Necessary Condition for Frequency Synchronization in Network Structures

Fumito Mori

Phys. Rev. Lett. 104, 108701 – Published 10 March 2010

Abstract

We present the necessary condition for complete frequency synchronization of phase-coupled oscillators in network structures. The surface area of a set of sites is defined as the number of links between the sites within the set and those outside the set. The necessary condition is that the surface area of any set of $c N$ ( $0 < c < 1$ ) oscillators in the $N$ -oscillator system must exceed $\sqrt{N}$ in the limit $N \to \infty$ . We also provide the necessary condition for macroscopic frequency synchronization. Thus, we identify networks in which one or both of the above mentioned types of synchronization do not occur.

Received 15 September 2009

DOI:https://doi.org/10.1103/PhysRevLett.104.108701

Authors & Affiliations

Fumito Mori^*

Department of Physics, Kyushu University, Fukuoka 812-8581, Japan

^*fmori@brain.riken.jp Address all correspondence to BTCC, Brain Science Institute, RIKEN, 2-1, Hirosawa, Wako-shi, Saitama 351-0198, Japan.

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Vol. 104, Iss. 10 — 12 March 2010

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Images

Figure 1
Definition of surface and surface area. $c N$ ( $0 < c < 1$ ) sites are chosen from $N$ sites in a network. The surface of a set ${c N}$ is defined as the set of links between the sites within ${c N}$ and those outside ${c N}$ . The surface area is defined as the number of links at the surface.Reuse & Permissions
Figure 2
Minimum surface area $S_{\min }$ for various networks. $S_{\min }$ is (a) 2 in a one-dimensinal lattice, (b) $2^{2} \sqrt{c N}$ in a square lattice, (c) $c N k p$ in a WS-type small-world network, and (d) 4 in the Sierpinski gasket.Reuse & Permissions
Figure 3
Schematic of the connected caveman graph consisting of $N / (k + 1)$ complete graphs or caves. A link in each cave is rewired to connect the cave with its neighbor.Reuse & Permissions
Figure 4
The order parameter $r$ for the Kuramoto-oscillator system on our network model for (a) $N^{0.5} < k < N^{0.6}$ and (b) $N^{0.2} < k < N^{0.3}$ .Reuse & Permissions

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