Funnelling Effect in Networks

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 D1 = 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) ∝ k−γ , D1 increases linearly with γ initially and then attains a constant value. It shows a power law behaviour, D1 ∝ N−ρ, with the number of nodes N where ρ is weakly dependent on γ for γ > 2.2. The latter variation is also independent of the number of searches. On stretched exponential networks with P (k) ∝ exp (−kδ), ρ is strongly dependent on δ. The funnelling distribution for a model social network, where the question of funnelling is most relevant, is also investigated.