A system model of three-body interactions in complex networks: consensus and conservation

Abstract

Networked complex systems in a wide range of physics, biology and social sciences involve synergy among multiple agents beyond pairwise interactions. Higher-order mathematical structures such as hypergraphs have been increasingly popular in modelling and analysis of complex dynamical behaviours. Here, we study a simple three-body consensus model, which favourably incorporates higher-order network interactions, higher-order dimensional states, the group reinforcement effect and the social homophily principle. The model features asymmetric roles of acting agents using modulating functions. We analytically establish sufficient conditions for nonlinear consensus and conservation of states for agents with both discrete-time and continuous-time dynamics. We show that higher-order interactions encoded in three-body edges give rise to consensus and conservation for systems with gravity-like and Heaviside-like modulating functions. Furthermore, we illustrate our theoretical results with numerical simulations and examine the system convergence time through a network depreciation process.

Engineering Emergence: A Survey on Control in the World of Complex Networks

“…In complex networks, consensus approaches are applied to clustering, connectivity, or network dynamics: managing stochastic fluctuations [215], robustness based on connectivity [216], positive edge constraints [217], behavioral consensus in evolutionary dynamics [218], observer-type protocols [219], nonlinear dynamics [220], three-body interactions [221], heterogeneous timescales [222], etc. When consensus requires that nodes in a complex network agree on aligning one of their internal states or parameters, temporarily or not, it becomes a case of synchronization (Section 3.2).…”

Constrained Consensus in State-Dependent Directed Multiagent Networks

“…We assume the function b satisfies one of the following conditions. The property of b in Assumption 1 is similar to the law of gravity as the modulation function will remain positive while decline as the distance increases [14]. Assumption 2 has a cut-off point.…”

“…Although state-dependent multiagent networks are included in a general framework of time-varying communication graphs [13], [14], they have specific characteristics that are worth deep investigation. In fact, the communication topology is unknown a priori and cannot be solely chosen by individual agents in a state-dependent multiagent network.…”

Higher-order random network models

“…In addition, the consensus dynamics over temporal hypergraphs were investigated [21]. Moreover, a three-body consensus dynamical model over hypergraphs was developed [22], which features asymmetric roles of interacting agents in a triangle. Simplicial complexes on n vertices [23] can also be used to develop higher-order graph models.…”