The Resistance Distance Is a Diffusion Distance on a Graph

Abstract

The resistance distance is a squared Euclidean metric on the vertices of a graph derived from the consideration of a graph as an electrical circuit. Its connection with the commute time of a random walker on the graph has made it particularly appealing for the analysis of networks. Here, we prove that the resistance distance is given by a difference of “mass concentrations” obtained at the vertices of a graph by a diffusive process. The nature of this diffusive process is characterized here by means of an operator corresponding to the matrix logarithm of a Perron-like matrix based on the pseudoinverse of the graph Laplacian. We prove also that this operator is indeed the Laplacian matrix of a signed version of the original graph, in which nonnearest neighbors’ “interactions” are also considered. In this way, the resistance distance is part of a family of squared Euclidean distances emerging from diffusive dynamics on graphs.

1. Introduction

The connections between graphs and electrical circuits can be traced back to the works of Gustav Kirchhoff in the XIX century [1,2]. One of the most enduring mathematical concepts introduced by Kirchhoff is the Laplacian matrix of a graph [3,4]. This structural representation of a graph is based on a diagonal matrix of vertex degrees—the number of incident edges to a vertex—and the pairwise adjacency relation between vertices in the graph. It has been proved to be relevant in algebraic graph theory [5,6,7] as well as in its applications to dynamical systems on graphs [8,9].

A concept strengthening the links between electrical circuits and graph theory is the effective resistance between a pair of points in the circuits [10]. The seminal paper by Klein and Randić [11] proving that the effective resistance is a distance between the corresponding pairs of vertices in a graph (also proved in the less-known paper [12]) triggered an explosion of research in this area [13,14,15,16,17]. A particularly important discovery was the fact that the resistance distance is equivalent to the commute time of a random walker on the graph [10,18,19,20,21]. This is possibly the reason for more recent uses of the resistance distance in the analysis of real-world networks [22,23,24,25,26,27]. However, as a navigational tool for large graphs, resistance distance is useless. It was proved mathematically by von Luxburg et al. [28] that “the commute distance converges to an expression that does not take into account the structure of the graph at all and that is completely meaningless as a distance function on the graph”.

Two other distances have found applications in the study of large graphs as well as in data sciences. The so-called “diffusion distance” proposed by Coifman [29,30,31,32] has found many applications in data analysis, while the “communicability distance” proposed by Estrada [33,34,35,36,37] has found applications in the study of complex networks. These distances, like the resistance distance, are squared Euclidean distances (SEDs), i.e., their square roots are Euclidean distances between pairs of vertices in the graph. When SEDs are represented in the form of a matrix, they give rise to the so-called Euclidean distance matrices (EDMs), which are widely studied in the intersection between linear algebra and geometry [38,39,40,41]. Recently, it has been shown that the diffusion and communicability distances correspond to metrics quantifying differences of mass concentrations at the vertices of a graph for conservative and nonconservative diffusive processes, respectively [42]. Due to the fact that diffusion is a ubiquitous dynamic in nature and man-made systems [43], it is intriguing to know whether the “resistance distance” is somehow related to such a family of processes, which would then belong to the general class of “diffusion distances”, such as the two ones previously mentioned.

Here, we prove that the resistance distance is a diffusion distance on a graph, meaning that it is a metric derived from an operator controlling a diffusive dynamics on the graph. Such an operator corresponds to the matrix logarithm of a Perron-like matrix based on the Moore–Penrose pseudoinverse of the graph Laplacian. We prove that such an operator is also a graph Laplacian representing a transformation of the graph into a weighted, signed complete graph. Therefore, our main finding here involves proving that the resistance distance is a member of the family of diffusive distances, such as Coifman “diffusion distance” and the “communicability distance” on graphs. Other properties of the operators defined here are also studied in this work, as well as those of the new diffusive distances.

2. Preliminaries

Here, we consider simple, undirected, connected graphs $G=\left(V,E\right)$ with $\#V=n$ vertices and $\#E=m$ edges. The degree of a vertex is the number of vertices adjacent to it. Let A be the adjacency matrix of G and let K be the diagonal matrix of vertex degrees. The standard graph Laplacian is defined as $\mathcal{L}:=K-A$. Let ${\mu }_j$ be an eigenvalue of $\mathcal{L}$. Then, $0={\mu }_1<{\mu }_2\le \dots \le {\mu }_n$ and we will designate by ${\psi }_{jv}$ the vth entry of the eigenvector corresponding to the jth eigenvalue of $\mathcal{L}$. The standard diffusion on a graph is expressed by the following equation:

$${\displaystyle \frac{dx\left(t\right)}{dt}}=-\gamma \mathcal{L}x\left(t\right),x\left(0\right)=x^0,$$

(1)

where $x\left(t\right)$ is a column vector of the mass concentrations at every vertex of the graph at time t, $\gamma$ is a parameter indicating the diffusivity of the system (hereafter referred to as $\gamma =1$), and $x\left(0\right)=x^0$ is the initial condition. The solution to the abstract Cauchy problem (ACP) is given by:

$$\begin{array}{cc}\hfill x\left(t\right)& =e^{-t\mathcal{L}}{x}^0\hfill \end{array},$$

(2)

where $e^M={\sum }_{k=0}^{\infty }{\displaystyle \frac{M^k}{k!}}$ is the matrix exponential of the generic matrix M. Using this process as a basis, Coifman [29,30,31,32] has defined a metric known as the diffusion distance, which is given by:

$$D_{vw}\left(t\right):={\left(e^{-t\mathcal{L}}\right)}_{vv}+{\left(e^{-t\mathcal{L}}\right)}_{ww}-2{\left(e^{-t\mathcal{L}}\right)}_{vw},t>0.$$

(3)

It is easy to prove that $D_{vw}\left(t>0\right)$ is a square Euclidean distance between the vertices of the graph. Let $\mathcal{L}=V\Lambda V^T$ where $\Lambda$ is a diagonal matrix of eigenvalues of $\mathcal{L}$ and V is an orthogonal matrix of eigenvectors. Then, if ${\phi }_i$ is the ith column of $V^T$, we have

$$\begin{array}{cc}\hfill D_{vw}& ={\left({\phi }_v-{\phi }_w\right)}^T{e}^{-t\Lambda }\left({\phi }_v-{\phi }_w\right)\hfill \\ \hfill & ={\left(e^{-\left(t\Lambda \right)/2}{\phi }_v-e^{-\left(t\Lambda \right)/2}{\phi }_w\right)}^T\left(e^{-\left(t\Lambda \right)/2}{\phi }_v-e^{-\left(t\Lambda \right)/2}{\phi }_w\right)\hfill \\ \hfill & ={\left(y_v-y_w\right)}^T\left(y_v-y_w\right)\hfill \\ \hfill & ={∥y_v-y_w∥}^2.\hfill \end{array}$$

(4)

Another type of graph Laplacian was defined by Lerman and Ghosh [44] as ${\mathcal{L}}_{\chi }:=\chi I-A$, where $\chi \ge 0$ is a parameter. Using it, we can define another diffusive process on the graph, which can be written as:

$${\displaystyle \frac{dx\left(t\right)}{dt}}=-{\mathcal{L}}_{\chi }x\left(t\right),x\left(0\right)=x^0,$$

(5)

with the solution given by $x\left(t\right)=e^{-t{\mathcal{L}}_{\chi }}{x}^0$. The diffusive process is not conservative in the sense that [42]:

$$\underset{t\to \infty }{lim}x\left(t\right)=\left\{\begin{array}{c}\left({\varsigma }_1^T{x}^0\right){\varsigma }_1{e}^{t\left({\lambda }_1-\chi \right)}=\infty \mathrm{for} \chi <{\lambda }_1\\ \left(\sum _j{x}_j^0{\varsigma }_1\left(j\right)\right){\varsigma }_1 \mathrm{for} \chi ={\lambda }_1\\ \left({\varsigma }_1^T{x}^0\right){\varsigma }_1{e}^{-t\left(\chi -{\lambda }_1\right)}=0 \mathrm{for} \chi >{\lambda }_1,\end{array}\right.$$

(6)

where ${\lambda }_j,{\varsigma }_j$ are the eigenpair of A. Then, it is clear that ${\sum }_{j=1}^n{x}_j\left(t>0\right)\ne {\sum }_{j=1}^n{x}_j^0$, while for the standard diffusion it is always true that ${\sum }_{j=1}^n{x}_j\left(t>0\right)={\sum }_{j=1}^n{x}_j^0$, such that the mass is conserved on the vertices of the graph.

When $\chi =0$, the SED

$${\xi }_{vw}\left(t\right):={\left(e^{tA}\right)}_{vv}+{\left(e^{tA}\right)}_{ww}-2{\left(e^{tA}\right)}_{vw},t>0,$$

(7)

is known as the squared communicability distance between the corresponding pair of vertices in the graph [35,36,42].

In a different context–the context of electrical circuits represented by graphs–another SED has been studied. Let us start by recalling that the Moore–Penrose pseudoinverse of a matrix M is the matrix $M^{+}$ which satisfies the following four conditions (see [45] for a review of this topic): (i) $M{M}^{+}M=M$; (ii) $M^{+}M{M}^{+}=M^{+}$; (iii) ${\left(M{M}^{+}\right)}^T=M{M}^{+}$; and (iv) ${\left(M^{+}M\right)}^T=M^{+}M$. Hereafter, we designate by ${\mathcal{L}}^{+}$ the Moore–Penrose pseudoinverse of the graph Laplacian. It is known that ([46,47]; see also Chapter 10 in [40]):

$${\mathcal{L}}^{+}={=(+{\displaystyle \frac{1}{n}}J)}^{-1}-{\displaystyle \frac{1}{n}}J,$$

(8)

where J is the all-ones $n\times n$ matrix.

The resistance distance ${\Omega }_{vw}$ between two vertices v and w of the graph G is then defined on the basis of the Moore–Penrose pseudoinverse of the Laplacian as follows:

$${\Omega }_{vw}:={\left({\mathcal{L}}^{+}\right)}_{vv}+{\left({\mathcal{L}}^{+}\right)}_{ww}-2{\left({\mathcal{L}}^{+}\right)}_{vw},$$

(9)

which corresponds to the effective resistance between the two points in an electrical circuit represented by G (see [10,17] for connections with electrical networks and [11] for a different proof that ${\Omega }_{vw}$ is a distance). Using a similar approach as for the standard diffusion distance, we can prove that ${\Omega }_{vw}$ is a SED.

Saxena et al. [48] have recently proposed a diffusion-like model based on the pseudoinverse of the graph Laplacian:

$${\displaystyle \frac{dx\left(t\right)}{dt}}=-{\mathcal{L}}^{+}x\left(t\right),x\left(0\right)=x^0,$$

(10)

where the solution to the ACP is given by:

$$\begin{array}{cc}\hfill x\left(t\right)& =e^{-t{\mathcal{L}}^{+}}{x}^0\hfill \end{array}.$$

(11)

The model is based on considering that the dynamics for the i-th vertex of the graph, which has degree equal to $k_i,$ evolves according to

$${\dot{x}}_i\left(t\right)=-{\displaystyle \frac{x_i\left(t\right)}{k_i}}+{\dot{x}}_j\left(t\right).$$

(12)

By considering all n vertices of the graph $G=\left(V,E\right)$ we get

$$\sum _{\left(i,j\right)\in E}\left[{\dot{x}}_i\left(t\right)-{\dot{x}}_j\left(t\right)\right]=-x_i\left(t\right),$$

(13)

which in matrix-vector form is written as

$$\mathcal{L}\dot{x}\left(t\right)=-x\left(t\right).$$

(14)

The previously mentioned model emerges by applying the Moore–Penrose pseudoinverse of $\mathcal{L}$ to both sides of the previous equation, where $\overline{\dot{x}}\left(t\right)={\displaystyle \frac{1}{n}}{J}_n\dot{x}\left(t\right)=\left({\displaystyle \frac{1}{n}}\sum {\dot{x}}_i\left(t\right)\right)\mathbf{1}=\mathbf{0}$ with $J_n$ being the all-ones matrix.

The question about whether ${\mathcal{L}}^{+}$ is a graph Laplacian or not have been debated. First, van Mieghem [49] has used a restrictive definition of a graph Laplacian as an $n\times n$ matrix which is (see for instance [50]): (i) positive semidefinite, (ii) of rank $n-1$, (iii) with zero row sum, and (iv) without positive non-diagonal elements (and all diagonal ones being positive). He concludes that ${\mathcal{L}}^{+}$ “is not always a Laplacian” because some of its off-diagonal elements can be positive (see pp. 133–134 in [49]).

A relaxation of this definition by Balaji & Bapat [39] extends the concept of graph Laplacian to those matrices which are positive semidefinite $n\times n$ of rank $n-1$ and with zero row sum. Therefore, many of the graph Laplacians in use, which may have non-positive out-diagonal entries, are well-defined according to this definition.

More recently, Estrada [51] has defined a graph Laplacian on the basis of its physical roots. That is, as the divergence $\left(\nabla ·\right)$ of the gradient $\nabla f$, where f is twice-differentiable, $\nabla \left({\nabla }^{T}f\right)$, so that for the graph we have:

Definition 1.

Let ${\nabla }^G$ be a matrix representing an incidence relation between the vertices and a fixed orientation of some other element of the graph, i.e., edges, other subsets of vertices, etc. Then, the matrix:

$$L_G={\nabla }^{G}W{\left({\nabla }^G\right)}^T,$$

(15)

is a graph Laplacian, where W is a diagonal matrix of weights of the elements of the graph which are incident to the vertices. $L_G$ can then be written as

$$\left(L_{G}f\right)\left(v\right):=\sum _{w\in V}\gamma \left(v,w\right)\left(f\left(v\right)-f\left(w\right)\right),$$

(16)

where $f\in {\mathbb{C}}^V$ and where the edge weights $\gamma \left(v,w\right)\in \mathbb{C}$.

Recall that the standard incidence matrix of a connected graph is the $n\times m$ matrix with elements given by:

$${\nabla }_{ij}=\left\{\begin{array}{cc}+1& \mathrm{if} v_i \mathrm{is} \mathrm{the} \mathrm{head} \mathrm{of} e_j;\\ -1& \mathrm{if} v_i \mathrm{is} \mathrm{the} \mathrm{tail} \mathrm{of} e_j;\\ 0& \mathrm{otherwise},\end{array}\right.$$

(17)

where we have made an arbitrary orientation to the edges of G, such that $L\left(G\right)=\nabla {\nabla }^T.$

Accordingly, the Moore–Penrose pseudoinverse of the standard graph Laplacian is a Laplacian matrix of the graph, as has been proved in the following result [51].

Proposition 1.

Let ${\mathcal{L}}^{+}\in {\mathbb{R}}^{n\times n}$ be the Moore-Penrose pseudoinverse of the graph Laplacian $\mathcal{L}$. Then, ${\mathcal{L}}^{+}$ is a graph Laplacian, which can be defined as $\tilde{\nabla }{\tilde{\nabla }}^T=:{\mathcal{L}}^{+}$, where

$$\tilde{\nabla }=U{\left({\sum }^{+}\right)}^T{V}^T,$$

(18)

such that $\nabla =U\sum V^T$ is the singular value decomposition of the standard graph incidence matrix and ${\sum }^{+}$ is the Moore–Penrose pseudoinverse of ∑.

We also recall the definition of matrix logarithm [52,53].

Definition 2.

Let $M\in {\mathbb{C}}^{n\times n}$ with no eigenvalues in ${\mathbb{R}}^{-}$. Then,

$$logM:=-\sum _{k=1}^{\infty }{\displaystyle \frac{{\left(I-M\right)}^k}{k}}$$

(19)

converges and $exp\left(log\left(M\right)\right)=M$. We call $log\left(M\right)$ the matrix logarithm of M.

3. Results

Let us then start by introducing the following operator: $P^{+}:=I-\epsilon {\mathcal{L}}^{+}$, where $0<\epsilon <{\mu }_2$. Notice that although we will consider a different approach here, $P^{+}$ can be thought of as the discrete-time version of (10) as it is defined for the standard Laplacian, i.e., $P:=I-\epsilon \mathcal{L},$ and which is known as the Perron operator [8]. Let us proceed by proving the following result.

Lemma 1.

Let ${\sigma }_j$ be an eigenvalue of $P^{+}$. Then,

$$1={\sigma }_1\ge \dots \ge {\sigma }_n>0.$$

(20)

Proof. 

First, let us write ${\mathcal{L}}^{+}=V{\Lambda }^{+}{V}^T$, where ${\Lambda }^{+}$ is the diagonal matrix of eigenvalues and V the matrix whose columns are orthonormalized eigenvectors of ${\mathcal{L}}^{+}$. Then, it can be easily shown that

$$\begin{array}{cc}\hfill P^{+}& =I-\epsilon \left(V{\Lambda }^{+}{V}^T\right)\hfill \\ \hfill & =V\left(I-\epsilon {\Lambda }^{+}\right)V^T,\hfill \end{array}$$

(21)

which implies that ${\sigma }_1=1$ and ${\sigma }_{j\ge 2}=1-{\displaystyle \frac{\epsilon }{{\mu }_{n-j+2}}}$. Therefore, because $0<\epsilon <{\mu }_2$, all the eigenvalues of $P^{+}$ are positive and bounded between zero and one. □

Proposition 2.

The matrix $Z:=-log{P}^{+}$ is a Laplacian matrix of the graph G.

Proof. 

Because $P^{+}$ has no eigenvalues in ${\mathbb{R}}^{-}$, then $Z:=-log{P}^{+}$ exists and it is unique. Let us write

$$\begin{array}{cc}\hfill Z& =-log\left(V\left(I-\epsilon {\Lambda }^{+}\right)V^T\right)\hfill \\ \hfill & =-Vlog\left(I-\epsilon {\Lambda }^{+}\right)V^T,\hfill \end{array}$$

(22)

which indicates that the eigenvalues ${\omega }_j$ of Z are ${\omega }_1=log{\sigma }_1=0$ and ${\omega }_{j\ge 2}=log{\sigma }_{j\ge 2}=-log\left(1-{\displaystyle \frac{\epsilon }{{\mu }_{n-j+2}}}\right)$ and because $0<\epsilon <{\mu }_2$, we have ${\omega }_{j\ge 2}=-log{\sigma }_{j\ge 2}>0$. Therefore, $Z:=-log{P}^{+}$ is a positive semidefined $n\times n$ matrix of rank $n-1$. We now prove that it has zero row-sum. For that, let us use the power-series representation:

$$-log\left(I-\epsilon {\mathcal{L}}^{+}\right):=\sum _{k=1}^{\infty }{\displaystyle \frac{{\epsilon }^k}{k}}{\left({\mathcal{L}}^{+}\right)}^k,$$

(23)

and write

$$\begin{array}{cc}\hfill \left(-log\left(I-\epsilon {\mathcal{L}}^{+}\right)\right)\mathbf{1}& =\sum _{k=1}^{\infty }{\displaystyle \frac{{\epsilon }^k}{k}}{\left({\mathcal{L}}^{+}\right)}^k\mathbf{1}\hfill \\ \hfill & =\sum _{k=1}^{\infty }{\displaystyle \frac{{\epsilon }^k}{k}}{\left({\mathcal{L}}^{+}\right)}^{k-1}\left({\mathcal{L}}^{+}\mathbf{1}\right).\hfill \end{array}$$

(24)

Because we have previously proved that ${\mathcal{L}}^{+}\mathbf{1}=\mathbf{0}$, then $\left(-log\left(I-\epsilon {\mathcal{L}}^{+}\right)\right)\mathbf{1}=\mathbf{0}$, and then Z is a Laplacian matrix of G. □

Remark 1.

The entries of Z are given by $Z_{vw}=-{\sum }_{j=2}^n{\psi }_{jv}{\psi }_{jw}log\left(1-{\displaystyle \frac{\epsilon }{{\mu }_j}}\right)$. Then, although $Z_{vv}=-{\sum }_{j=2}^n{\psi }_{jv}^{2}log\left(1-{\displaystyle \frac{\epsilon }{{\mu }_j}}\right)>0$ for every $v\in V,$ the nondiagonal entries are not necessarily negative for every pair of vertices $v,w$. That is, the largest contribution to $Z_{vw}$ comes from the smallest nontrivial eigenvalue of $\mathcal{L}$, i.e., the algebraic connectivity ${\mu }_2$ [54,55]. It is well known that the eigenvector associated with ${\mu }_2$, also known as the Fiedler vector, splits vertices of the graph into two subsets $V=V_1\cup V_2,$ such that $sign\left({\mu }_{2v}\right)\ne sign\left({\mu }_{2w}\right)$ if $v\in V_1$ and $w\in V_2$ (see [56]). Each of these two clusters are formed by vertices which are more tightly connected among them than with vertices in the other cluster. Therefore, let us consider two vertices v and w belonging to the same cluster, such that $sign\left({\mu }_{2v}\right)=sign\left({\mu }_{2w}\right)$; then, it is plausible that $Z_{vw}=-{\sum }_{j=2}^n{\psi }_{jv}{\psi }_{jw}log\left(1-{\displaystyle \frac{\epsilon }{{\mu }_j}}\right)>0$ (notice that $log\left(1-{\displaystyle \frac{\epsilon }{{\mu }_j}}\right)<0$). Contrastingly, for the standard Laplacian we have ${\mathcal{L}}_{vw}={\sum }_{j=2}^n{\psi }_{jv}{\psi }_{jw}{\mu }_j$, such that the contribution of ${\mu }_2$ is the smallest one. In this case, the largest contribution comes from ${\mu }_n$ whose associated eigenvector ${\psi }_n$ may have ${\psi }_{nv}{\psi }_{nw}<0$ if $\left(v,w\right)\in E$. For instance, Song and Wang [57] have proved that in a tree, for all $\left(v,w\right)\in E$, it is true that ${\psi }_{nv}{\psi }_{nw}<0$. Thus, ${\mathcal{L}}_{vw}<0$ as expected for nonnegatively weighted graphs. The fact that there are nonnegative out-diagonal entries in Z indicates that it corresponds to a representation of the graph in which some negative signs are assigned to some edges, transforming the graph into a signed one. Additionally, because $Z_{vw}\ne 0$ even for pairs $\left(v,w\right)\notin E$, the matrix Z also accounts for non-nearest neighbor interactions.

Let us now define a diffusive process taking place on the graph G whose dynamics is controlled by Z.

Definition 3.

Let $x\left(t\right)$ be a vector of mass concentrations at the vertices of G at time $t>0$. Then, the evolution of $x\left(t\right)$ with time on the graph is given by:

$${\displaystyle \frac{dx\left(t\right)}{dt}}=-Zx\left(t\right),$$

(25)

subjected to the initial condition $x\left(0\right)=x^0.$ Notice that we have defined $Z=-log{P}^{+}$ (with negative sign) to keep the tradition in graph theory of using a negative sign in the right-hand side of diffusion equations, like in (1). We will call this process “resistive diffusion” for reasons that will become evident later on.

Then, the solution to the ACP for the resistive diffusion equation is given by

$$\begin{array}{cc}\hfill x\left(t\right)& =e^{-tZ}{x}^0\hfill \\ \hfill & =e^{.tlog\left(I-\epsilon {\mathcal{L}}^{+}\right)}{x}^0\hfill \\ \hfill & ={\left(I-\epsilon {\mathcal{L}}^{+}\right)}^t{x}^0\hfill \\ \hfill & ={\left(P^{+}\right)}^t{x}^0.\hfill \end{array}$$

(26)

We then prove the following result.

Theorem 1.

The resistive diffusion on a graph G converges to the average of the mass concentrations at the vertices at time $t\to \infty$, i.e.,

$$\underset{t\to \infty }{lim}x\left(t\right)={\displaystyle \frac{1}{n}}\sum _{i=1}^n{x}_i^0.$$

(27)

Proof. 

Let us consider the limit ${\displaystyle \underset{t\to \infty }{lim}}x\left(t\right)$, which by expressing $P^{+}$ in term of its spectral decomposition is given by

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim}x\left(t\right)& =\underset{t\to \infty }{lim}V{\left(I-\epsilon {\Lambda }^{+}\right)}^t{V}^T{x}^0.\hfill \end{array}$$

(28)

Then, for sufficiently large t only the contributions from the largest eigenvalue ${\sigma }_1$ and its associated eigenvector ${\psi }_1$ survives, i.e.,

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim}x\left(t\right)& ={\sigma }_1^t{\psi }_1\left({\psi }_1^T{x}^0\right)\hfill \end{array}.$$

(29)

Because ${\sigma }_1=1$ and ${\psi }_1={\displaystyle \frac{1}{\sqrt{n}}}\mathbf{1}$, we have

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim}x\left(t\right)& ={\displaystyle \frac{1}{\sqrt{n}}}\mathbf{1}\left({\displaystyle \frac{1}{\sqrt{n}}}{\mathbf{1}}^T{x}^0\right),\hfill \end{array}$$

(30)

which proves the result. □

Lemma 2.

The resistive diffusion on a connected graph is always conservative.

Proof. 

Let us start by proving that: ${\mathbf{1}}^T{P}^{+}={\mathbf{1}}^T$:

$$\begin{array}{cc}\hfill {\mathbf{1}}^T{P}^{+}& ={\mathbf{1}}^T\left(I-\epsilon {\mathcal{L}}^{+}\right)\hfill \\ \hfill & ={\mathbf{1}}^T-\epsilon {\mathbf{1}}^T{\mathcal{L}}^{+}\hfill \\ \hfill =& {\mathbf{1}}^T.\hfill \end{array}$$

(31)

Therefore, let us take the sum of the entries of $x\left(t\right)$ at an arbitrary time t,

$$\begin{array}{cc}\hfill {\mathbf{1}}^{T}x\left(t\right)& ={\mathbf{1}}^T{\left(P^{+}\right)}^t{x}^0\hfill \\ \hfill & =\left({\mathbf{1}}^T{P}^{+}\right){\left(P^{+}\right)}^{t-1}{x}^0\hfill \\ \hfill & ={\mathbf{1}}^T{\left(P^{+}\right)}^{t-1}{x}^0\hfill \\ \hfill & =\dots \hfill \\ \hfill & ={\mathbf{1}}^T{x}^0,\hfill \end{array}$$

(32)

which proves that ${\sum }_{j=1}^n{x}_j={\sum }_{j=1}^n{x}_j^0$ for any t. □

Remark 2.

In comparison with the standard diffusion process, the resistive diffusion is very slow. Let us consider that all initial mass concentration is allocated at the vertex $v\in V$ and we obtain the value of the mass concentration that remains at this vertex at a time $t>0,$ $x_v\left(t\right)={\sum }_{j=2}^n{\psi }_{jv}^2{\left(1-{\displaystyle \frac{\epsilon }{{\mu }_j}}\right)}^t$, in contrast with that obtained for the standard diffusion: ${\tilde{x}}_v\left(t\right)={\sum }_{j=2}^n{\psi }_{jv}^2{e}^{-t{\mu }_j}.$ For very large lengths of time, i.e., $t\gg 1$, the dominant contribution to $x_v\left(t\right)$ is made by the largest Laplacian eigenvalue and its corresponding eigenvector: $x_v\left(t\right)={\psi }_{nv}^2{\left(1-{\displaystyle \frac{\epsilon }{{\mu }_n}}\right)}^t$, while for the standard diffusion it is made by the algebraic connectivity and the Fiedler vector: ${\tilde{x}}_v\left(t\right)={\psi }_{j2}^2{e}^{-t{\mu }_2}.$ While $e^{-t{\mu }_2}$ decays very quickly as $t\to \infty$, the convergence of ${\left(1-{\displaystyle \frac{\epsilon }{{\mu }_n}}\right)}^t$ is extremely slow due to the fact that ${\displaystyle \frac{\epsilon }{{\mu }_n}}$ is close to zero. Consequently, we consider appropriate the term “resistive diffusion” for this process. This is illustrated in the following example.

Example 1.

We consider a path graph (linear chain) of 101 vertices labeled in consecutive order from one end vertex, and in which we consider the standard and resistive diffusion processes with initial condition $x_{51}^0={\delta }_{jv}$, where ${\delta }_{jv}$ is the Kronecker delta function. Due to the fact that in the steady state $x_v\left(t\right)=x_w\left(t\right)$ for all $v,w\in V$, we can use the standard deviation of the entries of the vector $x\left(t\right)$, $std\left(x\left(t\right)\right)$, as a measure of the convergence of the diffusion to its steady state, i.e., ${\displaystyle \underset{t\to \infty }{lim}}std\left(x\left(t\right)\right)=0$. This is illustrated in Figure 1a, where it can be seen that the convergence of the resistive diffusion (continuous line) towards the steady state is extremely slow in comparison with the standard diffusion one (broken line). This is a consequence of the fact that the mass concentration is largely retained at the vertex 51 as explained in the previous remark. The mechanism used by the resistive diffusion to transfer mass to other vertices is very peculiar as can be seen in Figure 1b, where we plot the mass concentration at every vertex averaged over time for both types of diffusion. While in the standard diffusion we observe the typical Gaussian decay (broken line) in the case of the resistive diffusion, we observe negative peaks at the vertices connected with the one allocating the initial mass concentration. This indicates that due to the large stickiness of the mass at the vertex 51, the process “extracts” mass from its nearest neighbors even at the cost of leaving them with negative amounts at a relatively short time. The process then converges to the steady state where all the vertices have exactly the same (positive) mass concentration.

4. Resistance Distance as a Diffusion Distance

Let us consider the resistive diffusive process of Definition (3), whose solution is $x\left(t\right)={\left(P^{+}\right)}^t{x}^0.$ Then, let us consider the mass concentrations at the vertices v and w at a given time $t>0,$ which are given by:

$$x_v\left(t\right)=\sum _{j=1}^n{\left({\left(P^{+}\right)}^t\right)}_{vj}{x}_j^0,$$

(33)

and

$$x_w\left(t\right)=\sum _{j=1}^n{\left({\left(P^{+}\right)}^t\right)}_{wj}{x}_j^0.$$

(34)

Let us now consider the initial condition in which all the mass concentration is initially allocated at the vertex v. That is, let $x_j^0={\delta }_{jv}$ where ${\delta }_{jv}$ is the Kronecker delta function. Let us then calculate the difference in mass concentrations between the vertices v and w at a given time $t>0,$ which is the concentration that remained at the vertex v at time $t>0,$ and the mass concentration which has flowed to the vertex w. Namely, we have:

$${\mathcal{F}}_{vw|x_j^0={\delta }_{jv}}\left(t\right):=x_{v|x_j^0={\delta }_{jv}}\left(t\right)-x_{w|x_j^0={\delta }_{jv}}\left(t\right)={\left({\left(P^{+}\right)}^t\right)}_{vv}-{\left({\left(P^{+}\right)}^t\right)}_{vw}.$$

(35)

Similarly, if we now consider the initial condition in which all the mass concentration is allocated at the vertex w, we obtain the flow from w to v as follows:

$${\mathcal{F}}_{wv|x_j^0={\delta }_{jw}}\left(t\right):=x_{w|x_j^0={\delta }_{jw}}\left(t\right)-x_{v|x_j^0={\delta }_{jw}}\left(t\right)={\left({\left(P^{+}\right)}^t\right)}_{ww}-{\left({\left(P^{+}\right)}^t\right)}_{wv}.$$

(36)

As the graph is undirected, the matrix $P^{+}$ is symmetric, which implies that ${\left({\left(P^{+}\right)}^t\right)}_{vw}={\left({\left(P^{+}\right)}^t\right)}_{wv}.$ Therefore, the sum of the flows on both directions under the respective initial conditions is given by:

$${\mathcal{D}}_{vw}\left(t\right):={\mathcal{F}}_{vw|x_j^0={\delta }_{jv}}\left(t\right)+{\mathcal{F}}_{wv|x_j^0={\delta }_{jw}}={\left({\left(P^{+}\right)}^t\right)}_{vv}+{\left({\left(P^{+}\right)}^t\right)}_{ww}-2{\left({\left(P^{+}\right)}^t\right)}_{vw}.$$

(37)

We then have the following result.

Theorem 2.

For any two vertices in G and for any $t>0$, ${\mathcal{D}}_{vw}\left(t\right)$ is a square Euclidean distance between the corresponding vertices.

Proof. 

It is easy to realize that

$$\begin{array}{cc}\hfill {\mathcal{D}}_{vw}\left(t\right)& =\sum _{j=1}^n{\psi }_{jv}^2{\sigma }_j^t+\sum _{j=1}^n{\psi }_{jw}^2{\sigma }_j^t-2\sum _{j=1}^n{\psi }_{jv}{\psi }_{jw}{\sigma }_j^t\hfill \\ \hfill & =\sum _{j=1}^n{\sigma }_j^t{\left({\psi }_{jv}-{\psi }_{jw}\right)}^2.\hfill \end{array}$$

(38)

Then, let ${\phi }_v$ be the vth row of the matrix V in $P^{+}\left(t\right)=V{\left(I-\epsilon {\Lambda }^{+}\right)}^t{V}^T$, such that

$${\mathcal{D}}_{vw}\left(t\right)={\left({\phi }_v-{\phi }_w\right)}^T{\left(I-\epsilon {\Lambda }^{+}\right)}^t\left({\phi }_v-{\phi }_w\right).$$

(39)

Because $P^{+}\left(t\right)$ is positive-defined, it has a unique square root, such that we can write

$${\mathcal{D}}_{vw}\left(t\right)={\left({\phi }_v-{\phi }_w\right)}^T{\left(I-\epsilon {\Lambda }^{+}\right)}^{t/2}{\left(I-\epsilon {\Lambda }^{+}\right)}^{t/2}\left({\phi }_v-{\phi }_w\right),$$

(40)

and by grouping the terms we have

$$\begin{array}{cc}\hfill {\mathcal{D}}_{vw}\left(t\right)& ={\left({\left(I-\epsilon {\Lambda }^{+}\right)}^{t/2}\left({\phi }_v-{\phi }_w\right)\right)}^T{\left(I-\epsilon {\Lambda }^{+}\right)}^{t/2}\left({\phi }_v-{\phi }_w\right)\hfill \\ \hfill & ={\left({\left(I-\epsilon {\Lambda }^{+}\right)}^{t/2}{\phi }_v-{\left(I-\epsilon {\Lambda }^{+}\right)}^{t/2}{\phi }_w\right)}^T\left({\left(I-\epsilon {\Lambda }^{+}\right)}^{t/2}{\phi }_v-{\left(I-\epsilon {\Lambda }^{+}\right)}^{t/2}{\phi }_w\right).\hfill \end{array}$$

(41)

Let us then define $q_v\left(t\right):={\left(I-\epsilon {\Lambda }^{+}\right)}^{t/2}{\varrho }_v$ and finally write

$$\begin{array}{cc}\hfill {\mathcal{D}}_{vw}\left(t\right)& ={\left(q_v\left(t\right)-q_w\left(t\right)\right)}^T\left(q_v\left(t\right)-q_w\left(t\right)\right)\hfill \\ \hfill & ={∥q_v\left(t\right)-q_w\left(t\right)∥}^2\hfill \end{array},$$

(42)

which proves the result. □

Remark 3.

A EDM whose nondiagonal entries are the distances ${\mathcal{D}}_{vw}\left(t\right)$ can be obtained as follows:

$$D\left(t\right)=diag\left(P^{+}\left(t\right)\right){\mathbf{1}}^T+\mathbf{1}{\left(diag\left(P^{+}\left(t\right)\right)\right)}^T-2P^{+}\left(t\right),$$

(43)

where $diag\left(P^{+}\left(t\right)\right)$ is the column vector containing the diagonal entries of $P^{+}\left(t\right)$. In other words,

$$D_{vw}\left(t\right)={\left(P^{+}\left(t\right)\right)}_{vv}+{\left(P^{+}\left(t\right)\right)}_{ww}-2{\left(P^{+}\left(t\right)\right)}_{vw}.$$

(44)

The fact that the resistance distance is just a diffusive distance on the graph is given by the following Corollary of the previous Theorem.

Corollary 1.

The resistance distance ${\Omega }_{vw}$ between the vertices v and w is the special case of ${\mathcal{D}}_{vw}\left(t\right)$ for $t=1$, i.e., ${\Omega }_{vw}={\displaystyle \frac{2-{\mathcal{D}}_{vw}\left(t=1\right)}{\epsilon }}$.

Proof. 

Let us write ${\mathcal{D}}_{vw}\left(t\right)$ for $t=1$

$${\mathcal{D}}_{vw}\left(t=1\right)={\left({\phi }_v-{\phi }_w\right)}^T\left(I-\epsilon {\Lambda }^{+}\right)\left({\phi }_v-{\phi }_w\right).$$

(45)

Then, due to the orthonormalization of the eigenvectors, we have:

$$\begin{array}{cc}\hfill {\mathcal{D}}_{vw}\left(t=1\right)& ={\left({\phi }_v-{\phi }_w\right)}^T\left({\phi }_v-{\phi }_w\right)-{\left({\phi }_v-{\phi }_w\right)}^T\epsilon {\Lambda }^{+}\left({\phi }_v-{\phi }_w\right)\hfill \\ \hfill & ={\phi }_v^T{\phi }_v+{\phi }_w^T{\phi }_w-2{\phi }_v{\phi }_w-\epsilon {\left({\phi }_v-{\phi }_w\right)}^T{\Lambda }^{+}\left({\phi }_v-{\phi }_w\right)\hfill \\ \hfill & =2-\epsilon {\left({\phi }_v-{\phi }_w\right)}^T{\Lambda }^{+}\left({\phi }_v-{\phi }_w\right)\hfill \\ \hfill & =2-\epsilon {\Omega }_{vw},\hfill \end{array}$$

(46)

where ${\Omega }_{vw}$ is the resistance distance between the corresponding vertices. □

Remark 4.

In the context of a diffusion process on the graph G, the resistance distance can be written as

$$\begin{array}{cc}\hfill {\Omega }_{vw}& ={\displaystyle \frac{2-{\mathcal{D}}_{vw}\left(t=1\right)}{\epsilon }}\hfill \\ \hfill & ={\displaystyle \frac{2-\left({\mathcal{F}}_{vw|x_j^0={\delta }_{jv}}\left(t=1\right)+{\mathcal{F}}_{wv|x_j^0={\delta }_{jw}}\left(t=1\right)\right)}{\epsilon }}\hfill \\ \hfill & ={\displaystyle \frac{\left({\mathcal{F}}_{vw|x_j^0={\delta }_{jv}}\left(t=0\right)+{\mathcal{F}}_{wv|x_j^0={\delta }_{jw}}\left(t=0\right)\right)-\left({\mathcal{F}}_{vw|x_j^0={\delta }_{jv}}\left(t=1\right)+{\mathcal{F}}_{wv|x_j^0={\delta }_{jw}}\left(t=1\right)\right)}{\epsilon }}\hfill \\ \hfill & ={\displaystyle \frac{\left({\mathcal{F}}_{vw|x_j^0={\delta }_{jv}}\left(t=0\right)-{\mathcal{F}}_{vw|x_j^0={\delta }_{jv}}\left(t=1\right)\right)+\left({\mathcal{F}}_{wv|x_j^0={\delta }_{jw}}\left(t=0\right)-{\mathcal{F}}_{wv|x_j^0={\delta }_{jw}}\left(t=1\right)\right)}{\epsilon }},\hfill \end{array}$$

(47)

where ${\mathcal{F}}_{vw|x_j^0={\delta }_{jv}}\left(t=0\right)=1$ and ${\mathcal{F}}_{wv|x_j^0={\delta }_{jw}}\left(t=0\right)=1$ because we have initialized the processes by allocating all the mass concentration at the corresponding vertex. Therefore, the resistance distance is the difference of flow between the corresponding vertices at $t=1$ relative to the initial mass concentrations at those vertices.

Remark 5.

Using the expression for ${\mathcal{D}}_{vw}\left(t\right)$, we can see that

$${\mathcal{D}}_{vw}\left(2\right)={\left({\phi }_v-{\phi }_w\right)}^T{\left(I-\epsilon {\Lambda }^{+}\right)}^2\left({\phi }_v-\phi \right),$$

(48)

which can be expressed as

$${\mathcal{D}}_{vw}\left(2\right)=2-2\epsilon {\Omega }_{vw}+{\epsilon }^2{\left({\phi }_v-{\phi }_w\right)}^T{\left({\Lambda }^{+}\right)}^2\left({\phi }_v-{\phi }_w\right).$$

(49)

Therefore, the term ${\left({\phi }_v-{\phi }_w\right)}^T{\left({\Lambda }^{+}\right)}^2\left({\phi }_v-{\phi }_w\right)$ is the analogous to the resistance distance but in which the eigenvalues of ${\mathcal{L}}^{+}$ are squared. Thus, we can call ²${\Omega }_{vw}:={\sum }_{j=2}^n{\mu }_j^{-2}{\left({\psi }_{jv}-{\psi }_{jw}\right)}^2$ the second-power resistance distance and generalize it to: ^k${\Omega }_{vw}:={\left({\left({\mathcal{L}}^{+}\right)}^k\right)}_{vv}+{\left({\left({\mathcal{L}}^{+}\right)}^k\right)}_{ww}-2{\left({\left({\mathcal{L}}^{+}\right)}^k\right)}_{vw}.$

5. Conclusions

The main conclusion of this work is that the resistance distance, which is typically derived from a physical scenario related to representing a graph as an electrical circuit, can also be interpreted as a diffusion distance on the graph. It allows us to integrate this Euclidean distance into a family of diffusive distances, which include the Coifman diffusion and the communicability distances, as the most relevant cases. Such an integration allows for a better comparison between the different distances in the same graph context. Also, because the topological shortest path distance—the number of edges in the shortest path connecting two vertices in the graph—is identical to the resistance distance on trees, i.e., acyclic graphs, the current results give a physical interpretation for the shortest paths on a tree as a diffusive distance on a process controlled by the Z operator.