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Identifying power-law scaling in real networks - indicative of preferential attachment - has proved controversial. Critics argue that measuring the temporal evolution of a network directly is better than measuring the degree distribution…

Fractals represent one of the fundamental manifestations of complexity, and fractal networks serve as tools for characterizing and investigating the fractal structures and properties of large-scale systems. Higher-order networks have…

Combinatorics · Mathematics 2026-05-01 Lin Qi , Jiaxin Zhang

We introduce a family of one-dimensional geometric growth models, constructed iteratively by locally optimizing the tradeoffs between two competing metrics, and show that this family is equivalent to a family of preferential attachment…

Disordered Systems and Neural Networks · Physics 2007-05-23 N. Berger , C. Borgs , J. T. Chayes , R. M. D'Souza , R. D. Kleinberg

We provide an analytic expression for the quantity described in the title. Namely, we perform a preferential attachment growth process to generate a scale-free network. At each stage we add a new node with $m$ new links. Let $k$ denote the…

Social and Information Networks · Computer Science 2014-08-13 Michael Small

Correlation between nodes is found to be a common and important property in many complex networks. Here we investigate degree correlations of the Barabasi-Albert (BA) Scale-Free model with both analytical results and simulations, and find…

Statistical Mechanics · Physics 2009-11-10 Huang Zhuang-Xiong , Wang Xin-Ran , Zhu Han

A version of ``preferential attachment'' random graphs, corresponding to linear ``weights'' with random ``edge additions,'' which generalizes some previously considered models, is studied. This graph model is embedded in a continuous-time…

Probability · Mathematics 2007-05-23 K. B. Athreya , A. P. Ghosh , S. Sethuraman

We propose and study a model of scale-free growing networks that gives a degree distribution dominated by a power-law behavior with a model-dependent, hence tunable, exponent. The model represents a hybrid of the growing networks based on…

Disordered Systems and Neural Networks · Physics 2009-11-10 H. Y. Lee , H. Y. Chan , P. M. Hui

What is the underlying mechanism leading to power-law degree distributions of many natural and artificial networks is still at issue. We consider that scale-free networks emerges from self-organizing process, and such a evolving model is…

Statistical Mechanics · Physics 2007-05-23 Gang Yan , Tao Zhou , Ying-Di Jin , Zhong-Qian Fu

Based on measurements of the Internet topology data, we found out that there are two mechanisms which are necessary for the correct modeling of the Internet topology at the Autonomous Systems (AS) level: the Interactive Growth of new nodes…

Networking and Internet Architecture · Computer Science 2009-09-29 Shi Zhou , Raul J. Mondragon

Based on the concept and techniques of first-passage probability in Markov chain theory, this letter provides a rigorous proof for the existence of the steady-state degree distribution of the scale-free network generated by the…

Probability · Mathematics 2008-05-13 Zhenting Hou , Xiangxing Kong , Dinghua Shi , Guanrong Chen

We present analytical results for the effect of preferential node deletion on the structure of networks that evolve via node addition and preferential attachment. To this end, we consider a preferential-attachment-preferential-deletion…

Physics and Society · Physics 2025-06-24 Barak Budnick , Ofer Biham , Eytan Katzav

Many realistic networks are scale-free, with small characteristic path lengths, high clustering, and power law in their degree distribution. They can be obtained by dynamical networks in which a preferential attachment process takes place.…

Physics and Society · Physics 2017-03-13 Francesco Caravelli , Alioscia Hamma , Massimiliano Di Ventra

Preferential attachment is an appealing edge generating mechanism for modeling social networks. It provides both an intuitive description of network growth and an explanation for the observed power laws in degree distributions. However,…

Methodology · Statistics 2017-12-21 Phyllis Wan , Tiandong Wang , Richard A. Davis , Sidney I. Resnick

Motivated by widely observed examples in nature, society and software, where groups of already related nodes arrive together and attach to an existing network, we consider network growth via sequential attachment of linked node groups, or…

Statistical Mechanics · Physics 2009-04-30 Vladimir Filkov , Zachary M. Saul , Soumen Roy , Raissa M. D'Souza , Premkumar T. Devanbu

The preferential attachment (PA) model is a popular way of modeling dynamic social networks, such as collaboration networks. Assuming that the PA function takes a parametric form, we propose and study the maximum likelihood estimator of the…

Statistics Theory · Mathematics 2022-08-17 Fengnan Gao , Aad van der Vaart

Most of the networks observed in real life obey power-law degree distribution. It is hypothesized that the emergence of such a degree distribution is due to preferential attachment of the nodes. Barabasi-Albert model is a generative…

Social and Information Networks · Computer Science 2011-11-28 S. M. Vijay Mahantesh , Sudarshan Iyengar , M. Vijesh , Shruthi Nayak , Nikitha Shenoy

This paper analyzes key properties of networks generated by geometric preferential attachment. We establish that the expected number of triangles is proportional to that of the standard preferential attachment model, with a proportionality…

Probability · Mathematics 2025-11-25 Chenxu Feng , Yifan Li

We study an evolving spatial network in which sequentially arriving vertices are joined to existing vertices at random according to a rule that combines preference according to degree with preference according to spatial proximity. We…

Probability · Mathematics 2015-06-30 Jonathan Jordan , Andrew R. Wade

In order to better understand dynamical functions on amounts of natural and man-made complex systems, lots of researchers from a wide range of disciplines, covering statistic physics, mathematics, theoretical computer science, and so on,…

Social and Information Networks · Computer Science 2019-05-09 Fei Ma , Ding Wang , Ping Wang , Bing Yao

A network is formed using the $N$ sites of an one-dimensional lattice in the shape of a ring as nodes and each node with the initial degree $k_{in}=2$. $N$ links are then introduced to this network, each link starts from a distinct node,…

Statistical Mechanics · Physics 2009-11-07 G. Mukherjee , S. S. Manna