"A Bounded-Confidence Model of Opinion Dynamics on Hypergraphs"

A new paper of mine just came out in final form. Here are some details about it.

Title: A Bounded-Confidence Model of Opinion Dynamics on Hypergraphs

Authors: Abigail Hickok, Yacoub Kureh, Heather Z. Brooks, Michelle Feng, and Mason A. Porter

Abstract: People's opinions evolve with time as they interact with their friends, family, colleagues, and others. In the study of opinion dynamics on networks, one often encodes interactions between people in the form of dyadic relationships, but many social interactions in real life are polyadic (i.e., they involve three or more people). In this paper, we extend an asynchronous bounded-confidence model (BCM) on graphs, in which nodes are connected pairwise by edges, to an asynchronous BCM on hypergraphs, in which arbitrarily many nodes can be connected by a single hyperedge. We show that our hypergraph BCM converges to consensus for a wide range of initial conditions for the opinions of the nodes, including for nonuniform and asymmetric initial opinion distributions. We also show that, under suitable conditions, echo chambers can form on hypergraphs with community structure. We demonstrate that the opinions of nodes can sometimes jump from one opinion cluster to another in a single time step; this phenomenon (which we call ``opinion jumping") is not possible in standard dyadic BCMs. Additionally, we observe a phase transition in the convergence time of our BCM on a complete hypergraph when the variance $\sigma^2$ of the initial opinion distribution equals the confidence bound $c$. We prove that the convergence time grows at least exponentially fast with the number of nodes when $\sigma^2 > c$ and the initial opinions are normally distributed. Therefore, to determine the convergence properties of our hypergraph BCM when the variance and the number of hyperedges are both large, it is necessary to use analytical methods instead of relying only on Monte Carlo simulations.

"Graph-Theoretic" and "Graphical" Language in the Description of Marriages

This Onion article makes me realize that one can describe marriage laws very precisely using graph theory: allowing non-bipartite graphs, allowing hypergraphs, etc.

Clearly, the outcomes of the court cases ought to be written using graph-theoretic (or, to use a perhaps unfortunate pun, "graphical") language.

Note: I have no comment about self-edges.

"Cross-Linked Structure of Network Evolution"

A new paper of mine was published in final form today. In this paper, my coauthors and I use a structure called a "cross-link" that connects a pair of time-dependent edges based on the similarity of their temporal evolution. In our study, the time-dependent edges arise from similarity of temporal dynamics of different nodes. The basic idea is to try to tease out when sets of edges evolve separately and when there is co-evolution. In this paper, we consider time-dependent networks that we construct from time series from functional brain networks and from output of coupled Kuramoto oscillators. Here are the details of the paper.


Title: Cross-Linked Structure of Network Evolution

Authors: Danielle S. Bassett, Nicholas F. Wymbs, Mason A. Porter, Peter J. Mucha, and Scott T. Grafton

Abstract: We study the temporal co-variation of network co-evolution via the cross-link structure of networks, for which we take advantage of the formalism of hypergraphs to map cross-link structures back to network nodes. We investigate two sets of temporal network data in detail. In a network of coupled nonlinear oscillators, hyperedges that consist of network edges with temporally co-varying weights uncover the driving co-evolution patterns of edge weight dynamics both within and between oscillator communities. In the human brain, networks that represent temporal changes in brain activity during learning exhibit early co-evolution that then settles down with practice. Subsequent decreases in hyperedge size are consistent with emergence of an autonomous subgraph whose dynamics no longer depends on other parts of the network. Our results on real and synthetic networks give a poignant demonstration of the ability of cross-link structure to uncover unexpected co-evolution attributes in both real and synthetic dynamical systems. This, in turn, illustrates the utility of analyzing cross-links for investigating the structure of temporal networks.