Random graphs containing arbitrary distributions of subgraphs

In 2007 we introduced a general model of sparse random graphs with independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random graph by adding not only edges but also other small graphs. In other words, we first construct an inhomogeneous random hypergraph with independent hyperedges, and then replace each...

We introduce a set of iterative equations that exactly solves the size distribution of components on small arbitrary graphs after the random removal of edges. We also demonstrate how these equations can be used to predict the distribution of the node partitions (i.e., the constrained distribution of the size of each component) in undirected graphs. Besides opening the way to the theoretical prediction of percolation on arbitrary graphs of large but finite size, we show how ou...

We study the family of network models derived by requiring the expected properties of a graph ensemble to match a given set of measurements of a real-world network, while maximizing the entropy of the ensemble. Models of this type play the same role in the study of networks as is played by the Boltzmann distribution in classical statistical mechanics; they offer the best prediction of network properties subject to the constraints imposed by a given set of observations. We giv...

Consider a connected graph $G=(E,V)$ with $N=|V|$ vertices. The main purpose of this paper is to explore the question of uniform sampling of a subtree of $G$ with $n$ nodes, for some $n\leq N$ (the spanning tree case correspond to $n=N$, and is already deeply studied in the literature). We provide new asymptotically exact simulation methods using Markov chains for general connected graphs $G$, and any $n\leq N$. We highlight the case of the uniform subtree of $\mathbb{Z}^2$ w...

Percolation theory can be used to describe the structural properties of complex networks using the generating function formulation. This mapping assumes that the network is locally tree-like and does not contain short-range loops between neighbours. In this paper we use the generating function formulation to examine clustered networks that contain simple cycles and cliques of any order. We use the natural generalisation to the Molloy-Reed criterion for these networks to descr...

Designing algorithms that generate networks with a given degree sequence while varying both subgraph composition and distribution of subgraphs around nodes is an important but challenging research problem. Current algorithms lack control of key network parameters, the ability to specify to what subgraphs a node belongs to, come at a considerable complexity cost or, critically, sample from a limited ensemble of networks. To enable controlled investigations of the impact and ro...

Random graph models are frequently used as a controllable and versatile data source for experimental campaigns in various research fields. Generating such data-sets at scale is a non-trivial task as it requires design decisions typically spanning multiple areas of expertise. Challenges begin with the identification of relevant domain-specific network features, continue with the question of how to compile such features into a tractable model, and culminate in algorithmic detai...

We present and investigate an extension of the classical random graph to a general class of inhomogeneous random graph models, where vertices come in different types, and the probability of realizing an edge depends on the types of its terminal vertices. This approach provides a general framework for the analysis of a large class of models. The generic phase structure is derived using generating function techniques, and relations to other classes of models are pointed out.

It appeared recently that the classical random graph model used to represent real-world complex networks does not capture their main properties. Since then, various attempts have been made to provide accurate models. We study here a model which achieves the following challenges: it produces graphs which have the three main wanted properties (clustering, degree distribution, average distance), it is based on some real-world observations, and it is sufficiently simple to make i...

Desirable random graph models (RGMs) should (i) be tractable so that we can compute and control graph statistics, and (ii) generate realistic structures such as high clustering (i.e., high subgraph densities). A popular category of RGMs (e.g., Erdos-Renyi and stochastic Kronecker) outputs edge probabilities, and we need to realize (i.e., sample from) the edge probabilities to generate graphs. Typically, each edge (in)existence is assumed to be determined independently. Howeve...