Networks whose structure of connections evolves in time constitute a big challenge in the study of synchronization, in particular when the time scales for the evolution of the graph topology are comparable with (or even longer than) those pertinent to the units' dynamics. We here focus on networks with a slow-switching structure, and show that the necessary conditions for synchronization, i.e. the conditions for which synchronization is locally stable, are determined by the time average of the largest Lyapunov exponents of transverse modes of the switching topologies. Comparison between fast-and slow-switching networks allows elucidating that slow-switching processes prompt synchronization in the cases where the Master Stability Function is concave, whereas fast-switching schemes facilitate synchronization for convex curves. Moreover, the condition of slow-switching enables the introduction of a control strategy for inducing synchronization in networks with arbitrary structure and coupling strength, which is of evident relevance for broad applications in real world systems. Synchronization plays a crucial role in a wide spectrum of technological, biological, and social networks 1-4 . Even though studied intensively under various aspects [5] [6] [7] [8] [9] [10] [11] , yet a full clarification is needed on the effects of time-dependent coupling structures [12] [13] [14] [15] [16] , particularly for those circumstances for which the coupling strengths (or connection topologies) may switch in time. For the specific condition (termed as the blinking network case) for which the graph structure changes much more rapidly than the units' dynamics, a comprehensive theoretical framework was established, and applications were developed accordingly [17] [18] [19] . However, such an extreme situation does not properly match the generic case, as in the majority of real-world systems the time scale for the coupling evolution turns out to be in fact secular with respect to that of the individual dynamics. For instance, the time scales of synaptic plasticity (the basis of most models in neural circuits) is known to expand from milliseconds to days [20] [21] [22] [23] , and long term plasticity between neurons is actually thought to be the real clue for the higher level functioning of the brain, such as long term memory 24 . A further example is evolutionary processes leading to mutations in genetic networks, which evidently take place over timescales much longer than those related to genetic expressions. Moreover, based on minimal models of interacting binary state nodes, different time scales of coupling switching may even lead the network into distinct phases of collective behavior 25 . Synchronization in slow-switching networks calls still for a general theory, and for elucidation. Despite the importance of the issue, current studies are so far limited to either simple unit dynamics (e.g. linear system 26 ), or simple coupling dynamics (e.g. on-off coupling 27,28 ), or continuous switching processes 29 . In this paper, we provide instead the general criteria for stable synchronization in slowly varying networked systems, and demonstrate that the stability of synchronization is fully characterized by the largest Lyapunov exponents of the transverse modes associated to the varying structures. Comparison to the fast-switching case allows to show that enhancement of synchronization can be predicted from the profile of the master stability function (MSF). Moreover, a network controlling strategy can be sorted out from our results, which embeds a controlling structure into the system, predesigned as a standard one (i.e. regardless of the specific topology to be controlled), in a way that it may find applications even to systems whose structures are unknown. We start by considering a networked system with N identical systems, described as =
This fast-and slow-switching difference immediately leads to an interesting result that when the MSF is concave (convex), a synchronization is easier to be maintained for fast-(slow-) switching than the opposite case REF .
Synchronization in slowly switching networks of coupled oscillators
Scientific Reports
Scientific Reports
Computer Science Medicine