Funnelling effect in networks

TL;DR

This paper investigates the funnelling effect in network searches, revealing power law behaviors in funnelling capacity distributions across various network types and analyzing how network parameters influence this phenomenon.

Contribution

It introduces the funnelling capacity distribution in networks, analyzes its behavior in different network models, and explores how network parameters affect the funnelling effect.

Findings

Power law behavior in funnelling capacity distribution in random networks.

D_1 increases linearly with gamma initially, then saturates in scale-free networks.

Funnelling distribution varies with network parameters like gamma and delta.

Abstract

Funnelling effect, in the context of searching on networks, precisely indicates that the search takes place through a few specific nodes. We define the funnelling capacity $f$ of a node as the fraction of successful dynamic paths through it with a fixed target. The distribution $D(f)$ of the fraction of nodes with funnelling capacity $f$ shows a power law behaviour in random networks (with power law or stretched exponential degree distribution) for a considerable range of values of the parameters defining the networks. Specifically we study in detail $D_1=D(f=1)$, which is the quantity signifying the presence of nodes through which all the dynamical paths pass through. In scale free networks with degree distribution $P(k) \propto k^{-\gamma}$, $D_1$ increases linearly with $\gamma$ initially and then attains a constant value. It shows a power law behaviour, $D_1 \propto N^{-\rho}$, with the number of nodes $N$ where $\rho$ is weakly dependent on $\gamma$ for $\gamma > 2.2$. The latter variation is also independent of the number of searches. On stretched exponential networks with $P(k) \propto \exp{(-k^\delta)}$, $\rho$ is strongly dependent on $\delta$. The funnelling distribution for a model social network, where the question of funnelling is most relevant, is also investigated.