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| Different definitions have been given for the dimension of a [[complex network]] or [[graph theory|graph]]. For example, [[Metric dimension (graph theory)|metric dimension]] is defined in terms of the resolving set for a graph. Dimension has also been [[Fractal dimension on networks|defined]] based on the [[Minkowski–Bouligand dimension|box covering method]] applied to graphs.<ref name=goh>K.-I. Goh, G. Salvi, B. Kahng and D. Kim, Phys. Rev.
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| Lett. 96, 018701 (2006).</ref> Here we describe the definition based on the '''complex network zeta function'''.<ref name="Shankerb">{{cite journal|author=O. Shanker|year=2007|title=Graph Zeta Function and Dimension of Complex Network|journal=Modern Physics Letters B|volume= 21|pages=639–644|doi=10.1142/S0217984907013146 | issue=11|bibcode=2007MPLB...21..639S}}</ref> This generalises the definition based on the scaling property of the volume with distance.<ref name="Shankera">{{cite journal|author=O. Shanker|year=2007|title=Defining Dimension of a Complex Network |journal=Modern Physics Letters B|volume= 21|pages=321–326|doi=10.1142/S0217984907012773 | issue=6|bibcode=2007MPLB...21..321S}}</ref> The best definition depends on the application.
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| ==Definition==
| | All the same, it was Madame Dumont skills and professionalism in playing the game that made this French woman so well-liked and attractive to miners and employees of the city. In 1854 she constructed her own on line casino home in Nevada, California. People accept her as a lovely and gracious host of the house that blackjack players have a higher urge on aspiring to have a match with her.<br><br>Logic to win roulette machines tells us that by mapping an region of touched internet casino numbers, leaves the odds in favour of these that are untouched. We can now start to see the greater potential within our own sport and use this kind of a model as a winning gadget.<br><br>If you begin betting with 10 USD, your twenty fifth bet will be 1. Yes, you will have the probabilities of squaring all the losses made via the previous defeats but you can only live internet peaceful casino to see the sun if you get by the 7th or 8th hand. This then is a dubious fashion of betting.<br><br>For all too many people, the price is higher - financially, and in time and focus. Evidently over fifteen%twenty five of Canadian teenagers have at minimum a reasonable habit to what I contact bad gambling.<br><br>What are the chances of that happening? After a couple of changes, the modern roulette wheel was first introduced some time early 19th century. Roulette or, as some would like to refer to it, the King of On line casino video games really began in 18th century France when mathematician Blaise Pascal began tinkering and creating a perpetual movement machine.<br><br>To perform online roulette you have to visit any of the websites that offer online on line casino games or gambling. And you won't get distracted by the environment. In contrast to the casinos, you can't be produced to remain there for a lengthy time. Casinos are recognized to offer a great deal of enjoyment that these who visit them invest a lot of time. online roulette offers you much more manage over the game. You can perform it any time and if you want to stop, you can do so. It is quite different from that you go to and perform at land casinos. The online roulette does not provide you this kind of issues. The much more you remain the more you invest money. Additional, there are also other attractions and liquor that is served free of price.<br><br>Instead of waiting around for 5 rounds, wait for 7. You do not have do double the stakes, simply because winning will earn you 3 times the quantity you initially placed. If this technique is not producing good outcomes, try the much more sophisticated method of placing your bets on one:2 fields (rows or dozens). If you are betting on colours (crimson or black), adhere to this fundamental tactic: If you get, begin again at $1; if you lose, double your stakes. Your chances of successful with this tactic are more than ninety nine.<br><br>In Outlook, they're on the left aspect of the display. Be sure to clear out e-mails and folders you no lengthier need. Then review the folders you've set up in your e-mail system. What have you been keeping more than there?<br><br>This is the very exact same reason that casino s would instead allow you perform with chips than actual money. Yet as financial transactions are much more getting secure as technologies moves ahead, having to pay online would also turn out to be the norm in close to future. You are more connected to real cash than casino chips; therefore it's easier to invest. Some people are not comfortable paying on-line. This understandable, there is a distinction when you can bodily really feel the money, and understanding you have cash in the financial institution.<br><br>The Luxor Resort in conjunction with several other casinos offers what they call the One Club. The great factor about the 1 Club is that it can be used at so numerous different locations. Whilst you play the slots, you can rack up some fantastic comps such as meals, resort rooms or just plain bucks. You can sign up for the 1 Club for totally free and earn comp bucks while playing at Mandalay Bar, Luxor, Monte Carlo, or Circus Circus. You can use the comp dollars how ever you wish at any of the collaborating casinos.<br><br>With attractive payouts and phenomenal graphics, this slot can quickly become your favorite. The game is complete of unique bonuses that allow multiplying the winnings. The historic theme of historical occasions of King Arthur provides gamers a glimpse of life when kings dominated. The game is a favorite among the United kingdom online casino gamers simply because of its amazing top jackpot of 210,000 coins. Avalon Slot machine is included on the highlighted video games list at All Slots On line casino. |
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| One usually thinks of dimension for a set which is dense, like the points on a line, for example. Dimension makes sense in a discrete setting, like for graphs, only in the large system limit, as the size tends to infinity. For example, in Statistical Mechanics, one considers discrete points which are located on regular lattices of different dimensions. Such studies have been extended to arbitrary networks, and it is interesting to consider how the definition of dimension can be extended to cover these cases. A very simple and obvious way to extend the definition of dimension to arbitrary large networks is to consider how the volume (number of nodes within a given distance from a specified node) scales as the distance (shortest path connecting two nodes in the graph) is increased. For many systems arising in physics, this is indeed a useful approach. This definition of dimension could be put on a strong mathematical foundation, similar to the definition of Hausdorff dimension for continuous systems. The mathematically robust definition uses the concept of a zeta function for a graph. The complex network zeta function and the graph surface function were introduced to characterize large graphs. They have also been applied to study patterns in Language Analysis. In this section we will briefly review the definition of the functions and discuss further some of their properties which follow from the definition.
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| We denote by <math> \textstyle r_{ij} </math> the distance from node <math>\textstyle i</math> to node <math>\textstyle j</math>, i.e., the length of the shortest path connecting the first node to the second node. <math>\textstyle r_{ij}</math> is <math>\textstyle \infty</math> if there is no path from node <math>\textstyle i</math> to node <math>\textstyle j</math>. With this definition, the nodes of the complex network become points in a [[metric space]].<ref name="Shankerb"/> Simple generalisations of this definition can be studied, e.g., we could consider weighted edges. The graph surface function, <math>\textstyle S(r )</math>, is defined as the number of nodes which are exactly at a distance <math> \textstyle r </math> from a given node, averaged over all nodes of the network. The complex network zeta function <math>\textstyle \zeta_G ( \alpha )</math> is defined as
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| :<math> \zeta_G ( \alpha ) := \frac{1}{N}\sum_i \sum_{j\neq i }r^{-\alpha}_{ij}, </math>
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| where <math>\textstyle N</math> is the graph size, measured by the number of nodes. When <math>\textstyle \alpha</math> is zero all nodes contribute equally to the sum in the previous equation. This means that <math>\textstyle \zeta_{G}(0)</math> is <math>\textstyle N-1</math>, and it diverges when <math>\textstyle N \rightarrow \infty</math>. When the exponent <math>\textstyle \alpha</math> tends to infinity, the sum gets contributions only from the nearest neighbours of a node. The other terms tend to zero. Thus, <math>\textstyle \zeta_G ( \alpha )</math> tends to the average degree <math>\textstyle <k></math> for the graph as <math>\textstyle \alpha \rightarrow \infty</math>.
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| :<math> \langle k \rangle = \lim_{\alpha \rightarrow \infty} \zeta_G ( \alpha ). </math>
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| The need for taking an average over all nodes can be avoided by using the concept of supremum over nodes, which makes the concept much easier to apply for formally infinite graphs.<ref name="ShankerTCS">{{cite journal|author=O. Shanker|year=2010|title=Complex Network Dimension and Path Counts|journal=Theoretical Computer Science|volume= 411|pages=2454–2458|doi=10.1016/j.tcs.2010.02.013|issue=26–28}}</ref> The definition can be expressed as a weighted sum over the node distances. This gives the Dirichlet series relation
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| :<math> \zeta_G ( \alpha ) = \sum_r S(r)/r^{\alpha}. </math>
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| This definition has been used in the [[shortcut model]] to study several processes and their dependence on dimension.
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| ==Properties==
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| <math>\textstyle \zeta_G ( \alpha )</math> is a decreasing function of <math>\textstyle \alpha</math>, <math>\textstyle \zeta_G ( \alpha_1 ) > \zeta_G ( \alpha_2 )</math>, if <math>\textstyle \alpha_1 < \alpha_2</math>. If the average degree of the nodes (the mean coordination number for the graph) is finite, then there is exactly one value of <math>\textstyle \alpha</math>, <math>\textstyle \alpha_{transition}</math>, at which the complex network zeta function transitions from being infinite to being finite. This has been defined as the dimension of the complex network. If we add more edges to an existing graph, the distances between nodes will decrease. This results in an increase in the value of the complex network zeta function, since <math>\textstyle S(r)</math> will get pulled inward. If the new links connect remote parts of the system, i.e., if the distances change by amounts which do not remain finite as the graph size <math>\textstyle N \rightarrow \infty</math>, then the dimension tends to increase. For regular discrete ''d''-dimensional lattices <math>\textstyle \mathbf Z^d</math> with distance defined using 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 transition occurs at <math>\textstyle \alpha = d</math>. The definition of dimension using the complex network zeta function satisfies properties like monotonicity (a subset has a lower or the same dimension as its containing set), stability (a union of sets has the maximum dimension of the component sets forming the union) and Lipschitz invariance,<ref name=falconer>[[Kenneth Falconer (mathematician)|K. Falconer]], Fractal Geometry: Mathematical Foundations and Applications, Wiley, second edition, 2003</ref> provided the operations involved change the distances between nodes only by finite amounts as the graph size <math>\textstyle N</math> goes to <math>\textstyle \infty</math>. Algorithms to calculate the complex network zeta function have been presented.<ref name="Shankerc">{{cite 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|issue=7|bibcode=2008MPLB...22..459S}}</ref> | |
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| ==Values for discrete regular lattices==
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| For a one-dimensional regular lattice the graph surface function <math>\textstyle S_{1}(r)</math> is exactly two for all values of <math>\textstyle r</math> (there are two nearest neighbours, two next-nearest neighbours, and so on). Thus, the complex network zeta function <math>\textstyle \zeta_G ( \alpha )</math> is equal to <math>\textstyle 2\zeta(\alpha)</math>, where <math>\textstyle \zeta(\alpha)</math> is the usual Riemann zeta function. By choosing a given axis of the lattice and summing over cross-sections for the allowed range of distances along the chosen axis the recursion relation below can be derived
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| :<math> S_{d+1}(r) = 2+S_d (r)+2\sum^{r-1}_{i=1}S_d(i). </math>
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| From combinatorics the surface function for a regular lattice can be written<ref name="Shankerd">{{cite journal|author=O. Shanker|year=2008|title=Sharp dimension transition in a shortcut model |journal=J. Phys. A: Math. Theor. |volume= 41|pages=285001|doi=10.1088/1751-8113/41/28/285001|issue=28|bibcode=2008JPhA...41B5001S}}</ref> as
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| :<math> S_d(r) = \sum^{d-1}_{i=0}(-1)^{i}2^{d-i}{d \choose i} { d+r-i-1 \choose d-i-1 }. </math>
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| The following expression for the sum of positive integers raised to a given power <math>\textstyle k</math> will be useful to calculate the surface function for higher values of <math>\textstyle d</math>: | |
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| :<math> \sum^{r}_{i=1}i^k = \frac{r^{k+1}}{(k+1)}+ \frac{r^k}{2}+\sum^{\lfloor (k+1)/2 \rfloor}_{j=1}\frac{(-1)^{j+1}2\zeta(2j)k!r^{k+1-2j}}{(2\pi)^{2j}(k+1-2j)!}. </math>
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| Another formula for the sum of positive integers raised to a given power <math>\textstyle k</math> is
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| :<math> \sum^{n}_{k=1}\bigl(\begin{smallmatrix} n+1\ k \end{smallmatrix}\bigr)\sum^{r}_{i=1}i^{k} = (r+1)((r+1)^{n}-1). </math>
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| :<math>\textstyle S_{d}(r) \rightarrow O(2^{d}r^{d-1}/\Gamma(d))</math> as <math>\textstyle r\rightarrow \infty</math>.
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| The Complex network zeta function for some lattices is given below.
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| :<math>\textstyle d=1</math> : <math>\textstyle \zeta_G(\alpha)=2\zeta(\alpha)</math>
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| :<math>\textstyle d=2</math> : <math>\textstyle \zeta_G(\alpha)=4\zeta(\alpha-1)</math>
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| :<math>\textstyle d=3</math> : <math>\textstyle \zeta_G(\alpha)=4\zeta(\alpha-2)+2\zeta(\alpha</math>)
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| :<math>\textstyle d=4</math> : <math>\textstyle \zeta_G(\alpha)=\frac{8}{3}\zeta(\alpha-3)+\frac{16}{3}\zeta(\alpha-1)</math>
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| :<math>\textstyle r\rightarrow \infty</math> : <math>\textstyle \zeta_{G}(\alpha)=2^{d}\zeta(\alpha-d+1)/\Gamma(d)</math> (for <math>\alpha</math> near the transition point.)
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| ==Random graph zeta function==
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| Random graphs are networks having some number <math>\textstyle N</math> of vertices, in which each pair is connected with probability <math>\textstyle p</math>, or else the pair is disconnected. Random graphs have a diameter of two with probability approaching one, in the infinite limit (<math>\textstyle N \rightarrow \infty</math>). To see this, consider two nodes <math>\textstyle A</math> and <math>\textstyle B</math>. For any node <math>\textstyle C</math> different from <math>\textstyle A</math> or <math>\textstyle B</math>, the probability that <math>\textstyle C</math> is not simultaneously connected to both <math>\textstyle A</math> and <math>\textstyle B</math> is <math>\textstyle (1-p^2)</math>. Thus, the probability that none of the <math>\textstyle N-2</math> nodes provides a path of length <math>\textstyle 2</math> between nodes <math>\textstyle A</math> and <math>\textstyle B</math> is <math>\textstyle (1-p^2)^{N-2}</math>. This goes to zero as the system size goes to infinity, and hence most random graphs have their nodes connected by paths of length at most <math>\textstyle 2</math>. Also, the mean vertex degree will be <math>\textstyle p(N-1)</math>. For large random graphs almost all nodes are at a distance of one or two from any given node, <math>\textstyle S(1)</math> is <math>\textstyle p(N-1)</math>, <math>\textstyle S(2)</math> is <math>\textstyle (N-1)(1-p)</math>, and the graph zeta function is
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| :<math> \zeta_G ( \alpha ) = p(N-1) + (N-1)(1-p)2^{-\alpha}. </math>
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| ==References==
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| <references/>
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| [[Category:Graph theory]]
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| [[Category:Dimension]]
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| [[Category:Network theory]]
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