Narrow Results

In previous work, the limit structure of positive and negative finite threshold boolean networks without inputs (TBNs) over the complete digraph K_n was analyzed and an algorithm was presented for computing this structure in polynomial time. Those results are generalized in this paper to cover the case of arbitrary TBNs over K_n. Although the limit structure is now more complicated, containing, not only fixed-points and cycles of length 2, but possibly also cycles of arbitrary length, a simple algorithm is still available for its determination in polynomial time. Finally, the algorithm is generalized to cover the case of symmetric finite boolean networks over K_n.

Our numerical simulation first displays exponential increase of the mean number of attractors with N for K = 2,3,4 and 50≦ N ≦350. The mean length of attractors also increases exponentially with N for K = 2, but increases linearly with N for K = 3,4. We further found the larger the K the larger S/N value; which yields the results that the mean length and the mean number of attractors of critical random Boolean networks will decrease with larger K.

Evolution of cooperation has attracted considerable attention but so far no definitive answer exists. Probably each kind of problem has specific answers. This paper deals with evolution of cooperation in public goods games. We use random Boolean networks to formalize the non-local influence of K agents over a given agent. This formalism allows the representation of a variety of network regulation mechanisms by means of (i) different topologies and (ii) Boolean functions that do the regulation proper. However, random functions and connections do not necessarily lead to cooperation. Thus, it is necessary to find what kind of network structure is prone to promote cooperation. We employ an evolutionary approach to show that evolving the topologies and the random functions leads to much fitter structures.