Synchronization: An Obstacle to Identification of Network Topology

Abstract

Identification of network topology is an important issue for physics, biology, engineering, and other science disciplines. Under the application of an adaptive-feedback control algorithm, we explore a process through which the topology of an unknown dynamical network can be identified. Analytical results show that the key to guaranteeing successful topology identification is the existence of the condition that all coupling terms of the unknown network are linearly independent of the synchronization manifold between the drive (unknown) network and a response network. When complete synchronization is achieved in the unknown network, the linearly independent condition is no longer satisfied, making its topology unidentifiable. We also find that partial synchronization in the unknown network implies a part of topology being unidentifiable. The results can be extended to projection synchronization and some generalized synchronization. Furthermore, we show that when the network dynamical equation is stable, its topology can be identified subject to some limitations. Finally, a method of avoiding synchronization for a network is presented and verified by numerical simulations.

Repeated-drive adaptive feedback identification of network topologies

“…We assume that the topology of the network, i.e., the adjacency matrix C, is unknown, which could be determined based on the dynamics of X(t) = {x i (t)}. To do this, we first resort to the adaptive feedback control method [9,17,24]. One can design a drive-response system, where the drive system is described by Eqs.…”

“…The parameter k i is governed bẏ k i = e i 2 /2. The adjacency matrix of the response network d ij (t) evolves with time and adapts itself to the drive network d ij = −e T i H(y j ) [9,17,24]. When the response network is synchronized by the drive system, i.e., y i (t) = x i (t), the response adjacency matrix d ij approaches a steady one.…”

“…. ,N. However, the values of d * ij do not necessarily coincide with c ij when nodes in the network, for example, say, i and j , are synchronous to each other [x i (t) = x j (t)] [24]. When all nodes are not synchronized, the output functions H[x j (t)] are linearly independent of each other, and the coefficients…”

“…It can be found that d 14 (t) evolves to a value far from c 14 = 1. This indicates that synchronization among nodes in a network degenerates the linking information among nodes and thus hinders the identification of network topology [24]. Let us introduce the success rate, i.e., the probability of successful identification, P s (s).…”

“…It should be valuable if one can infer the network structure from very limited data. This often happens when there exists synchronization among nodes in a network, where the network dynamics becomes degenerated [24]. Most of the previous schemes cannot be successfully applied to topology detections in the presence of synchronous motion on a network.…”