Kirchhoff Index as a Measure of Edge Centrality in Weighted Networks: Nearly Linear Time Algorithms

Abstract

Estimating the relative importance of vertices and edges is a fundamental issue in the analysis of complex networks, and has found vast applications in various aspects, such as social networks, power grids, and biological networks. Most previous work focuses on metrics of vertex importance and methods for identifying powerful vertices, while related work for edges is much lesser, especially for weighted networks, due to the computational challenge. In this paper, we propose to use the well-known Kirchhoff index as the measure of edge centrality in weighted networks, called θ-Kirchhoff edge centrality. The Kirchhoff index of a network is defined as the sum of effective resistances over all vertex pairs. The centrality of an edge e is reflected in the increase of Kirchhoff index of the network when the edge e is partially deactivated, characterized by a parameter θ. We define two equivalent measures for θ-Kirchhoff edge centrality. Both are global metrics and have a better discriminating power than commonly used measures, based on local or partial structural information of networks, e.g. edge betweenness and spanning edge centrality. Despite the strong advantages of Kirchhoff index as a centrality measure and its wide applications, computing the exact value of Kirchhoff edge centrality for each edge in a graph is computationally demanding. To solve this problem, for each of the θ-Kirchhoff edge centrality metrics, we present an efficient algorithm to compute its -approximation for all the m edges in nearly linear time in m. The proposed θ-Kirchhoff edge centrality is the first global metric of edge importance that can be provably approximated in nearly-linear time. Moreover, according to the θ-Kirchhoff edge centrality, we present a θ-Kirchhoff vertex centrality measure, as well as a fast algorithm that can compute -approximate Kirchhoff vertex centrality for all the n vertices in nearly linear time in m.

Fully dynamic spectral vertex sparsifiers and applications

“…The power of vertex or edge graph sparsifiers, which preserve certain properties while reducing problem sizes, has long been studied in data structures [20,21]. Ideas from these results are central to recent works on offline maintenance for 3-connectivity [57], generating random spanning trees [18], and new notions of centrality for networks [43]. Our result is the first to maintain such vertex sparsifiers, specifically Schur complements, for general graphs in online settings.…”

Current Flow Group Closeness Centrality for Complex Networks?

“…Our investigation can be viewed as combining this line of current based centrality measures with the study of selecting groups of k vertices. For the former, a subset of the authors of this paper (Li and Zhang) recently demonstrated that current flow centrality measures for single edges can be computed provably efficiently [LZ18]. Our approximation algorithm in Section 7 is directly motivated by that routine.…”

Spectra, Hitting Times and Resistance Distances ofq- Subdivision Graphs

“…Kirchhoff index has found wide applications. For example, it can be used as measures of the overall connectedness of a network [45], the robustness of first-order consensus algorithm in noisy networks [46], as well as the edge centrality of complex networks [47]. In recent years, several modifications for Kirchhoff index have been proposed, including additive degree-Kirchhoff index [48] and multiplicative degree-Kirchhoff index [40].…”