Does dynamics reflect topology in directed networks?

– We present and analyze a topologically induced transition from ordered, synchronized to disordered dynamics in directed networks of oscillators. The analysis reveals where in the space of networks this transition occurs and its underlying mechanisms. If disordered, the dynamics of the units is precisely determined by the topology of the network and thus characteristic for it. We develop a method to predict the disordered dynamics from topology. The results suggest a new route towards understanding how the precise dynamics of the units of a directed network may encode information about its topology. Networks of interacting units prevail in a variety of systems, ranging from gene regulatory networks and neural networks to food webs and the world wide web [1, 2]. A fundamental question is: What kind of dynamics can we expect given a network of prescribed connection topology [3]? Even in networks of known dynamical units, known type of interactions between them and known topological details, it is hard to infer which kind of typical collective dynamics the network will display (cf. Refs. [3–6]). If parts of the network exhibit residual symmetries, such as permutation or translation invariance, some general properties of the dynamics can be deduced [6]. If, however, no symmetries remain, it is still an open question how topological factors can control network dynamics. See, e.g., [7–9] for some interesting recent approaches for phase oscillator networks of specific connectivities. In this Letter, we study directed networks of phase oscillators that exhibit a mechanism to synchronize and reveal general principles about how topology controls dynamics: Specifically, in networks with an invariant state of in-phase synchrony we analyze how the topology can control the units' dynamics in a neighborhood of synchrony. We find that such networks, depending on their coarse-scale topological properties, belong to one of two classes exhibiting very different long term dynamics: Networks of class I show in-phase synchrony in which each unit displays identical dynamics, independent of the unit's topological identity [10], i.e. independent of where in the network it is located. Networks of class II, instead of synchrony, show disordered dynamics. Here, together with the initial condition, the fine-scale topology precisely controls the dynamics of each unit. The dynamics therefore strongly depends on where the unit is located in the network – its topological identity. We develop a method to predict the disordered dynamics from the network's topology. Due to their topological origin, both the …