Probability measures on graph trajectories

1. Introduction

As is well known, Markov chains model random walks on graphs. Let $\Gamma$ be a directed graph. Its set of vertices $\Gamma_{0}$ represent the states of the system and its edges $\Gamma_{1}$ indicate transitions between states. There are two flavors of random walk: those in discrete time and those in continuous time. This note will consider the continuous time variant.

The dynamics of continuous time random walk are encoded by a master equation

p^{\prime}(t)=\mathbb{H}(t)p(t)\,,

where $\mathbb{H}(t)$ is a time dependent matrix of transition rates and $p(t)$ is a 1-parameter family of probability distributions on $\Gamma_{0}$ . The solutions to the equation describe the time evolution of probability. For vertices $i$ and $j$ , the matrix entry $\mathbb{H}(t)_{ij}$ is the instantaneous rate of change at time $t$ in jumping from state $i$ to state $j$ along the set of edges of $\Gamma$ having initial vertex $i$ and terminal vertex $j$ . The operator $\mathbb{H}$ is called the master operator; its off diagonal entries are non-negative and the sum of the entries in any column add to zero.

Given a continuous time Markov chain with state diagram $\Gamma$ , our goal here will be to construct a probability distribution on the space of trajectories in $\Gamma$ . By a trajectory in $\Gamma$ , we mean a path of contiguous edges equipped with jump times at each vertex of the path. Note that such a probability distribution amounts to a description of the stochastic process associated with the Markov chain.

Remark 1.1.

We apologize to the reader in advance for our somewhat unconventional treatment: two of us are algebraic topologists and one is a chemical physicist.

2. Preliminaries

For a set $T$ , let $\binom{T}{2}$ denote the set of its non-empty subsets of cardinality $2$ . An undirected graph consists of data

X:=(X_{0},X_{1},\delta)\,,

in which $X_{0}$ is the set of vertices, $X_{1}$ is the set of edges and

\delta\colon\!X_{1}\to\tbinom{X_{0}}{2}

is a function. We will always assume that $X$ is locally finite in the sense that the function $\delta$ is a finite-to-one. With this definition multiple edges connecting a pair of distinct vertices are permitted, but we do not permit loop edges, i.e., edges which connect a vertex to itself.

A directed graph $\Gamma$ is defined in a similar way, but where now $\delta$ is replaced by a function $d\colon\!\Gamma_{1}\to\Gamma_{0}(2)$ , where $\Gamma_{0}(2):=\Gamma_{0}\times\Gamma_{0}\setminus\Delta$ , i.e., the cartesian product with its diagonal deleted. We write $d=(d_{0},d_{1})$ , where $d_{i}\colon\!\Gamma\to\Gamma_{0}$ is is the function which assigns to a directed edge its source, respectively target. Note that the canonical map $\pi\colon\!\Gamma_{0}(2)\to\binom{\Gamma_{0}}{2}$ is a double cover, and the composition $\delta:=\pi\circ d$ defines the underlying undirected graph.

Example 2.1.

Given an undirected graph $X$ , we may construct its double. This is the directed graph

DX:=\Gamma=(\Gamma_{0},\Gamma_{1},d)\,,

in which $\Gamma_{0}=X_{0}$ and $\Gamma_{1}$ is the set of ordered pairs $(i,\alpha)\in X_{0}\times X_{1}$ in which $i\in\delta(\alpha)$ . The function $d\colon\!\Gamma_{1}\to\Gamma_{0}(2)$ is given by $d(i,\alpha)=(i,j)$ , where $\delta(\alpha)=\{i,j\}$ .

Remark 2.2.

Let $\mathcal{G}$ be the category of undirected graphs. An object is an undirected graph and a morphism $f\colon\!(G_{0},G_{1},\delta)\to(H_{0},H_{1},\delta^{\prime})$ consists of functions $f_{i}\colon\!G_{i}\to H_{i}$ , $i=0,1$ such that $\delta^{\prime}f_{1}(\alpha)=f_{0}(\delta\alpha)$ . Similarly, one has the category $\mathcal{G}^{+}$ of directed graphs. Then we have an adjoint functor pair

U\colon\!\mathcal{G}^{+}\leftrightarrows\mathcal{G}:D

where $U$ is the forgetful functor and $D$ is given by the double.

2.1. Markov chains

Let $\Gamma$ be a directed graph. A continuous time Markov chain with state diagram $\Gamma$ is an assignment of a continuous function

k_{\alpha}\colon\!\mathbb{R}\to[0,\infty)\,,

to each edge $\alpha\in\Gamma_{1}$ . The function $k_{\alpha}$ is called the transition rate of $\alpha$ . If $d(\alpha)=(i,j)$ , then $k_{\alpha}$ is to be interpreted as the instantaneous rate of change of probability in jumping from $i$ to $j$ along $\alpha$ .

Remark 2.3.

The foundational material on Markov chains can be found in the texts of Norris [Norris] and Stroock [STR14]. When the transition rates are constant, the Markov chain is said to be time homogeneous. When the rates are not constant, the chain is said to be time inhomogeneous.¹¹1In contrast with the homogeneous case, the literature on the inhomogeneous case is scant, with the known results making strong additional assumptions. The only foundational work we are of aware of that treats the time inhomogeneous case is Stroock’s text (cf. [STR14, §5.5.2]).

Remark 2.4.

The canonical map $\Gamma\to DU\Gamma$ is an embedding. Given a Markov chain on $\Gamma$ , one has a canonical extension to $DU\Gamma$ by defining the rates to be zero on those edges which aren’t in $\Gamma$ . The Markovian dynamics of the two chains coincide. From this standpoint, there is nothing to lose by assuming that $\Gamma=DX$ for some undirected graph $X$ .

If $\Gamma$ is infinite, we also require the following growth constraints.

Definition 2.5 (Rate Bound).

For each $t>0$ , there exists a constant $R$ , possibly depending on $t$ , such that

k_{\alpha}(s)\leq R

for $0\leq s\leq t$ and every $\alpha\in\Gamma_{1}$ .

Definition 2.6 (Degree Bound).

Let $\deg\colon\!\Gamma_{0}\to\mathbb{N}$ be the function which assigns to a vertex its degree, i.e., the number of edges meeting it. There is a positive integer $D$ such that

\deg(i)\leq D\,,\quad\text{for all }\quad i\in\Gamma_{0}\,.

Observe that when $\Gamma$ is a finite, both conditions hold automatically.

2.2. The master equation

Then the rates define a time dependent square matrix $\mathbb{H}=\mathbb{H}(t)$ , as follows. For $i\neq j$ , set

h_{ij}=\sum_{d(\alpha)=(i,j)}k_{\alpha}\,,

where the sum is interpreted as zero when $d^{-1}(i,j)$ is the empty set. Then the matrix entries of $\mathbb{H}$ are given by

\mathbb{H}_{ij}=\begin{cases}h_{ij}\,,\qquad&i\neq j\,;\\ -\sum_{\ell\neq i}h_{\ell i}\,,\quad\text{ if }&i=j\,,\end{cases}

where the indices range over $i,j\in\Gamma_{0}$ . The time dependent matrix $\mathbb{H}$ is called the master operator. Associated with $\mathbb{H}$ is a linear, first order ordinary differential equation

(1)

p^{\prime}(t)=\mathbb{H}p(t),\qquad

in which $p(t)$ is a one parameter family of (probability) distributions on the set of vertices $\Gamma_{0}$ . Equation (1) is called the (forward) Kolmogorov equation or the master equation [STR14, eqn. 5.5.2]. Its solutions describe the evolution of an initial distribution $p(0)$ .

Remark 2.7.

If $\Gamma=DX$ and the transition rates are constant with value 1, then $\mathbb{H}$ is the graph Laplacian of $X$ and (1) is a combinatorial version of the heat (diffusion) equation.

Remark 2.8.

The forward Kolmogorov equation is often written in the literature in adjoint form, i.e., as

q^{\prime}(t)=q(t)\mathbb{W}\,,

where $q(t)=p(t)^{*}$ and $\mathbb{W}=\mathbb{H}^{*}$ are the transposed matrices. The backward equation (which we will not consider here) is

q^{\prime}(t)=\mathbb{W}q(t)\,.

2.3. Trajectories

A path of length $n$ in $\Gamma$ consists of a sequence of edges

\alpha_{\bullet}:=(\alpha_{1},\dots,\alpha_{n})\,,

such that $d_{1}(\alpha_{k})=d_{0}(\alpha_{k+1})$ for $1\leq k<n$ . We let

i_{k}(\alpha_{\bullet}):=i_{k}

denote the $k$ -th vertex of the path, i.e., $i_{k}=d_{0}(\alpha_{k}))$ if $k\leq n$ and $i_{n+1}=d_{1}(\alpha_{n})$ .

A trajectory of length $n$ and duration $t>0$ is a pair

(\alpha_{\bullet},t_{\bullet})\,,

such that $\alpha_{\bullet}$ is a path of length $n$ and $t_{\bullet}=(t_{1},\dots,t_{n})$ is a sequence of real numbers satisfying

0\leq t_{1}\leq\cdots\leq t_{n}\leq t\,.

In what follows, it will be convenient to set

t_{0}:=0\quad\text{ and }\quad t_{n+1}=:t\,.

Remark 2.9.

For a vertex $i_{k}=i_{k}(\alpha_{\bullet})$ of the path $\alpha_{\bullet})$ , the number $t_{k}$ is called the jump time and the number $w_{k}:=t_{k}-t_{k-1}$ is called wait time.

3. The probability of a trajectory

Let $(\Gamma,k_{\bullet})$ be as in the previous section. Given a vertex $i\in\Gamma_{0}$ and an interval $[a,b]$ , the escape rate at $i$ is

u_{i}(a,b):=\exp\left(-\sum_{d_{1}(\alpha)=i}\int_{a}^{b}k_{\alpha}(s)\,ds\right)=\exp\left(\int_{a}^{b}h_{ii}(s)\,ds\right)\,.

Fix an initial probability distribution $q\colon\!\Gamma_{0}\to\mathbb{R}_{+}$ . For $j\in\Gamma_{0}$ , set $q_{j}:=q(j)$ .

Let

\mathcal{T}(\Gamma,n,t)

denote the set of trajectories of $\Gamma$ having length $n$ . Define a function

f\colon\!\mathcal{T}(\Gamma,n,t)\to\mathbb{R}_{+}

by the formula

	$\displaystyle f(\alpha_{\bullet},t_{\bullet})$	$\displaystyle=q_{i_{1}}u_{i_{1}}(0,t_{1})k_{\alpha_{1}}(t_{1})\cdots u_{i_{n}}(t_{n-1},t_{n})k_{\alpha_{n}}(t_{n})u_{i_{n+1}}(t_{n},t)\,,$
		$\displaystyle=q_{i_{1}}\prod_{m=1}^{n+1}u_{i_{m}}(t_{m-1},t_{m})\prod_{m=1}^{n}k_{\alpha_{m}}(t_{m})$

(compare [Bellac, eqn. 1.112] in the constant rate case).²²2The function $f$ is a discrete analogue of the Onsager-Machlup Lagrangian [Onsager-Machlup].

Consider the master equation

p^{\prime}(t)=\mathbb{H}p(t),\quad p(0)=q\,.

Let

\mathcal{P}(\Gamma,n)

denote the set of paths of length $n$ and let $\mathcal{P}^{i}(\Gamma,n)\subset\mathcal{P}(\Gamma,n)$ denote the subset of those paths which have terminus $i\in\Gamma_{0}$ .

Theorem 3.1.

The formal solution to the master equation is the vector valued function $p(t)$ whose component at $i\in\Gamma_{0}$ is given by the expression

p_{i}(t)=\sum_{n=0}^{\infty}\sum_{\scriptscriptstyle\alpha_{\bullet}\in\mathcal{P}^{i}(\Gamma,n)}\int_{0}^{t}\!\!\!\int_{0}^{t_{n}}\!\!\!\cdots\!\!\!\int_{0}^{t_{2}}f(\alpha_{\bullet},t_{\bullet})\,dt_{1}\cdots dt_{n}\,.

Proof.

Write $\mathbb{H}=A_{0}+A_{1}$ , where $A_{0}$ is the diagonal matrix with entries $h_{ii}$ . For $\epsilon>0$ , set $\mathbb{H}_{\epsilon}=A_{0}+\epsilon A_{1}$ . Consider the equation

(2)

\dot{p}=H_{\epsilon}p,\quad p(0)=q\,.

We seek a formal solution $p=p^{0}+\epsilon p^{1}+\epsilon^{2}p^{2}+\cdots$ with $p^{0}(0)=q$ and $p^{n}(0)=0$ for $n>0$ . Once such a solution is found, we set $\epsilon=1$ to obtain the formal solution to the master equation.

Expanding (2) in $\epsilon$ , we obtain the linear system

(3)

\dot{p}^{n}=A_{0}p^{n}+A_{1}p^{n-1}\,,\quad n=0,1,2,\dots

where by convention $p^{-1}:=0$ .

For $i\in\Gamma_{0}$ , the $i$ -th equation of the system is the first order linear differential equation

(4)

\dot{p}_{i}^{n}=h_{ii}p_{i}^{n}+\sum_{j\neq i}h_{ij}p_{j}^{n-1}\,.

If $n=0$ , the system is uncoupled and separation of variables gives

p_{i}^{0}=q_{i}e^{\int_{0}^{t}h_{ii}(t_{1})dt_{1}}=q_{i}u_{i}(0,t)\,.

For $n>0$ , the solution to (4) can be iteratively solved using the integrating factor. The first iteration gives

	$\displaystyle p_{i}^{n}(t)$	$\displaystyle=\sum_{j\neq i}\int_{0}^{t}e^{\int_{t_{n}}^{t}h_{ii}\,dt_{n-1}}h_{ij}(t_{n})p_{j}^{n-1}(t_{n})\,dt_{n}$
		$\displaystyle=\sum_{\alpha_{n}}\int_{0}^{t}u_{i_{n+1}}(t_{n},t)k_{\alpha_{n}}(t_{n})p_{i_{n}}^{n-1}(t_{n})\,dt_{n}\,.$

where the second sum is over all edges $\alpha_{n}$ with terminus $i=i_{n+1}$ and whose source is denoted in the integrand by $i_{n}$ . We then repeat the procedure using $p_{i}^{n-1}$ in place of $p_{i}^{n}$ , to obtain

p_{i}^{n}(t)=\sum_{\alpha_{\bullet}}\int_{0}^{t}\!\!\!\int_{0}^{t_{n}}u_{i_{n+1}}(t_{n},t)k_{\alpha_{n}}(t_{n})u_{i_{n}}(t_{n-1},t_{n})k_{\alpha_{n-1}}(t_{n-1})p_{i_{n-1}}^{n-2}(t_{n-1})\,dt_{n-1}dt_{n}\,,

where the sum is indexed over all paths of length two $\alpha_{\bullet}=(\alpha_{n-1},\alpha_{n})$ satisfying $d(\alpha_{n-1})=(i_{n-1},i_{n})$ , $d(\alpha_{n})=(i_{n},i_{n+1})$ , and $i_{n+1}=i$ . Applying this procedure a total of $n$ times results in the desired expression for $p^{n}_{i}(t)$ . ∎

Let $\mathcal{T}(\Gamma,t)$ denote the space of trajectories of duration $t$ of arbitrary length.

Corollary 3.2.

Let $p(t)$ denote the formal solution to the master equation. Then $p(t)$ is a probability distribution on $\Gamma_{0}$ for every $t\geq 0$ . In particular, the function $f$ is a probability density on $\mathcal{T}(\Gamma,t)$ .

Proof.

As the rate bound holds, there is a constant $C>0$ , independent of $n$ , such $f(\alpha_{\bullet},t_{\bullet})\leq C^{n}$ for all trajectories of length $n$ . As the degree bound holds, there is global bound $D$ on the degree function, so the number of paths of length $n$ terminating at a vertex $i$ is at most $D^{n}$ . Consequently,

0\leq\sum_{\alpha_{\bullet}}\int_{0}^{t}\!\!\!\int_{0}^{t_{n}}\!\!\!\cdots\!\!\!\int_{0}^{t_{2}}f(\alpha_{\bullet},t_{\bullet})\,dt_{1}\cdots dt_{n}\leq\frac{(CDt)^{n}}{n!}\,,

where the sum ranges over paths of length $n$ with terminus $i$ . Here, we have used the fact that $t^{n}/n!$ is the volume of the $n$ -simplex $0\leq t_{1}\leq\cdots t_{n}\leq t$ . By the comparison test, $\sum_{n}p^{n}_{i}(t)$ converges. Therefore $p(t)=\sum_{n}p^{n}(t)$ also converges.

Let $\mathbf{1}\colon\!\Gamma_{0}\to\mathbb{R}$ be the row vector which is identically one at every vertex. It will suffice to show that $\mathbf{1}\cdot p(t)=1$ . Observe that $\mathbf{1}\cdot\mathbb{H}=0$ , since the entries of $\mathbb{H}$ in any column add to zero. Then for all $t$ we have

\frac{d}{dt}(\mathbf{1}\cdot p(t))=\mathbf{1}\cdot p^{\prime}(t)=\mathbf{1}\cdot\mathbb{H}p(t)=0\,.

Consequently, $\mathbf{1}\cdot p(t)$ is a constant. But $\mathbf{1}\cdot p(0)=1$ , hence $\mathbf{1}\cdot p(t)=1$ for all $t$ . ∎

4. Fundamental solutions

Consider the master equation with initial distribution $p(0)=\delta_{i}$ for a fixed vertex $i$ , where $\delta_{i}(j)=\delta_{ij}$ is the Kronecker delta function. The solution to this equation is called a fundamental solution and will be denoted by $u(i,t)$ .

In this case the density $f$ is supported on the set of trajectories $(\alpha_{\bullet},t_{\bullet})$ with initial vertex $i$ . Let $\mathcal{T}_{i}(\Gamma,t)\subset\mathcal{T}(\Gamma,t)$ denote the subspace of trajectories which start at the vertex $i$ .

Corollary 4.1.

With respect to this assumption, the function $f$ is a probability density on $\mathcal{T}_{i}(\Gamma,t)$ .

Remark 4.2.

The general solution $p(t)$ to the master equation with initial condition $p(0)=q$ is obtained from the fundamental solutions using the identity

(5)

p(t)=\sum_{j\in\Gamma_{0}}q_{j}u(j,t)\,.

Definition 4.3.

Define the propagator $K\colon\!\Gamma_{0}\times\Gamma_{0}\times\mathbb{R}_{+}\to\mathbb{R}_{+}$ by

K(i,j,t)=u_{j}(i,t)\,,

i.e., the probability of the set of trajectories of duration $t$ which start at vertex $i$ and terminate at vertex $j$ . Note the initial condition $K(i,j,0)=\delta_{ij}$ .

Setting $\psi(x,t):=p_{x}(t)$ , equation (5) becomes

\psi(y,t)=\sum_{x\in\Gamma_{0}}K(x,y,t)\psi(x,0)=\int_{x\in\Gamma_{0}}K(x,y,t)\psi(x,0)\,,

which is familiar to the physics literature. By Theorem 3.1, we obtain the path integral representation

K(x,y,t)=\sum_{n=0}^{\infty}\sum_{\scriptscriptstyle\alpha_{\bullet}\in\mathcal{P}^{y}_{x}(\Gamma,n)}\int_{0}^{t}\!\!\!\int_{0}^{t_{n}}\!\!\!\cdots\!\!\!\int_{0}^{t_{2}}f(\alpha_{\bullet},t_{\bullet})\,dt_{1}\cdots dt_{n}\,,

where $\mathcal{P}^{y}_{x}(\Gamma,n)$ is the set of paths of length $n$ which start at $x$ and terminate at $y$ . Furthermore, the series converges if $\Gamma$ satisfies the rate and degree bounds.

Example 4.4.

Let $X$ be an $r$ -regular graph, i.e., the number of edges meeting each vertex is $r$ . Assume that the rates $k_{\bullet}$ are constant with value one. Then the master operator $\mathbb{H}$ is the negative of the graph Laplacian. In this instance elementary to check that $f(\alpha_{\bullet},t_{\bullet})=\delta_{i}e^{-rt}$ . Let $\pi_{n}(i,j)$ be the number of paths of length $n$ in $\Gamma=DX$ from $i$ to $j$ . Then a straightforward calculation shows

K(i,j,t)=e^{-rt}\sum_{n=0}^{\infty}\pi_{n}(i,j)\frac{t^{n}}{n!}\,.

In this case, $K$ is the combinatorial heat kernel.

As $\Gamma$ is $r$ -regular, there are precisely $r^{n}$ paths of length $n$ which start at $i$ . Set

\phi_{n}(i,j):=\frac{\pi_{n}(i,j)}{r^{n}}\,.

Then $\phi_{n}(i,j)$ is the probability of the set of paths (not trajectories), which end at $j$ after $n$ -jumps, given that such paths start at $i$ (where the probability of jumping across an edge meeting any vertex is $1/r$ ).

Let

P(n)=\frac{(rt)^{n}e^{-rt}}{n!}\,.

Then $P$ is the Poisson probability mass function with parameter $\lambda=rt$ . Consequently,

K(i,j,t)=\sum_{n=0}^{\infty}\phi_{n}(i,j)P(n)={\mathbb{E}}[\Phi_{ij}]\,,

is the (Poisson) expected value of the random variable $\Phi_{ij}(n):=\phi_{n}(i,j)$ . Summarizing, the continuous time random walk on an $r$ -regular graph with uniform rate $1/r$ may be thought of as a discrete time random walk subordinated to a Poisson process (cf. [Feller, chap. X§7]).

Probability measures on graph trajectories

Abstract.

1. Introduction

Remark 1.1.

2. Preliminaries

Example 2.1.

Remark 2.2.

2.1. Markov chains

Remark 2.3.

Remark 2.4.

Definition 2.5 (Rate Bound).

Definition 2.6 (Degree Bound).

2.2. The master equation

Remark 2.7.

Remark 2.8.

2.3. Trajectories

Remark 2.9.

3. The probability of a trajectory

Theorem 3.1.

Proof.

Corollary 3.2.

Proof.

4. Fundamental solutions

Corollary 4.1.

Remark 4.2.

Definition 4.3.

Example 4.4.

References