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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">
| | One of many main trends in real estate currently is people selling their domiciles via the process called FSBO. Frequently this is because of the theory that they can save thousands of dollars in commissions. However this is really incorrect. What"s left out of the advertisements for FSBO may be the fact that these 1000s of preserved fee pounds are generally consumed and realized by doing the projects that are seen to by an agent. A realtor is a professional home seller. ask yourself this: if you needed to have a cavity filled would you go see an electrician? No? Then why would you allow a non-professional to market your most valuable property? <br><br>Realtors offer services that aren"t only designed to help relieve your property sale but services that are designed to exclusively protect your investment. One of the most valuable items that a real estate agent can provide is marketing. Unlike attempting to sell on your own, an agent has use of a huge number of advertising methods. I learned about [http://www.newswire.net/article/upsell/index/id/81585 plano realtor, top realtor plano texas, top real estate agent plano texas, just listed plano texas, plano real estate agent, richardson realtor, allen realtor, frisco realtor, mckinney realtor, north dallas realtor,] by browsing books in the library. Often the front lines which is an already-established web site. Furthermore to listing your home on the local MLS, a realtor may have your own site that is designed to display their entries. This really is where many virginia homes are first viewed by prospective customers. Additional marketing is normally done through newspapers and other print media together with a number of information sheets and brochures that are available 24/7. <br><br>Still another aspect of home marketing the place where a realty is available in handy is in the connections they keep with other property professionals. So that you can most readily useful reach the buying market a realtor can put much effort into marketing a house to other local agents. Home selling is really about positive exposure for the house under consideration and only a realtor provides that amount of exposure. There is a reason a large portion of FSBO"s fundamentally find yourself listing with a realtor as a way to have the coverage and value they deserve..New Method, LLC<br>14 Inverness Drive East C-108<br>Englewood, CO 80112<br><br>If you enjoyed this write-up and you would certainly such as to receive more info relating to health insurance plan; [http://www.migente.com/raggedterminolo15 continue reading this], kindly check out our webpage. |
| {{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==
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| | |
| 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
| |
| |author=O. Shanker
| |
| |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]]
| |
| |volume=20 |issue= 11|pages=649
| |
| |doi=10.1142/S0217984906011128
| |
| |bibcode = 2006MPLB...20..649S }}</ref><ref name="ref1">
| |
| {{cite journal
| |
| |author=D. Ruelle
| |
| |year=1968
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| |title=Statistical mechanics of a one-dimensional lattice gas
| |
| |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">
| |
| {{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
| |
| |journal=[[Communications in Mathematical Physics]]
| |
| |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">
| |
| {{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
| |
| |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">
| |
| {{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]]
| |
| |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
| |
| |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
| |
| }}</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.
| |
| | |
| ==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">
| |
| {{cite journal
| |
| |author=O. Shanker
| |
| |year=2008
| |
| |title=Algorithms for Fractal Dimension Calculation
| |
| |journal=[[Modern Physics Letters B]]
| |
| |volume=22 |issue= 07|pages=459–466
| |
| |doi=10.1142/S0217984908015048
| |
| |bibcode = 2008MPLB...22..459S }}</ref> 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.<ref name="refJPAOS">
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| {{cite journal
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| |author=O. Shanker
| |
| |year=2008
| |
| |title=Sharp dimension transition in a shortcut model
| |
| |journal=[[J. Phys. A]]
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| |volume=41 |issue= 28|pages=285001
| |
| |doi=10.1088/1751-8113/41/28/285001
| |
| |bibcode = 2008JPhA...41B5001S }}</ref>
| |
| | |
| ==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 <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.
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| [[Category:Graph theory]]
| |
| [[Category:Networks]]
| |
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