On the interaction of adaptive timescales on networks

The dynamics of real-world systems often involve multiple processes that influence system state. The timescales that these processes operate on may be separated by orders of magnitude or may coincide closely. Where timescales are not separable, the way that they relate to each other will be important for understanding system dynamics. In this paper, we present a short overview of how modellers have dealt with multiple timescales and introduce a definition to formalise conditions under which timescales are separable. We investigate timescale separation in a simple model, consisting of a network of nodes on which two processes act. The first process updates the values taken by the network’s nodes, tending to move a node’s value towards that of its neighbours. The second process influences the topology of the network, by rewiring edges such that they tend to more often lie between similar individuals. We show that the behaviour of the system when timescales are separated is very different from the case where they are mixed. When the timescales of the two processes are mixed, the ratio of the rates of the two processes determines the systems equilibrium state. We go on to explore the impact of heterogeneity in the system’s timescales, i.e., where some nodes may update their value and/or neighbourhood faster than others, demonstrating that it can have a significant impact on the equilibrium behaviour of the model.