On the Complexity of the Classification of Synchronizing Graphs

This article deals with the general ideas of almost global synchronization of Kuramoto coupled oscillators and synchronizing graphs. It reviews the main existing results and gives some new results about the complexity of the problem. It is proved that any connected graph can be transformed into a synchronized one by making suitable groups of twin vertices. As a corollary it is deduced that any connected graph is the induced subgraph of a synchronizing graph. This implies a big structural complexity of synchronizability. Finally the former is applied to find a two integer parameter family G(a,b) of connected graphs such that if b is the k-th power of 10, the synchronizability of G(a,b) is equivalent to find the k-th digit in the expansion in base 10 of the square root of 2. Thus, the complexity of classify G(a,b) is of the same order than the computation of square root of 2. This is the first result so far about the computational complexity of the synchronizability problem.

[1]  Mark W. Spong,et al.  On Exponential Synchronization of Kuramoto Oscillators , 2009, IEEE Transactions on Automatic Control.

[2]  S. Strogatz.  From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators , 2000 .

[3]  Eduardo Alberto Canale,et al.  Global Synchronization Properties for Different Classes of Underlying Interconnection Graphs for Kuramoto Coupled Oscillators , 2009, FGIT.

[4]  Eduardo Alberto Canale,et al.  Global properties of Kuramoto bidirectionally coupled oscillators in a ring structure , 2009, 2009 IEEE Control Applications, (CCA) & Intelligent Control, (ISIC).

[5]  Enrique Mallada,et al.  Synchronization of phase-coupled oscillators with arbitrary topology , 2010, Proceedings of the 2010 American Control Conference.

[6]  Max Donath,et al.  American Control Conference , 1993 .

[7]  Pablo Monzon,et al.  On the Characterization of Families of Synchronizing Graphs for Kuramoto Coupled Oscillators , 2009 .

[8]  Chang Lin,et al.  The mathematical research for the Kuramoto model of the describing neuronal synchrony in the brain , 2009 .

[9]  N. Biggs.  Algebraic Graph Theory: COLOURING PROBLEMS , 1974 .

[10]  A. Jadbabaie,et al.  On the stability of the Kuramoto model of coupled nonlinear oscillators , 2005, Proceedings of the 2004 American Control Conference.

[11]  Eduardo Alberto Canale,et al.  Gluing Kuramoto coupled oscillators networks , 2007, 2007 46th IEEE Conference on Decision and Control.

[12]  李幼升,et al.  Ph , 1989 .

[13]  F. Paganini,et al.  Global considerations on the Kuramoto model of sinusoidally coupled oscillators , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[14]  Huzihiro Araki,et al.  International Symposium on Mathematical Problems in Theoretical Physics , 1975 .

[15]  A. Winfree.  The geometry of biological time , 1991 .

[16]  Fgit,et al.  Future Generation Information Technology, First International Conference, FGIT 2009, Jeju Island, Korea, December 10-12, 2009. Proceedings , 2009, FGIT.