Higher order clustering coefficients in Barabasi-Albert networks

TL;DR

This paper introduces higher order clustering coefficients for random networks, revealing that in Barabási-Albert networks, local neighborhoods are more tightly connected than the global network, and small clustering coefficients are due to randomness.

Contribution

The paper defines higher order clustering coefficients and applies them to analyze local connectivity in BA networks, uncovering new insights about neighborhood structure.

Findings

Average shortest path in a node's neighborhood is smaller than in the whole network.

Small clustering coefficients in large BA networks are due to randomness.

Neighborhood connectivity depends only on the network parameter m.

Abstract

Higher order clustering coefficients $C(x)$ are introduced for random networks. The coefficients express probabilities that the shortest distance between any two nearest neighbours of a certain vertex $i$ equals $x$, when one neglects all paths crossing the node $i$. Using $C(x)$ we found that in the Barab\'{a}si-Albert (BA) model the average shortest path length in a node's neighbourhood is smaller than the equivalent quantity of the whole network and the remainder depends only on the network parameter $m$. Our results show that small values of the standard clustering coefficient in large BA networks are due to random character of the nearest neighbourhood of vertices in such networks.