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| An important question in [[statistical mechanics]] is the dependence of model behaviour on the dimension of the system. The '''shortcut model'''<ref name="Shankerb">
| | They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. It is very easy to customize plugins according to the needs of a particular business. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed. <br><br>Creating a website from scratch can be such a pain. When you write a new post, you'll see a small bar that goes across the text input area. A Wordpress plugin is a software that you can install into your Wordpress site. You can up your site's rank with the search engines by simply taking a bit of time with your site. Now a days it has since evolved into a fully capable CMS platform which make it, the best platform in the world for performing online business. <br><br>It is very easy to install Word - Press blog or website. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. You can now search through the thousands of available plugins to add all kinds of functionality to your Word - Press site. Enough automated blog posts plus a system keeps you and your clients happy. If you enjoyed this information and you would such as to receive even more details concerning [http://www.dune2.h2m.ru/profile/jucaple wordpress backup] kindly browse through our web site. For any web design and development assignment, this is definitely one of the key concerns, specifically for online retail outlets as well as e-commerce websites. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. Specialty about our themes are that they are easy to load, compatible with latest wordpress version and are also SEO friendly. Word - Press is the most popular open source content management system (CMS) in the world today. This includes enriching the content with proper key words, tactfully defining the tags and URL. <br><br>As a open source platform Wordpress offers distinctive ready to use themes for free along with custom theme support and easy customization. As a website owner, you can easily manage CMS-based website in a pretty easy and convenient style. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. It is a fact that Smartphone using online customers do not waste much of their time in struggling with drop down menus. Likewise, professional publishers with a multi author and editor setup often find that Word - Press lack basic user and role management capabilities. |
| {{cite journal
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| |author=O. Shanker
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| |year=2007
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| |title=Graph Zeta Function and Dimension of Complex Network
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| |journal=[[Modern Physics Letters B]]
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| |volume=21 |issue=11 |pages=639–644
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| |doi=10.1142/S0217984907013146
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| |bibcode = 2007MPLB...21..639S }}</ref><ref name="Shankera">
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| {{cite journal
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| |author=O. Shanker
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| |year=2007
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| |title=Defining Dimension of a Complex Network
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| |journal=[[Modern Physics Letters B]]
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| |volume=21 |issue=6 |pages=321–326
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| |doi=10.1142/S0217984907012773
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| |bibcode = 2007MPLB...21..321S }}</ref> was introduced in the course of studying this dependence. The model interpolates between discrete regular lattices of integer dimension.
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| | |
| ==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.<ref name="os">
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| {{cite journal
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| |author=O. Shanker
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| |year=2006
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| |title=Long range 1-d potential at border of thermodynamic limit
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| |journal=[[Modern Physics Letters B]]
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| |volume=20 |issue= 11|pages=649
| |
| |doi=10.1142/S0217984906011128
| |
| |bibcode = 2006MPLB...20..649S }}</ref><ref name="ref1">
| |
| {{cite journal
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| |author=D. Ruelle
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| |year=1968
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| |title=Statistical mechanics of a one-dimensional lattice gas
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| |journal=[[Communications in Mathematical Physics]]
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| |volume=9 |issue= 4|pages=267
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| |doi=10.1007/BF01654281
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| |bibcode = 1968CMaPh...9..267R }}</ref><ref name="ref2">
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| {{cite journal
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| |author=F. Dyson
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| |year=1969
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| |title=Existence of a phase-transition in a one-dimensional Ising ferromagnet
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| |journal=[[Communications in Mathematical Physics]]
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| |volume=12 |issue= 2|pages=91
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| |doi=10.1007/BF01645907
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| |bibcode = 1969CMaPh..12...91D }}</ref><ref name="ref3">
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| {{cite journal
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| |author=J. Frohlich and T. Spencer
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| |year=1982
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| |title=The phase transition in the one-dimensional Ising Model with 1/r<sup>2</sup> interaction energy
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| |journal=[[Communications in Mathematical Physics]]
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| |volume=84 |issue= |pages=87
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| |doi=10.1007/BF01208373
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| |bibcode = 1982CMaPh..84...87F }}</ref><ref name="ref4">
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| {{cite journal
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| |author=M. Aizenman, J.T. Chayes, L. Chayes, C.M. Newman
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| |year=1988
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| |title=Discontinuity of the magnetization in one-dimensional 1/{{!}}x−y{{!}}<sup>2</sup> Ising and Potts models
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| |journal=[[Journal of Statistical Physics]]
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| |volume=50 |issue= |pages=1
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| |doi=10.1007/BF01022985
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| |bibcode = 1988JSP....50....1A }}</ref><ref name="ref5">
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| {{cite journal
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| |author=J.Z. Imbrie, C.M. Newman
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| |year=1988
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| |title=An intermediate phase with slow decay of correlations in one dimensional 1/{{!}}x−y{{!}}<sup>2</sup> percolation, Ising and Potts models
| |
| |journal=[[Communications in Mathematical Physics]]
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| |volume=118 |issue= 2|pages=303
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| |doi=10.1007/BF01218582
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| |bibcode = 1988CMaPh.118..303I }}</ref><ref name="ref9">
| |
| {{cite journal
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| |author=E. Luijten and H.W.J. Blöte
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| |year=1995
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| |title=Monte Carlo method for spin models with long-range interactions
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| |journal=[[International Journal of Modern Physics C]]
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| |volume=6 |issue= 3|pages=359
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| |doi=10.1142/S0129183195000265
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| |bibcode = 1995IJMPC...6..359L }}</ref><ref name="ref10">
| |
| {{cite journal
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| |author=R.H. Swendson and J.-S. Wang
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| |year=1987
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| |title=Nonuniversal critical dynamics in Monte Carlo simulations
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| |journal=[[Physical Review Letters]]
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| |volume=58 |issue= 2|pages=86
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| |doi=10.1103/PhysRevLett.58.86
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| |bibcode=1987PhRvL..58...86S
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| }}</ref><ref name="ref11">
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| {{cite journal
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| |author=U. Wolff
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| |year=1989
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| |title=Collective Monte Carlo Updating for Spin Systems
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| |journal=[[Physical Review Letters]]
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| |volume=62 |issue= 4|pages=361–364
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| |doi=10.1103/PhysRevLett.62.361
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| |pmid=10040213
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| |bibcode=1989PhRvL..62..361W
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| }}</ref> In recent years the study has been extended from regular lattices to [[complex network]]s. The shortcut model has been used in studying several processes and their dependence on dimension.
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| ==Dimension of complex network==
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| Usually, dimension is defined based on the scaling exponent of some property in the appropriate limit. One property one could use <ref name="Shankera"/> is the scaling of volume with distance. For regular lattices <math>\textstyle \mathbf Z^d</math> the number of nodes <math>\textstyle j</math> within a distance <math>\textstyle r(i,j)</math> of node <math>\textstyle i</math> scales as <math>\textstyle r(i,j)^d</math>.
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| 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 <math>\textstyle r(i,j)</math> between nodes <math>\textstyle i</math> and <math>\textstyle j</math> as the length of the shortest path connecting the nodes.
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| For complex networks one can define the volume as the number of nodes <math>\textstyle j</math> within a distance <math>\textstyle r(i,j)</math> of node <math>\textstyle i</math>, averaged over <math>\textstyle i</math>, and the dimension may be defined as the exponent which determines the scaling behaviour of the volume with distance. For a vector <math>\textstyle \vec{n}=(n_1,\dots, n_d)\in \mathbf Z^d</math>, where <math>\textstyle d</math> is a positive integer, the Euclidean norm <math>\textstyle \|\vec{n}\|</math> is defined as the Euclidean distance from the origin to <math>\textstyle \vec{n}</math>, i.e.,
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| :<math> \|\vec{n}\|=\sqrt{n_1^2+\cdots + n_d^2}. </math>
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| However, the definition which generalises to complex networks is the <math>\textstyle L^1</math> norm,
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| :<math> \|\vec{n}\|_1=\|n_1\|+\cdots +\|n_d\|. </math>
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| The scaling properties hold for both the Euclidean norm and the <math>\textstyle L^1</math> norm. The scaling relation is
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| :<math> V(r) = kr^d, </math>
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| where d is not necessarily an integer for complex networks. <math>\textstyle k</math> is a geometric constant which depends on the complex network. If the scaling relation Eqn. holds, then one can also define the surface area <math>\textstyle S(r)</math> as the number of nodes which are exactly at a distance <math>\textstyle r</math> from a given node, and <math>\textstyle S(r)</math> scales as
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| :<math> S(r) = kdr^{d-1}. </math>
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| A definition based on the [[complex network zeta function]]<ref name="Shankerb"/> generalises the definition based on the scaling property of the volume with distance<ref name="Shankera"/> and puts it on a mathematically robust footing.
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| ==Shortcut model==
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| 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 <math>\textstyle N</math> vertices with periodic boundary conditions. Each vertex is joined to its neighbors on either side, which results in a system with <math>\textstyle N</math> edges. The network is extended by taking each node in turn and, with probability <math>\textstyle p</math>, adding an edge to a new location <math>\textstyle m</math> nodes distant.
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| The rewiring process allows the model to interpolate between a one-dimensional regular lattice and a two-dimensional regular lattice. When the rewiring probability <math>\textstyle p=0</math>, we have a one-dimensional regular lattice of size <math>\textstyle N</math>. When <math>\textstyle p=1</math>, every node is connected to a new location and the graph is essentially a two-dimensional lattice with <math>\textstyle m</math> and <math>\textstyle N/m</math> nodes in each direction. For <math>\textstyle p</math> between <math>\textstyle 0</math> and <math>\textstyle 1</math>, we have a graph which interpolates between the one and two dimensional regular lattices. The graphs we study are parametrized by
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| :<math> \text{size} = N,\,</math>
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| :<math> \text{shortcut distance} = m,\,</math>
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| :<math> \text{rewiring probability} = p.\,</math>
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| ==Application to extensiveness of power law potential==
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| One application using the above definition of dimension was to the
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| extensiveness of statistical mechanics systems with a power law potential where the interaction varies with the distance <math>\textstyle r</math> as <math>\textstyle 1/r^\alpha</math>. In one dimension the system properties like the free energy do not behave extensively when <math>\textstyle 0\leq\alpha\leq1</math>, i.e., they increase faster than N as <math>\textstyle N\rightarrow\infty</math>, where N is the number of spins in the system.
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| Consider the Ising model with the Hamiltonian (with N spins)
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| :<math> H=-\frac{1}{2}\sum_{i,j}J(r(i,j))s_{i}s_{j} </math>
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| where <math>\textstyle s_{i}</math> are the spin variables, <math>\textstyle r(i,j)</math> is the distance between node <math>\textstyle i</math> and node <math>\textstyle j</math>, and <math>\textstyle J(r(i,j))</math> are the couplings between the spins. When the <math>\textstyle J(r(i,j))</math> have the behaviour <math>\textstyle 1/r^\alpha</math>, we have the power law potential. For a general complex network the condition on the exponent <math>\textstyle \alpha</math> which preserves extensivity of the Hamiltonian was studied. At zero temperature, the energy per spin is proportional to
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| :<math> \rho=\sum_{i,j}J(r(i,j)), </math>
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| and hence extensivity requires that <math>\textstyle \rho</math> be finite. For a general complex network <math>\textstyle \rho</math> is proportional to the [[Riemann zeta function]] <math>\textstyle \zeta ( \alpha - d + 1)</math>. Thus, for the potential to be extensive, one requires | |
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| :<math> \alpha > d.\, </math>
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| 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.<ref name="Shankerc">
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| {{cite journal
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| |author=O. Shanker
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| |year=2008
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| |title=Algorithms for Fractal Dimension Calculation
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| |journal=[[Modern Physics Letters B]]
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| |volume=22 |issue= 07|pages=459–466
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| |doi=10.1142/S0217984908015048
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| |bibcode = 2008MPLB...22..459S }}</ref> The sharp transition in the dimension has been explained in terms of the combinatorially large
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| number of available paths for points separated by distances large compared to 1.<ref name="refJPAOS">
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| {{cite journal
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| |author=O. Shanker
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| |year=2008
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| |title=Sharp dimension transition in a shortcut model
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| |journal=[[J. Phys. A]]
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| |volume=41 |issue= 28|pages=285001
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| |doi=10.1088/1751-8113/41/28/285001
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| |bibcode = 2008JPhA...41B5001S }}</ref>
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| ==Conclusion==
| |
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| 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 <math>\textstyle p</math>. 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.
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| ==References==
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| {{reflist}}
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| [[Category:Graph theory]]
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| [[Category:Networks]]
| |
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