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Partial Phase Cohesiveness in Networks of Kuramoto Oscillator Networks

Yuzhen Qin, Yu Kawano, Oscar Portoles, Ming Cao

TL;DR

It is analytically shown that partial synchronization can be induced by strong regional connections in coupled subnetworks of Kuramoto oscillators, the first known criterion that is applicable to non-complete graphs.

Abstract

Partial, instead of complete, synchronization has been widely observed in various networks including, in particular, brain networks. Motivated by data from human brain functional networks, in this technical note, we analytically show that partial synchronization can be induced by strong regional connections in coupled subnetworks of Kuramoto oscillators. To quantify the required strength of regional connections, we first obtain a critical value for the algebraic connectivity of the corresponding subnetwork using the incremental 2 norm. We then introduce the concept of the generalized complement graph, and obtain another condition on the weighted nodal degree by using the incremental infinity norm. Under these two conditions, regions of attraction for partial phase cohesiveness are estimated in the forms of the incremental 2 and infinity norms, respectively. Our result based on the incremental infinity norm is the first known criterion that is applicable to non-complete graphs. Numerical simulations are performed on a two-level network to illustrate our theoretical results; more importantly, we use real anatomical brain network data to show how our results may reveal the interplay between anatomical structure and empirical patterns of synchrony.

Partial Phase Cohesiveness in Networks of Kuramoto Oscillator Networks

TL;DR

It is analytically shown that partial synchronization can be induced by strong regional connections in coupled subnetworks of Kuramoto oscillators, the first known criterion that is applicable to non-complete graphs.

Abstract

Partial, instead of complete, synchronization has been widely observed in various networks including, in particular, brain networks. Motivated by data from human brain functional networks, in this technical note, we analytically show that partial synchronization can be induced by strong regional connections in coupled subnetworks of Kuramoto oscillators. To quantify the required strength of regional connections, we first obtain a critical value for the algebraic connectivity of the corresponding subnetwork using the incremental 2 norm. We then introduce the concept of the generalized complement graph, and obtain another condition on the weighted nodal degree by using the incremental infinity norm. Under these two conditions, regions of attraction for partial phase cohesiveness are estimated in the forms of the incremental 2 and infinity norms, respectively. Our result based on the incremental infinity norm is the first known criterion that is applicable to non-complete graphs. Numerical simulations are performed on a two-level network to illustrate our theoretical results; more importantly, we use real anatomical brain network data to show how our results may reveal the interplay between anatomical structure and empirical patterns of synchrony.

Paper Structure

This paper contains 8 sections, 8 theorems, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Incremental $2$-Norm
  4. Incremental $\infty$-Norm
  5. Main Results
  6. Comparisons
  7. Numerical Examples
  8. Concluding Remarks

Key Result

Theorem 1

Assume that the algebraic connectivity of $\mathcal{G}_{r}$ is greater than the critical value specified by Then, each of the following equations has a unique solution, $\gamma_s\in[0,\pi/2)$ and $\gamma_m\in(\pi/2,\pi]$, respectively, where ${\rm \;sinc} (\eta) = \sin(\eta)/\eta$ for any $\eta \in \mathbb S^1$. Furthermore, the following statements hold:

Figures (2)

  • Figure 1: (a) The network structure considered in Example \ref{['exam:partial']}; (b) the interconnection structure: each oscillator in a community is connected to exact one oscillator in another; (c) the trajectory of $\|B_c^\top x(t)\|_\infty$, where $x=[\theta_j^p]_{10\times 1}, j=\mathcal{T}_{5}, p=2,3$; (d) the magnitude $r$ of the order parameter evaluated on other regions ($1, 4,5$ and $6$).
  • Figure 2: (a) the anatomical brain network visualized by BrainNet Viewer xia2013brainnet, edges only of weights larger than $0.15$ are shown for clarity; (b) the maximum phase difference (absolute value) of the oscillators in $9,30,33$, where $x=[\theta_j^p]_{30\times 1}, j\in\mathcal{T}_{10}, p=9,30,33$; (c) the magnitude $r$ of the global order parameter; (d) the magnitude $r$ evaluated on the regions $2$ and $23$.

Theorems & Definitions (13)

  • Remark 1
  • Definition 1
  • Theorem 1
  • Definition 2
  • Proposition 1
  • Theorem 2
  • Proposition 2
  • Proposition 3
  • Corollary 1
  • Corollary 2
  • ...and 3 more