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Register a new userGrowth fluctuation in preferential attachment dynamics
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Authors
Yasuhiro Hashimoto
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In the Yule-Simon process, selection of words follows the preferential attachment mechanism, resulting in the power-law growth in the cumulative number of individual word occurrences. This is derived using mean-field approximation, assuming a continuum limit of both the time and number of word occurrences. However, time and word occurrences are inherently discrete in the process, and it is natural to assume that the cumulative number of word occurrences has a certain fluctuation around the average behavior predicted by the mean-field approximation. We derive the exact and approximate forms of the probability distribution of such fluctuation analytically and confirm that those probability distributions are well supported by the numerical experiments.
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We introduce a two-dimensional growth model where every new site is located, at a distance $r$ from the barycenter of the pre-existing graph, according to the probability law $1/r^{2+alpha_G} (alpha_G ge 0)$, and is attached to (only) one pre-existing site with a probability $propto k_i/r^{alpha_A}_i (alpha_A ge 0$; $k_i$ is the number of links of the $i^{th}$ site of the pre-existing graph, and $r_i$ its distance to the new site). Then we numerically determine that the probability distribution for a site to have $k$ links is asymptotically given, for all values of $alpha_G$, by $P(k) propto e_q^{-k/kappa}$, where $e_q^x equiv [1+(1-q)x]^{1/(1-q)}$ is the function naturally emerging within nonextensive statistical mechanics. The entropic index is numerically given (at least for $alpha_A$ not too large) by $q = 1+(1/3) e^{-0.526 alpha_A}$, and the characteristic number of links by $kappa simeq 0.1+0.08 alpha_A$. The $alpha_A=0$ particular case belongs to the same universality class to which the Barabasi-Albert model belongs. In addition to this, we have numerically studied the rate at which the average number of links $<k_i>$ increases with the scaled time $t/i$; asymptotically, $<k_i > propto (t/i)^beta$, the exponent being close to $beta={1/2}(1-alpha_A)$ for $0 le alpha_A le 1$, and zero otherwise. The present results reinforce the conjecture that the microscopic dynamics of nonextensive systems typically build (for instance, in Gibbs $Gamma$-space for Hamiltonian systems) a scale-free network.
We introduce a network growth model in which the preferential attachment probability includes the fitness vertex and the Euclidean distance between nodes. We grow a planar network around its barycenter. Each new site is fixed in space by obeying a power law distribution.
We present a simple model of network growth and solve it by writing down the dynamic equations for its macroscopic characteristics like the degree distribution and degree correlations. This allows us to study carefully the percolation transition using a generating functions theory. The model considers a network with a fixed number of nodes wherein links are introduced using degree-dependent linking probabilities $p_k$. To illustrate the techniques and support our findings using Monte-Carlo simulations, we introduce the exemplary linking rule $p_k$ proportional to $k^{-alpha}$, with $alpha$ between -1 and plus infinity. This parameter may be used to interpolate between different regimes. For negative $alpha$, links are most likely attached to high-degree nodes. On the other hand, in case $alpha>0$, nodes with low degrees are connected and the model asymptotically approaches a process undergoing explosive percolation.
In this article we presented a brief study of the main network models with growth and preferential attachment. Such models are interesting because they present several characteristics of real systems. We started with the classical model proposed by Barabasi and Albert: nodes are added to the network connecting preferably to other nodes that are more connected. We also presented models that consider more representative elements from social perspectives, such as the homophily between the vertices or the fitness that each node has to build connections. Furthermore, we showed a version of these models including the Euclidean distance between the nodes as a preferential attachment rule. Our objective is to investigate the basic properties of these networks as distribution of connectivity, degree correlation, shortest path, cluster coefficient and how these characteristics are affected by the preferential attachment rules. Finally, we also provided a comparison of these synthetic networks with real ones. We found that characteristics as homophily, fitness and geographic distance are significant preferential attachment rules to modeling real networks. These rules can change the degree distribution form of these synthetic network models and make them more suitable to model real networks.
A message passing algorithm is derived for recovering communities within a graph generated by a variation of the Barab{a}si-Albert preferential attachment model. The estimator is assumed to know the arrival times, or order of attachment, of the vertices. The derivation of the algorithm is based on belief propagation under an independence assumption. Two precursors to the message passing algorithm are analyzed: the first is a degree thresholding (DT) algorithm and the second is an algorithm based on the arrival times of the children (C) of a given vertex, where the children of a given vertex are the vertices that attached to it. Comparison of the performance of the algorithms shows it is beneficial to know the arrival times, not just the number, of the children. The probability of correct classification of a vertex is asymptotically determined by the fraction of vertices arriving before it. Two extensions of Algorithm C are given: the first is based on joint likelihood of the children of a fixed set of vertices; it can sometimes be used to seed the message passing algorithm. The second is the message passing algorithm. Simulation results are given.
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