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In this paper, we study several high-order network-influence-propagationframeworks and their connection to the classical network diffusion frameworkssuch as the triggering model and the general threshold model. In one framework,we use hyperedges to represent many-to-one influence -- the collectiveinfluence of a group of nodes on another node -- and define the hypergraphtriggering model as a natural extension to the classical triggering model. Inanother framework, we use monotone Boolean functions to capture the diverselogic underlying many-to-one influence behaviors, and extend the triggeringmodel to the Boolean-function triggering model. We prove that theBoolean-function triggering model, even with refined details of influencelogic, is equivalent to the hypergraph triggering model, and both areequivalent to the general threshold model. Moreover, the general thresholdmodel is optimal in the number of parameters, among all models with the sameexpressive power. We further extend these three equivalent models byintroducing correlations among influence propagations on different nodes.Surprisingly, we discover that while the correlated hypergraph-based model isstill equivalent to the correlated Boolean-function-based model, the correlatedgeneral threshold model is more restrictive than the two high-order models. Ourstudy sheds light on high-order network-influence propagations by providing newinsight into the group influence behaviors in existing models, as well asdiverse modeling tools for understanding influence propagations in networks.

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