Transients Versus Attractors in Complex Networks

Understanding and predicting the behavior of complex multiagent systems like brain or ecological food net requires new approaches and paradigms. Traditional analyses based on just asymptotic results of behavior as time goes to infinity, or on straightforward mathematical images that can accommodate only fixed points or limit cycles do not tell much about these systems. To obtain sensible dynamical models of natural phenomena, such as the reproducible order observed in ecological, cognitive or behavioral experiments, one cannot afford to neglect the transient dynamics of the underlying complex network. In disclosing such dynamical mechanisms, the focus of interest must be on reproducible or, even, structurally stable transients. In this tutorial, we formulate the Winnerless Competition (WLC) principle that induces robust transient dynamics in open complex networks. The main point of WLC principle is the transformation of the acquired information into ensemble (spatio)-temporal output via intrinsic transient dynamics of the network. Such encoding provides a reproducible transient response, whose geometrical image in phase space is a stable heteroclinic sequence. We compile a diverse list of natural phenomena which can be rigorously modeled by the WLC. Together with the experimental and numerical results of the networks with different levels of complexity, we evaluate the robustness and reproducibility of the WLC dynamics and discuss the advantages of future possible application of the discussed approach.