Complex Networks from Simple Rules

Complex networks are all around us, and they can be generated by simple mechanisms. Understanding what kinds of networks can be produced by following simple rules is therefore of great importance. This issue is investigated by studying the dynamics of extremely simple systems where a “writer” moves around a network, modifying it in a way that depends upon the writer’s surroundings. Each vertex in the network has three edges incident upon it, which are colored red, blue, and green. This edge coloring is done to provide a way for the writer to orient its movement. The dynamics of a space of 3888 of these colored trinet automata systems are explored. A large variety of behavior is found, ranging from the very simple to the very complex. Our systems are studied using simulations (with appropriate visualization techniques) and selected rules are analyzed mathematically. An empirical classification scheme is arrived at, which reveals a lot about the kinds of dynamics and networks that can be generated by these systems.

[1]  Przemyslaw Prusinkiewicz,et al.  The Algorithmic Beauty of Plants , 1990, The Virtual Laboratory.

[2]  Alexander G. D. Lamb.  Dense Graphs, Node Sets, and Riders: Toward a Foundation for Particle Physics without Continuum Mathematics , 2010, Complex Syst..

[3]  A. Turing.  On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[4]  Mike Mannion,et al.  Complex systems , 1997, Proceedings International Conference and Workshop on Engineering of Computer-Based Systems.

[5]  Chris Cannings,et al.  Some Models of Reproducing Graphs: III Game Based Reproduction , 2010 .

[6]  Tommaso Bolognesi.  Planar Trinet Dynamics with Two Rewrite Rules , 2008, Complex Syst..

[7]  Edward Fredkin.  Five big questions with pretty simple answers , 2004, IBM J. Res. Dev..

[8]  Chris Cannings,et al.  Some Models of Reproducing Graphs: I Pure Reproduction , 2010 .

[9]  K. Tomita,et al.  Graph automata: natural expression of self-reproduction , 2002 .

[10]  Chris Cannings,et al.  Some Models of Reproducing Graphs: II Age Capped Vertices , 2010 .

[11]  Eric Gottlieb,et al.  Color-induced subgraphs of Grünbaum colorings of triangulations of the sphere , 2004, Australas. J Comb..

[12]  J. Yorke,et al.  Chaos: An Introduction to Dynamical Systems , 1997 .

[13]  Tommaso Bolognesi,et al.  Causal Sets from Simple Models of Computation , 2010, Int. J. Unconv. Comput..

[14]  S. Wolfram.  Statistical mechanics of cellular automata , 1983 .

[15]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[16]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[17]  Francesc Comellas,et al.  Complex Networks: Deterministic Models. , 2007 .

[18]  C. Danforth,et al.  Dynamic structure of networks updated according to simple, local rules. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Chris Cannings,et al.  Games on graphs that grow deterministically , 2009, 2009 International Conference on Game Theory for Networks.

[20]  Pedro Pina,et al.  Polygonal terrains on Mars: A contribution to their geometric and topological characterization , 2008 .

[21]  D. Weaire,et al.  Topological changes in a two-dimensional foam cluster , 2003, The European physical journal. E, Soft matter.

[22]  Hector Zenil.  Randomness Through Computation: Some Answers, More Questions , 2011 .

[23]  Stephen Wolfram,et al.  A New Kind of Science , 2003, Artificial Life.