Absorbing random-walk centrality: Theory and algorithms

TL;DR

This paper introduces a new graph centrality measure based on absorbing random walks, analyzes its properties, proves NP-hardness, and proposes efficient algorithms for large-scale computation.

Contribution

It formally defines the $k$ absorbing random-walk centrality, proves its NP-hardness, and develops spectral clustering and personalized PageRank algorithms for scalable solutions.

Findings

The problem is NP-hard.

The objective function is monotone and supermodular.

Efficient algorithms outperform naive approaches on large datasets.

Abstract

We study a new notion of graph centrality based on absorbing random walks. Given a graph $G=(V,E)$ and a set of query nodes $Q\subseteq V$, we aim to identify the $k$ most central nodes in $G$ with respect to $Q$. Specifically, we consider central nodes to be absorbing for random walks that start at the query nodes $Q$. The goal is to find the set of $k$ central nodes that minimizes the expected length of a random walk until absorption. The proposed measure, which we call $k$ absorbing random-walk centrality, favors diverse sets, as it is beneficial to place the $k$ absorbing nodes in different parts of the graph so as to "intercept" random walks that start from different query nodes. Although similar problem definitions have been considered in the literature, e.g., in information-retrieval settings where the goal is to diversify web-search results, in this paper we study the problem formally and prove some of its properties. We show that the problem is NP-hard, while the objective function is monotone and supermodular, implying that a greedy algorithm provides solutions with an approximation guarantee. On the other hand, the greedy algorithm involves expensive matrix operations that make it prohibitive to employ on large datasets. To confront this challenge, we develop more efficient algorithms based on spectral clustering and on personalized PageRank.