Path Cohomology of Locally Finite Digraphs,Hodge's Theorem and the $p$-Lazy Random Walk

TL;DR

This paper extends the theory of Markov chains on digraphs by introducing a cohomology framework, defining new Laplace operators, proving a Hodge Decomposition, and linking the $p$-Lazy Random Walk's behavior to these spectral properties.

Contribution

It develops a cohomology theory for infinite, locally finite digraphs, introduces new Laplace operators, and connects these to the asymptotic behavior of a novel $p$-Lazy Random Walk.

Findings

Defined cohomology for infinite digraphs in arbitrary dimensions

Proved Hodge Decomposition Theorem for the new Laplacians

Linked the spectrum of Laplacians to the mixing time of the $p$-Lazy Random Walk

Abstract

The study of Markov chains on discrete spaces, such as digraphs, has captivated mathematicians in recent decades due to its interconnectedness with topology, geometry, dynamics, spectral theory, and differential equations. Furthermore, extensive exploration of these multifaceted relationships has been pursued for their practical utility in diverse fields, including machine learning and image segmentation. In recent times, these interrelations have been generalized to higher dimensions within the framework of finite-dimensional simplicial complexes. In this paper, we embark on a further extension of these concepts. Initially, we introduce a cohomology of infinite (though locally finite) digraphs in arbitrary dimensions. Subsequently, in the latter portion of this manuscript, we define a fresh family of Laplace operators and conduct an examination of their spectrum, culminating in the proof of the Hodge Decomposition Theorem within this framework. Finally, we conclude by presenting a Markov chain, the $p$-Lazy Random Walk, whose asymptotic behavior is intrinsically linked to these cohomologies, while its mixing time is related to the the spectrum of our Laplace operators. This development opens doors to numerous unexplored questions, particularly regarding potential generalizations of the Ollivier-Ricci curvature to this topology and these Laplacians.

Equations

$$l^2({\Lambda }_p):=\{f:{\Lambda }_p\to K;\sum _{x\in {\Lambda }_p}{f}^2(x)<\infty \}.$$

$$⟨f,g⟩=\sum _{x_{\in }{\Lambda }_p}f(x)g(x).$$

$$df(i_0\cdots i_p)=\sum _{q=0}^p(-1{)}^{q}f(i_0\cdots \widehat{i_q}\cdots i_p),$$

$$\partial f(i_0\cdots i_p)=\sum _{k\in V}\sum _{q=0}^{p+1}(-1{)}^{q}f(i_0\cdots i_{q-1}k{i}_q\cdots i_p);$$

$$d^{2}f(i_0\cdots i_p)=\sum _{q=0}^p(-1{)}^{q}df(i_0\cdots \widehat{i_q}\cdots i_p)$$

$$=\sum _{q=0}^p(-1{)}^q\left(\sum _{r=0}^{q-1}(-1{)}^{r}f(i_0\cdots \widehat{i_r}\cdots \widehat{i_q}\cdots i_p)+\sum _{r=q+1}^p(-1{)}^{r-1}f(i_0\cdots \widehat{i_q}\cdots \widehat{i_r}\cdots i_p)\right)$$

$$\sum _{0\le r<q\le p}(-1{)}^{q+r}f(i_0\cdots \widehat{i_r}\cdots \widehat{i_q}\cdots i_p)-\sum _{0\le q<r\le p}(-1{)}^{q+r}f(i_0\cdots \widehat{i_q}\cdots \widehat{i_r}\cdots i_p).$$

$${\Vert df\Vert }^2=\sum _{i_0,\cdots ,i_p\in V}df(i_0\cdots i_p{)}^2=\sum _{i_0,\cdots ,i_p\in V}{\left(\sum _{q=0}^p(-1{)}^{q}f(i_0\cdots \widehat{i_q}\cdots i_p)\right)}^2$$

$$=\sum _{i_0,\cdots ,i_p\in V}\sum _{q=0}^p\sum _{r=0}^p(-1{)}^{q+r}f(i_0\cdots \widehat{i_q}\cdots i_p)f(i_0\cdots \widehat{i_r}\cdots i_p)$$

$$=\sum _{i_0,\cdots ,i_p\in V}\sum _{q=0}^p(f(i_0\cdots \widehat{i_q}\cdots i_p{)}^2+\sum _{r<p}(-1{)}^{q+r}f(i_0\cdots \widehat{i_r}\cdots i_q\cdots i_p)f(i_0\cdots i_r\cdots \widehat{i_q}\cdots i_p)+$$

$$+\sum _{r>p}(-1{)}^{q+r-1}f(i_0\cdots \widehat{i_q}\cdots i_r\cdots i_p)f(i_0\cdots i_q\cdots \widehat{i_r}\cdots i_p)).$$

$${\Vert df\Vert }^2=\sum _{i_0,\cdots ,i_p\in V}\sum _{q=0}^{p}f(i_0\cdots \widehat{i_q}\cdots i_p{)}^2={\Vert f\Vert }^2<\infty .$$

$$⟨df,g⟩$$

$$=\sum _{i_0\cdots i_p\in {\Lambda }_p}\sum _{q=0}^p(-1{)}^{q}f(i_0\cdots \widehat{i_q}\cdots i_p)g(i_0\cdots i_p)$$

$$=\sum _{i_0\cdots i_{p-1}\in {\Lambda }_{p-1}}\sum _{q=0}^p\sum _{k\in V}(-1{)}^{q}f(i_0\cdots i_{p-1})g(i_0\cdots i_{q-1}k{i}_q\cdots i_{p-1})$$

$$=\sum _{i_0\cdots i_{p-1}\in {\Lambda }_p}f(i_0\cdots i_{p-1})\partial g(i_0\cdots i_{p-1})=⟨f,\partial g⟩.$$

$${\mathcal{A}}^p(V):=\{f\in l^2({\Lambda }_p):f\text{ is allowed}\}.$$

$${\Omega }^p(V):=\{f\in {\mathcal{A}}^p:df\in {\mathcal{A}}^{p+1}\text{ and }\partial f\in {\mathcal{A}}^{p-1}\}.$$

$$0\to {\Omega }^0\stackrel{d^0}{\to }\cdots \stackrel{d^{p-2}}{\to }{\Omega }^{p-1}\stackrel{d^{p-1}}{\to }{\Omega }^p\stackrel{d^p}{\to }\cdots$$

$$0\to {\Omega }^0\stackrel{{\partial }_0}{\leftarrow }\cdots \stackrel{{\partial }_{p-2}}{\leftarrow }{\Omega }^{p-1}\stackrel{{\partial }_{p-1}}{\leftarrow }{\Omega }^p\stackrel{{\partial }_p}{\leftarrow }\cdots$$

$$H_n:=\frac{\ker d^n}{\mathrm{Im}{d}^{n-1}}.$$

$$\Delta =d\delta +\delta d;$$

$$\Delta :=d\partial +\partial d.$$

$$\Delta f(x)=\partial df(x)=\sum _{k\sim x}(df(kx)-df(xk))=\sum _{k\sim x}(f(x)-f(k)-f(k)+f(x))=2\sum _{k\sim x}(f(x)-f(k)).$$

$$\Delta f(x)=2(m(x)f(x)-\sum _{y\sim x}f(y)).$$

$$⟨\Delta f,f⟩=⟨(d\partial +\partial d)f,f⟩=⟨df,df⟩+⟨\partial f,\partial f⟩={\Vert df\Vert }^2+{\Vert \partial f\Vert }^2.$$

$$\Delta f=0.$$

$$H^p:=\{f\in {\Omega }^p:\Delta f=0\}$$

$${\Vert \partial f\Vert }^2=⟨\partial f,\partial f⟩=⟨f,d\partial f⟩=0,$$

$$⟨f,dg⟩=⟨\partial f,g⟩=0.$$

$${\Lambda }^p=\Delta ({\Lambda }^p)\oplus H^p=d\partial ({\Lambda }^p)\oplus \partial d({\Lambda }^p)\oplus H^p=d({\Lambda }^p)\oplus \partial ({\Lambda }^p)\oplus H^p.$$

$$f=H(f)+\Delta G(f)=H(f)+d\partial G(f)+\partial dG(f).$$

Full text

1. 1 Path Cohomology of Digraphs
2. 2 The Discrete Laplacian and Harmonic Forms
3. 3 The Hodge Theorem
4. 4 The Heat Equation
5. 5 Elementary Paths Orientation
6. 6 The $p$-lazy Random Walk and Expectation Process

1 Path Cohomology of Digraphs

In this section we present the definition of path cohomology for a locally finite digraph $G=(V,E)$ with coefficients in field $\operatorname{\mathbb{K}}$ as a generalization of [2]; in which the theory was developed to finite digraphs. We point out that the name cohomology is due to the fact that it is dual to the homology path theory, also presented in [2] as well as in [3]. We also emphasize that it is done in analogy to the de Rham cohomology, for differentiable manifolds.

Definition 1.1**.**

A digraphs $G=(V,E)$ is said to be locally finite if, for each vertex $v\in V$, $v$ figures in at most a finite number of edges in $E$.

Every digraph here mentioned is assumed to be locally finite.

Definition 1.2**.**

For each non-negative integer $p$, a $p$-form on the set $V$ of vertices is any application on the sequences $\{i_{k}\}_{k=0}^{p}$ of $p+1$ vertices over the field $\operatorname{\mathbb{K}}$.

We denote the sequences $\{i_{k}\}_{k=0}^{p}$ simply by $i_{0}\cdots i_{p}$, without delimiters between the vertices. They are also called elementary $p$-path over $V$.

Definition 1.3**.**

The space of all formal $\operatorname{\mathbb{K}}$-linear combination of all elementary $p$-path over $V$ is denoted by $\Lambda_{p}=\Lambda_{p}(V)$. Its elements are called $p$-paths on $V$.

It is easy to see that the $\operatorname{\mathbb{K}}$-linear space of all $p$-forms is dual to $\Lambda_{p}$, and it is denoted by $\Lambda^{p}=\Lambda^{p}(V)$.

We will restrict ourselves to the Hilbert space of the square-integrable forms, i.e.,

[TABLE]

with the canonical inner product

[TABLE]

Definition 1.4**.**

Define the linear operators $d:\Lambda^{p-1}\to\Lambda^{p}$, exterior differential, and $\partial:\Lambda^{p}\to\Lambda^{p-1}$, boundary operator, by

[TABLE]

[TABLE]

in which $\widehat{i_{q}}$ means the omission of the index $i_{q}$.

Lemma 1.5**.**

We have that $d^{2}=0$.

Proof.

We have by $p\leq 1$ that

[TABLE]

[TABLE]

As $(-1)^{q}(-1)^{r-1}=(-1)^{q+r-1}=-(-1)^{q+r}$, we have that the RHS of the equality above is equal to

[TABLE]

By switching $q$ and $r$ in the last sum we see that the twos sums cancel out, whence $d^{2}f(i_{0}\cdots i_{p})=0$. ∎

Definition 1.6**.**

The forms in $\ker d$ are said to be closed. And those in $\operatorname{Im}d$ are said to be exact.

Proposition 1.7**.**

If $f\in l^{2}(\Lambda_{p-1})$ then $df\in l^{2}(\Lambda_{p})$.

Proof.

Observe that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Inverting the indices in the latter sum, the last two sums cancel wach other. This way:

[TABLE]

∎

Proposition 1.8**.**

the operators $d$ and $\partial$ are adjoint with respect to the fixed inner product.

Proof.

Given $f$ a $(p-1)$-form and $g$ a $p$-form on $G$, with $p>0$,

[TABLE]

∎

Corollary 1.9**.**

$\partial^{2}\equiv 0$.

We say that an elementary $p$-path $i_{0}\cdots i_{p}$ is allowed on $G$ if $(i_{k},i_{k+1})\in E$ for $k=1,\cdots,{p-1}$. Otherwise it is said to be not allowed.

A $p$-form is said to be allowed if it is null on every $\operatorname{\mathbb{K}}$-linear combination of not allowed elementary $p$-paths.

Definition 1.10**.**

For any non-negative integer $p$, the space of $p$-allowed forms is the subspace of $l^{2}(\Lambda_{p})$,

[TABLE]

It is clear that f being allowed does not mean that $df$ or $\partial f$ is also allowed.

We will then restrict ourselves to the following space

[TABLE]

This way we have the chain complexes

[TABLE]

and

[TABLE]

Definition 1.11**.**

We define over the complex 5 the path cohomologies by

[TABLE]

It is well defined by Lemma 1.5.

2 The Discrete Laplacian and Harmonic Forms

In the de Rham Theory one can define, over the spaces of forms, the Hodge Laplacian by

[TABLE]

in which $d$ is the exterior differential and $\delta$ is the codifferential. This Laplacian is globally defined and it is strongly related to the manifold topology. Our goal in this section is to define a Laplacian over digraphs, which is a generalization of the classic discrete Laplacian and present some basic results.

Definition 2.1**.**

The Laplacian (or Laplace operator) is the linear operator over $\Omega^{p}$ given by

[TABLE]

Proposition 2.2**.**

The Laplacian is self-adjoint.

Proof.

It follows directly from the fact that $d$ and $\partial$ are adjoint. ∎

Example 2.3**.**

Let $G=(V,E)$ be a locally finite digraph and $f\in\Omega^{0}$, that is, its domain is a subset of the vertices set $V$. Then,

[TABLE]

Therefore,

[TABLE]

This way, the Laplacian here defined is a multiple of the classical discrete Laplacian (see [7]).

Proposition 1**.**

Let $f$ be a $p$-form. Then, $\Delta f=0$ iff $df=0$ and $\partial f=0$.

Proof 2.4**.**

Clearly $df=0$ and $\partial f=0$ imply $\Delta\varphi=0$. Now,

[TABLE]

This way $\Delta f=0$ means that $df=0$ and $\partial f=0$.

Definition 2.5**.**

A $p$-form $f$ is said to harmonic if

[TABLE]

And we set the space of harmonic $p$-forms as

[TABLE]

Lemma 2.6**.**

$f\in H^{p}$* iff $f\in(\operatorname{Im}d)^{\perp}\cap\ker d.$*

Proof 2.7**.**

Suppose That $f\in(\operatorname{Im}d)^{\perp}\cap\ker d$. Then $df=0$ and

[TABLE]

because $f\in(\operatorname{Im}d)^{\perp}$. Then, by Proposition 1, $\Delta f=0$.

Now suppose that $\Delta f=0$. Then, again, by Proposition 1, $f\in\ker d\cap\ker\partial$. This way

[TABLE]

So $f\in(\operatorname{Im}d)^{\perp}\cap\ker d.$

3 The Hodge Theorem

Our goal is to prove the following theorem to locally finite digraphs, under certain hypotheses; which will be dealt later.

Theorem 3.1** (Hodge Theorem).**

Let $G$ be a locally finite and connected digraph such that there are $x_{0}\in V$ and $C\geq 0$ with $\Delta d(\cdot,x_{0})\geq-C$. Then every path cohomology class has a unique representative that minimizes the norm. This is called the harmonic representative.

We will deal with this theorem in analogy to what is done in [1] (chapter 8) to the Hodge Theorem on manifolds. It follows – as will be shown – from the following theorem; which will be proved in the next section.

Theorem 3.2**.**

There are linear operators $H$ (harmonic projection) and $G$ (Green’s operator) on $\Lambda^{p}$, for all $p$, that are characterized by the following properties:

$H(\alpha)$* is harmonic;* 2. 2.

$G(\alpha)$* is orthogonal to the space of harmonic forms;* 3. 3.

$\alpha=H(\alpha)+\Delta G(\alpha)$.

Corollary 3.3**.**

There is an orthogonal direct sum

[TABLE]

We now show prove the Theorem 3.1 as a consequence of Theorem 3.2.

Proof 3.4**.**

Let $f$ be an exact form. By Theorem 3.2 it can be written

[TABLE]

Since $H(f)$ is harmonic,

[TABLE]

Furthermore,

[TABLE]

Then

[TABLE]

Thus $f=H(f)+d\partial G(f)$ is cohomologous to the harmonic form $H(f)$.

For uniqueness, let $f_{1}$ and $f_{2}$ be two harmonic forms in the same cohomology class that differ by the exact form $dg$, i.e., $f_{1}-f_{2}=dg$. Then

[TABLE]

given the fact that harmonic forms are in $\ker\partial$. Therefore $dg=0$ and $f_{1}=f_{2}$.

4 The Heat Equation

In this section we will prove the Theorem 3.2. The pictorial idea behind this proof is that a given initial temperature, given by a form $u_{0}$, should spread through the digraph “uniformly”; i.e., to $u_{0}$ should converge to a harmonic form. To do so we shall solve the Cauchy Problem (CP):

[TABLE]

on $[0,T[\times V$ with initial condition $u_{0}\in\Omega^{p}$.

A map $p:]0,\infty[\times V^{p+1}\times V^{p+1}\to\mathbb{R}$ is said to be fundamental solution to the heat equation,

[TABLE]

if for any bounded initial condition $u_{0}\in\Omega^{p}$, the function

[TABLE]

is differentiable in $t$, satisfies the heat equation, and if for any allowed path $x$,

[TABLE]

We shall, for now on, consider the usual metric on connected digraphs. That is, $d(x,x)=0$ and, if $x\neq y$, there is a finite number of vertices $x=x_{0}\sim x_{1}\sim\cdots\sim x_{k}=y\in V$ which connect $x$ and $y$. Then, $d(x,y)$ is the smallest number $k$ of such vertices.

Finally, the following theorem takes place. It is stated on [7] to [math]-forms, only. But, with minor modifications, it its proof holds to $p$-forms for any non-negative integer $p$.

Theorem 4.1**.**

Let $G=(V,E)$ be a locally finite and connected digraph such that there are $x_{0}\in V$ and $C\geq 0$ with $\Delta d(\cdot,x_{0})\geq-C$. Then the following holds true:

• •

There is a unique fundamental solution $p:]0,\infty[\times V^{p+1}\times V^{p+1}\to\mathbb{R}$ of the heat equation.

• •

$G$* is stochastically complete, i.e.*

[TABLE]

for any $t>0$ and any allowed path $x$.

• •

For every $u_{0}\in l^{1}$ and the corresponding bounded solution $u$ of (CP) we have

[TABLE]

for any $t>0$.

We will then divide the proof of 3.2 in the following propositions.

Proposition 2**.**

If $u(t,x)$ is a general solution of the heat equation, then $\left\lVert u(t,x)\right\rVert^{2}$ is (nonstrictly) decreasing.

Proof 4.2**.**

[TABLE]

Let

[TABLE]

with $p$ as in Theorem 4.1. This is, as we have seen, the only solution to (CP).

Proposition 3**.**

The semigroup property $T_{t_{1}+t_{2}}=T_{t_{1}}T_{t_{2}}$ holds.

Proof 4.3**.**

It holds because $u(t_{1}+t_{2},x)$ can be obtained by solving the heating equation with intial condition $u(t_{2},x)$ and then evaluating it at $t_{1}$.

Proposition 4**.**

$T_{t}$* is formally self-adjoint.*

Proof 4.4**.**

As

[TABLE]

[TABLE]

we can write $\langle T_{t}\alpha,T_{\tau}\beta\rangle=g(t+\tau)$. Thus,

[TABLE]

Proposition 5**.**

$T_{t}\alpha$* converges to a harmonic form $H(\alpha)$.*

Proof 4.5**.**

We have

[TABLE]

[TABLE]

Applying 3 and then 4, we have

[TABLE]

Therefore,

[TABLE]

By (2), $\left\lVert T_{t}\alpha\right\rVert^{2}$ converges. Thus, $\left\lVert T_{t+2h}\alpha-T_{t}\alpha\right\rVert^{2}$ can be taken arbitrarily small. As $l^{2}$ is complete, $T_{t}\alpha$ converges, in $l^{2}$, to a form $H(\alpha)\in l^{2}$.

To prove that $H(\alpha)$ is indeed harmonic, one just need to take $\tau>0$ and note that the relation $T_{t}\alpha=T_{\tau}T_{t-\tau}\alpha$ implies, taking the limit $t\to\infty$, that $T_{\tau}H(\alpha)=H(\alpha)$. Then $T_{t}H(\alpha)$ is constant on $t$. Therefore,

[TABLE]

Thus, $H(\alpha)$ is indeed harmonic.

Furthermore,

[TABLE]

Then, $H$ is self-adjoint.

Proposition 6**.**

$T_{t}$* is compact for any $t$.*

Proof 4.6**.**

Let $\{\phi_{i}\}_{i}$ be an enumerable orthonormal basis of $l^{2}(\Lambda^{p})$. Then $\phi_{i}(x)\phi_{j}(y)$ is also an enumerable orthonormal basis of $l^{2}(\Lambda^{p}\times\Lambda^{p})$. Define then

[TABLE]

so that

[TABLE]

Now, we define

[TABLE]

and

[TABLE]

This way, the image of $T_{n}$ is a finite dimensional subspace of $l^{2}(\Lambda^{p})$.

As every operator whose image is finite dimensional is compact, $T_{n}$ is compact.

Also, we have

[TABLE]

[TABLE]

[TABLE]

As $\left\lVert p\right\rVert^{2}_{l^{2}}=\sum_{i,j=1}^{\infty}|p_{i,j}|^{2}<\infty$, the sum above tends to zero as $n\to\infty$. Thus, $T_{n}\to T_{t}$ with respect to operator norm. Therefore we can evoke the theorem below.

Theorem 4.7**.**

Let $A:X\to Y$ be a linear map and $A_{n}\in\mathcal{L}(X,Y)$ be a sequence of compact linear maps such that $A_{n}\xrightarrow{n\to\infty}A$ with respect to operator norm. Then $A$ is compact.

Its proof can be found in [4], page 408.

Proposition 7**.**

The Green’s operator, given by the integral,

[TABLE]

is well defined. Moreover, the following equality takes place

[TABLE]

Further, $G(f)$ is orthogonal to the space of harmonic forms for any $f$.

Proof 4.8**.**

To show that the Green’s operator is well define we must show that $\left\lVert T_{t}f(x)-Hf(x)\right\rVert$ decay rapidly enough.

It is a well known fact that if $T$ is a compact self adjoint operator on a Hilbert space, $V$, and

[TABLE]

then either $m(T)$ or $-m(T)$ is the greatest eigenvalue of $T$ and

[TABLE]

holds true.

This way, the maximum of the variational problem:

[TABLE]

has solution. We call this maximum by $\beta(t)$ and the correspondent maximum value by $\lambda(t)$.

By semigroup property, $T_{2t}\beta=\lambda^{2}(t)\beta$. On the other hand, $T_{2t}\beta=\lambda(2t)\beta$. Thus, $\lambda(t+t)=\lambda(t)\cdot\lambda(t)$. Therefore the function $\lambda(t)$ is multiplicative and then $\lambda(t)=e^{-\lambda_{1}t}$, with $\lambda_{1}>0$. Moreover, the minus sign is given by the fact that $T_{t}\beta$ converges to $H\beta=0$ as $t\to\infty$.

As $\left\lVert T_{t}\alpha-H\alpha\right\rVert=\left\lVert T_{t}(\alpha-H\alpha)\right\rVert\leq\left\lVert T_{t}\beta\right\rVert$, we have $||T_{t}\alpha-H\alpha||\leq e^{-\lambda_{1}t}\left\lVert\beta\right\rVert\leq e^{-\lambda_{1}t}$. Therefore it decays rapidly enough.

Now,

[TABLE]

At last, let $g$ be a harmonic form. Then

[TABLE]

5 Elementary Paths Orientation

Definition 5.1**.**

Let $G=(V,E)$ be a digraph. The $d$-paths of the form $\pm e_{i_{0}\cdots i_{d}}$ are called oriented elementary $d$-path of $G$ whenever $e_{i_{0}\cdots i_{d}}$ is an elementary $d$-path of $G$.

Furthermore we say that $e_{i_{0}\cdots i_{d}}$ and $-e_{i_{0}\cdots i_{d}}$ have opposite orientations. And we denote the set of all oriented elementary $d$-paths on $G$ by $G_{d}^{\pm}$. The set of elementary $d$-paths will be denoted by $G_{d}^{+}$ and the one of those with reverted orientation by $G_{d}^{-}$.

Definition 5.2**.**

Given a elementary $d$-path $v=e_{i_{0}\cdots i_{d}}$ on the digraph $G=(V,E)$, we say that an elementary $d$-path $w$ is a neighbour of $v$ if for some $k\in V$ and for some $r,q\in\{0,1,\cdots,p\}$

[TABLE]

Furthermore, we say that a sub-digraph $G^{\prime}$ of $G$ is a $d$-component of $G$ if for any two elementary $d$-paths $v,v^{\prime}$ there exists a chain of $d$-paths $v=v_{0}\sim v_{1}\sim\cdots\sim v_{n}=\pm v^{\prime}$.

If $G$ is a $d$-component itself we say that $G$ is $d$-connected.

Definition 5.3**.**

Given an elementary allowed and oriented elementary $d$-path $v$ we define as its valence the $d$-form

[TABLE]

We will, for now on, restrict ourselves to a fixed $d$-connected and $d$-regular digraph $G$ of limited valence; i.e., there is a integer $N>0$ such that $m_{0}(v)\leq N$ for any $v\in V$. That implies that for any $d>0$ there is a $M>0$ such that $m(v)\leq M$ for any $v\in G_{d}^{\pm}$.

We will then fix the following inner product over $\Omega^{k}$:

[TABLE]

where $\omega:G_{\bullet}^{\pm}\to]0,\infty[$ is the weight function given by

[TABLE]

With respect to this inner product explicit computations show that the adjoint of $\partial$ is the $d$ operator given by

[TABLE]

We highlight that the valence $m$ is bounded – so it is well defined.

Thus, explicitly,

[TABLE]

and

[TABLE]

We define then the upper and the lower Laplacian respectively by:

[TABLE]

[TABLE]

It is clear that all facts proven to the Laplacian trough this paper holds for the normalized Laplacian $\Delta:=\Delta^{+}+\Delta^{-}.$

Proposition 8**.**

For any elementary $d$-path $i_{0}\cdots i_{d}$ in $G$,

[TABLE]

and

[TABLE]

Proof 5.4**.**

Explicit calculations give:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

Proposition 9**.**

The spectrum of $\Delta^{+}$ is contained in $[0,M+1]$.

Proof 5.5**.**

Assume that $\lambda f(v)=(\Delta^{+}f)(v)$. As $f\in l^{2}$, $|f(w)|$ is limited and, therefore, has a maximum point $u\in G_{d}^{\pm}$ (say).

By Proposition 8,

[TABLE]

Thus,

[TABLE]

(since $m(u)\leq M$).

6 The $p$-lazy Random Walk and Expectation Process

Definition 6.1**.**

The $p$-lazy random walk on $G$ starting on the $d$-elementary path $v^{*}$ is the Markov chain on $\Omega_{d}(G)$ with transition probabilities

[TABLE]

We denote the probability that the random walk which starts at $v$ reaches $w$ at time $n$ by $\textbf{p}_{n}^{v}(w)$.

Definition 6.2**.**

For $d\geq 2$, the expectation process on $G$ starting at $v$ is the sequence of $d$-forms $\{\mathcal{E}_{n}^{v}\}_{n=0}^{\infty}$ defined by

[TABLE]

The name “expectation” is due to the fact that for any $d$-form $f$,

[TABLE]

Moreover, we have

[TABLE]

Then we have the following definition.

Definition 6.3**.**

We define the transition operator by

[TABLE]

Proposition 10**.**

The transition operator is given by

[TABLE]

Where $I$ denotes the identity.

So that

[TABLE]

Proof 6.4**.**

It follows trivially from Proposition 8,

Proposition 11**.**

The spectrum of $A$ is contained in $[p-M(1-p),1]$.

Proof 6.5**.**

By Proposition 10 we have that

[TABLE]

But the last equality is equivalent to

[TABLE]

By Proposition 9 we have that

[TABLE]

Thus $\lambda\in[p-M(1-p),1]$.

Proposition 12**.**

There is a positive integer $K$ s.t. the expectation process satisfies

[TABLE]

Proof 6.6**.**

We have that

[TABLE]

And $\left\lVert\mathcal{E}_{0}^{v}\right\rVert=\left\lVert\mathbbm{1}_{v}\right\rVert=\frac{1}{\sqrt{m(v)}}\leq 1$; where

[TABLE]

Then, by Proposition 11,

[TABLE]

For the lower bound, let $v=i_{0}\cdots i_{d}$, fix $0\leq q\leq d$ and define

[TABLE]

So $f\in\ker\Delta^{+}$, since $f\in\operatorname{Im}d$, and $\left\lVert f\right\rVert^{2}=\sum_{w\sim v}m(w)$. Note that, as $G$ has bounded valence, there is a positive integer $K$ s.t. $\left\lVert f\right\rVert^{2}\leq K$. Since $\Delta^{+}$ decomposes w.r.t the orthogonal sum $\Omega^{d}=\ker\Delta^{+}\oplus\operatorname{Im}\Delta^{+}$ so does $A$. Thus,

[TABLE]

Proposition 13**.**

The expectation process $\{\mathcal{E}^{v}_{n}\}_{n}$ converges to $\mathcal{E}_{\infty}^{v}:=\operatorname{proj}_{\ker\Delta^{+}}(\mathbbm{1}_{v})$. So if the homology of $G$ is trivial, $\mathcal{E}_{\infty}^{v}$ is exact.

Proof 6.7**.**

Since $A$ decomposes w.r.t. the orthogonal sum $\Omega^{d}=\ker\Delta^{+}\oplus\operatorname{Im}\Delta^{+}$ and $A|_{\ker\Delta^{+}}=I|_{\ker\Delta^{+}}$, this means that $A^{n}$ converges to the orthogonal projection $\operatorname{proj}_{\ker\Delta^{+}}$. Thus

[TABLE]

So $\mathcal{E}_{\infty}^{v}$ is closed. Therefore, if the homology of $G$ is trivial then it is exact.