## Funnelling effect in networks

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Sen, Parongama

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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.

Comment: Talk given in Complex2009, Shanghai; some results reported earlier in arXiv:0801.0370

Comment: Talk given in Complex2009, Shanghai; some results reported earlier in arXiv:0801.0370

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Physics - Physics and Society