Degree-dependent intervertex separation in complex networks

Details Degree-dependent intervertex separation in complex networks Degree-dependent intervertex separation in complex networks

2004-11-22

arxiv.org/abs/cond-mat/0411526v2

Phys. Rev. E 73, 056122 (2006)

Physics

Condensed Matter

Statistical Mechanics

8 pages, 3 figures

Scientific paper

We study the mean length $\ell(k)$ of the shortest paths between a vertex of degree $k$ and other vertices in growing networks, where correlations are essential. In a number of deterministic scale-free networks we observe a power-law correction to a logarithmic dependence, $\ell(k) = A\ln [N/k^{(\gamma-1)/2}] - C k^{\gamma-1}/N + ...$ in a wide range of network sizes. Here $N$ is the number of vertices in the network, $\gamma$ is the degree distribution exponent, and the coefficients $A$ and $C$ depend on a network. We compare this law with a corresponding $\ell(k)$ dependence obtained for random scale-free networks growing through the preferential attachment mechanism. In stochastic and deterministic growing trees with an exponential degree distribution, we observe a linear dependence on degree, $\ell(k) \cong A\ln N - C k$. We compare our findings for growing networks with those for uncorrelated graphs.

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