Complex Network Topologies and Synchronization

Abstract

Synchronization in networks with different topologies is studied. We show that for a large class of oscillators there exist two classes of networks; class-A: networks for which the condition of stable synchronous state is sigmagamma <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> > a, and class-B: networks for which this condition reads gamma <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> /gamma <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> < b, where a and b are constants that depend on local dynamics, synchronous state and the coupling matrix, but not on the Laplacian matrix of the graph describing the topology of the network. Here gamma <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> = 0 < gamma <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> les... les gamma <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> are the eigenvalues of the Laplacian matrix, where N is the order of the graph. Synchronization in networks whose topology is described by classical random graphs and power-law random graphs when N rarr infin is investigated in detail