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This article reviews and evaluates models of network evolution based on the notion of structural diversity. We show that diversity is an underlying theme of three principles of network evolution: the preferential attachment model,…

Social and Information Networks · Computer Science 2020-09-22 Jérôme Kunegis

We include complex connectivity structures and heterogeneity in models of multilayer networks or multilayer hypergraphs growing by preferential attachment. We consider the most generic connectivity structure, where the probability of…

Disordered Systems and Neural Networks · Physics 2025-05-26 Francesco Di Lauro , Luca Ferretti

Preferential attachment --- by which new nodes attach to existing nodes with probability proportional to the existing nodes' degree --- has become the standard growth model for scale-free networks, where the asymptotic probability of a node…

Adaptation and Self-Organizing Systems · Physics 2014-11-12 Michael Small , Yingying Li , Thomas Stemler , Kevin Judd

Many complex natural and physical systems exhibit patterns of interconnection that conform, approximately, to a network structure referred to as scale-free. Preferential attachment is one of many algorithms that have been introduced to…

Physics and Society · Physics 2014-11-18 Linjun Zhang , Michael Small , Kevin Judd

We introduce a growing network model in which a new node attaches to a randomly-selected node, as well as to all ancestors of the target node. This mechanism produces a sparse, ultra-small network where the average node degree grows…

Statistical Mechanics · Physics 2009-11-10 P. L. Krapivsky , S. Redner

There are diverse mechanisms driving the evolution of social networks. A key open question dealing with understanding their evolution is: How various preferential linking mechanisms produce networks with different features? In this paper we…

Physics and Society · Physics 2015-06-12 Haibo Hu , Jinli Guo , Xuan Liu

We consider a preferential attachment random graph with self-reinforcement. Each time a new vertex comes in, it attaches itself to an old vertex with a probability that is proportional to the sum of the degrees of that old vertex at all…

Probability · Mathematics 2025-07-29 Yogesh Dahiya , Frank den Hollander

We study the growth of a directed network, in which the growth is constrained by the cost of adding links to the existing nodes. We propose a new preferential-attachment scheme, in which a new node attaches to an existing node i with…

Statistical Mechanics · Physics 2007-05-23 Volkan Sevim , Per Arne Rikvold

Generated networks are widely used in network-based research as a convenient simulation environment. Generating universal networks that more accurately reflect real-world patterns is a cornerstone task. This study proposes a vari-linear…

Physics and Society · Physics 2026-04-27 Jinhu Ren , Linyuan Lü

This paper provides time-dependent expressions for the expected degree distribution of a given network that is subject to growth, as a function of time. We consider both uniform attachment, where incoming nodes form links to existing nodes…

Statistical Mechanics · Physics 2013-12-16 Babak Fotouhi , Michael G. Rabbat

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

We study the growth of bipartite networks in which the number of nodes in one of the partitions is kept fixed while the other partition is allowed to grow. We study random and preferential attachment as well as combination of both. We…

Disordered Systems and Neural Networks · Physics 2009-11-13 Fernando Peruani , Monojit Choudhury , Animesh Mukherjee , Niloy Ganguly

We consider an evolving preferential attachment random graph model where at discrete times a new node is attached to an old node, selected with probability proportional to a superlinear function of its degree. For such schemes, it is known…

Probability · Mathematics 2017-04-20 Sunder Sethuraman , Shankar C. Venkataramani

Complex networks in different areas exhibit degree distributions with heavy upper tail. A preferential attachment mechanism in a growth process produces a graph with this feature. We herein investigate a variant of the simple preferential…

Probability · Mathematics 2018-04-18 Angelica Pachon , Laura Sacerdote , Shuyi Yang

We study properties of some standard network models when the population is split into two types and the connection pattern between the types is varied. The studied models are generalizations of the Erd\H{o}s-R\'{e}nyi graph, the…

Probability · Mathematics 2017-07-17 Maria Deijfen , Robert Fitzner

We provide a framework for modeling social network formation through conditional multinomial logit models from discrete choice and random utility theory, in which each new edge is viewed as a "choice" made by a node to connect to another…

Social and Information Networks · Computer Science 2020-05-22 Jan Overgoor , Austin R. Benson , Johan Ugander

We highlight intriguing features of complex networks that are grown by \emph{redirection}. In this mechanism, a target node is chosen uniformly at random from the pre-existing network nodes and the new node attaches either to this initial…

Physics and Society · Physics 2025-01-14 P. L. Krapivsky , S. Redner

We propose a simple preferential attachment model of growing network using the complementary probability of Barab\'asi-Albert (BA) model, i.e., $\Pi(k_i) \propto 1-\frac{k_i}{\sum_j k_j}$. In this network, new nodes are preferentially…

Physics and Society · Physics 2016-01-20 A. Lachgar , A. Achahbar

We introduce a new type of preferential attachment tree that includes choices in its evolution, like with Achlioptas processes. At each step in the growth of the graph, a new vertex is introduced. Two possible neighbor vertices are selected…

Probability · Mathematics 2014-02-18 Yury Malyshkin , Elliot Paquette

We investigate the degree sequences of geometric preferential attachment graphs in general compact metric spaces. We show that, under certain conditions on the attractiveness function, the behaviour of the degree sequence is similar to that…

Probability · Mathematics 2012-08-27 Jonathan Jordan