Emergent bipartiteness in a society of knights and knaves

We propose a simple model of a social network based on the so-called knights-and-knaves puzzles. The model describes the formation of networks between two classes of agents where links are formed by agents introducing their neighbors to others of their own class. We show that if the proportion of knights and knaves is within a certain range, the network self-organizes to a perfectly bipartite state. However, if the excess of one of the two classes is greater than a threshold value, bipartiteness is not observed. We offer a detailed theoretical analysis of the behavior of the model, investigate its behavior in the thermodynamic limit and argue that it provides a simple example of a topology-driven model whose behavior is strongly reminiscent of first-order phase transitions far from equilibrium.

[1]  S. Kauffman.  Metabolic stability and epigenesis in randomly constructed genetic nets. , 1969, Journal of theoretical biology.

[2]  D. Thouless.  Introduction to Phase Transitions and Critical Phenomena , 1972 .

[3]  D. Rand,et al.  Correlation models for childhood epidemics , 1997, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[4]  Thilo Gross,et al.  Epidemic dynamics on an adaptive network. , 2005, Physical review letters.

[5]  E. Ising.  Beitrag zur Theorie des Ferromagnetismus , 1925 .

[6]  Alain Barrat,et al.  Who's talking first? Consensus or lack thereof in coevolving opinion formation models. , 2007, Physical review letters.

[7]  S Redner,et al.  Dynamics of social balance on networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Thilo Gross,et al.  Robust oscillations in SIS epidemics on adaptive networks: Coarse graining by automated moment closure , 2008 .

[9]  Thilo Gross,et al.  Patterns of cooperation: fairness and coordination in networks of interacting agents , 2009, 0902.2954.

[10]  Maxi San Miguel,et al.  Conservation laws for voter-like models on random directed networks , 2009, 0902.1769.

[11]  D J PRICE,et al.  NETWORKS OF SCIENTIFIC PAPERS. , 1965, Science.

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

[13]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[14]  Tang,et al.  Self-Organized Criticality: An Explanation of 1/f Noise , 2011 .

[15]  P. L. Krapivsky,et al.  Social balance on networks : The dynamics of friendship and enmity , 2006 .

[16]  Thilo Gross,et al.  Adaptive coevolutionary networks: a review , 2007, Journal of The Royal Society Interface.