Mechanism for linear preferential attachment in growing networks

The topology of large social, technical and biological networks such as the World Wide Web or protein interaction networks has caught considerable attention in the past few years (reviewed in Newman 2003 ), and analysis of the structure of such networks revealed that many of them can be classified as broad-tailed, scale-free-like networks, since their vertex connectivities follow approximately a power-law. Preferential attachment of new vertices to highly connected vertices is commonly seen as the main mechanism that can generate scale-free connectivity in growing networks ( Watts 2004 ). Here, we propose a new model that can generate broad-tailed networks even in the absence of network growth, by not only adding vertices, but also selectively eliminating vertices with a probability that is inversely related to the sum of their first- and second order connectivity.

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Coexistence in Preferential Attachment Networks

Combinatorics Probability Computing ◽

10.1017/s0963548315000383 ◽

2016 ◽

Vol 25 (6) ◽

pp. 797-822 ◽

Cited By ~ 9

Author(s):

TONĆI ANTUNOVIĆ ◽

ELCHANAN MOSSEL ◽

MIKLÓS Z. RÁCZ

Keyword(s):

Real World ◽

Preferential Attachment ◽

Theoretical Models ◽

Qualitative Feature ◽

Large Networks ◽

New Model ◽

Growing Networks ◽

Attachment Model

We introduce a new model of competition on growing networks. This extends the preferential attachment model, with the key property that node choices evolve simultaneously with the network. When a new node joins the network, it chooses neighbours by preferential attachment, and selects its type based on the number of initial neighbours of each type. The model is analysed in detail, and in particular, we determine the possible proportions of the various types in the limit of large networks. An important qualitative feature we find is that, in contrast to many current theoretical models, often several competitors will coexist. This matches empirical observations in many real-world networks.

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Temporal Behavior of Local Characteristics in Complex Networks with Preferential Attachment-Based Growth

Symmetry ◽

10.3390/sym13091567 ◽

2021 ◽

Vol 13 (9) ◽

pp. 1567

Author(s):

Sergei Sidorov ◽

Sergei Mironov ◽

Nina Agafonova ◽

Dmitry Kadomtsev

Keyword(s):

Preferential Attachment ◽

Dynamic Processes ◽

Total Degree ◽

Temporal Behavior ◽

Local Characteristics ◽

Growing Networks ◽

Order Of Magnitude ◽

Expected Values ◽

Fixed Node ◽

Skewness And Kurtosis

The study of temporal behavior of local characteristics in complex growing networks makes it possible to more accurately understand the processes caused by the development of interconnections and links between parts of the complex system that occur as a result of its growth. The spatial position of an element of the system, determined on the basis of connections with its other elements, is constantly changing as the result of these dynamic processes. In this paper, we examine two non-stationary Markov stochastic processes related to the evolution of Barabási–Albert networks: the first describes the dynamics of the degree of a fixed node in the network, and the second is related to the dynamics of the total degree of its neighbors. We evaluate the temporal behavior of some characteristics of the distributions of these two random variables, which are associated with higher-order moments, including their variation, skewness, and kurtosis. The analysis shows that both distributions have a variation coefficient close to 1, positive skewness, and a kurtosis greater than 3. This means that both distributions have huge standard deviations that are of the same order of magnitude as the expected values. Moreover, they are asymmetric with fat right-hand tails.

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ESTIMATION OF POWER-LAW EXPONENT OF DEGREE DISTRIBUTION USING MEAN VERTEX DEGREE

Modern Physics Letters B ◽

10.1142/s0217984909020230 ◽

2009 ◽

Vol 23 (17) ◽

pp. 2073-2088 ◽

Cited By ~ 7

Author(s):

NOBUTOSHI IKEDA

Keyword(s):

Numerical Simulations ◽

Power Law ◽

Degree Distribution ◽

Minimum Degree ◽

Preferential Attachment ◽

Random Walkers ◽

Power Law Exponent ◽

Growing Networks ◽

The Mean ◽

Ordinary Integral

The power-law exponent γ describing the form of the degree distribution of networks is related to the mean degree of the networks. This relation provides a useful method to estimate this exponent because mean degrees can easily be obtained by the number of vertices and edges in the networks. We improved the relation obtained only by ordinary integral approximation, and examined its availability for number of vertices, minimum degree considered, and range of γ, taking also into consideration the number of vertices with minimum degree. As a result, we were able to estimate slight deviations of γ from 3, which are usually observed in numerical simulations of growing networks with linear preferential attachment. Furthermore, using this method, we were also able to predict to what extent γ changed by joining pre-existing vertices in growing networks or by imposing restrictions to prevent the gaining of new edges for certain vertices. For cases where γ < 2, we estimated the power-law exponent of degree distributions of networks formed by traces of random walkers from the increased rate of vertices with created edges.

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