An important question in statistical mechanics is the dependence of model behaviour on the dimension of the system. The shortcut model was introduced in the course of studying this dependence. The model interpolates between discrete regular lattices of integer dimension.
Contents
Introduction
The behaviour of different processes on discrete regular lattices have been studied quite extensively. They show a rich diversity of behaviour, including a non-trivial dependence on the dimension of the regular lattice. In recent years the study has been extended from regular lattices to complex networks. The shortcut model has been used in studying several processes and their dependence on dimension.
Dimension of complex network
Usually, dimension is defined based on the scaling exponent of some property in the appropriate limit. One property one could use is the scaling of volume with distance. For regular lattices
For systems which arise in physical problems one usually can identify some physical space relations among the vertices. Nodes which are linked directly will have more influence on each other than nodes which are separated by several links. Thus, one could define the distance
For complex networks one can define the volume as the number of nodes
However, the definition which generalises to complex networks is the
The scaling properties hold for both the Euclidean norm and the
where d is not necessarily an integer for complex networks.
A definition based on the complex network zeta function generalises the definition based on the scaling property of the volume with distance and puts it on a mathematically robust footing.
Shortcut model
The shortcut model starts with a network built on a one-dimensional regular lattice. One then adds edges to create shortcuts that join remote parts of the lattice to one another. The starting network is a one-dimensional lattice of
The rewiring process allows the model to interpolate between a one-dimensional regular lattice and a two-dimensional regular lattice. When the rewiring probability
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
Consider the Ising model with the Hamiltonian (with N spins)
where
and hence extensivity requires that
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 processes studied include the behaviour of the power law potential as a function of the dimension, the behaviour of self-avoiding random walks, and the scaling of the mean path length. It may be useful to compare the shortcut model with the small-world network, since the definitions have a lot of similarity. In the small-world network also one starts with a regular lattice and adds shortcuts with probability