[Paper Review] Consensus dynamics on temporal hypergraphs

Neuhäuser, Leonie, Lambiotte, Renaud+1 more|arXiv (Cornell University)|Sep 10, 2021

Opinion Dynamics and Social Influence48 citations

TL;DR

This paper investigates consensus dynamics on temporal hypergraphs—networks with time-varying, multi-way interactions—comparing them to projections that flatten time or multi-way structure. It shows that 3-way interactions slow convergence compared to pairwise interactions, and in nonlinear dynamics, early-activating groups gain a 'first-mover advantage,' altering the final consensus value in ways not captured by static or pairwise projections.

ABSTRACT

We investigate consensus dynamics on temporal hypergraphs that encode network systems with time-dependent, multi-way interactions. We compare this dynamics with that on appropriate projections of this higher-order network representation that flatten the temporal, the multi-way component, or both. For linear average consensus dynamics, we find that the convergence of a randomly switching time-varying system with multi-way interactions is slower than the convergence of the corresponding system with pairwise interactions, which in turn exhibits a slower convergence rate than a consensus dynamics on the corresponding static network. We then consider a nonlinear consensus dynamics model in the temporal setting. Here we find that in addition to an effect on the convergence speed, the final consensus value of the temporal system can differ strongly from the consensus on the aggregated, static hypergraph. In particular we observe a first-mover advantage in the consensus formation process: If there is a local majority opinion in the hyperedges that are active early on, the majority in these first-mover groups has a higher influence on the final consensus value - a behaviour that is not observable in this form in projections of the temporal hypergraph.

Motivation & Objective

• To understand how temporal ordering and multi-way interactions jointly affect consensus dynamics in higher-order networks.
• To compare consensus dynamics on temporal hypergraphs with their projections into static, pairwise, or aggregated networks.
• To investigate whether temporal ordering and higher-order interactions lead to distinct dynamical outcomes beyond convergence speed.
• To explore the emergence of non-trivial effects—such as first-mover advantage—under nonlinear consensus dynamics on temporal hypergraphs.

Proposed method

• Formalizes consensus dynamics on temporal hypergraphs using a time-switched system model with switching interaction matrices.
• Applies linear consensus dynamics on hypergraphs, leveraging known results to rewrite them as equivalent weighted pairwise dynamics for comparison.
• Extends a nonlinear consensus model from [16] to temporal hypergraphs to study nonlinearity effects on convergence and final consensus values.
• Employs simulation-based analysis on synthetic hypergraphs with two clusters connected by 3-edges to compare dynamics across temporal, aggregated, and reduced projections.
• Uses relaxation time and convergence speed metrics to quantify dynamical differences between temporal and projected systems.
• Analyzes the influence of interaction ordering by varying which group forms the local majority in early-activating hyperedges.

Experimental results

Research questions

• RQ1 How does the presence of multi-way interactions affect the convergence speed of consensus dynamics in time-varying networks compared to pairwise interactions?
• RQ2 Does temporal ordering of hyperedges lead to qualitative differences in the final consensus value, especially under nonlinear dynamics?
• RQ3 Can a 'first-mover advantage' emerge in consensus formation when early-activating hyperedges contain a local majority, and is this effect absent in standard network projections?
• RQ4 How do the combined effects of temporal dynamics and higher-order interactions alter consensus outcomes compared to static or pairwise network models?
• RQ5 To what extent does the timescale of interactions influence the convergence behavior and final consensus value in temporal hypergraphs?

Key findings

• Linear consensus dynamics on temporal hypergraphs converge more slowly than on corresponding pairwise or static networks, with 3-way interactions introducing an additional slowdown beyond temporal effects.
• In nonlinear consensus dynamics, the final consensus value differs significantly from that on the aggregated static hypergraph due to temporal ordering of hyperedges.
• A first-mover advantage emerges: groups that are in local majority in early-activating hyperedges exert disproportionately higher influence on the final consensus value.
• This first-mover effect is absent in both the aggregated hypergraph and the reduced pairwise network, indicating it is uniquely driven by the temporal ordering of multi-way interactions.
• The convergence speed is faster when the initial configuration is highly asymmetric and when the orientation of 3-edges aligns with the first-mover group.
• The system's behavior becomes increasingly similar to the aggregated dynamics as interaction timescales shorten, indicating a convergence to static behavior under fast dynamics.