# Persistent Homology of Convection Cycles in Network Flows

Minh Quang Le<EMAIL_ADDRESS>Department of Mathematics, University at
Buffalo, State University of New York, Buffalo, NY 14260, USA  Dane Taylor
<EMAIL_ADDRESS>Department of Mathematics, University at Buffalo, State
University of New York, Buffalo, NY 14260, USA

###### Abstract

Convection is a well-studied topic in fluid dynamics, yet it is less
understood in the context of networks flows. Here, we incorporate techniques
from topological data analysis (namely, persistent homology) to automate the
detection and characterization of convective/cyclic/chiral flows over
networks, particularly those that arise for irreversible Markov chains (MCs).
As two applications, we study convection cycles arising under the PageRank
algorithm, and we investigate chiral edges flows for a stochastic model of a
bi-monomer’s configuration dynamics. Our experiments highlight how system
parameters—e.g., the teleportation rate for PageRank and the transition rates
of external and internal state changes for a monomer—can act as _homology
regularizers_ of convection, which we summarize with _persistence barcodes_
and _homological bifurcation diagrams._ Our approach establishes a new
connection between the study of convection cycles and homology, the branch of
mathematics that formally studies cycles, which has diverse potential
applications throughout the sciences and engineering.

###### pacs:

89.75.Hc, 02.50.Ga, 87.15.hj,84.35.+i

##  I Introduction

One of the main goals of topological data analysis (TDA) is to characterize
the structure of an object—usually a point cloud—through its topological
features. In particular, persistent homology edelsbrunner2010computational ;
otter2017roadmap  is a family of techniques that detect and summarize
multiscale topological features and has been applied to a wide variety of
applications including timeseries data khasawneh2016chatter ; perea2015sliding
, image processing xin1994topology , machine learning motta2019hyperparameter
, and artificial intelligence gabella2019topology ; liu2019scalable .
Complementing the study of point-cloud data, another line of research involves
utilizing the TDA toolset to study complex systems, for which applications
include the analysis of spreading processes over social networks
taylor2015topological , network neuroscience petri2014homological ;
giusti2015clique ; kilic2022simplicial , mechanical-force networks
kondic2012topology , jamming in granular material kramar2013persistence ,
molecular structure liang1998analytical , and DNA folding
ichinomiya2020protein . In this paper, we employ techniques from TDA to study
Markov chains (MCs), which provide a foundation to numerous areas of science
and engineering including queuing theory kendall1953stochastic , population
dynamics kingman1969markov , as well as statistical (and machine learning)
models that rely on Markov chain Monte Carlo gilks1995introducing , hidden
Markov models tierney1994markov , and Markov decision process
parr1998reinforcement .

We utilize the mathematical framework of persistent homology to automate the
detection (and summarize the multiscale properties) of convection cycles that
arise for the stationary flows of irreversible MCs. Notably, while convection
cycles have been extensively studied in fluid dynamics, they are less
understood in the context of flows over networks. For example, it was recently
observed that the coupling together of reversible MCs can give rise to an
irreversible MC with convection cycles that are an emergent property
taylor2015topological . Emergent convection cycles have also been recently
found to describe the phenomenon of “chiral edge flows” tang2021topology ,
providing new insights into the quantum Hall effect, configurational dynamics
of monomers, and biological (e.g., circadian) rhythms. Given the inherent
prevalence of convection cycles in MCs and other network flows, it is
important that we place their study on a stronger mathematical, computational,
and theoretical footing.

We study convection using a branch of mathematics called _homology_ and the
related field _computational homology_ kaczynski2006computational . Both are
concerned with studying the absence/presence of $k$-dimensional “holes” (and
their connectivity) within a topological space such as a simplicial complex.
Importantly, cycles on a graph are 1-dimensional (1D) holes, and so persistent
homology is a natural fit to analyze convection cycles. We construct
filtrations of graphs by including edges according to the stationary flows
across them (which is done in descending order so that the last edges to be
included are those with the smallest stationary flows), and we summarize the
persistent homology of the filtered graphs’ associated clique complexes. See
Fig. 1 for a graph and its associated clique complex. Computationally, we
implement these techniques by building on a popular TDA framework called Gudhi
gudhi:urm , which we adapt to implement _edge-value clique (EVC) filtrations_
of scalar-functions that are defined over the the edges of a graph.

We apply this technique to two applications. First, we study convection cycles
arising under the PageRank algorithm page1999pagerank , examining the role of
the _teleportation parameter_. Second, we study chiral edge flows that emerge
for a 4-state model that describes the configurational changes of a bi-monomer
tang2021topology , examining the roles of the external and internal transition
rates. These parameters significantly affect convection cycles arising for
these respective applications, and we show that they act as “homology
regularizers” of convection. We introduce “homological bifurcation diagrams”
to summarize these effects. Our methods provide mathematically principled (and
automated) tools to gain a deeper understanding of the structural patterns of
convection on networks, and they are expected to be useful to myriad
applications across the physical, social, biological and computational
sciences.

The remainder of this paper is organized as follows: We present background
information in Sec. II, our methodology in Sec. III, applications in Sec. IV,
and a discussion in Sec. V.

##  II Background Information

Here, we present introductory material about simplicial complexes and homology
(Sec. II.1), persistent homology of graphs (Sec. II.2), and discrete-time MCs
(Sec. II.3).

###  II.1 Simplicial complexes (SCs) and their homology

We first define an _undirected graph_ $G=\\{\mathcal{V},\mathcal{E}\\}$, where
$\mathcal{V}=\\{1,\dots,N_{0}\\}$ is a set of $N_{0}$ vertices and
$\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$ is a set of edges. Note that
each vertex is specified by a single index $i\in V$, and each edge is
specified by an unordered pair $(i,j)\in\mathcal{V}\times\mathcal{V}$. More
generally, we define a $(k+1)$-tuple of vertices
$\sigma=(i_{0},i_{1},\dots,i_{k})$ as a $k$-dimensional simplex, or
$k$-_simplex_ 111The notion of “dimension” is more interpretable for the case
of a non-abstract simplicial complex, for which the vertices correspond to
locations in a Euclidean metric space. In that case, every $k$-simplex is
defined as the $k$-dimensional surface that is contained by its faces, which
themselves are $(k-1)$ dimensional surfaces. For example, a 2-simplex is a
triangle defined as the interior of 3 line segments (which are the 3
1-dimensional cofaces of the 2-simplex).. Vertices and edges are equivalent to
0-simplices and 1-simplices, respectively. An abstract SC is a set of
simplices of possibly different dimensions, and it is a generalization of an
undirected graph. It is also a type of _hypergraph_ with a constraint on which
simplices can exist. That is, for any $k$-simplex $(i_{0},i_{1},\dots,i_{k})$
in an SC, its _faces_ are the $(k-1)$ simplices in which one of the of indices
is omitted (e.g., $i_{1}$ is omitted to yield $(i_{0},i_{2},\dots,i_{k})$).
The _cofaces_ of a $(k-1)$-simplex are the $k$-simplices for which it is a
face. Note that the faces of an edge $(i,j)$ are the vertices $i$ and $j$, and
likewise, $(i,j)$ is a coface of each of these vertices.

With these definitions, we state the two restrictions that are required for a
SC: (i) for any face, its faces must be included in the SC; and (ii) the
intersection of any two faces is either a face of both, or it is an empty set.
The dimension of an SC is the maximum dimension of its simplices, and an
undirected graph is a 1-dimensional SC—it contains 0-simplices and
1-simplices, and for any edge $(i,j)$, the vertices $i$ and $j$ must exist. We
will focus on a particular type of SC that can be generated from a graph and
is called a _clique complex_. A clique complex $K(G)$ of a graph $G$ is the SC
in which there is a 1-to-1 correspondence between the $(k+1)$-cliques in the
graph and the $k$-simplices in the SC. (Recall that an _n-clique_ is a
complete subgraph on $n+1$ vertices of a graph.) Given this 1-to-1
correspondence, the map from $G$ to $K(G)$ is invertible, and $G$ can be
recovered as the $1$-skeleton of $K(G)$. (A _$k$ -skeleton_ of a SC is the SC
that is obtained after removing all simplices having dimensions that are
greater than $k$.)

Figure 1: A graph and an associated simplicial complex (SC). (A) Graph
$\mathcal{G}$ with $N=7$ vertices and $M={\color[rgb]{0,0,0}10}$ undirected,
unweighted edges. (B) The corresponding _clique complex_ $\mathcal{S}$, where
each $k$-clique gives rise to a $(k-1)$-simplex. Each triangle (i.e.,
$3$-clique) gives rise to a $2$-simplex (see shaded triangles). The SC is a
topological space and has associated vector spaces. Consider the space
$\mathbb{R}^{7}$ of real-valued functions defined over the vertices. Since
there is just 1 connected component, the SC’s 0-dimensional (0D) homology is a
1D subspace of $\mathbb{R}^{7}$. Similarly, there are two “homological”
1-cycles that are not a boundary of a 2-simplex, and so the 1D homology is a
2D subspace of $\mathbb{R}^{10}$ (i.e., the space of real-valued functions
defined over the ten edges).  Figure 2:  Persistent homology of a graph
according to an _edge-value clique (EVC) filtration_. (A) An undirected graph
${G}(\mathcal{V},\mathcal{E})$ with a scalar function
$f:\mathcal{E}\to\mathbb{R}$ defined over the edges $\mathcal{E}$. We apply an
EVC filtration to the graph by considering a monotonically decreasing
_filtration parameter_ $\epsilon=(0,5]$, and by considering a filtered
sequence of graphs ${G}(\mathcal{V},\mathcal{E}_{\epsilon})$, where
$\mathcal{E}_{\epsilon}=\\{(i,j)|f(i,j)>\epsilon\\}$ is the subset of edges
for which $f(i,j)$ are larger than a threshold $\epsilon$. (B) Visualization
of the graphs’ associated clique complexes $K_{\epsilon}\equiv
K({G}(\mathcal{V},\mathcal{E}_{\epsilon}))$ for several $\epsilon$. (C) A
_persistence barcode_ summarizes how the 0-dimensional (red) and 1-dimensional
(blue) homological $k$-cycles of $K_{\epsilon}$ change with decreasing
$\epsilon$. The arrows highlight two events: at $\epsilon=\epsilon_{b}$, a
homological 1-cycle involving four edges is _born_ ; at
$\epsilon=\epsilon_{d}$, the 1-cycle _dies_ , since it is “filled in” by a
1-simplex and two 2-simplices.

We next discuss _simplicial homology_ , which will lead to a formal definition
of “homological” cycles. To this end, we consider vector spaces defined over
the $k$-simplices in a SC. A _$k$ -chain_,
$\sum_{n=1}^{N_{k}}\alpha_{n}\sigma_{n}$, is a linear combination of
$k$-simplices $\\{\sigma_{n}\\}$ with weights $\\{\alpha_{k}\\}$. (Note that
$N_{0}$ and $N_{1}$ are the numbers of vertices and edges, respectively.) If a
SC contains $N_{k}$ different $k$-simplices, then the vector space of
$k$-chains is $N_{k}$-dimensional, and it is isomorphic to
$\mathbb{R}^{N_{k}}$ if one assumes $\alpha_{k}\in\mathbb{R}$. We now consider
a simplicial map $f:X_{k}\to X_{k-1}$ between $X_{k}$, which is a SC of
dimension $k$, and $X_{k-1}$, which is a SC of dimension $k-1$ that contains
the faces of simplices in $X_{k}$. Considering the vector space $C_{k}$ of
$k$-chains defined over $k$-simplices in $X_{k}$ and vector space $C_{k-1}$ of
$(k-1)$-chains defined over their cofaces in $X_{k-1}$, we define the linear
_boundary map_ $\partial_{k}:C_{k}\longrightarrow C_{k-1}$, where the action
of $\partial_{k}$ on any $k$-simplex is given by

$\displaystyle\partial_{k}(i_{0},...,i_{k})=\sum_{j=0}^{k}(-1)^{j}(i_{0},...,i_{j-1},i_{j+1},...,i_{k}).$
(1)

The boundary map allows one to relate vectors in $C_{k}$ to those in
$C_{k-1}$. For example, the _boundary_ of a 2-simplex (i.e., triangle)
$(i,j,k)$ is the signed combination if the associated edges,
$\partial_{2}(i,j,k)=(j,k)-(i,k)+(i,j)$. Notably, the boundary of any closed
path is zero, which yields an algebraic definition of a $k$-cycle: any
$k$-chain that lies within the subspace $Z_{k}$, where
$Z_{k}=\text{ker}(\partial_{k})\subseteq C_{k}$ is the _vector space of
$k$-cycles_.

Notably, $k$-cycles can arise for different reasons, and we distinguish two
types. The boundary map satisfies the property
$\partial_{k}\circ\partial_{k+1}=0$, which essentially states that the
boundary of a boundary is zero. [For the triangle,
$\partial_{1}\circ\partial_{2}(i,j,k)=\partial_{1}(j,k)-\partial_{1}(i,k)+\partial_{1}(i,j)=(k-j)-(k-i)+(j-i)=0$.]
Thus we define $B_{k}=\text{image}(\partial_{k+1})$ as the _subspace of
$(k+1)$-boundaries_, and it follows that $B_{k}\subseteq Z_{k}$. In other
words, some cycles arise simply because they are boundaries of
$(k+1)$-simplices For example, observe in Fig. 1(B) that there are two
“triangular” cycles that exist around the two 2-simplices, but that there are
two other cycles that also exist. The _$k$ -th simplicial homology_ is defined
as the quotient space $H_{k}=Z_{k}/B_{k}$, and it represents the subspace of
$k$-dimensional cycles (i.e., $k$-cycles) that do not arise simply as the
boundary of a $(k+1)$-simplex.

The $k$-th simplicical homology can be represented by the span of _homology
generators_ , which are a linearly independent set of $k$-chains that span
$H_{n}$ and represent the associated $k$-cycles. The number of linearly
independent homology generators is called a _Betti number_

$\displaystyle\beta_{k}=\dim H_{k}=\dim(Z_{k})-\dim(B_{k}).$ (2)

Informally, $\beta_{0}$ is the number of connected components; $\beta_{1}$ is
the number of 1-dimensional cycles or “loops” (that is, not including the
triangular boundaries of 2-simplices); and $\beta_{2}$ is the number of
2-dimensional holes or “voids” (e.g., the interior of a triangulated sphere).
For the SC shown in Fig. 1(B), $\beta_{0}=1$ since there’s one connected
component, and $\beta_{1}=2$ since there are two cycles that are not simply
the boundaries of $2$-simplices.

By formulating $k$-cycles algebraically, one can consider the linear
dependence and independence of $k$-cycles. As such, one can not only identify
cycles, but also investigate the relations/connectivity between cycles, which
we find to be instrumental for understanding pattern formation for cycle. We
also highlight that a given homological $k$-cycle can potentially have more
than one homological generator. Such generators are said to be _homologically
equivalent_ , and they can be obtained by considering linear combinations of
$k$-cycles (including both homological $k$-cycles and boundaries). We will
later show that this complicates the investigation of convection cycles
through the lens of homological $k$-cycles.

###  II.2 Persistent homology of scalar functions defined over edges

One of the greatest tools of topological data analysis is the study of
_persistent homology_ edelsbrunner2010computational ; otter2017roadmap . Here,
we examine how the homology of a topological object changes as it undergoes a
_filtration_ to yield a monotonically increasing sequence $X_{0}\subseteq
X_{1}\subseteq X_{1}\subseteq...$ (e.g., of simplicial complexes
$\\{X_{t}\\}$). We consider filtrations in which one has a scalar function
$f:\mathcal{E}\longrightarrow\mathbb{R}$ over the edges, and each edge
$(i,j)\in\mathcal{E}$ is retained/removed according to $f(i,j)$. The values
$f(i,j)$ could be edge weights for a weighted graph, but in general they can
encode any scalar property. We visualize such a graph and the values $f(i,j)$
in Fig. 2(A).

We call the process an _edge-value clique (EVC) filtration_ , and we construct
it as follows. Given a graph $G(\mathcal{V},\mathcal{E})$ and a _filtration
function_ $f$, we define the subsets
$\mathcal{E}_{\epsilon}=\\{(i,j)|f(i,j)>\epsilon\\}$ in which one retains
edges only for which $f(i,j)$ is sufficiently large. Note that the subsets
$\\{\mathcal{E}_{\epsilon}\\}$ are non-decreasing as $\epsilon$ decreases
(i.e., $\mathcal{E}_{\epsilon}\subseteq\mathcal{E}_{\epsilon^{\prime}}$ for
any $\epsilon^{\prime}<\epsilon$). We must specify a range over which to
decrease $\epsilon\in(\epsilon_{B},\epsilon_{A}]$, and in practice we assume
$\epsilon_{A}>\max_{(i,j)\in\mathcal{E}}f(i,j)$ and
$\epsilon_{B}<\min_{(i,j)\in\mathcal{E}}f(i,j)$. It then follows that
$\mathcal{E}_{\epsilon}=\emptyset$ is an empty set of edges when
$\epsilon\geq\epsilon_{A}$, and $\mathcal{E}_{\epsilon}=\mathcal{E}$ (i.e.,
all edges are retained) when $\epsilon\leq\epsilon_{B}$. See minh_code  for
our codebase that implements EVC filtrations by adapting the TDA framework
called Gudhi gudhi:urm , and which reproduces the results of this paper.

Figure 3:  Stationary distribution, edge flows, and convection cycles for an
irreversible MC. We study a discrete-time random walk over a directed,
weighted graph that resembles the undirected graph in Fig. 2, except that the
edges are now either directed or bidirectional. (Recall that undirected graphs
give rise to reversible MCs that lack convection cycles.) (A) The color of
each vertex indicates the _stationary distribution_ $\pi_{i}$ of random
walkers at each vertex $i$. (B) Edge colors indicate the _stationary flows_
$F_{ij}=\pi_{i}P_{ij}$ across edges, i.e., the stationary fraction of random
walkers that traverse each directed edge. (C) _Flow imbalances_
$\Delta_{ij}=(F_{ij}-F_{ji})$ manifest as a pattern of _convection cycles_. By
construction, $\Delta_{ij}=-\Delta_{ji}$, and we use arrows to indicate the
directions of imbalance, e.g., $i\rightarrow j$ if $\Delta_{ij}>0$.

In Fig. 2(B), we visualize a sequence of filtered clique complexes
$\\{K_{\epsilon}\\}$ that are associated with the filtered graphs
$\\{G_{\epsilon}\\}$ that are defined with the edge sets
$\\{\mathcal{E}_{\epsilon}\\}$. In Fig. 2(C), we summarize the persistent
homology of $\\{K_{\epsilon}\\}$ in a _persistence barcode_ , which reveals
how homology changes with $\epsilon$. Observe that when $\epsilon$ is
sufficiently large, $K_{\epsilon}$ contains vertices but no edges. On the
other hand, when $\epsilon$ decreases to be sufficiently small, then
$K_{\epsilon}$ recovers the original clique complex [recall Fig. 1(B)]. The
values of $\epsilon$ that were used to create Fig. 2(B) are indicated by the
vertical dotted lines in Fig. 2(C).

Each horizontal bar in the persistence barcode shown in Fig. 2(C) indicates
the _lifetime_ of a homological 1-cycle—that is, the values of $\epsilon$ for
which it exists. The red and blue bars reflect $0$-homology and $1$-homology
respectively. The dimensions of the homology spaces (i.e., Betti numbers) can
be found by counting the number of homological 1-cycles at any particular
$\epsilon$. For example, one can observe that $\beta_{1}=0$ when
$\epsilon=3.5$, $\beta_{1}=1$ when $\epsilon=2.5$, and $\beta_{1}=2$ when
$\epsilon=1.5$. Clearly, the homological $1$-cycles are undergoing
bifurcations as $\epsilon$ varies. A persistence barcode is convenient to
identify for each generator: the value $\epsilon_{b}$ of $\epsilon$ when it is
“born” (i.e., the homological $k$-cycle does not exist when
$\epsilon>\epsilon_{b}$); the value $\epsilon_{d}$ of $\epsilon$ when it
“dies” (i.e., the homological $k$-cycle does not exist when
$\epsilon<\epsilon_{d}$); its _lifetime_ $(\epsilon_{d},\epsilon_{b}]$; and
_lifespan_ $|\epsilon_{d}-\epsilon_{b}|$. A cycle’s lifespan quantifies its
persistence under the filtration, and it is often interpreted as a proxy for
the cycle’s significance (although short-lifetime cycles can also be important
in certain contexts).

###  II.3 Convection cycles for irreversible Markov chains (MCs)

We will apply persistence homology to study convection cycles in irreversible
MCs lovasz1993random , which we now briefly summarize. A discrete-time MC is a
“memoryless” random process in which for time steps $t=0,1,2,\dots$ , the
system state $S_{t}\in\mathcal{V}$ satisfies the Markov property
${\text{P}}[S_{t+1}=i|S_{0}=i_{0},...,S_{t}=i_{t}]={\text{P}}[S_{t+1}=i|S_{t}=i_{t}]$,
which implies that the probability of a state occurring at the next time step
only depends on the current state and not earlier states. In our case, we
consider MCs that correspond to a random walk on a (possibly) weighted and
directed graph having an _adjacency matrix_ ${\bf A}$ in which
$A_{ij}\in\mathbb{R}$ is nonzero if $(i,j)$ is an edge, $(i,j)\in\mathcal{E}$,
and $A_{ij}=0$ otherwise. We similarly define a _transition matrix_ , ${\bf
P}={\bf D}^{-1}{\bf A}$, where ${\bf D}$ is a diagonal matrix with entries
that encode the (possibly) weighted vertex degrees ${\bf
D}_{ii}=\sum_{j}A_{ij}$. For directed graphs, each $(i,j)$ is considered to be
an ordered pair, and each ${\bf D}_{ii}$ encodes the out-degree of vertex $i$.
Each matrix element $P_{ij}$ gives the probability for a random walk to
transition from vertex $i$ to $j$. Letting $x_{i}(t)$ denote the probability
that the system is in state $i$ (or equivalently, the probability that a
random walker is at vertex $i$) at time $t$, one can utilize the Markov
property to obtain the linear discrete-time system
$x_{j}(t+1)=\sum_{i}x_{i}(t)P_{ij}$. By defining ${\bf
x}(t)=[x_{1}(t),\dots,x_{N_{0}}]^{T}$, one equivalently has

$\displaystyle{\bf x}(t+1)^{T}={\bf x}(t)^{T}{\bf P}.$ (3)

Since ${\bf x}(t)$ is a vector of probabilities, we assume that it is
normalized in 1-norm, $\sum_{i}x_{i}(t)=1$.

Herein, we focus on network flows after a system reaches a _stationary state_
, in which case ${\bf x}(t)$ converges to a limiting vector
$\pi=\lim_{t\to\infty}{\bf x}(t)$ that satisfies the eigenvalue equation
$\pi^{T}=\pi^{T}{\bf P}$. By construction, $\pi$ is a vector of probabilities
and contains nonnegative entries. Furthermore, as a row-stochastic matrix,
${\bf P}$ has an eigenvalue equal to one (i.e., the largest eigenvalue) and
its right dominant eigenvector is the vector containing 1’s as entries. Our
assumption of convergence requires that matrix ${\bf P}$ is an irreducible and
aperiodic bapat1998max  or that the initial condition ${\bf x}(0)$ lies in a
converging subspace. In the stationary state, the _stationary flow_ across
each edge $(i,j)$ per time step is given by

$\displaystyle F_{ij}=\pi_{i}P_{ij}.$ (4)

Figure 4: Persistent homology of convection cycles. (A) Visualization of an
EVC filtration applied to flow imbalances arising for the irreversible MC
shown in Fig. 3, and we use the magnitude $|\Delta_{ij}|$ of flow imbalance as
the filtration function $f:\mathcal{E}\to\mathbb{R}$. We indicate flow
imbalances’ directions with arrows, noting that the clique complexes that are
constructed by the filtration are undirected, since the filtration does not
incorporate information about edge directions. (B) Persistence barcodes for
homological 1-cycles. Observe that the 1-cycle that first appears dies before
the other 1-cycles are born.

We study _convection cycles_ using an approach that was developed in
taylor2020multiplex . Specifically, for each edge we define the stationary
_flow imbalance_

$\displaystyle\Delta_{ij}=F_{ij}-F_{ji}.$ (5)

By construction, $\Delta_{ji}=-\Delta_{ij}$, and we say that the _imbalance
direction_ is from $i$ to $j$ when $\Delta_{ij}>0$. Importantly, the defining
feature of a _reversible MC_ is that $\Delta_{ij}=0$ for all $i$ and $j$. That
is, the directional flows match $\pi_{i}P_{ij}=\pi_{j}P_{ji}$ for any edge
$(i,j)$. This is the case for any undirected graph, since in this case ${\bf
A}={\bf A}^{T}$, and it follows that $\pi_{i}=D_{ii}/\sum_{j}D_{jj}$. In
contrast, an _irreversible MC_ yields asymmetric stationary flows and
$\Delta_{ij}$ is nonzero for some edges.  To formally define convection
cycles, we consider a new graph $G_{\Delta}(\mathcal{V},\mathcal{E}_{\Delta})$
such that each positive value $\Delta_{ij}$ gives rise to a directed edge
$(i,j,\Delta_{ij})\in\mathcal{E}_{\Delta}$ having weight $\Delta_{ij}$. We
then define a _convection cycle_ to be any non-intersecting closed path in
$G_{\Delta}(\mathcal{V},\mathcal{E}_{\Delta})$.

In Fig. 3, we illustrate for an example MC how flow imbalances manifest as a
pattern of convection cycles. In Figs. 3(A), 3(B), and Fig. 3(C), we use edge
colors to indicate the stationary distribution $\pi$, stationary edge flows
$F_{ij}$, and flow imbalances $\Delta_{ij}$, respectively. Observe that some
of the arrows in Figs. 3(A)–(B) are bidirectional, since some of the graph’s
edges are bidirectional. In contrast, the arrows in Fig. 3(C) are exclusively
directed since they now indicate the directions of flow imbalances. There
exists an edge $i\to j$ only if $\Delta_{ij}>0$, which also implies $j\to i$
is not an edge since $\Delta_{ji}=-\Delta_{ij}$. Observe in Fig. 3(C) that
this yields five convection cycles. In Sec. III.2, we will further discuss
these convection cycles and their relation to homological $1$-cycles.

Before continuing, we highlight that convection cycles revealed through flow
imbalances taylor2020multiplex  do not take into account the probability of
transitioning to or away from a convection cycle, and so they are not
necessarily “cyclic traps.” That is, the presence of a convection cycle does
not imply that it is unlikely for a random walker to leave (or move in an
opposite direction as) the cycle. For example, observe in Fig. 3 that the
counter-clockwise flow around convection cycle $A\to B\to C\to D$ is
approximately 0.025, yet there is a flow of approximately $0.02$ that leaves
the cycle at node D, and a flow of approximately $0.15$ moves in the opposite
direction from node C to B. Future research will likely uncover complementary
notions of convection with different advantages/disadvantages, and our
proposed techniques using persistent homology can likely be similarly
extended.

##  III Homological analyses of convection

We now employ persistent homology to automate the detection, characterization,
and summarization of the homological patterns of convection cycles. In Sec.
III.1, we study the MC that was presented in Fig. 3. In Sec. III.2, we discuss
the relation between convection cycles and homological $1$-cycles.

Figure 5:  Relation between convection cycles and homological 1-cycles. (A)
The flow imbalances shown in Fig. 3(C) give rise to five convection cycles,
which we label I–V. (B) Persistent homology using EVC filtrations applied to a
network of flow imbalances reveals three _homological 1-cycles_ , which we
label (i)–(iii). Each homological 1-cycle represents a “1-dimensional hole”
and can be represented by one or more _homological generator_ (recall Sec.
II.1). Observe that there is a one-to-one correspondence between convection
cycles I and II and homological 1-cycles (i) and (ii). In contrast, there are
three homologically equivalent generators for 1-cycle (iii) as shown. Two of
the generators correespond to convection cycles III and IV. The third
generator does not correspond to a convection cycle, because the edge
directions are not consistently in the same orientation (i.e., always
clockwise or counter-clockwise).

###  III.1 Persistent homology of convection cycles

Recall from Sec. II.2 that EVC filtrations were defined for an undirected
graph with a scalar function defined on the edges. Therefore, given an MC
corresponding to a (potentially) directed and weighted graph, we first
consider the associated undirected graph. Then we study homology under an EVC
filtration in which the filtration function
$f:\mathcal{E}\longrightarrow\mathbb{R}$ is given by the magnitudes of the
flow imbalances

$\displaystyle f(i,j)=|\Delta_{ij}|.$ (6)

In this way, the persistent homology that is revealed corresponds to the
convection cycles that arise under flow imbalances.

In Fig. 4, we visualize persistence barcodes for an EVC filtration associated
with the convection cycles shown in Fig. 3(C). Note that this figure is
analogous to Fig. 2(C), where we had previously chosen the filtration function
$f(i,j)$ to be the edge weights. Since we now use a different function $f$,
the cycles now have different births, deaths, lifetimes and lifespans.
Interestingly, the 1-cycle involving vertices $\\{A,B,C,D\\}$ is now born and
dies before the other two 1-cycles are born. While there is an obvious
connection between the EVC homology of a graph induced by edge weights and
that which is induced by convective flows, this relation remains unclear and
should be explored in future work.

We note that one could also construct EVC filtrations by increasing $\epsilon$
and retaining edges $(i,j)$ for which $|\Delta_{ij}|$ is smaller than
$\epsilon$. In Appendix A, we provide an example illustrating why EVC
filtrations with decreasing $\epsilon$ are superior to those with increasing
$\epsilon$ for the goal of studying convection cycles. In particular, EVC
filtrations that decrease $\epsilon$ focus on 1-cycles that are associated
with large-flow convection cycles (i.e., large values of $|\Delta_{ij}|$),
which we consider to be the ones that are more significant. In contrast, EVC
filtrations that increase $\epsilon$ focus on 1-cycles that are associated
with small-flow convection cycles (i.e., small values of $|\Delta_{ij}|$),
which we consider to be less significant.

###  III.2 Comparing convection cycles and homological 1-cycles

We propose to study pattern formation for convection cycles using persistent
homology techniques for homological 1-cycles; however, one should keep in mind
that these are two different notions for cycles. Homological $1$-cycles are
1-dimensional holes for a topological space, and $k$-cycles generalize to
higher dimensional by representing higher-dimensional holes (Sec. II.1). In
contrast, we define convection cycles to be closed non-backtracking paths in a
directed graph that encodes flow imbalances (Sec. II.3). In this section, we
will clarify the relationship between homological $k$-cycles and convection
cycles, thereby revealing the capabilities and disadvantages of existing
persistent homology techniques for studying convection cycles. Continuing with
the previous example [see Figs. 3-4], we find that flow imbalances give rise
to five convection cycles, which we enumerate I–V and visualize in Fig. 5(A).
In contrast, we identify three homological 1-cycles using persistent homology
with EVC filtrations, which we enumerate (i)–(iii) and visualize in Fig. 5(B).

Observe that there is a one-to-one correspondence between convection cycles I
and II and homological 1-cycles (i) and (ii). Also observe that homological
1-cycle (iii) has three homologically equivalent generators, and any of them
can be used to represent the 1-cycle (which again, is defined as a
1-dimensional hole). Each subsequent generator can be obtained via a
_topological retraction_ in which a 2-simplex is collapsed down onto one of
its edges. Interestingly, the first two homological generators for 1-cycle
(iii) correspond to convection cycles III and IV. In contrast, the third
generator corresponds to a loop that is not a convection cycle, since the
flow-imbalances’ directions do not point in a consistent direction along the
cycle (i.e., clockwise or counter-clockwise). Finally, observe that convection
cycle V is a boundary of a 2-simplex, and it therefore does not contribute to
the 1-dimensional simplicial homology.

Thus, it is important to not misinterpret one notion of cycle for the other.
At the same time, our findings in Fig. 5 also highlight that there is a need
for new persistent homology techniques that cater specifically to convection
cycles and directed graphs. For example, if one were to omit the 2-simplex
that involves vertices A, B and D from the clique complexes that arise under
an EVC filtration, then convection cycle V would coincide with a homological
1-cycle. However, the aim of this paper is not to develop new methods for
persistent homology. Instead, we proposed to begin this pursuit by studying
convection cycles using existing methods for persistent homology. Even though
there is not an exact one-to-one match between convection cycles and
homological 1-cycles, because they are closely related, we find that
persistent homology can effectively detect and summarize convection cycles’
patterns.

##  IV Applications

In this section, we apply our approach to two applications. In Sec. IV.1, we
study MCs arising for the Google PageRank algorithm, exploring how convection
cycles are effected by the teleportation parameter $\alpha$. In Sec. IV.2, we
study a type of emergent convection cycle called a chiral edge flow.

Figure 6:  Persistent homology of convection cycles arising under PageRank. We
study the MC associated with PageRank for the directed graph from Fig. 3(A)
under several choices of $\alpha$. (A) Flow imbalances $\Delta_{ij}(\alpha)$
give rise to convection cycles. For clarity, we do not visualize flow
imbalances for transitions due to teleportation. (B) Their homology changes
with $\alpha$, which is summarized by persistence barcodes.  Figure 7:
Bifurcation diagram summarizes homological changes onset by $\alpha$. (A)
Birth and death times $\epsilon_{b,d}$ of 1-cycles arising under PageRank
versus $\alpha$. (B) For comparison, we depict the vertices’ PageRanks
$\pi_{i}(\alpha)$. Vertical dashed lines near $\alpha^{*}=0.54$ highlight that
there are three cycles when $\alpha>\alpha^{*}$, but only two when
$\alpha<\alpha^{*}$.

###  IV.1 Teleportation is a homology regularizer for PageRank

We now study the persistent homology of convection cycles arising for the
PageRank algorithm page1999pagerank ; langville2006updating , which is a
popular technique to rank the importance of vertices in graphs. It has been
applied to numerous applications (see survey gleich2015pagerank ), but most
notably, for many years it was utilized by Google to rank website and
facilitate web search. The PageRank of a vertex $i$ is given by the stationary
density $\pi_{i}(\alpha)$ of MC having a transition matrix of the form

$\displaystyle{\bf P}(\alpha)$ $\displaystyle\equiv\alpha{\bf
P}+(1-\alpha)N^{-1}{\bf 1}{\bf 1}^{T},$ (7)

where ${\bf P}$ is the transition matrix described in Sec. II.3 and
$\alpha\in(0,1)$ is the _teleportation parameter_. As $\alpha\to 1$, the
second term vanishes and ${\bf P}(\alpha)\to{\bf P}$. Usually, $\alpha$ is
chosen to be near 1 (often 0.85) so that the second term can be considered as
a small perturbation that improves the mathematical characteristics of ${\bf
P}$—or more formally, it is a “‘regularization” of matrix ${\bf P}$. In
particular, when $\alpha\in(0,1)$ the matrix ${\bf P}(\alpha)$ is guaranteed
to be irreducible, aperiodic and positive, and the Perron-Frobenius theorem
bapat1998max  ensures that its dominant left eigenvector $\pi$ is unique and
has positive entries (i.e., $\pi_{i}(\alpha)>0$ for all $i$). In other words,
the PageRanks are well-defined for all vertices.

We now show that the introduction of teleportation also regularizes the
homology of convection cycles. In this experiment, we construct EVC
filtrations with the filtration function $f(i,j)=|\Delta_{ij}(\alpha)|$, which
now depends on $\alpha$. In Fig. 6(A), we illustrate for several choices of
$\alpha$ the flow imbalances that arise under PageRank, which we apply to the
graph from Fig. 3(A). In Fig. 6(B), we visualize their associated persistence
barcodes, which we create using EVC filtrations. Note that the choice
$\alpha=1$ recovers the transition matrix, stationary distribution, flow
imbalances, and persistence barcodes that were were previously studied in
Figs. 3 and 4.

Observe that the homological patterns of convection cycles significantly
change with $\alpha$. For example, when $\alpha$ is sufficiently small, the
homological 1-cycle $\\{A,B,C,D\\}$ vanishes—it is “washed out” by the
introduction of teleportation. In other word, $\alpha$ is a _homology
regularizer_. This is further illustrated in Fig. 7(A), where we plot the
birth and death times of homological 1-cycles versus $\alpha$. For comparison,
we also plot the PageRanks $\pi_{i}(\alpha)$ in Fig. 7(B). The vertical line
highlights that one of the 1-cycles vanishes when $\alpha$ decreases
(approximately) below $\alpha^{*}=0.54$

In Appendix A, we present additional experiments that explore convection
cycles arising under PageRank with $\alpha=0.8$. We show that homological
1-cycles arising for EVC filtrations with decreasing filtration parameter
$\epsilon$ reveal patterns for large-flow convection cycles. In contrast, when
EVC filtrations are constructed with increasing $\epsilon$, we find that the
resulting homological 1-cycles relate to small-flow convection cycles, and in
particular, those involving low probability teleportation transitions.

Figure 8:  MC model for configuration dynamics of a monomer. The system is
contains two monomers of sizes $s_{1}$ and $s_{2}$, respectively, giving the
external state $(s_{1},s_{2})$. Moreover, there are four internal states: a,
b, c, and d. Transitions involving changes to external and internal states
occur at rates $\gamma_{ex}$ and $\gamma_{in}$, respectively.  Figure 9:
Persistent homology chiral edge flow. (A) Flow imbalances $\Delta_{ij}$
between external states for the bi-monomer shown in Fig. 8 with
$\gamma_{in}=0.01$ and different $\gamma_{ex}$. In the limit
$\gamma_{ex}\gg\gamma_{in}$ tang2021topology , there is an emergence of a
chiral edge flow, i.e., a convection cycle around the lattice’s outer
boundary. (B) The corresponding persistence barcodes capture the emergence of
this prominent convection cycle and other convection cycles within the
lattice.

###  IV.2 Persistent homology of chiral edge flows

Our second application investigates homological patterns of convection cycles
that arise for an MC that models the stochastic configuration dynamics of two
monomers. We adopt the same notation as in tang2021topology , which motivated
our experiment. The monomer configuration (i.e., “external state”) is given by
the number of monomers of each type, $(s_{1},s_{2})$, whereas the “internal
state” is one of four possibilities: a, b, c, or d. Transitions that involve a
change of internal state occur at rate $\gamma_{in}$, whereas transitions
between involving external states (i.e., the addition or removal of a monomer)
occur at rate $\gamma_{ex}$. The resulting MC can be visualized as a
2-dimensional lattice, which we visualize in Fig. 8.

In Fig. 9(A), we visualize flow imbalances for transitions between the
external states. We fix $\gamma_{in}=0.01$ and consider several $\gamma_{ex}$.
Observe that as $\gamma_{ex}$ increases, a large counter-clockwise convection
cycle emerges on the boundary (i.e., “edge”) of the lattice. This type of
convection cycle is called a _chiral edge flow_ , and such convection cycles
have important implications for the quantum Hall effect, biological rhythms,
and the dynamics of monomers tang2021topology . In Fig. 9(B), we visualize
persistence barcodes for EVCs filtrations constructed using the method that we
described in Sec. III.1. The chiral edge cycle corresponds to the 1-cycle with
having the largest lifespan, and its homology becomes more persistent (i.e.,
prominent) in the limit $\gamma_{ex}\gg\gamma_{in}$.

##  V Discussion

In this paper, we examined the patterns of convection cycles that arise under
irreversible Markov chains (MCs) from the perspective of persistent homology.
Our approach required formalizing a type of filtration (EVC filtration) for
scalar functions that are defined on the edges of a graph, and we studied
convection cycles by choosing the filtration function to be an MC’s flow
imbalances in the stationary state. Because Markov chains are crucial to so
many diverse applications, we expect our methods to be broadly applicable
across the sciences and engineering. Herein, we highlighted two such
applications: the PageRank algorithm for centrality analysis and chiral edge
flows that arise for the configuration dynamics of monomers. Our experiments
revealed how system properties can act as homology regularizers of convection
cycles, and we introduced homological bifurcation diagrams to summarize these
changes. This approach automates the detection, summary, and examination of
convection cycles over networks, places it on stronger mathematical and
computational foundations, and paves the way for further investigation into
convective flows on networks.

Additionally, our work highlights the need for new persistent homology methods
to study convection cycles as well as other functions and signals defined on
directed graphs. In Sec. III.2, we discuss the relation between convection
cycles and homological 1-cycles, and we showed that these are two closely
related, but notably different, notions of cycles. Sometimes there is a one-
to-one correspondence between these cycles, and sometimes the relation is more
complicated, due in part to the fact that a given homological k-cycle can be
equivalently represented by possibly more than one homological generator. Such
generators may or may not correspond to convection cycles. Moreover,
convection cycles can also correspond to the boundaries of 2-simplices, and as
such, they will not be identified via the traditional tools of persistent
homology. Developing persistent homology techniques that cater to convection
cycles, and which specifically account for edge directions, remains an
important open challenge for the applied mathematics and physics communities.

Our work opens up several other new lines of research that are also worth
noting. First, convection cycles were recently found to be an emergent
property of multiplex Markov chains taylor2020multiplex  in which a set of
(intralayer) Markov chains are coupled together by another set of (interlayer)
Markov chains. It would be interesting to employ persistent homology to gain a
deeper understanding of this phenomenon. Second, chiral edge flows are known
to be important to other applications including the quantum Hall effect and
biological rhythms tang2021topology , and future work could utilize our
methods to investigate these exciting applications. Notably, our methods can
reveal convection cycles that exists in addition to a chiral edge flow, which
may lead to new insights for these applications and other applications (e.g.,
reinforcement learning) that rely on irreversible Markov chains.

See minh_code  for a codebase that reproduces our results and can be used to
study the persistent homology for convection cycles arising for other
applications.

Figure 10: Comparing EVC filtrations with decreasing and increasing filtration
parameter $\epsilon$. Extending our study in Sec. IV.1 that uses persistent
homology to study convection cycles arising for a MC under the PageRank
algorithm with $\alpha=0.8$, we now study homological $1$-cycles obtained via
two different EVC filrations. (A) Similar to our results in Fig. 6, we
construct EVC filtrations by including edges for which
$|\Delta_{ij}|>\epsilon$ while decreasing $\epsilon$. Observe that the
1-cycles reveal large-flow convection cycles that are associated with large
values of $|\Delta_{ij}|$. (B) For comparison, we construct EVC filtrations by
including weighted edges $|\Delta_{ij}|<\epsilon$ while increasing $\epsilon$.
Observe that these 1-cycles now correspond to small-flow convection cycles
that are associated with small values of $|\Delta_{ij}|$. They primarily
describe low-probability transitions that occur due to teleportation. In this
work we focus on EVC filtrations with decreasing $\epsilon$, since we consider
high-flow convection cycles to be the ones that are most important.

##  Appendix A Convection cycles are better revealed by filtrations that
decrease the filtration parameter $\epsilon$ versus increase $\epsilon$

In Sec. II.2, we defined EVC filtrations in which one decreases a filtration
parameter $\epsilon$, retaining edges for which $f(i,j)>\epsilon$. Our
numerical experiments that study convection cycles using persistent homology
use this approach and let the filtration be given by the flow imbalances
$f(i,j)=|\Delta_{ij}|$. By decreasing $\epsilon$, the cycles that are first
revealed correspond to large-flow convection cycles, which we consider to be
the ones that are more significant. One could also construct EVC filtrations
by increasing $\epsilon$ and retaining edges for which $f(i,j)<\epsilon$.
Here, we show that this latter filtration reveals 1-cycles that relate to
small-flow convection cycles, which we consider to be insignificant.

In Fig. 10, we study EVC filtrations applied to flow imbalances arising under
the PageRank algorithm with $\alpha=0.8$ for the same MC that we investigated
in Sec. IV.1. In Fig. 10(A) and Fig. 10(B), we illustrate EVC filtrations with
decreasing and increasing $\epsilon$, respectively. Observe in Fig. 10(A) that
the 1-cycles revealed by decreasing $\epsilon$ correspond to large-flow
convection cycles. In contract, observe in Fig. 10(B) that the 1-cycles
revealed by increasing $\epsilon$ are small-flow cycles that relate to low-
probability transitions that occur due to teleportation.

###### Acknowledgements.

MQL and DT were supported in part by the National Science Foundation
(DMS-2052720 and EDT-1551069) and the Simons Foundation (grant #578333).

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