Distribution of local magnetisations in random networks of automata

The authors derive an integral equation which gives the distribution P(m) of local magnetisations for a random network of automata (Kauffman model). They show how from this distribution one can recover the distances between configurations and the fraction of spins in the stable core (i.e. the spins which do not depend on the initial conditions). A numerical solution of the integral equation shows that P(m) has singularities at m=0, +or-1/2 and +or-1.

[1]  M. Mézard,et al.  Mean-field theory of randomly frustrated systems with finite connectivity , 1987 .

[2]  H. Orland.  Mean-field theory for optimization problems , 1985 .

[3]  Kanter,et al.  Mean-field theory of spin-glasses with finite coordination number. , 1987, Physical review letters.

[4]  A. Bray,et al.  Phase diagrams for dilute spin glasses , 1985 .

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

[6]  Bernard Derrida,et al.  Multivalley structure in Kauffman's model: analogy with spin glasses , 1986 .

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

[8]  P. Mottishaw,et al.  Replica Symmetry Breaking and the Spin-Glass on a Bethe Lattice , 1987 .

[9]  B. Derrida,et al.  Evolution of overlaps between configurations in random Boolean networks , 1986 .

[10]  H. Hilhorst,et al.  On the approach of the stationary state in Kauffman's random Boolean network , 1987 .

[11]  D. R. Bowman,et al.  Spin-glass theory in the Bethe approximation: Insights and problems , 1982 .

[12]  Bernard Derrida,et al.  Statistical properties of randomly broken objects and of multivalley structures in disordered systems , 1987 .

[13]  B. Derrida.  Dynamical phase transition in nonsymmetric spin glasses , 1987 .

[14]  S. Kauffman.  The large scale structure and dynamics of gene control circuits: an ensemble approach. , 1974, Journal of theoretical biology.

[15]  M. Thouless.  Spin-glass on a Bethe lattice. , 1986, Physical review letters.

[16]  Replica symmetric solutions in the Ising spin glass: the tree approximation , 1987 .

[17]  E. Gardner,et al.  An Exactly Solvable Asymmetric Neural Network Model , 1987 .