The distribution of shortest path lengths in a class of node duplication network models

TL;DR

This paper analytically derives the distribution of shortest path lengths in a node duplication network model, revealing small-world properties and scale-free degree distribution, relevant for social and citation networks.

Contribution

It provides an exact analytical solution for the shortest path length distribution in a node duplication network model, highlighting its small-world and scale-free characteristics.

Findings

Shortest path length distribution derived analytically

Mean distance and diameter scale as ln t

Network exhibits small-world properties

Abstract

We present analytical results for the distribution of shortest path lengths (DSPL) in a network growth model which evolves by node duplication (ND). The model captures essential properties of the structure and growth dynamics of social networks, acquaintance networks and scientific citation networks, where duplication mechanisms play a major role. Starting from an initial seed network, at each time step a random node, referred to as a mother node, is selected for duplication. Its daughter node is added to the network, forming a link to the mother node, and with probability $p$ to each one of its neighbors. The degree distribution of the resulting network turns out to follow a power-law distribution, thus the ND network is a scale-free network. To calculate the DSPL we derive a master equation for the time evolution of the probability $P_t(L=\ell)$, $\ell=1,2,\dots$, where $L$ is the distance between a pair of nodes and $t$ is the time. Finding an exact analytical solution of the master equation, we obtain a closed form expression for $P_t(L=\ell)$. The mean distance, $\langle L \rangle_t$, and the diameter, $\Delta_t$, are found to scale like $\ln t$, namely the ND network is a small world network. The variance of the DSPL is also found to scale like $\ln t$. Interestingly, the mean distance and the diameter exhibit properties of a small world network, rather than the ultrasmall world network behavior observed in other scale-free networks, in which $\langle L \rangle_t \sim \ln \ln t$.