Optimal Intervention on Weighted Networks via Edge Centrality

Dongyue Li, Tina Eliassi-Rad, Hongyang R. Zhang

Abstract: Suppose there is a spreading process propagating on a weighted graph. Denote the graph's weight matrix as W. How would we reduce the number of nodes affected during the process? This question appears in recent studies about counterfactual outcomes of implementing edge-weight interventions on mobility networks (Chang et al. (2021)). A practical algorithm to reduce infections is by removing edges with the highest edge centrality, defined as the product of two adjacent nodes’ eigen- scores (Tong et al. (2012)). In this work, we design edge-weight reduction algorithms on static and time- varying weighted networks with theoretical guarantees. First, we prove that edge centrality equals the gradient of the largest eigenvalue of WW⊤ (over W) and generalize the gradient for the largest r eigenvalues of WW⊤. Second, we design a Frank-Wolfe algorithm for finding the optimal edge-weight reduction to shrink the largest r eigenvalues of WW⊤ under any reduction budget. Third, we extend our algorithm to time-varying networks with guaranteed optimality. We perform a detailed empirical study to validate our approach. Our algorithm significantly reduces the number of infections compared with existing methods on eleven weighted networks. Further, we illustrate several properties of our algorithm: the benefit of choosing r, fast convergence to the optimum, and a linear-scale runtime per iteration.