A fractal analysis of interconnection complexity

The emergent, collective properties of computer interconnections are shown to be characterized by a noninteger dimension D_i , which is, in general, different from the system´s Euclidean dimension. This dimension characterizes the properties of a fractal support, or substrate, on which interconnections are placed to provide communication throughout the system. The interconnection support also acts as a host for a multifractal spectrum of interconnection distribution processes which characterize the change in connectivity in moving from the backplane to the transistor level. The properties of fractal systems are investigated by attempting to minimize their total wire length using a simulated annealing algorithm. Systems whose interconnection dimension is approximately equal to their Euclidean dimension are shown to possess minimum wire length arrangements. These results are then interpreted in terms of a geometrical temperature T _i=1/D_i. This analysis indicates that the system passes through a phase transition at T_i≈1/2 and that attainable system temperatures are bounded by 1/3⩽T_i⩽1. The consequences for simulated annealing are discussed