Gradient Networks

TL;DR

This paper introduces gradient networks formed by local scalar field gradients on substrate networks, derives their in-degree distribution for Erdős-Rényi graphs, and explores implications for network congestion and the emergence of scale-free structures.

Contribution

It provides an exact in-degree distribution formula for gradient networks on Erdős-Rényi graphs and links gradient network properties to network congestion and the evolution of scale-free networks.

Findings

In-degree distribution follows a power law R(l)~1/l for Erdős-Rényi graphs.

Scale-free networks are less prone to congestion compared to random graphs.

Gradient network analysis explains the spontaneous emergence of scale-free structures.

Abstract

We define gradient networks as directed graphs formed by local gradients of a scalar field distributed on the nodes of a substrate network G. We derive an exact expression for the in-degree distribution of the gradient network when the substrate is a binomial (Erdos-Renyi) random graph, G(N,p). Using this expression we show that the in-degree distribution R(l) of gradient graphs on G(N,p) obeys the power law R(l)~1/l for arbitrary, i.i.d. random scalar fields. We then relate gradient graphs to congestion tendency in network flows and show that while random graphs become maximally congested in the large network size limit, scale-free networks are not, forming fairly efficient substrates for transport. Combining this with other constraints, such as uniform edge cost, we obtain a plausible argument in form of a selection principle, for why a number of spontaneously evolved massive networks are scale-free. This paper also presents detailed derivations of the results recently reported in Nature, vol. 428, pp. 716 (2004).