Random Networks Tossing Biased Coins

We propose a random walks based model to generate complex networks. Many authors studied and developed different methods and tools to analyze complex networks by random walk processes. Just to cite a few, random walks have been adopted to perform community detection, exploration tasks and to study temporal networks. Moreover, they have been used also to generate scale-free networks. In this work, we define a random walker that plays the role of "edges-generator". In particula...

The focus of this work is on estimation of the in-degree distribution in directed networks from sampling network nodes or edges. A number of sampling schemes are considered, including random sampling with and without replacement, and several approaches based on random walks with possible jumps. When sampling nodes, it is assumed that only the out-edges of that node are visible, that is, the in-degree of that node is not observed. The suggested estimation of the in-degree dist...

The poster presents an analytic formalism describing metric properties of undirected random graphs with arbitrary degree distributions and statistically uncorrelated (i.e. randomly connected) vertices. The formalism allows to calculate the main network characteristics like: the position of the phase transition at which a giant component first forms, the mean component size below the phase transition, the size of the giant component and the average path length above the phase ...

Complex networks describe a wide range of systems in nature and society, much quoted examples including the cell, a network of chemicals linked by chemical reactions, or the Internet, a network of routers and computers connected by physical links. While traditionally these systems were modeled as random graphs, it is increasingly recognized that the topology and evolution of real networks is governed by robust organizing principles. Here we review the recent advances in the f...

The interactions between the components of complex networks are often directed. Proper modeling of such systems frequently requires the construction of ensembles of digraphs with a given sequence of in- and out-degrees. As the number of simple labeled graphs with a given degree sequence is typically very large even for short sequences, sampling methods are needed for statistical studies. Currently, there are two main classes of methods that generate samples. One of the existi...

In the last 15 years, statistical physics has been a very successful framework to model complex networks. On the theoretical side, this approach has brought novel insights into a variety of physical phenomena, such as self-organisation, scale invariance, emergence of mixed distributions and ensemble non-equivalence, that display unconventional features on heterogeneous networks. At the same time, thanks to their deep connection with information theory, statistical physics and...

The past two decades have seen significant successes in our understanding of complex networked systems, from the mapping of real-world social, biological and technological networks to the establishment of generative models recovering their observed macroscopic patterns. These advances, however, are restricted to pairwise interactions, captured by dyadic links, and provide limited insight into higher-order structure, in which a group of several components represents the basic ...

In many networks of scientific interest we know that the link between any pair of vertices conforms to a specific probability, such as the link probability in the Barab\'asi-Albert scale-free networks. Here we demonstrate how the distributions of link probabilities can be utilized to generate various complex networks simply and effectively. We focus in particular on the problem of complex network generation and develop a straightforward paradigm by using the strategy of verte...

We propose a simple random process inducing various types of random graphs and the scale free random graphs among others. The model is of a threshold nature and differs from the preferential attachment approach discussed in the literature before. The degree statistics of a random graph in our model is governed by the control parameter $\eta$ stirring the pure exponential statistics for the degree distribution (at $\eta=0,$ when a threshold is changed each time a new edge ad...

We use mathematical methods from the theory of tailored random graphs to study systematically the effects of sampling on topological features of large biological signalling networks. Our aim in doing so is to increase our quantitative understanding of the relation between true biological networks and the imperfect and often biased samples of these networks that are reported in public data repositories and used by biomedical scientists. We derive exact explicit formulae for de...