Scale free topology as an effective feedback system

Biological networks are often heterogeneous in their connectivity pattern, with degree distributions featuring a heavy tail of highly connected hubs. The implications of this heterogeneity on dynamical properties are a topic of much interest. Here we introduce a novel approach to analyze such networks the lumped hub approximation. Based on the observation that in finite networks a small number of hubs have a disproportionate effect on the entire system, we construct an approximation by lumping these nodes into a single effective hub, and replacing the rest by a homogeneous bulk. We use this approximation to study dynamics of networks with scale-free degree distributions, focusing on their probability of convergence to fixed points. We find that the approximation preserves convergence statistics over a wide range of settings. Our mapping provides a parametrization of scale free topology which is predictive at the ensemble level and also retains properties of individual realizations. Specifically for outgoing scale-free distributions, the role of the effective hub on the network can be elucidated by feedback analysis. We show that outgoing hubs have an organizing role that can drive the network to convergence, in analogy to suppression of chaos by an external drive. In contrast, incoming hubs have no such property, resulting in a marked difference between the behavior of networks with outgoing vs. incoming scale free degree distribution. Combining feedback analysis with mean field theory predicts a transition between convergent and divergent dynamics which is corroborated by numerical simulations. Our results show how interpreting topology as a feedback circuit can provide novel insights on dynamics. Furthermore, we highlight the effect of a handful of outlying hubs, rather than of the connectivity distribution law as a whole, on network dynamics.

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