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We bring rigor to the vibrant activity of detecting power laws in empirical degree distributions in real-world networks. We first provide a rigorous definition of power-law distributions, equivalent to the definition of regularly varying…

Physics and Society · Physics 2019-10-23 Ivan Voitalov , Pim van der Hoorn , Remco van der Hofstad , Dmitri Krioukov

A random network model which allows for tunable, quite general forms of clustering, degree correlation and degree distribution is defined. The model is an extension of the configuration model, in which stubs (half-edges) are paired to form…

Probability · Mathematics 2012-07-31 Frank Ball , Tom Britton , David Sirl

We demonstrate how sophisticated graph properties, such as small distances and scale-free degree distributions, arise naturally from a reinforcement mechanism on layered graphs. Every node is assigned an a-priori i.i.d. fitness with…

Probability · Mathematics 2020-06-02 Markus Heydenreich , Christian Hirsch

Large real-life complex networks are often modeled by various random graph constructions and hundreds of further references therein. In many cases it is not at all clear how the modeling strength of differently generated random graph model…

Data Structures and Algorithms · Computer Science 2020-09-01 András Faragó , Rupei Xu

We present an algorithm for generating random networks with arbitrary degree distribution and Clustering (frequency of triadic closure). We use this algorithm to generate networks with exponential, power law, and poisson degree…

Statistical Mechanics · Physics 2009-11-10 Erik Volz

Unlike the well-studied models of growing networks, where the dominant dynamics consist of insertions of new nodes and connections, and rewiring of existing links, we study {\em ad hoc} networks, where one also has to contend with rapid and…

Disordered Systems and Neural Networks · Physics 2009-11-10 Nima Sarshar , Vwani Roychowdhury

We introduce a general stochastic model for the spread of rumours, and derive mean-field equations that describe the dynamics of the model on complex social networks (in particular those mediated by the Internet). We use analytical and…

Physics and Society · Physics 2009-11-13 Maziar Nekovee , Y. Moreno , G. Bianconi , M. Marsili

We find that scale-free random networks are excellently modeled by a deterministic graph. This graph has a discrete degree distribution (degree is the number of connections of a vertex) which is characterized by a power-law with exponent…

Statistical Mechanics · Physics 2009-11-07 S. N. Dorogovtsev , A. V. Goltsev , J. F. F. Mendes

Degree distribution, or equivalently called degree sequence, has been commonly used to be one of most significant measures for studying a large number of complex networks with which some well-known results have been obtained. By contrast,…

Physics and Society · Physics 2020-02-19 Fei Ma , Xiaoming Wang , Ping Wang

The control and risk assessment in complex information systems require to take into account extremes arising from nodes with large node degrees. Various sampling techniques like a Page Rank random walk, a Metropolis-Hastings Markov chain…

Statistics Theory · Mathematics 2015-02-18 Natalia Markovich

Network growth is currently explained through mechanisms that rely on node prestige measures, such as degree or fitness. In many real networks those who create and connect nodes do not know the prestige values of existing nodes, but only…

Disordered Systems and Neural Networks · Physics 2007-05-23 Santo Fortunato , Alessandro Flammini , Filippo Menczer

We provide a general framework for analyzing degree correlations between nodes separated by more than one step (i.e., beyond nearest neighbors) in complex networks. One probability and four conditional probabilities are introduced to fully…

Physics and Society · Physics 2018-06-20 Yuka Fujiki , Taro Takaguchi , Kousuke Yakubo

Many real world networks, such as social networks, are primarily formed through local interactions between agents. Additionally, in contrast with common network models, social and biological networks exhibit a high degree of clustering.…

Physics and Society · Physics 2015-03-10 Navid Dianati , Nima Dehmamy

Maximum entropy null models of networks come in different flavors that depend on the type of constraints under which entropy is maximized. If the constraints are on degree sequences or distributions, we are dealing with configuration…

Physics and Society · Physics 2020-10-30 Ivan Voitalov , Pim van der Hoorn , Maksim Kitsak , Fragkiskos Papadopoulos , Dmitri Krioukov

We apply a Bethe-Peierls approach to statistical-mechanics models defined on random networks of arbitrary degree distribution and arbitrary correlations between the degrees of neighboring vertices. Using the NP-hard optimization problem of…

Disordered Systems and Neural Networks · Physics 2009-11-07 Alexei Vazquez , Martin Weigt

We introduce a random intersection graph process aimed at modeling sparse evolving affiliation networks that admit tunable (power law) degree distribution and assortativity and clustering coefficients. We show the asymptotic degree…

Probability · Mathematics 2013-01-24 Mindaugas Bloznelis , Michal Karonski

We study the properties of the giant connected component in random graphs with arbitrary degree distribution. We concentrate on the degree-degree correlations. We show that the adjoining nodes in the giant connected component are correlated…

Statistical Mechanics · Physics 2010-05-11 Piotr Bialas , Andrzej K. Oleś

In this paper, we empirically investigate correlations among four centrality measures, originated from the social science, of various complex networks. For each network, we compute the centrality measures, from which the partial correlation…

Physics and Society · Physics 2007-05-23 Chang-Yong Lee

The probability distribution of number of ties of an individual in a social network follows a scale-free power-law. However, how this distribution arises has not been conclusively demonstrated in direct analyses of people's actions in…

Physics and Society · Physics 2014-08-25 Lev Muchnik , Sen Pei , Lucas C. Parra , Saulo D. S. Reis , Jose S. Andrade, , Shlomo Havlin , Hernan A. Makse

We propose a possible relation between complex networks and gravity. Our guide in our proposal is the power-law distribution of the node degree in network theory and the information approach to gravity. The established bridge may allow us…

General Physics · Physics 2012-11-30 J. A. Nieto