Impact of Community Structure on Cascades

TL;DR

This paper analyzes how community structure influences the spread of behaviors in networks, providing mathematical models, conditions for widespread adoption, and a new seeding strategy that challenges traditional approaches.

Contribution

It introduces a differential-equation-based approximation for cascades on structured networks and proposes a gradient heuristic for optimal seeding.

Findings

Mean-field equations accurately model adoption dynamics.

Contagion can occur from infinitesimal initial sets under certain conditions.

Heuristic seeding outperforms high-degree node seeding in experiments.

Abstract

We study cascades under the threshold model on sparse random graphs with community structure. In this model, individuals adopt the new behavior based on how many neighbors have already chosen it. Specifically, we consider the permanent adoption model wherein individuals that have adopted the new behavior (or opinion) cannot change their state. We present a differential-equation-based tight approximation to the stochastic process of adoption and prove the validity of the mean-field equations. In addition, we characterize both necessary and sufficient conditions for contagion to happen no matter how small the set of initial adopters is. Finally, we study the problem of optimum seeding given budget constraints and propose a gradient-based heuristic seeding strategy. Our algorithm, numerically, dispels commonly held beliefs in the literature that suggest the best seeding strategy is to seed over the vertices with the highest number of neighbors.