Graph properties, graph limits and entropy

We study the relation between the growth rate of a graph property and the entropy of the graph limits that arise from graphs with that property. In particular, for hereditary classes we obtain a new description of the colouring number, which by well-known results describes the rate of growth. We study also random graphs and their entropies. We show, for example, that if a hereditary property has a unique limiting graphon with maximal entropy, then a random graph with this property, selected uniformly at random from all such graphs with a given order, converges to this maximizing graphon as the order tends to infinity.Comment: 24 page

Graph properties, graph limits, and entropy

We study the relation between the growth rate of a graph property and the entropy of the graph limits that arise from graphs with that property. In particular, for hereditary classes we obtain a new description of the coloring number, which by well-known results describes the rate of growth. We study also random graphs and their entropies. We show, for example, that if a hereditary property has a unique limiting graphon with maximal entropy, then a random graph with this property, selected uniformly at random from all such graphs with a given order, converges to this maximizing graphon as the order tends to infinity. © 2017 Wiley Periodicals, Inc

"Euboean" Pottery from Early Iron Age Eretria in the Light of the Neutron Activation Analysis

We analyze complexity in spatial network ensembles through the lens of graph entropy. Mathematically, we model a spatial network as a soft random geometric graph, i.e., a graph with two sources of randomness, namely nodes located randomly in space and links formed independently between pairs of nodes with probability given by a specified function (the "pair connection function") of their mutual distance. We consider the general case where randomness arises in node positions as well as pairwise connections (i.e., for a given pair distance, the corresponding edge state is a random variable). Classical random geometric graph and exponential graph models can be recovered in certain limits. We derive a simple bound for the entropy of a spatial network ensemble and calculate the conditional entropy of an ensemble given the node location distribution for hard and soft (probabilistic) pair connection functions. Under this formalism, we derive the connection function that yields maximum entropy under general constraints. Finally, we apply our analytical framework to study two practical examples: ad hoc wireless networks and the US flight network. Through the study of these examples, we illustrate that both exhibit properties that are indicative of nearly maximally entropic ensembles.Comment: 7 pages, 4 figure

Cut distance identifying graphon parameters over weak* limits

The theory of graphons comes with the so-called cut norm and the derived cut distance. The cut norm is finer than the weak* topology. Dole\v{z}al and Hladk\'y [Cut-norm and entropy minimization over weak* limits, J. Combin. Theory Ser. B 137 (2019), 232-263] showed, that given a sequence of graphons, a cut distance accumulation graphon can be pinpointed in the set of weak* accumulation points as a minimizer of the entropy. Motivated by this, we study graphon parameters with the property that their minimizers or maximizers identify cut distance accumulation points over the set of weak* accumulation points. We call such parameters cut distance identifying. Of particular importance are cut distance identifying parameters coming from subgraph densities, t(H,*). This concept is closely related to the emerging field of graph norms, and the notions of the step Sidorenko property and the step forcing property introduced by Kr\'al, Martins, Pach and Wrochna [The step Sidorenko property and non-norming edge-transitive graphs, J. Combin. Theory Ser. A 162 (2019), 34-54]. We prove that a connected graph is weakly norming if and only if it is step Sidorenko, and that if a graph is norming then it is step forcing. Further, we study convexity properties of cut distance identifying graphon parameters, and find a way to identify cut distance limits using spectra of graphons. We also show that continuous cut distance identifying graphon parameters have the "pumping property", and thus can be used in the proof of the the Frieze-Kannan regularity lemma.Comment: 48 pages, 3 figures. Correction when treating disconnected norming graphs, and a new section 3.2 on index pumping in the regularity lemm

Breaking of ensemble equivalence for dense random graphs under a single constraint

Two ensembles are often used to model random graphs subject to constraints: the microcanonical ensemble (= hard constraint) and the canonical ensemble (= soft constraint). It is said that breaking of ensemble equivalence (BEE) occurs when the specific relative entropy of the two ensembles does not vanish as the size of the graph tends to infinity. The latter means that it matters for the scaling properties of the graph whether the constraint is met for every single realisation of the graph or only holds as an ensemble average. In the literature, it was found that BEE is the rule rather than the exception for two classes: sparse random graphs when the number of constraints is of the order of the number of vertices and dense random graphs when there are two or more constraints that are frustrated. In the present paper we establish BEE for a third class: dense random graphs with a single constraint, namely, on the density of a given finite simple graph. We show that BEE occurs only in a certain range of choices for the density and the number of edges of the simple graph, which we refer to as the BEE-phase. We show that, in part of the BEE-phase, there is a gap between the scaling limits of the averages of the maximal eigenvalue of the adjacency matrix of the random graph under the two ensembles, a property that is referred to as spectral signature of BEE. Proofs are based on an analysis of the variational formula on the space of graphons for the limiting specific relative entropy derived by Den Hollander et al. (2018), in combination with an identification of the minimising graphons and replica symmetry arguments. We show that in the replica symmetric region of the BEE-phase, as the size of the graph tends to infinity, the microcanonical ensemble behaves like an Erd\H{o}s-R\'enyi random graph, while the canonical ensemble behaves like a mixture of two Erd\H{o}s-R\'enyi random graphs.Comment: 21 pages, 4 figures. arXiv admin note: text overlap with arXiv:1807.0775

On the Typical Structure of Graphs in a Monotone Property

Given a graph property $\mathcal{P}$, it is interesting to determine the typical structure of graphs that satisfy $\mathcal{P}$. In this paper, we consider monotone properties, that is, properties that are closed under taking subgraphs. Using results from the theory of graph limits, we show that if $\mathcal{P}$ is a monotone property and $r$ is the largest integer for which every $r$-colorable graph satisfies $\mathcal{P}$, then almost every graph with $\mathcal{P}$ is close to being a balanced $r$-partite graph.Comment: 5 page

Hitting Time of the Von Neumann Entropy for Networks Undergoing Rewiring

A random graph model defines a distribution over graphs, and this distribution induces a distribution over certain measurements of graphs, such as the Von Neumann entropy. Interestingly, the Von Neumann entropy of an empirical network typically has a very small probability to be drawn from the distribution over the Von Neumann entropy of graphs generated by the Erdős–Rényi random graph model with the same number of vertices and edges. It has been shown that Erdős–Rényi random graph model may be inappropriate for modeling certain properties of real-world networks such as the small-world property [22] and scale-free property [2], and the Von Neumann entropy provides yet another, complementary way to measure how real-world networks differ from Erdős–Rényi random graphs. Subjecting the network to a random rewiring process offers another approach to measure how far it is from an Erdős–Rényi random graph. In particular, it can be shown using Markov chain theory that the ensemble of networks after many non-degree-preserving rewirings limits to the ensemble of Erdős–Rényi random graphs. In this paper, we develop a connection between these two approaches by studying the Von Neumann entropy of networks undergoing rewiring. More specifically, we are interested in the number of rewiring times needed until the Von Neumann entropy of the rewired graph is larger than some quantile of the distribution over the Von Neumann entropy of Erdős–Rényi random networks, as it can be used to quantify the difference between any given network and networks generated by the Erdős–Rényi random graph model. However, performing a large number of simulations to compute the expected number of rewiring times is not computationally efficient, and we apply matrix perturbation methods to derive an estimation by using a small perturbation to the adjacency matrix to approximate one random rewiring of a graph. The estimation can be computed directly for a given network without performing any numerical simulation.Bachelor of Scienc