Degree-dependent intervertex separation in complex networks

TL;DR

This paper investigates how the average shortest path length from a vertex depends on its degree in various growing networks, revealing different scaling behaviors influenced by network correlations and degree distributions.

Contribution

It introduces a degree-dependent shortest path length law for deterministic and stochastic networks, highlighting the effects of correlations and degree distributions.

Findings

Power-law correction to logarithmic dependence in scale-free networks

Linear degree dependence in exponential degree distribution trees

Comparison of growing networks with uncorrelated graphs

Abstract

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.