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 show that interpreting topology as a feedback circuit can provide novel insights on dynamics. 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, which acts as a feedback loop with the rest of the nodes. 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, 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. Furthermore, they highlight the effect of a handful of outlying hubs, rather than of the connectivity distribution law as a whole, on network dynamics.Author summary: Nature abounds with complex networks of interacting elements—from the proteins in our cells, through neural networks in our brains, to species interacting in ecosystems. In all of these fields, the relation between network structure and dynamics is an important research question. A recurring feature of natural networks is their heterogeneous structure: individual elements exhibit a huge diversity of connectivity patterns, which complicates the understanding of network dynamics. To address this problem, we devised a simplified approximation for complex structured networks which captures their dynamical properties. Separating out the largest “hubs”—a small number of nodes with disproportionately high connectivity—we represent them by a single node linked to the rest of the network. This enables us to borrow concepts from control theory, where a system’s output is linked back to itself forming a feedback loop. In this analogy, hubs in heterogeneous networks implement a feedback circuit with the rest of the network. The analogy reveals how these hubs can coordinate the network and drive it more easily towards stable states. Our approach enables analyzing dynamical properties of heterogeneous networks, which is difficult to achieve with existing techniques. It is potentially applicable to many fields where heterogeneous networks are important.