Eigenvalue confinement and spectral gap for random simplicial complexes

We consider the adjacency operator of the Linial-Meshulam model for random simplicial complexes on $n$ vertices, where each $d$-cell is added independently with probability $p$ to the complete $(d-1)$-skeleton. Under the assumption $np(1-p) \gg \log^4 n$, we prove that the spectral gap between the $\binom{n-1}{d}$ smallest eigenvalues and the remaining $\binom{n-1}{d-1}$ eigenvalues is $np - 2\sqrt{dnp(1-p)} \, (1 + o(1))$ with high probability. This estimate follows from a more general result on eigenvalue confinement. In addition, we prove that the global distribution of the eigenvalues is asymptotically given by the semicircle law. The main ingredient of the proof is a F\"uredi-Koml\'os-type argument for random simplicial complexes, which may be regarded as sparse random matrix models with dependent entries.Comment: 29 pages, 6 figure

On the spectrum of Random Simplicial Complexes in Thermodynamic Regime

Linial-Meshulam complex is a random simplicial complex on $n$ vertices with a complete $(d-1)$-dimensional skeleton and $d$-simplices occurring independently with probability p. Linial-Meshulam complex is one of the most studied generalizations of the Erd\H{o}s-R{\'e}nyi random graph in higher dimensions. In this paper, we discuss the spectrum of adjacency matrices of the Linial-Meshulam complex when $np \rightarrow \lambda$. We prove the existence of a non-random limiting spectral distribution(LSD) and show that the LSD of signed and unsigned adjacency matrices of Linial-Meshulam complex are reflections of each other. We also show that the LSD is unsymmetric around zero, unbounded and under the normalization $1/\sqrt{\lambda d}$, converges to standard semicircle law as $\lambda \rightarrow \infty$. In the later part of the paper, we derive the local weak limit of the line graph of the Linial-Meshulam complex and study its consequence on the continuous part of the LSD.Comment: 32 Pages, 7 Figures, 1 Tabl

Contagion on Complex Systems: Structure and Dynamics

Complex systems are important when representing empirical systems in that they can model the underlying structure of interactions. Accounting for this structure can offer important insights for empirical systems such as social networks, biological processes, social phenomena, opinion formation, and many other examples. Pairwise networks are a representation of complex systems comprising a collection of entities (nodes) and pairwise interactions between entities (edges). Hypergraphs are a generalization of pairwise networks where interactions are no longer constrained to be between two nodes, but rather can be of arbitrary size. Modeling dynamics on hypergraphs can uncover rich behavior that one might not see if the dynamics simply occurred on a pairwise network. We focus on the interplay between the structure of a complex system, a particular dynamical process, and the resulting dynamical behavior. In the context of hypergraphs, we explain the effects that degree heterogeneity, assortative mixing, and community structure have on a simple hypergraph contagion model. Likewise, for pairwise networks, we explore both types of structure; structure in the underlying contact network and varying heterogeneity in the infection model. We examine the effect that representing inherently multiplex data (relationships of different types) with uniplex networks (relationships of a single type) has on the resulting dynamical behavior. We present two open source software libraries: (1) XGI, a package for representing complex systems with group interactions and (2) HyperContagion, a package for simulating hypergraph contagion, both of which can be used by the growing community of researchers studying higher-order interactions

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

Random walks and diffusion on networks

Random walks are ubiquitous in the sciences, and they are interesting from both theoretical and practical perspectives. They are one of the most fundamental types of stochastic processes; can be used to model numerous phenomena, including diffusion, interactions, and opinions among humans and animals; and can be used to extract information about important entities or dense groups of entities in a network. Random walks have been studied for many decades on both regular lattices and (especially in the last couple of decades) on networks with a variety of structures. In the present article, we survey the theory and applications of random walks on networks, restricting ourselves to simple cases of single and non-adaptive random walkers. We distinguish three main types of random walks: discrete-time random walks, node-centric continuous-time random walks, and edge-centric continuous-time random walks. We first briefly survey random walks on a line, and then we consider random walks on various types of networks. We extensively discuss applications of random walks, including ranking of nodes (e.g., PageRank), community detection, respondent-driven sampling, and opinion models such as voter models