Cliques in rank-1 random graphs: the role of inhomogeneity

We study the asymptotic behavior of the clique number in rank-1 inhomogeneous random graphs, where edge probabilities between vertices are roughly proportional to the product of their vertex weights. We show that the clique number is concentrated on at most two consecutive integers, for which we provide an expression. Interestingly, the order of the clique number is primarily determined by the overall edge density, with the inhomogeneity only affecting multiplicative constants or adding at most a $\log\log(n)$ multiplicative factor. For sparse enough graphs the clique number is always bounded and the effect of inhomogeneity completely vanishes.Comment: 29 page

Clustering Spectrum of scale-free networks

Real-world networks often have power-law degrees and scale-free properties such as ultra-small distances and ultra-fast information spreading. In this paper, we study a third universal property: three-point correlations that suppress the creation of triangles and signal the presence of hierarchy. We quantify this property in terms of $\bar c(k)$, the probability that two neighbors of a degree-$k$ node are neighbors themselves. We investigate how the clustering spectrum $k\mapsto\bar c(k)$ scales with $k$ in the hidden variable model and show that $c(k)$ follows a {\it universal curve} that consists of three $k$-ranges where $\bar c(k)$ remains flat, starts declining, and eventually settles on a power law $\bar c(k)\sim k^{-\alpha}$ with $\alpha$ depending on the power law of the degree distribution. We test these results against ten contemporary real-world networks and explain analytically why the universal curve properties only reveal themselves in large networks

Pattern theorems, ratio limit theorems and Gumbel maximal clusters for random fields

We study occurrences of patterns on clusters of size n in random fields on Z^d. We prove that for a given pattern, there is a constant a>0 such that the probability that this pattern occurs at most an times on a cluster of size n is exponentially small. Moreover, for random fields obeying a certain Markov property, we show that the ratio between the numbers of occurrences of two distinct patterns on a cluster is concentrated around a constant value. This leads to an elegant and simple proof of the ratio limit theorem for these random fields, which states that the ratio of the probabilities that the cluster of the origin has sizes n+1 and n converges as n tends to infinity. Implications for the maximal cluster in a finite box are discussed.Comment: 23 pages, 2 figure

Scale-free percolation

Abstract We formulate and study a model for inhomogeneous long-range percolation on Zd. Each vertex x¿Zd is assigned a non-negative weight Wx, where (Wx)x¿Zd are i.i.d. random variables. Conditionally on the weights, and given two parameters a,¿>0, the edges are independent and the probability that there is an edge between x and y is given by pxy=1-exp{-¿WxWy/|x-y|a}. The parameter ¿ is the percolation parameter, while a describes the long-range nature of the model. We focus on the degree distribution in the resulting graph, on whether there exists an infinite component and on graph distance between remote pairs of vertices. First, we show that the tail behavior of the degree distribution is related to the tail behavior of the weight distribution. When the tail of the distribution of Wx is regularly varying with exponent t-1, then the tail of the degree distribution is regularly varying with exponent ¿=a(t-1)/d. The parameter ¿ turns out to be crucial for the behavior of the model. Conditions on the weight distribution and ¿ are formulated for the existence of a critical value ¿c¿(0,8) such that the graph contains an infinite component when ¿>¿c and no infinite component when ¿0, les arêtes sont indépendantes et la probabilité qu’il existe un lien entre x et y est pxy=1-exp{-¿WxWy/|x-y|a}. Le paramètre ¿ est le paramètre de percolation tandis que a caractérise la portée des interactions. Nous étudierons la distribution des degrés dans le graphe résultant et l’existence éventuelle d’une composante infinie ainsi que la distance de graphe entre deux sites éloignés. Nous montrons d’abord que la queue de la distribution des degrés est liée à la queue de la distribution des poids. Quand la queue de la distribution de Wx est à variation régulière d’indice t-1, alors la queue de la distribution des degrés est à variation régulière d’indice ¿=a(t-1)/d. Le paramètre ¿ s’avère crucial pour décrire le modèle. Des conditions sur la distribution des poids et de ¿ sont formulées pour l’existence d’une valeur critique ¿c¿(0,8) telle que le graphe contienne une composante infinie quand ¿>¿c et aucune composante infinie quand

On the random structure of behavioural transition systems

Random graphs have the property that they are very predictable. Even by exploring a small part reliable observations are possible regarding their structure and size. An unfortunate observation is that standard models for random graphs, such as the Erdös-Rényi model, do not reflect the structure of the graphs that we find in behavioural modelling. In this paper we propose an alternative model, which we show to be a better reflection of ‘real’ state spaces. We show how we can use this structure to predict the size of state spaces, and we show that in this model software bugs are much easier to find than in the more standard random graph models. Not only gives this theoretical evidence that testing might be more effective than thought by some, but it also gives means to quantify the amount of residual errors based on a limited number of test runs

Ising models on power-law random graphs

We study a ferromagnetic Ising model on random graphs with a power-law degree distribution and compute the thermodynamic limit of the pressure when the mean degree is finite (degree exponent $\tau>2$), for which the random graph has a tree-like structure. For this, we adapt and simplify an analysis by Dembo and Montanari, which assumes finite variance degrees ($\tau>3$). We further identify the thermodynamic limits of various physical quantities, such as the magnetization and the internal energy

The structure of typical clusters in large sparse random configurations

The initial purpose of this work is to provide a probabilistic explanation of a recent result on a version of Smoluchowski's coagulation equations in which the number of aggregations is limited. The latter models the deterministic evolution of concentrations of particles in a medium where particles coalesce pairwise as time passes and each particle can only perform a given number of aggregations. Under appropriate assumptions, the concentrations of particles converge as time tends to infinity to some measure which bears a striking resemblance with the distribution of the total population of a Galton-Watson process started from two ancestors. Roughly speaking, the configuration model is a stochastic construction which aims at producing a typical graph on a set of vertices with pre-described degrees. Specifically, one attaches to each vertex a certain number of stubs, and then join pairwise the stubs uniformly at random to create edges between vertices. In this work, we use the configuration model as the stochastic counterpart of Smoluchowski's coagulation equations with limited aggregations. We establish a hydrodynamical type limit theorem for the empirical measure of the shapes of clusters in the configuration model when the number of vertices tends to $\infty$. The limit is given in terms of the distribution of a Galton-Watson process started with two ancestors

Random walk on the high-dimensional IIC

We study the asymptotic behavior the exit times of random walk from Euclidean balls around the origin of the incipient infinite cluster in a manner inspired by [26]. We do this by obtaining bounds on the effective resistance between the origin and the boundary of these Euclidean balls. We show that the geometric properties of long-range percolation clusters are significantly different from those of finite-range clusters. We also study the behavior of random walk on the backbone of the IIC and we prove that the Alexander-Orbach conjecture holds for the incipient infinite cluster in high dimensions, both for long-range percolation and for finite-range percolation