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.

People's opinions evolve over 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 boundedconfidence model (BCM) on graphs, in which nodes are connected pairwise by edges, to hypergraphs. We show that our hypergraph BCM converges to consensus under a wide range of initial conditions for the opinions of the nodes. We show that, under suitable conditions, echo chambers can form on hypergraphs with community structure. We also observe that the opinions of individuals can sometimes jump from one opinion cluster to another in a single time step, a phenomenon (which we call "opinion jumping") that is not possible in standard dyadic BCMs. We also show that there is a phase transition in the convergence time on the complete hypergraph when the variance σ 2 of the initial opinion distribution equals the confidence bound c. 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.

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It is common to use networks to encode the architecture of interactions between entities in complex systems in the physical, biological, social, and information sciences. Moreover, to study the large-scale behavior of complex systems, it is important to study mesoscale structures in networks as building blocks that influence such behavior [17,43]. In this paper, we present a new approach for describing low-rank mesoscale structure in networks, and we illustrate our approach using several synthetic network models and empirical friendship, collaboration, and protein-protein interaction (PPI) networks. We find that these networks possess a relatively small number of 'latent motifs' that together can successfully approximate most subnetworks at a fixed mesoscale. We use an algorithm that we call "network dictionary learning" (NDL) [30], which combines a network sampling method [29] and nonnegative matrix factorization [19,30], to learn the latent motifs of a given network. The ability to encode a network using a set of latent motifs has a wide range of applications to network-analysis tasks, such as comparison, denoising, and edge inference. Additionally, using our new network denoising and reconstruction (NDR) algorithm, we demonstrate how to denoise a corrupted network by using only the latent motifs that one learns directly from the corrupted networks.

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