Critical dynamics of randomly assembled and diluted threshold networks

The dynamical behavior of a class of randomly assembled networks of binary threshold units subject to random deletion of connections is studied based on the annealed approximation suitable in the thermodynamic limit. The dynamical phase diagram is constructed for several forms of the probability density distribution of nonvanishing connection strengths. The family of power-law distribution functions rho0(x)=(1-alpha)/(2|x|alpha) is found to play a special role in expanding the domain of stable, ordered dynamics at the expense of the disordered, "chaotic" phase. Relationships with other recent studies of the dynamics of complex networks allowing for variable in-degree of the units are explored. The relevance of the pruning of network connections to neural modeling and developmental neurobiology is discussed.