Optimal Spread in Network Consensus Models

TL;DR

This paper develops a method to identify node subsets in a network that maximize the speed of information spread, using a submodular rank function and properties of the spread dynamics to guarantee near-optimal solutions.

Contribution

It introduces a novel submodular rank function and a greedy-based approach to find near-optimal spreading sets with theoretical performance guarantees.

Findings

The proposed method guarantees at least (1-1/e) of the optimal solution's rank.

Performance bounds improve when the solution exceeds the greedy solution's rank.

The approach balances solution quality and computational effort based on the curvature of the spread function.

Abstract

In a model of network communication based on a random walk in an undirected graph, what subset of nodes (subject to constraints on the set size), enable the fastest spread of information? The dynamics of spread is described by a process dual to the movement from informed to uninformed nodes. In this setting, an optimal set $A$ minimizes the sum of the expected first hitting times $F(A)$, of random walks that start at nodes outside the set. In this paper,the problem is reformulated so that the search for solutions is restricted to a class of optimal and "near" optimal subsets of the graph. We introduce a submodular, non-decreasing rank function $\rho$, that permits some comparison between the solution obtained by the classical greedy algorithm and one obtained by our methods. The supermodularity and non-increasing properties of $F$ are used to show that the rank of our solution is at least $(1-\frac{1}{e})$ times the rank of the optimal set. When the solution has a higher rank than the greedy solution this constant can be improved to $(1-\frac{1}{e})(1+\chi)$ where $\chi >0$ is determined a posteriori. The method requires the evaluation of $F$ for sets of some fixed cardinality $m$, where $m$ is much smaller than the cardinality of the optimal set. When $F$ has forward elemental curvature $\kappa$, we can provide a rough description of the trade-off between solution quality and computational effort $m$ in terms of $\kappa$.