Adaptive networks appear in many biological applications. They combine topological evolution of the network with dynamics in the network nodes. Recently, the dynamics of adaptive networks has been investigated in a number of parallel studies from different fields, ranging from genomics to game theory. Here we review these recent developments and show that they can be viewed from a unique angle. We demonstrate that all these studies are characterized by common themes, most prominently: complex dynamics and robust topological self-organization based on simple local rules.

Untangling Food Webs: The factors affecting the stability of food webs are important in conservation and ecological restoration. Gross et al. (p. 747 ) used a generalized modeling approach to evaluate billions of replicates of food webs in order to reveal the properties that stabilize (or destabilize) food webs. Variability in the strength of trophic links between predator and prey strength affected stability in different ways depending on the size of the web—stabilizing only in relatively small food webs and destabilizing in larger ones. Universal topological rules were extracted for the patterns of network links that enhance food-web stability.

Information processing in complex systems is often found to be maximally efficient close to critical states associated with phase transitions. It is therefore conceivable that also neural information processing operates close to criticality. This is further supported by the observation of power-law distributions, which are a hallmark of phase transitions. An important open question is how neural networks could remain close to a critical point while undergoing a continual change in the course of development, adaptation, learning, and more. An influential contribution was made by Bornholdt and Rohlf, introducing a generic mechanism of robust self-organized criticality in adaptive networks. Here, we address the question whether this mechanism is relevant for real neural networks. We show in a realistic model that spike-time-dependent synaptic plasticity can self-organize neural networks robustly toward criticality. Our model reproduces several empirical observations and makes testable predictions on the distribution of synaptic strength, relating them to the critical state of the network. These results suggest that the interplay between dynamics and topology may be essential for neural information processing.

We present a universal approach to the investigation of the dynamics in generalized models. In these models the processes that are taken into account are not restricted to specific functional forms. Therefore a single generalized models can describe a class of systems which share a similar structure. Despite this generality, the proposed approach allows us to study the dynamical properties of generalized models efficiently in the framework of local bifurcation theory. The approach is based on a normalization procedure that is used to identify natural parameters of the system. The Jacobian in a steady state is then derived as a function of these parameters. The analytical computation of local bifurcations using computer algebra reveals conditions for the local asymptotic stability of steady states and provides certain insights on the global dynamics of the system. The proposed approach yields a close connection between modelling and nonlinear dynamics. We illustrate the investigation of generalized models by considering examples from three different disciplines of science: a socioeconomic model of dynastic cycles in china, a model for a coupled laser system and a general ecological food web.

In adaptive networks, fragmentation transitions have been observed in which the network breaks into disconnected components. We present an analytical approach for calculating the transition point in general adaptive network models. Using the example of an adaptive voter model, we demonstrate that the proposed approach yields good agreement with numerical results.