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Fluctuations in Network Dynamics

By Peyman Khorsand

Taken from: “Fluctuations in Network Dynamics”, by M. Argollo de Menezes and A.-L. Barabasi

The flow through a node in a network is time dependent. This time dependence can be partly be described by mean flow, f_{\mathrm i}(t) and the fluctuation around this mean \sigma_{\mathrm i}(t).
In most natural networks, random, scale-free or small-world there is a functional dependence between mean and fluctuation of the flow through the nodes. In this paper it is claimed that the relation between f and \sigma in different networks (only scale-free and random network) fall into two different categories and characterized by their \alpha-exponent

\sigma \sim \langle f \rangle^{\alpha}

In all the networks they studied \alpha is equal to 1/2 or 1. In the networks with \alpha=1/2 the internal noise is responsible for flow fluctuations, while it was claimed that in networks with \alpha=1 the external noise is responsible for the fluctuations. Two different models are proposed.

Model 1: At any time step, W number of walkers are placed randomly on the network nodes, the preform M step walks.

Model 2: At any time step W random pairs of nodes are selected and they are connected through the shortest path between them (degeneracy problem is not discussed).

In both cases we observe \alpha=1/2 if W is fixed and \alpha=1 if W has large fluctuations. In general the fluctuations on a given nodes can be decompose into internal and external components
\sigma_{\mathrm i}^2=(\sigma_{\mathrm i}^{\mathrm {int}})^2 +(\sigma_{\mathrm i}^{\mathrm {ext}})^2
\sigma_{\mathrm i}^2 = a_{\mathrm i}^2 \langle f_{\mathrm i} \rangle +\left[ {\sigma_{\mathrm {dr}}\over \langle W(t)\rangle } \langle f_{\mathrm i} \rangle \right]^2

where \sigma_{\mathrm{dr}}=\sigma_{\mathrm{dr}}(\Delta W), represent the external driving force in the noise magnitude. Moreover, by increasing \Delta W a transition form \alpha =1 /2 to \alpha=1 behavior can be seen (some issues regarding the fitting process should be considered.).

In conclusion: “The \alpha=1/2 captures an endogenous behavior determined by the system’s internal fluctuations” while “The \alpha=1 exponent describes driven systems, in which the fluctuations of individual nodes are dominated by the time dependent changes in the external driving forces.”

This entry was posted on October 11, 2008 at 9:40 am and is filed under Stochastic Processes. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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