On Top and Beyond Networks

Today we tend to see networks wherever we look. In this talk, I would like to suggest two directions in which the network picture can be extended. One is related to the fact that most networks have a function, serve some purpose, which is either spontaneously evolved or purposefully designed. The efficiency of this function depends on the whole structure, topology, geometry, etc.: so there is a functional defined over the network. Furthermore, there is a success criterion (some extremal principle) associated with this functional. The functional may be very far from a simple convex one, it can display a complex phase space landscape. The extrema of this functional may define another network („on top of the underlying one”) which is in a highly nontrivial, indirect relationship with the original network, and which can carry a slow dynamics, punctuated equilibria, bifurcations, sensitivity to initial conditions, etc. The other extension consists in realizing that in most cases there is a background field acting on the nodes of the network that cannot be described in terms of a binary interaction. Its effect may be subtle yet profound, and very hard to model. These ideas will be illustrated by several examples from spin glasses to banking.