en>GlennLawyer: corrected definition of betweenness centrality

2014-08-28T07:17:38Z

corrected definition of betweenness centrality

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129.34.20.23 at 15:26, 6 February 2014

2014-02-06T15:26:03Z

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en>JosephGrimaldi: writing needs to be improved, first step

2013-12-21T16:35:43Z

writing needs to be improved, first step

← Older revision		Revision as of 18:35, 21 December 2013
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	~~Myrtle Benny~~ is ~~how I~~'~~m known~~ as and ~~I really feel comfy when individuals use~~ the ~~full~~ name. ~~One~~ of ~~over~~ the ~~counter std test (~~[~~http~~://~~www~~.~~cam4teens~~.~~com~~/~~blog~~/~~84472 Suggested Reading~~]) ~~things he enjoys most~~ is ~~ice skating~~ but he is ~~having difficulties~~ to ~~find~~ time ~~for~~ it. ~~Years ago we moved~~ to ~~North Dakota~~ and ~~I love every day living here~~. ~~He utilized~~ to be ~~unemployed but now he~~ is a ~~meter reader~~.		In the study of [[scale-free network]]s, a '''copying mechanism''' is a process by which such a network can form and grow, by means of repeated steps in which nodes are duplicated with mutations from existing nodes. Several variations of copying mechanisms have been studied. In the '''general copying model''', a growing network starts as a small initial graph and, at each time step, a new vertex is added with a given number ''k'' of new outgoing edges. As a result of a stochastic selection, the neighbors of the new vertex are either chosen randomly among the existing vertices, or one existing vertex is randomly selected and ''k'' of its neighbors are ‘copied’ as heads of the new edges.<ref name=kogias>{{citation\|first=A.\|last=Kogias\|journal=Int. J. Web Engineering and Technology\|volume=2\|issue=1\|year=2005}}.</ref>

			== Motivation ==
			Copying mechanisms for modeling growth of the [[world wide web]] are motivated by the following intuition:

			*Some web page authors will note an interesting but novel commonality between certain pages, and will link to pages exhibiting this commonality; pages created with this motivation are modeled by a random choice among existing pages.

			*Most authors, on the other hand, will be interested in certain already-represented topics, and will collect together links to pages about these topics. Pages created in this way can be modeled by node copying.

			Those are the growth and [[preferential attachment]] properties of the networks.

			== Description ==
			For the simple case, nodes are never deleted. At each step we create a new node
			with a single edge emanating from it. Let u be a page chosen
			uniformly at random from the pages in existence before this
			step.

			(I) With probability <math>p</math>, the only parameter of the model,
			the new edge points to u.

			(II) With probability <math>1-p</math>,
			the new edge points to the destination of u's (sole) out-link;
			the new node attains its edge by copying.

			The second process increases the probability of high-degree nodes' receiving new incoming edges. In fact, since u is selected randomly, the probability that a webpage with degree <math>k</math>
			will
			receive a new hyperlink is proportional
			with
			<math>(1-p)k</math>, indicating that the copying mechanism effectively amounts to a
			linear preferential attachment. Kumar et al. prove that the expectation of the incoming [[degree distribution]]
			is <math>P(k_{in})=k^{-(2-p)/(1-p)}</math>,
			thus <math>P(k)</math> follows a [[power-law]] with an exponent which varies between 2 (for
			<math>p\rightarrow 0</math>) and <math>\infty</math> (for <math>p\rightarrow 1</math>).

			Above is the linear growth copying model. Since the web is currently growing exponentially,
			there is the exponential growth copying model. At each step a new epoch of vertices arrives whose size is a constant
			fraction of the current graph. Each of these vertices may
			link only to vertices from previous epochs.

			The evolving models above are by no means complete. They
			can be extended in several ways. First of all, the tails in the
			models are either static, chosen uniformly from the new
			vertices, or chosen from the existing vertices proportional
			to their out-degrees. This process could be made more sophisticated
			to account for the observed deviations of the
			out-degree distribution from the [[power-law]] [[Probability distribution\|distribution]].
			Similarly, the models can be extended to include death processes,
			which cause vertices and edges to disappear as time
			evolves. A number of other extensions are possible, but we
			seek to determine the properties of this simple model, in order
			to understand which extensions are necessary to capture
			the complexity of the web.

			==Examples==
			===Undirected network models===
			====Protein interaction networks====
			Vazquez proposed a growing graph based on duplication modeling protein interactions. At every time step a prototype is chosen randomly. With probability q edges of the prototype are copied. With probability p an edge to the prototype is created.<ref name=Vazquez>A. Vazquez, A. Flammini, A. Maritan, A. Vespignani, Modeling of protein interaction networks, arXiv:cond-mat/0108043.</ref>

			====Proteome networks====
			<!-- Unsourced image removed: [[Image:Sole1.JPG\|right\|320px\|thumb\|'''Fig. 1:''' Growing network by duplication of nodes. First (a) duplication occurs after randomly selecting a node (arrow). The links from the newly created node (white) now can experience deletion (b) and new links can be created (c); these events occur with probabilities δ and α respectively.<ref name=Sole>R. V. Sole, R. Pastor-Satorras, E. Smith, T. B. Kepler, A model of large-scale proteome evolution, arXiv.org:cond-mat/0207311.</ref>]] -->
			Sole proposed a growing graph initialized with a 5-ring substrate. At every time step a new node is added and a
			prototype is chosen at random. The prototype's edges are copied with a probability δ. Furthermore, random nodes are connected to the newly introduced node with probability α= β/N, where δ and β are given parameters in (0,1) and N is the number of total nodes at the considered time step. (see fig. 1).<ref name=Sole>R. V. Sole, R. Pastor-Satorras, E. Smith, T. B. Kepler, A model of large-scale proteome evolution, arXiv.org:cond-mat/0207311.</ref>

			===Directed network models===
			====Biological networks====
			Middendorf-Ziv (MZ) proposed a growing directed graph modeling biological network dynamics. A prototype is chosen at random and duplicated. The prototype or progenitor node has edges pruned with probability β and edges added with probability α<<β. Based loosely on the undirected protein network model of Sole et al.<ref name="Sole"/>

			====WWW networks and citation networks====
			Vazquez proposed a growth model based on a recursive 'copying' mechanism, continuing to 2nd nearest neighbors, 3rd nearest neighbors etc. The authors call it a 'random walk' mechanism.).<ref name=VazquezII>A. Vazquez, Knowing a network by walking on it: emergence of scaling, arXiv:cond-mat/0006132.</ref>

			====Growing network with copying (GNC)====
			<!-- Missing image removed: [[Image:GNC2.JPG\|left\|320px\|thumb\|'''Fig. 2:''' Illustration of the growing network with copying (GNC). The time order of the nodes is indicated. Initial links are solid and links to ancestor nodes are dashed. Later links partially obscure earlier links. The new node initially attaches to random target node 4, as well as to its ancestors, 1 and 3.<ref name=Krapivsky>Krapivsky, P. L., and Redner, S., Network growth by copying, Phys. Rev. E 71 036118 (2005).</ref>
			]] -->

			Krapivsky and Redner proposed a new growing network model, which grows by adding nodes one at a time. A newly introduced node randomly selects a target node and links to it, as well as to all ancestor nodes of the target node (Fig. 2). If the target node is the initial root node, no additional links are generated by the copying mechanism. If the newly introduced node were to always choose the root node as the target, a star graph would be generated. On the other hand, if the target node is always the most recent one in the network, all previous nodes are ancestors of the target and the copying mechanism would give a complete graph. Correspondingly, the total number of links <math>L_N</math> in a network of N nodes can range from N−1 (star graph) to N(N−1)/2 (complete graph). Notice also that the number of outgoing links from each new node (the out-degree) can range between 1 and the current number of nodes.<ref name=Krapivsky>Krapivsky, P. L., and Redner, S., Network growth by copying, Phys. Rev. E 71 036118 (2005).</ref>

			==Notes==
			{{reflist}}

			==References==

			*Kleinberg, J. M., R. Kumar, P. Raghavan, S. Rajagopalan, and A. Tomkins, 1999, Proceedings of the International Conference on Combinatorics and Computing.

			*Kumar, R., P. Raghavan, S. Rajagopalan, D. Sivakumar, A. S. Tomkins and [[Eli Upfal\|E. Upfal]], 2000a, Proceedings of the 19th Symposium on Principles of Database Systems.

			*Kumar, R., P. Raghavan, S. Rajagopalan, D. Sivakumar, A. S. Tomkins and [[Eli Upfal\|E. Upfal]], 2000b, Proceedings of the 41st IEEE Symposium on Foundations of Computer Science.

			{{DEFAULTSORT:Copying Mechanism}}
			[[Category:Graph theory]]

en>Skatalites at 10:32, 30 March 2012

2012-03-30T10:32:11Z

New page

Myrtle Benny is how I'm known as and I really feel comfy when individuals use the full name. One of over the counter std test ([http://www.cam4teens.com/blog/84472 Suggested Reading]) things he enjoys most is ice skating but he is having difficulties to find time for it. Years ago we moved to North Dakota and I love every day living here. He utilized to be unemployed but now he is a meter reader.

Girvan–Newman algorithm - Revision history

en>GlennLawyer: corrected definition of betweenness centrality

129.34.20.23 at 15:26, 6 February 2014

en>JosephGrimaldi: writing needs to be improved, first step

en>Skatalites at 10:32, 30 March 2012