On Topological Minors in Random Simplicial Complexes

For random graphs, the containment problem considers the probability that a binomial random graph $G(n,p)$ contains a given graph as a substructure. When asking for the graph as a topological minor, i.e., for a copy of a subdivision of the given graph, it is well-known that the (sharp) threshold is at $p=1/n$. We consider a natural analogue of this question for higher-dimensional random complexes $X^k(n,p)$, first studied by Cohen, Costa, Farber and Kappeler for $k=2$. Improving previous results, we show that $p=\Theta(1/\sqrt{n})$ is the (coarse) threshold for containing a subdivision of any fixed complete $2$-complex. For higher dimensions $k>2$, we get that $p=O(n^{-1/k})$ is an upper bound for the threshold probability of containing a subdivision of a fixed $k$-dimensional complex.Comment: 15 page

Riemannian-geometric entropy for measuring network complexity

A central issue of the science of complex systems is the quantitative characterization of complexity. In the present work we address this issue by resorting to information geometry. Actually we propose a constructive way to associate to a - in principle any - network a differentiable object (a Riemannian manifold) whose volume is used to define an entropy. The effectiveness of the latter to measure networks complexity is successfully proved through its capability of detecting a classical phase transition occurring in both random graphs and scale--free networks, as well as of characterizing small Exponential random graphs, Configuration Models and real networks.Comment: 15 pages, 3 figure

Information geometric methods for complexity

Research on the use of information geometry (IG) in modern physics has witnessed significant advances recently. In this review article, we report on the utilization of IG methods to define measures of complexity in both classical and, whenever available, quantum physical settings. A paradigmatic example of a dramatic change in complexity is given by phase transitions (PTs). Hence we review both global and local aspects of PTs described in terms of the scalar curvature of the parameter manifold and the components of the metric tensor, respectively. We also report on the behavior of geodesic paths on the parameter manifold used to gain insight into the dynamics of PTs. Going further, we survey measures of complexity arising in the geometric framework. In particular, we quantify complexity of networks in terms of the Riemannian volume of the parameter space of a statistical manifold associated with a given network. We are also concerned with complexity measures that account for the interactions of a given number of parts of a system that cannot be described in terms of a smaller number of parts of the system. Finally, we investigate complexity measures of entropic motion on curved statistical manifolds that arise from a probabilistic description of physical systems in the presence of limited information. The Kullback-Leibler divergence, the distance to an exponential family and volumes of curved parameter manifolds, are examples of essential IG notions exploited in our discussion of complexity. We conclude by discussing strengths, limits, and possible future applications of IG methods to the physics of complexity.Comment: review article, 60 pages, no figure