Random graphs as models of networks

The degree distributions of complex networks are usually considered to be power law. However, it is not the case for a large number of them. We thus propose a new model able to build random growing networks with (almost) any wanted degree distribution. The degree distribution can either be theoretical or extracted from a real-world network. The main idea is to invert the recurrence equation commonly used to compute the degree distribution in order to find a convenient attachm...

Random graph (RG) models play a central role in the complex networks analysis. They help to understand, control, and predict phenomena occurring, for instance, in social networks, biological networks, the Internet, etc. Despite a large number of RG models presented in the literature, there are few concepts underlying them. Instead of trying to classify a wide variety of very dispersed models, we capture and describe concepts they exploit considering preferential attachment,...

In the last decades, the study of models for large real-world networks has been a very popular and active area of research. A reasonable model should not only replicate all the structural properties that are observed in real world networks (for example, heavy tailed degree distributions, high clustering and small diameter), but it should also be amenable to mathematical analysis. There are plenty of models that succeed in the first task but are hard to analyze rigorously. On ...

1. Basic constructions. 2. Equilibrium and nonequilibrium networks. 3. Equilibrium uncorrelated networks. 4. Nonequilibrium nongrowing scale-free nets. 5. Types of correlations. 6. When pair correlations are important. 7. When loops are important. 8. Pair degree-degree correlations in growing networks. 9. How to construct an equilibrium net with given degree-degree correlations. 10. How to construct a growing scale-free net with a given clustering (towards a real-space renorm...

A model of correlated random networks is examined, i.e. networks with correlations between the degrees of neighboring nodes. These nodes do not necessarily have to be direct neighbors, the maximum range of the correlations can be arbitrarily chosen. Two different methods for the creation of such networks are presented: one of them is a generalization of a well-known algorithm by Maslov and Sneppen. The percolation threshold for the model is calculated and the result is tested...

It appeared recently that the underlying degree distribution of networks may play a crucial role concerning their robustness. Empiric and analytic results have been obtained, based on asymptotic and mean-field approximations. Previous work insisted on the fact that power-law degree distributions induce high resilience to random failure but high sensitivity to attack strategies, while Poisson degree distributions are quite sensitive in both cases. Then, much work has been done...

Random networks are widely used to model complex networks and research their properties. In order to get a good approximation of complex networks encountered in various disciplines of science, the ability to tune various statistical properties of random networks is very important. In this manuscript we present an algorithm which is able to construct arbitrarily degree-degree correlated networks with adjustable degree-dependent clustering. We verify the algorithm by using empi...

In this paper, a random clique network model to mimic the large clustering coefficient and the modular structure that exist in many real complex networks, such as social networks, artificial networks, and protein interaction networks, is introduced by combining the random selection rule of the Erd\"os and R\'enyi (ER) model and the concept of cliques. We find that random clique networks having a small average degree differ from the ER network in that they have a large cluster...

Random network models play a prominent role in modeling, analyzing and understanding complex phenomena on real-life networks. However, a key property of networks is often neglected: many real-world networks exhibit spatial structure, the tendency of a node to select neighbors with a probability depending on physical distance. Here, we introduce a class of random spatial networks (RSNs) which generalizes many existing random network models but adds spatial structure. In these ...

Online social networks are a dominant medium in everyday life to stay in contact with friends and to share information. In Twitter, users can connect with other users by following them, who in turn can follow back. In recent years, researchers studied several properties of social networks and designed random graph models to describe them. Many of these approaches either focus on the generation of undirected graphs or on the creation of directed graphs without modeling the dep...