On a preferential attachment and generalized Pólya’s urn model

Collevecchio, Andrea; Cotar, Codina; LiCalzi, Marco

doi:10.1214/12-aap869

Cited by 44 publications

(47 citation statements)

References 23 publications

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“…An important feature of preferential attachment models is an asymptotic power-law degree distribution: the fraction P (k) of vertices in the network with degree k, goes as P (k) ∼ k −γ , with γ > 0, for large values of k. Real world modeling instances motivated the proposal of generalizations of the Barabási-Albert model, see e.g. [3,6,7,10,16,17,18,20,22]. A common characteristic of many of these models is the presence of the same attachment rule for all the nodes of the network.…”

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confidence: 99%

Scale-free behavior of networks with the copresence of preferential and uniform attachment rules

Pachón

Sacerdote

Yang

2018

Physica D: Nonlinear Phenomena

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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 attachment model, whose modifications are interesting for two main reasons: to analyze more realistic models and to study the robustness of the scale free behavior of the degree distribution.We introduce and study a model which takes into account two different attachment rules: a preferential attachment mechanism (with probability 1 − p) that stresses the rich get richer system, and a uniform choice (with probability p) for the most recent nodes. The latter highlights a trend to select one of the last added nodes when no information is available. The recent nodes can be either a given fixed number or a proportion (αn) of the total number of existing nodes. In the first case, we prove that this model exhibits an asymptotically power-law degree distribution. The same result is then illustrated through simulations in the second case. When the window of recent nodes has constant size, we herein prove that the presence of the uniform rule delays the starting time from which the asymptotic regime starts to hold.The mean number of nodes of degree k and the asymptotic degree distribution are also determined analytically. Finally, a sensitivity analysis on the parameters of the model is performed.Other models for different real world networks request the use of different growth paradigms and do not present the scale free property. For example, some networks exhibit the small-world phenomenon, in which sub-networks have connections between almost any two nodes within them. Furthermore, most pairs of nodes are connected by at least one short path. On the other hand, one of the most studied models, the Erdös-Rényi random graph, does not exhibit either the power-law behavior for the degree distribution of its nodes, nor the small-world phenomenon [4,12,13,14,15]. Mathematically, the growth of networks can be modeled through random graph processes, i.e. a family (G t ) t∈N of random graphs (defined on a common probability space), where t is interpreted as time. Different features of the model are then described as properties of the corresponding random graph process. In particular, the interest often focuses on the degree, deg(v, t), of a vertex v at time t, that is on the total number of incoming and (or) outgoing edges to and (or) from v, respectively. In this framework, new nodes of the Barabási-Albert model link with higher probabilities with nodes of higher degree. An important feature of preferential attachment models is an asymptotic power-law degree distribution: the fraction P (k) of vertices in the network with degree k, goes as P (k) ∼ k −γ , with γ > 0, for large values of k. Real world modeling instances motivated the proposal of generalizations of the Barabási-Albert model, see e.g. [3,6,7,10,16,17,18,20,22]. A common characteristic of many of these models is the presence of the...

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confidence: 99%

Scale-free behavior of networks with the copresence of preferential and uniform attachment rules

Pachón

Sacerdote

Yang

2018

Physica D: Nonlinear Phenomena

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“…[32]). The best known example of reinforced stochastic process is the standard Eggenberger-Pòlya urn [21,29], which has been widely studied and generalized (some recent variants can be found in [4,5,8,10,12,14,23,24,27]).…”

Section: Framework Model and Motivationsmentioning

confidence: 99%

Interacting reinforced stochastic processes: Statistical inference based on the weighted empirical means

Aletti¹,

Crimaldi²,

Ghiglietti³

2020

Bernoulli

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This work deals with a system of interacting reinforced stochastic processes, where each process X j = (Xn,j )n is located at a vertex j of a finite weighted direct graph, and it can be interpreted as the sequence of "actions" adopted by an agent j of the network. The interaction among the dynamics of these processes depends on the weighted adjacency matrix W associated to the underlying graph: indeed, the probability that an agent j chooses a certain action depends on its personal "inclination" Zn,j and on the inclinations Z n,h , with h = j, of the other agents according to the entries of W . The best known example of reinforced stochastic process is the Pòlya urn.The present paper characterizes the asymptotic behavior of the weighted empirical means Nn,j = n k=1 q n,k X k,j , proving their almost sure synchronization and some central limit theorems in the sense of stable convergence. By means of a more sophisticated decomposition of the considered processes adopted here, these findings complete and improve some asymptotic results for the personal inclinations Z j = (Zn,j )n and for the empirical means X j = ( n k=1 X k,j /n)n given in recent papers (e.g. [1,2,18]). Our work is motivated by the aim to understand how the different rates of convergence of the involved stochastic processes combine and, from an applicative point of view, by the construction of confidence intervals for the common limit inclination of the agents and of a test statistics to make inference on the matrix W , based on the weighted empirical means. In particular, we answer a research question posed in [1].

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“…Any weight w satisfying condition (6) is called super-linear (or strong), and the corresponding ERRW is called strongly reinforced walk.…”

Section: Edge-reinforced Random Walkmentioning

confidence: 99%

“…Our main result for edge-reinforced random walks is Theorem 1.1 Let G be an infinite connected graph of bounded degree. If w satisfies (6) and (8), the edge-reinforced random walk on G traverses a random attracting edge at all large times a.s., that is P G (G ∞ has only one edge) = 1.…”

Section: Edge-reinforced Random Walkmentioning

confidence: 99%

Edge- and vertex-reinforced random walks with super-linear reinforcement on infinite graphs

Cotar¹,

Thacker²

2017

Ann. Probab.

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In this paper we introduce a new simple but powerful general technique for the study of edge-and vertex-reinforced processes with super-linear reinforcement, based on the use of order statistics for the number of edge, respectively of vertex, traversals. The technique relies on upper bound estimates for the number of edge traversals, proved in a different context by Cotar and Limic [Ann. Appl. Probab. (2009)] for finite graphs with edge reinforcement. We apply our new method both to edge-and to vertex-reinforced random walks with super-linear reinforcement on arbitrary infinite connected graphs of bounded degree. We stress that, unlike all previous results for processes with super-linear reinforcement, we make no other assumption on the graphs.For edge-reinforced random walks, we complete the results of Limic and Tarrès [Ann. Probab. (2013)]. We show that on any infinite connected graph of bounded degree, with reinforcement weight function w taken from a general class of reciprocally summable reinforcement weight functions, the walk traverses two random neighbouring attracting vertices at all large times.

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On a preferential attachment and generalized Pólya’s urn model

Cited by 44 publications

References 23 publications

Scale-free behavior of networks with the copresence of preferential and uniform attachment rules

Scale-free behavior of networks with the copresence of preferential and uniform attachment rules

Interacting reinforced stochastic processes: Statistical inference based on the weighted empirical means

Edge- and vertex-reinforced random walks with super-linear reinforcement on infinite graphs

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