Exponential Random Simplicial Complexes

Abstract

Exponential random graph models have attracted significant research attention over the past decades. These models are maximum-entropy ensembles under the constraints that the expected values of a set of graph observables are equal to given values. Here we extend these maximum-entropy ensembles to random simplicial complexes, which are more adequate and versatile constructions to model complex systems in many applications. We show that many random simplicial complex models considered in the literature can be casted as maximum-entropy ensembles under certain constraints. We introduce and analyze the most general random simplicial complex ensemble $\mathbf{\Delta }$ with statistically independent simplices. Our analysis is simplified by the observation that any distribution $\mathbb{P}(O)$ on any collection of objects $\mathcal{O}=\{O\}$, including graphs and simplicial complexes, is maximum-entropy under the constraint that the expected value of $-\ln \mathbb{P}(O)$ is equal to the entropy of the distribution. With the help of this observation, we prove that ensemble $\mathbf{\Delta }$ is maximum-entropy under two types of constraints that fix the expected numbers of simplices and their boundaries.

22 pages, 6 figures

References in corpus (5)

• Emergent Complex Network Geometry
• Complex Quantum Network Geometries: Evolution and Phase Transitions
• Complex Quantum Network Manifolds in Dimension $d>2$ are Scale-Free
• Homological Domination in Large Random Simplicial Complexes
• Large random simplicial complexes, I

Cited by in corpus (24)

• Networks beyond pairwise interactions: structure and dynamics
• Simplicial models of social contagion
• The Statistical Physics of Real-World Networks
• What are higher-order networks?
• Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes
• Emergent Hyperbolic Network Geometry
• Network geometry with flavor: from complexity to quantum geometry
• Weighted Growing Simplicial Complexes
• Construction of and efficient sampling from the simplicial configuration model
• Network Geometry and Complexity
• Complex network view of evolving manifolds
• Dense Power-law Networks and Simplicial Complexes
• Complex Quantum Network Manifolds in Dimension $d>2$ are Scale-Free
• Emergent Network Modularity
• Random Simplicial Complexes: Models and Phenomena
• Aggregation models on hypergraphs
• Entropy of labeled versus unlabeled networks
• Hamiltonian Dynamics of Preferential Attachment
• Random Recursive Hypergraphs
• Structure of international trade hypergraphs
• Statistical physics of exchangeable sparse simple networks, multiplex networks and simplicial complexes
• Community detection in hypergraphs through hyperedge percolation
• Gelation in input-driven aggregation
• Novel Algorithms for Sampling Abstract Simplicial Complexes