On the Equivalence Between High-Order Network-Influence Frameworks: General-Threshold, Hypergraph-Triggering, and Logic-Triggering Models

TL;DR

This paper establishes the theoretical equivalence of several high-order influence propagation models in networks, including hypergraph, Boolean logic, and threshold models, and explores their extensions with correlations.

Contribution

It proves the equivalence of hypergraph, Boolean-function, and threshold influence models, and analyzes their extensions with correlated influences, providing new insights into high-order network influence behaviors.

Findings

Boolean-function and hypergraph models are equivalent.

All three models are equivalent to the general threshold model.

Correlated hypergraph and Boolean models remain equivalent, unlike the threshold model.

Abstract

In this paper, we study several high-order network-influence-propagation frameworks and their connection to the classical network diffusion frameworks such as the triggering model and the general threshold model. In one framework, we use hyperedges to represent many-to-one influence -- the collective influence of a group of nodes on another node -- and define the hypergraph triggering model as a natural extension to the classical triggering model. In another framework, we use monotone Boolean functions to capture the diverse logic underlying many-to-one influence behaviors, and extend the triggering model to the Boolean-function triggering model. We prove that the Boolean-function triggering model, even with refined details of influence logic, is equivalent to the hypergraph triggering model, and both are equivalent to the general threshold model. Moreover, the general threshold model is optimal in the number of parameters, among all models with the same expressive power. We further extend these three equivalent models by introducing correlations among influence propagations on different nodes. Surprisingly, we discover that while the correlated hypergraph-based model is still equivalent to the correlated Boolean-function-based model, the correlated general threshold model is more restrictive than the two high-order models. Our study sheds light on high-order network-influence propagations by providing new insight into the group influence behaviors in existing models, as well as diverse modeling tools for understanding influence propagations in networks.