A numerical study of the critical line of Kauffman networks

Kauffman networks were introduced in 1969 as a model of genetic regulatory systems. One of the most striking successes of this model is its ability to reproduce, for a critical value of its parameters, the observed scaling laws of the average cell replication time and of the average number of cell types in a given organism vs. the number of genes. Yet, the numerical evidence for such scaling laws in the model is still unsatisfactory, and restricted to a particular critical point, while we expect that the scaling behaviour is universal along the critical line. In this paper we try to sharpen the evidence for the scaling behaviour of critical systems, carrying on a detailed numerical investigation of their properties. We measure the length of the cycles (which in the model represents the period of cell cycles) and their number (which represents the number of cell types) for a point of the critical line different from the only one previously studied. Our results seem to confirm that such quantities scale as radicalN for all critical systems, at least for lengths and numbers small enough. On the other hand, we found that their probability distributions are very broad (power-law like) and become broader with system size. This means that there is an effective scale of the length and of the number of cycles that increases much faster than radicalN, and in the infinite size limit the biological analogy found by Kauffman may be lost. A numerical study of the modular structure of critical networks supports this conclusion. The implications of this fact for the biological interpretation of the model are briefly discussed. Finally, we found that the typical weight of the attraction basins tends to zero as a power law in the infinite size limit, with an exponent which seems to be universal along the whole critical line.