Rate equation approach for correlations in growing network models

Alain Barrat and Romualdo Pastor-Satorras

Phys. Rev. E 71, 036127 – Published 22 March 2005

Abstract

We propose a rate equation approach to compute two vertex correlations in scale-free growing network models based on the preferential attachment mechanism. The formalism, based on previous work of Szabó et al. [Phys. Rev. E. 67, 056102 (2002)] for the clustering spectrum, measuring three vertex correlations, is based on a rate equation in the continuous degree and time approximation for the average degree of the nearest neighbors of vertices of degree $k$ , with an appropriate boundary condition. We study the properties of both two and three vertex correlations for linear preferential attachment models, and also for a model yielding a large clustering coefficient. The expressions obtained are checked by means of extensive numerical simulations. The rate equation proposed can be generalized to more sophisticated growing network models, and also extended to deal with related correlation measures. As an example, we consider the case of a recently proposed model of weighted networks, for which we are able to compute a weighted two vertex correlation function, taking into account the strength of the interactions between connected vertices.

Received 25 October 2004

DOI:https://doi.org/10.1103/PhysRevE.71.036127

Authors & Affiliations

Alain Barrat¹ and Romualdo Pastor-Satorras²

¹Laboratoire de Physique Théorique (UMR du CNRS 8627), Bâtiment 210, Université de Paris-Sud 91405 Orsay, France
²Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Campus Nord B4, 08034 Barcelona, Spain

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Vol. 71, Iss. 3 — March 2005

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Images

Figure 1
Average degree for the nearest neighbors of the vertices of degree $k$ , ${\bar{k}}_{NN} (k, N)$ , for the LPA model for $m = 4$ , with $a = - 2$ (a) and $- 3$ (b). Symbols correspond to the different system sizes $N = 10^{4}$ (엯), $3 \times 10^{4}$ (◻), $10^{5}$ (◇), and $10^{6}$ (▵). Top plots: Raw data. Middle plots: Data rescaled by the size prefactor $N^{1 - 2 β}$ . Bottom plots: Data rescaled by the size prefactor with logarithmic corrections. The dashed lines represent a power-law decay with exponent $- 2 + 1 ∕ β$ .Reuse & Permissions
Figure 2
Clustering spectrum $\bar{c} (k, N)$ for the LPA model for $m = 4$ , with $a = - 2$ (a) and $- 3$ (b). Symbols correspond to the different system sizes $N = 10^{4}$ (엯), $N = 3 \times 10^{4}$ (◻), $N = 10^{5}$ (◇), and $N = 10^{6}$ (▵). Top plots: Raw data. Middle plots: Data rescaled by the size prefactor $N^{2 - 2 β}$ , corresponding to large $k$ . Bottom plots: Data rescaled by the size prefactor $N^{2 - 2 β} ∕ \ln (N)$ , corresponding to small $k$ . The full and dashed lines represent power-law decays with exponent $- 2 + 1 ∕ β$ and $- 4 + 2 ∕ β$ , respectively.Reuse & Permissions
Figure 3
Average degree for the nearest neighbors of the vertices of degree $k$ , ${\bar{k}}_{NN} (k, N)$ (a) and clustering spectrum $\bar{c} (k, N)$ (b) for the LPA with $m = 4$ and positive values of $a$ . Symbols correspond to the different system sizes $N = 10^{4}$ (엯), $N = 10^{5}$ (◻), and $N = 10^{6}$ (◇). Top plots: $a = 2$ . Middle plots: $a = 5$ . Bottom plots: $a = 10$ . Data for $\bar{c} (k, N)$ have been rescaled by the theoretical size prefactor $N$ . The dashed lines represent a power-law behavior with exponent $- 2 + 1 ∕ β$ .Reuse & Permissions
Figure 4
(a) Clustering spectrum $\bar{c} (k, N)$ for the generalized DMS model. The top plot corresponds to a system size $N = 10^{4}$ ; the bottom plot is for $N = 10^{6}$ . Symbols correspond to different values of the average degree $m = 2$ (엯), 4 (◻), and 8 (◇). The dashed lines are the theoretical predictions given by Eq. (62). (b) Average clustering coefficient $C (m)$ as a function of $m$ , for networks of size $N = 10^{5}$ . The full line represents the theoretical prediction Eq. (64). The dashed line marks the theoretical value for $m = 2$ , $C (2) = 2 π^{2} - 19$ .Reuse & Permissions
Figure 5
Average degree for the nearest neighbors of the vertices of degree $k$ , ${\bar{k}}_{NN} (k, N)$ , for the weighted growing model for $m = 4$ , with $δ = 2$ (a) and 5 (b). Symbols correspond to the different system sizes $N = 3 \times 10^{3}$ (엯), $N = 10^{4}$ (◻), $N = 3 \times 10^{4}$ (◇), and $N = 10^{5}$ (▵). Top plot: Raw data. Middle plot: Data rescaled by the size prefactor $N^{1 - 2 β}$ . Bottom plot: Data rescaled by the size prefactor with logarithmic corrections. The dashed lines represent a power-law decay with exponent $- 2 + 1 ∕ β$ . The middle plots display also the values of $N^{1 - 2 β} {\bar{k}}_{NN}^{w} (k, N)$ (filled symbols), which collapse onto a horizontal line, in agreement with the analytical prediction Eq. (78).Reuse & Permissions

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