty interior. It follows,
$N_{1}=N\cap{{S}}^{({m,r,d})}_{\mathbb{R}}({[\mathcal{F}_{n+1}]},{[u_{n+1}]},{\pi})$
is analytical almost everywhere and $N_{1}$ has $d_{N}-1$ dimensions.

Hence, we conclude that ${S_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u,\pi)}$ is
analytical almost everywhere. ∎

###  8.5 Non-increasing number of connected components of
${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$

One cornerstone of the main result is proof that there is no new connected
component in
${{S}_{\mathbb{R}}^{(m+1,r,d)}(\mathcal{F},u)}\setminus{{S}_{\mathbb{R}}^{(m+1,r,d)}(\mathcal{F},u)}$.
Hence, the number of connected components of
${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$ is not increased when $m$ is
increased to $m+1$. To prove this, we need two definitions.

######  Definition 8.12.

Let $m\leq m^{\prime}$. Let $T^{-(m,m^{\prime})}$ be a set of the level sets
of
$t(\mathcal{F},\cdot):\mathcal{W}^{({m^{\prime},r,d})}_{\mathbb{R}}\to\mathbb{R}^{|\mathcal{F}|}$
when $u\in T^{(m)}(\mathcal{F})$. More precisely,

$T^{-(m,m^{\prime})}(\mathcal{F})=\\{x\in\mathcal{W}^{({m^{\prime},r,d})}_{\mathbb{R}}\mid
t(\mathcal{F},x)\in T^{(m)}(\mathcal{F})\\}.$

Hence it is clear that
$t(\mathcal{F},T^{-(m,m^{\prime})}(\mathcal{F}))=T^{(m)}(\mathcal{F})$ for all
$m\leq m^{\prime}$.

######  Definition 8.13 (The Aggregated Feasible Region).

Let $M^{(m)}(\mathcal{F})\subset\mathbb{R}^{n(m,r,d)+m-1+|\mathcal{F}|}$ be
the aggregated region of ${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$ defined
by

$M^{(m)}(\mathcal{F})=\\{(x,z)\in\mathcal{W}^{({m,r,d})}_{\mathbb{R}}\times\mathbb{R}^{|\mathcal{F}|}:\forall
i\in[|\mathcal{F}|]\quad t(\mathcal{F}_{i},x)-z_{i}=0\\}$

######  Remark 8.14.

Note that $M^{(m)}(\mathcal{F})$ is the graph of the marginal map, thus
$M^{(m)}(\mathcal{F})$ is connected. Hence, the regular and critical points of
the marginal map are arbitrarily close in
${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$ when $u$ varies in
$T^{(m)}(\mathcal{F})$.

######  Theorem 8.15.

Let $\mathcal{F}$ be a set of independent graphs and let
$u\in\Omega(\mathcal{F})$ and let $m\geq m_{0}(\mathcal{F},u)$. Then there is
no new connected component
$N\subset{{S}_{\mathbb{R}}^{(m+1,r,d)}(\mathcal{F},u)}\setminus{{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$.
Hence the number of connected components of
${{S}_{\mathbb{R}}^{(m+1,r,d)}(\mathcal{F},u)}$ is not higher than the number
of connected components of ${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$.

###  8.6 The space of critical points of $f_{s}(\cdot)$ on
${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$

Recall that a point $(A,\pi)\in{{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$ is
a critical point of $f(\cdot,\pi)$ if there is
$\beta\in\mathbb{R}^{|\mathcal{F}|}$ such that

$\nabla f_{s}((A,\pi))=\sum_{i=1}^{|\mathcal{F}|}\beta_{i}\nabla
t(\mathcal{F}_{i},(A,\pi))$

Here we study the critical points of all
${{S}_{\mathbb{R}}^{(m,r,d)}(\mathcal{F},u)}$ when $u$ ranges over all the
feasible combinations of subgraph densities in $T^{(m)}(\mathcal{F})$. Then,
the spaces of critical points are defined by

######  Definition 8.16.

Let $\mathcal{F}$ be a set of independent simple graphs. Let $f_{s}(\cdot)$ be
a smooth function defined in (8.1) and let $m$ be a positive integer such that
$n(m,r,d)>|\mathcal{F}|$ then

$\displaystyle Cr^{(m)}(\mathcal{F},f_{s})$ $\displaystyle=$
$\displaystyle\\{((A,\pi),\beta)\in\mathcal{W}^{({m,r,d})}_{\mathbb{R}}\times\mathbb{R}^{|\mathcal{F}|}\mid
t(\mathcal{F},(A,\pi))\in Reg(\mathcal{F})$ $\displaystyle\nabla
f_{s}-\sum_{i=1}^{|\mathcal{F}|}\beta_{i}\nabla
t(\mathcal{F}_{i})\big{|}_{(A,\pi)}=0\\}.$

####  8.6.1 The split map on critical points of $f_{s}$

Here we prove that the split map
$\theta(\cdot,\lambda,k):\mathcal{W}^{({m,r,d})}_{\mathbb{R}}\to\mathcal{W}^{({m+1,r,d})}_{\mathbb{R}}$
preserves the critical points of density functions $f_{s}$ i.e.