Statistical Mechanics of Simplicial Complexes

Abstract

Simplicial complexes are a generalization of networks that can encode many-body interactions between more than two nodes. Whereas networks represent the interactions between the parts of a complex system using nodes and links, the simplices in a simplicial complex represent interactions between any number of nodes. In a number of applications these many-body interactions have been shown to carry important information about the complex system. Furthermore, the representation of systems as simplicial complexes has made it possible to characterise their structure in ways not possible with network representations, including using new tools inspired by algebraic topology or geometry. However, the use of simplicial complexes as a network science tool is still new and there remains an urgent need for a theoretical framework that can allow us to interpret the highly complex, high-dimensional data associated with real simplicial complexes. How do the new measures of structure that are being introduced relate to each other? And what can they tell us about the evolution or function of the simplicial complex? How can we best model simplicial complexes based on incomplete information? And what constitutes `interesting' or `signi cant' structure? In this thesis we tackle these problems through the development of stochastic models of simplicial complexes that can help us to disentangle the interactions, dependences and correlations between the structural properties of simplicial complexes and their evolution. First, we propose two maximum entropy models of a simplicial complex with given generalised degrees of the nodes (the number of simplices of a given dimension that a node participates in). These models have a clear use as null models for simplicial complexes as they are the most statistically appropriate models for simplicial complexes given knowledge of the generalized degrees. Importantly, they allow for a statistically rigorous understanding of the implications of particular choices of the generalized degrees for the structure of simplicial complexes and dynamics taking place upon them. Second, we propose a model of a simplicial complex that is weighted and growing. This model follows in the tradition of growing network models that seek to characterize the relations between simple mechanisms of growth (of the network) and reinforcement (of the weights) and their structural properties. The model exhibits a very rich variety of topologies and weight distributions for different values of the model parameters and different dimensions of simplices. Remarkably each of these distributions and scalings could be exhibited simultaneously within a single simplicial complex for faces of different dimension. The model shows that simple, plausible mechanisms of growth and reinforcement in simplicial complexes can produce a broad range of topologies and distributions, and shows the important role that dimension plays in determining these properties. Third, we propose a modelling framework for producing weighted networks and simplicial complexes which are both dense and scale-free. The growth mechanisms of the models contained within our framework are analogous to the Pitman-Yor process, a `ball-in-the-box' process wellknown among probability theorists for producing power-laws with exponent 2 (1; 2]. Our framework demonstrates the diffculty of producing a simple network which is both dense and scale-free. By relaxing the requirement for the network to be simple, either by a direction to the link or by reinterpreting the weight of a link as the number of multilinks between two nodes, we found that it was easy to create scale-free networks and simplicial complexes with tunable dense exponent 2 (1; 2].