Comparison of synchronizability in temporal networks with small-world and random topologies

We present a case study of how the topology of temporal networks affects their ability to synchronize. Specifically, we investigate chaotic pendula coupled through two distinct topologies of temporal networks: small-world and random. Both temporal networks evolve from a nearest-neighbor ring structure with identical edge-rewiring probabilities and timescales. The chaotic pendula are initialized with the same distribution of initial conditions. Here, the edge-rewiring probability refers to the likelihood that one endpoint of an edge is disconnected and then reconnected to a different, randomly chosen node-with self-connections and duplicate edges excluded. The timescale denotes the interval between successive network-rewiring events. The objective is to determine which temporal topology (small-world or random) more effectively facilitates synchronization under the same edge-rewiring parameters. Our results show that synchronization occurs only when the edge-rewiring probability exceeds a critical threshold specific to each network type. For networks with short timescales, temporal random networks achieve synchronization at lower edge-rewiring probabilities and at a faster rate. In contrast, when edge-rewiring probabilities are higher, temporal small-world networks exhibit better synchronization performance over longer timescales. When the edge-rewiring probability reaches one, both network types exhibit comparable synchronization levels. Additionally, we analyze the synchronizability of these temporal networks at short timescales using the master-stability function and proposed the edge-variation rate for analyzing synchronizability at longer timescales. Both analytical approaches provide approximate predictions that align with our numerical results.