The walk distances in graphs are defined as the result of appropriate transformations of the $\sum_{k=0}^\infty(tA)^k$ proximity measures, where $A$ is the weighted adjacency matrix of a connected weighted graph and $t$ is a sufficiently small positive parameter. The walk distances are graph-geodetic, moreover, they converge to the shortest path distance and to the so-called long walk distance as the parameter $t$ approaches its limiting values. In this paper, simple expressions for the long walk distance are obtained. They involve the generalized inverse, minors, and inverses of submatrices of the symmetric irreducible singular M-matrix ${\cal L}=\rho I-A,$ where $\rho$ is the Perron root of $A.$
Deep Dive into Simple expressions for the long walk distance.
The walk distances in graphs are defined as the result of appropriate transformations of the $\sum_{k=0}^\infty(tA)^k$ proximity measures, where $A$ is the weighted adjacency matrix of a connected weighted graph and $t$ is a sufficiently small positive parameter. The walk distances are graph-geodetic, moreover, they converge to the shortest path distance and to the so-called long walk distance as the parameter $t$ approaches its limiting values. In this paper, simple expressions for the long walk distance are obtained. They involve the generalized inverse, minors, and inverses of submatrices of the symmetric irreducible singular M-matrix ${\cal L}=\rho I-A,$ where $\rho$ is the Perron root of $A.$
The walk distances for graph vertices are a parametric family of graph distances proposed in [4]. Along with their modifications they generalize [5] the logarithmic forest distances [3], resistance distance, shortest path distance, and the weighted shortest path distance. The walk distances are graph-geodetic: for a distance 1 d(i, j) on the vertices of a graph G this means that d(i, j) + d(j, k) = d(i, k) if and only if every path in G connecting i and k visits j.
The long walk distance, d LW (i, j), is obtained by letting the parameter of the walk distances go to one of its limiting values (approaching the other limiting value leads to the shortest path distance). A number of expressions for d LW (i, j) are given in [5]. In this paper, we obtain simple expressions in terms of the matrix L = ρI -A, where A is the weighted adjacency matrix of a connected weighted graph G on n vertices and ρ is the Perron root of A. L is a symmetric irreducible singular M-matrix, so rank L = n -1 and L is positive semidefinite. In [5], L was called the para-Laplacian matrix of G. The expressions presented in this paper generalize some classical expressions for the resistance distance (cf. [2]). They enable one to conclude that the long walk distance can be considered as the counterpart of the resistance distance obtained by replacing the Laplacian matrix L = diag(A1)-A and the vector 1 (of n ones) which spans Ker L with the para-Laplacian matrix L and the eigenvector of A spanning Ker L. If A has constant row sums, then L = L and these distances coincide.
In the graph definitions we mainly follow [7]. Let G be a weighted multigraph (a weighted graph where multiple edges are allowed) with vertex set V (G) = V, |V | = n > 1, and edge set E(G). Loops are allowed; throughout the paper we assume that G is connected. For brevity, we will call G a graph. For i, j ∈ V, let n ij ∈ {0, 1, . . .} be the number of edges incident to both i and j in G; for every q ∈ {1, . . . , n ij }, w q ij > 0 is the weight of the qth edge of this type. Let
we set a ij = 0) and A = (a ij ) n×n ; A is the symmetric weighted adjacency matrix of G. In this paper, all matrix entries are indexed by the vertices of G.
The weighted Laplacian matrix of G is L = diag(A1)-A, where 1 is the vector of n ones. For v 0 , v m ∈ V, a v 0 → v m path (simple path) in G is an alternating sequence of vertices and edges v 0 , e 1 , v 1 , . . . , e m , v m where all vertices are distinct and each e i is an edge incident to v i-1 and v i . The unique v 0 → v 0 path is the “sequence” v 0 having no edges.
Similarly, a v 0 → v m walk in G is an arbitrary alternating sequence of vertices and edges v 0 , e 1 , v 1 , . . . , e m , v m where each e i is incident to v i-1 and v i . The length of a walk is the number m of its edges (including loops and repeated edges). The weight of a walk is the product of the m weights of its edges. The weight of a set of walks is the sum of the weights of its elements. By definition, for any vertex v 0 , there is one v 0 → v 0 walk v 0 with length 0 and weight 1.
Let r ij be the weight of the set R ij of all i → j walks in G, provided that this weight is finite. R = R(G) = (r ij ) n×n will be referred to as the matrix of the walk weights.
By d r (i, j) we denote the resistance distance between i and j, i.e., the effective resistance between i and j in the resistive network whose line conductances equal the edge weights w q ij in G. The resistance distance has several expressions via the generalized inverse, minors, and inverses of the submatrices of the weighted Laplacian matrix of G (see, e.g., [1,9] or the papers by Subak-Sharpe and Styan published in the 60s and cited in [5]).
) holds if and only if every path in G connecting i and k contains j.
Graph-geodetic functions can also be called bottleneck additive or cutpoint additive.
For any t > 0, consider the graph tG obtained from G by multiplying all edge weights by t. If the matrix of the walk weights of tG, R
where I denotes the identity matrix of appropriate dimension.
By assumption, G is connected, while its edge weights are positive, so R t is positive whenever it exists. The walk distances can be introduced as follows.
Definition 2. For a connected graph G, the walk distances on V (G) are the functions d t (i, j) : V (G)×V (G) → R and the functions positively proportional to them, where
.
(1)
Regarding the existence (finiteness) of R t , since G is connected, A is irreducible, so the Perron-Frobenius theory of nonnegative matrices provides the following result.
Lemma 2. For any weighted adjacency matrix A of a connected graph G, the series R t = ∞ k=0 (tA) k with t > 0 converges to (I -tA) -1 if and only if t < ρ -1 , where ρ = ρ(A) is the spectral radius of A. Moreover, ρ is an eigenvalue of A; as such ρ has multiplicity 1 and a positive eigenvector.
Thus, the walk distance d t (i, j) with parameter t exists if and only if 0 < t < ρ -1 . A topological interpretation of
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