Heterogeneity in Oscillator Networks: Are Smaller Worlds Easier to Synchronize?

Abstract

Small-world and scale-free networks are known to be more easily synchronized than regular lattices, which is usually attributed to the smaller network distance between oscillators. Surprisingly, we find that networks with a homogeneous distribution of connectivity are more synchronizable than heterogeneous ones, even though the average network distance is larger. We present numerical computations and analytical estimates on synchronizability of the network in terms of its heterogeneity parameters. Our results suggest that some degree of homogeneity is expected in naturally evolved structures, such as neural networks, where synchronizability is desirable.

Network synchronization: Spectral versus statistical properties

“…Many recent papers have investigated various facets of this relation. For example, some papers have reported correlations between network synchronizability and degree homogeneity [9][10][11], clustering coefficient [12], degree correlations [13], average degree, degree distribution, and so on [14]. In some cases the observed correlations can point in opposite directions; for instance, [9] finds that increasing the degree homogeneity improves synchronizability, whereas [14] and [13] report cases of better synchronizability for decreased homogeneity.…”

“…For example, some papers have reported correlations between network synchronizability and degree homogeneity [9][10][11], clustering coefficient [12], degree correlations [13], average degree, degree distribution, and so on [14]. In some cases the observed correlations can point in opposite directions; for instance, [9] finds that increasing the degree homogeneity improves synchronizability, whereas [14] and [13] report cases of better synchronizability for decreased homogeneity. Similarly, adding a few shortcut links to a sparse lattice is known to decrease the characteristic path length and improve synchronizability at the same time [15,16], although another study showed that better synchronization can result despite increased average distance [9].…”

Synchronization processes in complex networks

“…Recently, due to the realization that many networks in nature have complex topologies, these studies have been extended to systems where the pattern of connections is local but not necessarily regular [23,24,25,26,27,28,29,30,31,32]. Usually, due to the complexity of the analysis some further assumptions have been introduced.…”

“…The spectral information of the Laplacian matrix has been used to understand the structure of complex networks [41], and in particular to detect the community structure [42,43] (also the spectral analysis of the modularity matrix [44] can be used to this end). Recent studies have also focused on the spectral information of the Laplacian matrix and the synchronization dynamics [23,24,25,26,27,28,29,30]. The common approach is to take advantage of the master stability equation [45] to determine the relation between the relative stability of the synchronized state (via the ratio λ N /λ 2 ) and the heterogeneity of the topology, although sometimes some language abuse appears and authors talk about better or worse synchonizability instead of stability of the synchronized state.…”

Firing Patterns in Coupled Systems of Inhibitory Neurons

“…Since the average path length becomes short in small-world networks, synchronization is achieved more easily than in a regular lattice [10][11][12][13]. However, this is not the only condition; synchronizability also depends on the network size, the degree distribution (distribution of the number of links), the clustering coefficient, and so forth [14][15][16][17]. These studies are mainly based on the idea that the connections between cells promote in-phase synchronization.…”