Published: May 24, 2024

Dane Taylor, Department of Mathematics and Statistics, University of Wyoming

Consensus processes over networks: Past, present, and future

Models for consensus---that is, the reaching of agreement---have been developed, e.g., to study how group decisions are collectively made within social networks, how groups of animals collectively move, and how decentralized machine-learning (ML) algorithms train on dispersed data. In this talk, I will review applications and theories for consensus processes over networks and then describe ongoing work to extend these models. First, I will discuss the important role that network structure plays in shaping consensus dynamics, including results for how the presence of community structure may or may not impact the convergence rate for decentralized ML. Motivated by emerging scenarios in which collective decisions are made by human-AI teams, I will study consensus over a system comprised of 2 asymmetrically coupled networks, which are used to model a social network supported by a network of AI agents. I will present theory for when collective decisions are obtained optimally fast (i.e., with a maximal convergence rate) and when the resulting decisions are cooperative (i.e., the final state reflects the initial states of both networks). Time permitting, I will present another generalization in which consensus is formulated for “higher-order” networks with multiway interactions encoded by a simplicial complex (i.e., as opposed to a graph that encodes pairwise interactions). This discussion will touch on a few related mathematical areas including differential equations, random matrix theory, spectral perturbation/optimization, and algebraic topology/homology.

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