Shortcut model


Shortcut model

An important question in statistical mechanics is the dependence of model behaviour on the dimension of the system. The shortcut modelcite journal|author=Shanker, O.|year=2007|title=Graph Zeta Function and Dimension of Complex Network|journal=Modern Physics Letters B|volume= 21(11)|pages=639–644|doi=10.1142/S0217984907013146] cite journal|author=Shanker, O.|year=2007|title=Defining Dimension of a Complex Network |journal=Modern Physics Letters B|volume= 21(6)|pages=321–326|doi=10.1142/S0217984907012773] 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 O. Shanker, "Mod. Phys. Lett." B20, 649 (2006).

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] M. Aizenman, J. T. Chayes, L. Chayes and C. M. Newman, "J. Stat. Phys." 50, 1 (1988).

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] R. H. Swendson and J.-S. Wang, "Phys. Rev. Lett." 58, 86 (1987).

] U. Wolff, "Phys. Rev. Lett." 62, 361 (1989).

] . 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 extstyle mathbf Z^d the number of nodes extstyle j within a distance extstyle r(i,j) of node extstyle i scales as extstyle r(i,j)^d.

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 extstyle r(i,j) between nodes extstyle i and extstyle j as the length of the shortest path connecting the nodes.

For complex networks one can define the volume as the number of nodes extstyle j within a distance extstyle r(i,j) of node extstyle i, averaged over extstyle i, and the dimension may be defined as the exponent which determines the scaling behaviour of the volume with distance. For a vector extstyle vec{n}=(n_1,dots, n_d)in mathbf Z^d, where extstyle d is a positive integer, the Euclidean norm extstyle |vec{n}| is defined as the Euclidean distance from the origin to extstyle vec{n}, i.e.,

: |vec{n}|=sqrt{n_1^2+cdots + n_d^2}.

However, the definition which generalises to complex networks is the extstyle L^1 norm,

: |vec{n}|_1=|n_1|+cdots +|n_d|.

The scaling properties hold for both the Euclidean norm and the extstyle L^1 norm. The scaling relation is

: V(r) = kr^d,

where d is not necessarily an integer for comlex networks. extstyle k is a geometric constant which depends on the complex network. If the scaling relation Eqn. holds, then one can also define the surface area extstyle S(r) as the number of nodes which are exactly at a distance extstyle r from a given node, and extstyle S(r) scales as

: S(r) = kdr^{d-1}.

A definition based on the complex network zeta functioncite journal|author=Shanker, O.|year=2007|title=Graph Zeta Function and Dimension of Complex Network|journal=Modern Physics Letters B|volume= 21(11)|pages=639–644|doi=10.1142/S0217984907013146] generalises the definition based on the scaling property of the volume with distancecite journal|author=Shanker, O.|year=2007|title=Defining Dimension of a Complex Network |journal=Modern Physics Letters B|volume= 21(6)|pages=321–326|doi=10.1142/S0217984907012773] and puts it on a mathematically robust footing.

hortcut 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 extstyle N vertices with periodic boundary conditions. Each vertex is joined to its neighbors on either side, which results in a system with extstyle N edges. The network is extended by taking each node in turn and, with probability extstyle p, adding an edge to a new location extstyle m 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 extstyle p=0, we have a one-dimensional regular lattice of size extstyle N. When extstyle p=1, every node is connected to a new location and the graph is essentially a two-dimensional lattice with extstyle m and extstyle N/m nodes in each direction. For extstyle p between extstyle 0 and extstyle 1, we have a graph which interpolates between the one and two dimensional regular lattices. The graphs we study are parametrized by

: ext{size} = N,,: ext{shortcut distance} = m,,: ext{rewiring probability} = p.,

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 extstyle r as extstyle 1/r^alpha. In one dimension the system properties like the free energy do not behave extensively when extstyle 0leqalphaleq1, i.e., they increase faster than N as extstyle N ightarrowinfty, where N is the number of spins in the system.

Consider the Ising model with the Hamiltonian (with N spins)

: H=-frac{1}{2}sum_{i,j}J(r(i,j))s_{i}s_{j}

where extstyle s_{i} are the spin variables, extstyle r(i,j) is the distance between node extstyle i and node extstyle j, and extstyle J(r(i,j)) are the couplings between the spins. When the extstyle J(r(i,j)) have the behaviour extstyle 1/r^alpha, we have the power law potential. For a general complex network the condition on the exponent extstyle alpha which preserves extensivity of the Hamiltonian was studied. At zero temperature, the energy per spin is proportional to

: ho=sum_{i,j}J(r(i,j)),

and hence extensivity requires that extstyle ho be finite. For a general complex network extstyle ho is proportional to the Riemann zeta function extstyle zeta ( alpha - d + 1). Thus, for the potential to be extensive, one requires

: alpha > d.,

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 zerocite journal|author=O. Shanker, |year=2008|title=Algorithms for Fractal Dimension Calculation|journal=Modern Physics Letters B|volume= 22|pages=459–466|doi=10.1142/S0217984908015048] . 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 latticeand adds shortcuts with probability extstyle p. 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 randomlychosen node. As a result, the small world model tends to a randomgraph rather than a two-dimensional graph as the shortcut probabilityis increased.

References


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