Explaining Snapshots of Network Diffusions: Structural and Hardness Results

TL;DR

This paper investigates the feasibility of diffusion snapshots in social networks under various deterministic Linear Threshold Model variations, revealing complexity results and structural insights.

Contribution

It introduces the inverse problem of snapshot feasibility, analyzes its computational hardness across different dynamics, and provides structural understanding of social network diffusions.

Findings

Sequential dynamics with deactivations are computationally harder than other cases.

The problem is trivial on cliques for sequential dynamics.

Hardness results are established for all considered variations.

Abstract

Much research has been done on studying the diffusion of ideas or technologies on social networks including the \textit{Influence Maximization} problem and many of its variations. Here, we investigate a type of inverse problem. Given a snapshot of the diffusion process, we seek to understand if the snapshot is feasible for a given dynamic, i.e., whether there is a limited number of nodes whose initial adoption can result in the snapshot in finite time. While similar questions have been considered for epidemic dynamics, here, we consider this problem for variations of the deterministic Linear Threshold Model, which is more appropriate for modeling strategic agents. Specifically, we consider both sequential and simultaneous dynamics when deactivations are allowed and when they are not. Even though we show hardness results for all variations we consider, we show that the case of sequential dynamics with deactivations allowed is significantly harder than all others. In contrast, sequential dynamics make the problem trivial on cliques even though it's complexity for simultaneous dynamics is unknown. We complement our hardness results with structural insights that can help better understand diffusions of social networks under various dynamics.