create shortcuts that join remote parts of the lattice to one another. The starting network is a one-dimensional lattice of vertices with periodic boundary conditions. Each vertex is joined to its neighbors on either side, which results in a system with edges. The network is extended by taking each node in turn and, with probability , adding an edge to a new location nodes distant. The rewiring process allows the model to interpolate between a one-dimensional regular lattice and a two-dimensional regular lattice. When the rewiring probability , we have a one-dimensional regular lattice of size . When , every node is connected to a new location and the graph is essentially a two-dimensional lattice with and nodes in each direction. For between and , we have a graph which interpolates between the one and two dimensional regular lattices. The graphs we study are parametrized by Application to extensiveness of power law potential One application using the above definition of dimension was to the extensiveness of statistical mechanics systems with a power law potential where the interaction varies with the distance as . In one dimension the system properties like the free energy do not behave extensively when , i.e., they increase faster than N as , where N is the number of spins in the system. Consider the Ising model with the Hamiltonian (with N spins) where are the spin variables, is the distance between node and node , and are the couplings between the spins. When the have the behaviour , we have the power law potential. For a general complex network the condition on the exponent which preserves extensivity of the Hamiltonian was studied. At zero temperature, the energy per spin is proportional to and hence extensivity requires that be finite. For a general complex network is proportional to the Riemann zeta function . Thus, for the potential to be extensive, one requires Other processes which have been studied are self-avoiding random walks, and the scaling of the mean path length with the network size. These studies lead to the interesting result that the dimension transitions sharply as the shortcut probability increases from zero. The sharp transition in the dimension has been explained in terms of the combinatorially large number of available paths for points separated by distances large compared to 1. Conclusion The shortcut model is useful for studying the dimension dependence of different processes. The