Method used in statistical mechanics
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.
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 formula_1 the number of nodes formula_2 within a distance formula_3 of node formula_4 scales as formula_5.
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 formula_3 between nodes formula_4 and formula_2 as the length of the shortest path connecting the nodes.
For complex networks one can define the volume as the number of nodes formula_2 within a distance formula_3 of node formula_4, averaged over formula_4, and the dimension may be defined as the exponent which determines the scaling behaviour of the volume with distance. For a vector formula_13, where formula_14 is a positive integer, the Euclidean norm formula_15 is defined as the Euclidean distance from the origin to formula_16, i.e.,
formula_17
However, the definition which generalises to complex networks is the formula_18 norm,
formula_19
The scaling properties hold for both the Euclidean norm and the formula_18 norm. The scaling relation is
formula_21
where d is not necessarily an integer for complex networks. formula_22 is a geometric constant which depends on the complex network. If the scaling relation Eqn. holds, then one can also define the surface area formula_23 as the number of nodes which are exactly at a distance formula_24 from a given node, and formula_23 scales as
formula_26
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 formula_27 vertices with periodic boundary conditions. Each vertex is joined to its neighbors on either side, which results in a system with formula_27 edges. The network is extended by taking each node in turn and, with probability formula_29, adding an edge to a new location formula_30 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 formula_31, we have a one-dimensional regular lattice of size formula_27. When formula_33, every node is connected to a new location and the graph is essentially a two-dimensional lattice with formula_30 and formula_35 nodes in each direction. For formula_29 between formula_37 and formula_38, we have a graph which interpolates between the one and two dimensional regular lattices. The graphs we study are parametrized by
formula_39
formula_40
formula_41
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 formula_24 as formula_43. In one dimension the system properties like the free energy do not behave extensively when formula_44, i.e., they increase faster than N as formula_45, where N is the number of spins in the system.
Consider the Ising model with the Hamiltonian (with N spins)
formula_46
where formula_47 are the spin variables, formula_3 is the distance between node formula_4 and node formula_2, and formula_51 are the couplings between the spins. When the formula_51 have the behaviour formula_43, we have the power law potential. For a general complex network the condition on the exponent formula_54 which preserves extensivity of the Hamiltonian was studied. At zero temperature, the energy per spin is proportional to
formula_55
and hence extensivity requires that formula_56 be finite. For a general complex network formula_56 is proportional to the Riemann zeta function formula_58. Thus, for the potential to be extensive, one requires
formula_59
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 formula_29. However, the shortcuts are not constrained to connect to a node a fixed distance ahead. Instead, the other end of the shortcut can connect to any randomly chosen node. As a result, the small world model tends to a random graph rather than a two-dimensional graph as the shortcut probability is increased.
References.
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