Experiments in the dynamics of phase coupled oscillators when applied to graph colouring

This paper examines the capacity of networks of phase coupled oscillators to coordinate activity in a parallel, distributed fashion. To benchmark these networks of oscillators, we present empirical results from a study of the capacity of such networks to colour graphs. We generalise the update equation of Aihara et al. (2006) to an equation that can be applied to graphs requiring multiple colours. We find that our simple multi-phase model can colour some types of graphs, especially complete graphs and complete k-partite graphs with equal or a near equal number of vertices in each partition. A surprising empirical result is that the effectiveness of the approach appears to be more dependent upon the topology of the graph than the size of the graph.

[1]  Steven H. Strogatz,et al.  Sync: The Emerging Science of Spontaneous Order , 2003 .

[2]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[3]  K. Aihara,et al.  Periodic rhythm and anti-phase synchronization in calling behaviors of Japanese rain frogs , 2022 .

[4]  Chai Wah Wu,et al.  Synchronization in Coupled Chaotic Circuits and Systems , 2002 .

[5]  J. J. Hopfield,et al.  “Neural” computation of decisions in optimization problems , 1985, Biological Cybernetics.

[6]  Guy Theraulaz,et al.  Self-Organization in Biological Systems , 2001, Princeton studies in complexity.

[7]  G B Ermentrout,et al.  Beyond a pacemaker's entrainment limit: phase walk-through. , 1984, The American journal of physiology.

[8]  Heidi M. Rockwood,et al.  Huygens's clocks , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[9]  P. Libby.  The Scientific American , 1881, Nature.

[10]  J. Buck,et al.  Synchronous fireflies. , 1976, Scientific American.

[11]  L. Amzel,et al.  Making the right moves. , 2007, Structure.

[12]  Radhika Nagpal,et al.  Firefly-inspired sensor network synchronicity with realistic radio effects , 2005, SenSys '05.

[13]  James Copland,et al.  PROCEEDINGS OF THE ROYAL SOCIETY. , 2022 .

[14]  I-Jeng Wang,et al.  Decentralized synchronization protocols with nearest neighbor communication , 2004, SenSys '04.

[15]  Frank E. Hanson,et al.  Control of flashing in fireflies , 1981, Journal of comparative physiology.

[16]  Anna Scaglione,et al.  Time synchronization and reach-back communications with pulse-coupled oscillators for UWB wireless ad hoc networks , 2003, IEEE Conference on Ultra Wideband Systems and Technologies, 2003.

[17]  Paolo Santi,et al.  Self‐organization in Communication Networks , 2006 .

[18]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[19]  Stefan J. Johansson,et al.  On Coordination in Multi-agent Systems , 2002 .

[20]  Chris Arney.  Sync: The Emerging Science of Spontaneous Order , 2007 .

[21]  Frank E. Hanson,et al.  Control of flashing in fireflies , 2004, Journal of comparative physiology.

[22]  Yu-Ping Tian,et al.  Coordination of Multi-Agent Systems with Communication Delays , 2008 .

[23]  S. Strogatz,et al.  Synchronization of pulse-coupled biological oscillators , 1990 .