128.237.196.221: Much simpler and more elegant algebraic proof

2014-11-19T20:39:28Z

Much simpler and more elegant algebraic proof

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50.165.159.205 at 03:02, 10 February 2014

2014-02-10T03:02:05Z

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en>IronGargoyle: Reverted edits by B0NA13 (talk) to last revision by Peter Horn (HG)

2014-01-13T22:03:06Z

Reverted edits by B0NA13 (talk) to last revision by Peter Horn (HG)

← Older revision		Revision as of 00:03, 14 January 2014
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	~~Claude~~ is ~~her title~~ and ~~she completely digs~~ that ~~name~~. ~~Interviewing~~ is ~~how I make~~ a ~~residing~~ and it'~~s some thing I truly enjoy~~. ~~Kansas~~ is ~~where her home is but she requirements~~ to ~~move because of her family~~. ~~Playing croquet~~ is ~~some thing I will by no means give up~~.<br><br>~~Also visit my~~ web ~~site~~ ... [http://~~Zipperquick~~.~~com~~/~~profile/jusmxc extended car warranty~~]		[[File:Enwiki-degree-distribution.png\|thumb\|right\|320px\|In/out degree distribution for Wikipedia's hyperlink graph (logarithmic scales)]]
			In the study of [[graph (mathematics)\|graphs]] and [[complex network\|networks]], the [[degree (graph theory)\|degree]] of a node in a network is the number of connections it has to other nodes and the '''degree distribution''' is the [[probability distribution]] of these degrees over the whole network.

			==Definition==
			The [[degree (graph theory)\|degree]] of a node in a network (sometimes referred to incorrectly as the [[Connectivity (graph theory)\|connectivity]]) is the number of connections or [[Edge_(graph_theory)#Graph\|edges]] the node has to other nodes. If a network is [[directed graph\|directed]], meaning that edges point in one direction from one node to another node, then nodes have two different degrees, the in-degree, which is the number of incoming edges, and the out-degree, which is the number of outgoing edges.

			The degree distribution ''P''(''k'') of a network is then defined to be the fraction of nodes in the network with degree ''k''. Thus if there are ''n'' nodes in total in a network and ''n''<sub>''k''</sub> of them have degree ''k'', we have ''P''(''k'') = ''n''<sub>''k''</sub>/''n''.

			The same information is also sometimes presented in the form of a ''cumulative degree distribution'', the fraction of nodes with degree greater than or equal to ''k''.

			==Observed degree distributions==
			The degree distribution is very important in studying both real networks, such as the [[Internet]] and [[social networks]], and theoretical networks. The simplest network model, for example, the (Bernoulli) [[random graph]], in which each of ''n'' nodes is connected (or not) with independent probability ''p'' (or 1 − ''p''), has a [[binomial distribution]] of degrees:

			:<math>
			P(k) = {n-1\choose k} p^k (1 - p)^{n-1-k},
			</math>

			(or [[Poisson distribution\|Poisson]] in the limit of large ''n''). Most networks in the real world, however, have degree distributions very different from this. Most are highly right-skewed, meaning that a large majority of nodes have low degree but a small number, known as "hubs", have high degree. Some networks, notably the Internet, the [[world wide web]], and some social networks are found to have degree distributions that approximately follow a [[power law]]: ''P''(''k'') ~ ''k''<sup>−''γ''</sup>, where ''γ'' is a constant. Such networks are called [[scale-free networks]] and have attracted particular attention for their structural and dynamical properties.

			==See also==

			* [[Graph theory]]
			* [[Complex network]]
			* [[Scale-free network]]
			* [[Random graph]]

			== References ==

			* {{cite journal
			\| last = Albert
			\| first = R.
			\| coauthors = Barabasi, A.-L.
			\| title = Statistical mechanics of complex networks
			\| journal = Reviews of Modern Physics
			\| volume = 74
			\| pages = 47–97
			\| year = 2002
			\| arxiv = cond-mat/0106096
			\| doi = 10.1103/RevModPhys.74.47
			\| bibcode=2002RvMP...74...47A
			}}
			* {{cite journal
			\| last = Dorogovtsev
			\| first = S.
			\| coauthors = Mendes, J. F. F.
			\| title = Evolution of networks
			\| journal = Advances in Physics
			\| volume = 51
			\| pages = 1079–1187
			\| year = 2002
			\| arxiv = cond-mat/0106144
			\| doi = 10.1080/00018730110112519
			\| issue = 4
			\|bibcode = 2002AdPhy..51.1079D }}
			* {{cite journal
			\| last = Newman
			\| first = M. E. J.
			\| title = The structure and function of complex networks
			\| journal = SIAM Review
			\| volume = 45
			\| pages = 167–256
			\| year = 2003
			\| url = http://siamdl.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SIREAD000045000002000167000001
			\| doi = 10.1137/S003614450342480
			\| issue = 2
			\|arxiv = cond-mat/0303516 \|bibcode = 2003SIAMR..45..167N }}
			* {{cite book
			\| title= Complex Networks: Structure, Robustness and Function
			\| authors= [[Shlomo Havlin]] and Reuven Cohen
			\| year= 2010
			\| url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php
			\| publisher= Cambridge University Press
			}}

			[[Category:Graph theory]]
			[[Category:Graph invariants]]
			[[Category:Network theory]]

en>ClueBot NG: Reverting possible vandalism by 142.58.132.230 to version by Dinamik-bot. False positive? Report it. Thanks, ClueBot NG. (1165195) (Bot)

2012-08-08T21:04:36Z

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Claude is her title and she completely digs that name. Interviewing is how I make a residing and it's some thing I truly enjoy. Kansas is where her home is but she requirements to move because of her family. Playing croquet is some thing I will by no means give up.<br><br>Also visit my web site ... [http://Zipperquick.com/profile/jusmxc extended car warranty]

Pascal's rule - Revision history

128.237.196.221: Much simpler and more elegant algebraic proof

50.165.159.205 at 03:02, 10 February 2014

en>IronGargoyle: Reverted edits by B0NA13 (talk) to last revision by Peter Horn (HG)

en>ClueBot NG: Reverting possible vandalism by 142.58.132.230 to version by Dinamik-bot. False positive? Report it. Thanks, ClueBot NG. (1165195) (Bot)