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Register a new userDegree-dependent network growth: From preferential attachment to explosive percolation
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Added by
Bert Van Schaeybroeck
Publication date
2013
fields
Physics
and research's language is
English
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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.
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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.
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.
There has been a considerable amount of interest in recent years on the robustness of networks to failures. Many previous studies have concentrated on the effects of node and edge removals on the connectivity structure of a static network; the networks are considered to be static in the sense that no compensatory measures are allowed for recovery of the original structure. Real world networks such as the world wide web, however, are not static and experience a considerable amount of turnover, where nodes and edges are both added and deleted. Considering degree-based node removals, we examine the possibility of preserving networks from these types of disruptions. We recover the original degree distribution by allowing the network to react to the attack by introducing new nodes and attaching their edges via specially tailored schemes. We focus particularly on the case of non-uniform failures, a subject that has received little attention in the context of evolving networks. Using a combination of analytical techniques and numerical simulations, we demonstrate how to preserve the exact degree distribution of the studied networks from various forms of attack.
Global degree/strength based preferential attachment is widely used as an evolution mechanism of networks. But it is hard to believe that any individual can get global information and shape the network architecture based on it. In this paper, it is found that the global preferential attachment emerges from the local interaction models, including distance-dependent preferential attachment (DDPA) evolving model of weighted networks(M. Li et al, New Journal of Physics 8 (2006) 72), acquaintance network model(J. Davidsen et al, Phys. Rev. Lett. 88 (2002) 128701) and connecting nearest-neighbor(CNN) model(A. Vazquez, Phys. Rev. E 67 (2003) 056104). For DDPA model and CNN model, the attachment rate depends linearly on the degree or strength, while for acquaintance network model, the dependence follows a sublinear power law. It implies that for the evolution of social networks, local contact could be more fundamental than the presumed global preferential attachment. This is onsistent with the result observed in the evolution of empirical email networks.
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