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| {{Network Science}}
| | The best method to distinguish a person's ideal size is to have a thorough learning of the waist-to-hip ratio plus body mass index. Although this may be difficult to deal with initially, especially when the person becomes aware that they are not within the normal range. Acceptance is the first thing a individual requires to overcome, and then immediate action is taken. This is a easy overview of why you need to obtain a healthy weight chart.<br><br>Being tall inside the dating scene is a benefit. For me, my problem with standing out like a sore thumb rapidly diminished, because I enjoyed the additional attention plus getting noticed by all guys, placing me found on the front lines. Speaking of 'lines', the basketball line was annoying, yet it was a perfect initially line for guys to approach me with and I began not minding it because much.<br><br>But before we all breathe a sigh of relief plus down a cheeseburger, bag of potato chips, plus a carton of moose tracks ice cream (yum), the research waist to height ratio leader did state that the research didn't prove that weight was harmless. Being overweight or overweight still carries an improved risk of heart condition, diabetes plus certain kinds of cancers.<br><br>Waking up this morning I was so hungry however, I decided to stick to the diet. After my grapefruit breakfast, I am going on another bike ride. Well my stomach hurt so much it was hard to consume my lunch. I am thus hungry. I went over to my sisters home plus they you having pizza for dinner. I was tempted to consume it, but I didnt. Now Im getting willing to consume dinner. Im thus happy for dinner because I feel which this really is the best dinner from the four days. I got a hamburger patty plus stewed tomatoes. I then dipped the patty into the stewed tomatoes, it tasted like ketchup found on the hamburger. Tonights dinner had filled me up, that is superior considering I dont like to go to bed with belly cramps again. Im going to bed now and it feels wise to be full.<br><br>Here's an amusing thought: many each NFL linebacker is classified because fat, as are many additional expert athletes. Many elderly folks with rather small muscle qualify because "healthy weight". Those are the skinny-fat folks which the post above references plus it doesn't affect only the elderly.<br><br>According to the post "What Would the Real Barbie Look Like" inside the BBC News Magazine on Friday March 2, 2009 when Barbie were 5'6" then her waist will be 20", her hips 29" plus bust a mere 27". The closest real persons to people measurements are Victoria Beckham with a 23" waist plus Brigitte Bardot with a 20" waist. Both females are rather famous, rather skinny and not a average. According to the measurements of [http://safedietplansforwomen.com/waist-to-height-ratio waist height ratio] a Barbie figure would be accomplished by one in 100,000 ladies and a Ken figure can be attained inside 1 in fifty guys.<br><br>In order to calculate the ideal shoulder size we first have to know a ideal waist size. BUT in purchase to recognize the ideal waist size we have to begin with the height. The calculation goes from a height to the waist to the shoulders.<br><br>Utilizing weight resistance exercises to build muscle plus excellent strength interval cardio training to minimize body fat is not anything unique plus all qualified personal trainers should be capable to plan a advantageous system for you. Plus modern advances inside nutrition, you need to be capable to your desired bodyshape sooner instead of later. So plan the workouts and nutrition with a clear objective of getting a swimmer's physique plus not that of the bodybuilder plus you'll shortly get the admirable guy professional swimmer's build. |
| A '''scale-free network''' is a [[complex network|network]] whose [[degree distribution]] follows a [[power law]], at least asymptotically. That is, the fraction ''P''(''k'') of nodes in the network having ''k'' connections to other nodes goes for large values of ''k'' as
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| :<math>
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| P(k) \ \sim \ k^\boldsymbol{-\gamma}
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| </math>
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| where <math>\gamma</math> is a parameter whose value is typically in the range 2 < <math>\gamma</math> < 3, although occasionally it may lie outside these bounds.<ref>{{Cite doi|10.1073/pnas.0610245104}}</ref><ref>{{Cite doi|10.1007/s10955-013-0749-1}}</ref>
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| Many networks are conjectured to be scale-free, including World Wide Web links, biological networks, and social networks, although the scientific community is still discussing these claims as more sophisticated data analysis techniques become available.<ref name="Clauset">{{Cite journal
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| | doi = 10.1137/070710111
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| | last = Clauset
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| | first = Aaron
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| | coauthors = Cosma Rohilla Shalizi, M. E. J Newman
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| | title = Power-law distributions in empirical data
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| | journal = 0706.1062
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| | date = 2007-06-07
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| | arxiv = 0706.1062
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| |bibcode = 2009SIAMR..51..661C }}</ref> [[Preferential attachment]] and the [[fitness model (network theory)|fitness model]] have been proposed as mechanisms to explain conjectured power law degree distributions in real networks.
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| ==History==
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| In studies of the networks of citations between scientific papers, [[Derek J. de Solla Price|Derek de Solla Price]] showed in 1965 that the number of links to papers—i.e., the number of citations they receive—had a [[heavy-tailed distribution]] following a [[Pareto distribution]] or [[power law]], and thus that the citation network is scale-free. He did not however use the term "scale-free network", which was not coined until some decades later. In a later paper in 1976, Price also proposed a mechanism to explain the occurrence of power laws in citation networks, which he called "cumulative advantage" but which is today more commonly known under the name [[preferential attachment]].
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| Recent interest in scale-free networks started in 1999 with work by [[Albert-László Barabási]] and colleagues at the [[University of Notre Dame]] who mapped the topology of a portion of the World Wide Web,<ref>{{cite journal
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| |last1=Barabási |first1=Albert-László |authorlink1=Albert-László Barabási
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| |last2=Albert |first2=Réka.
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| |title=Emergence of scaling in random networks
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| |journal=[[Science (journal)|Science]]
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| |volume=286 |issue=5439 |pages=509–512 |date=October 15, 1999
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| |arxiv=cond-mat/9910332
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| |doi=10.1126/science.286.5439.509
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| |mr=2091634 |pmid=10521342
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| |bibcode = 1999Sci...286..509B }}</ref> finding that some nodes, which they called "hubs", had many more connections than others and that the network as a whole had a power-law distribution of the number of links connecting to a node. After finding that a few other networks, including some social and biological networks, also had heavy-tailed degree distributions, Barabási and collaborators coined the term "scale-free network" to describe the class of networks that exhibit a power-law degree distribution. Amaral et al. showed that most of the real-world networks can be classified into two large categories according to the decay of degree distribution P(k) for large k.
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| Barabási and Albert proposed a generative mechanism to explain the appearance of power-law distributions, which they called "[[preferential attachment]]" and which is essentially the same as that proposed by Price. Analytic solutions for this mechanism (also similar to the solution of Price) were presented in 2000 by Dorogovtsev, [[José Fernando Ferreira Mendes|Mendes]] and Samukhin <ref>{{Cite doi|10.1103/PhysRevLett.85.4633}}</ref> and independently by Krapivsky, [[Sidney Redner|Redner]], and Leyvraz, and later rigorously proved by mathematician [[Béla Bollobás]].<ref>{{Cite doi|10.1002/rsa.1009}}</ref> Notably, however, this mechanism only produces a specific subset of networks in the scale-free class, and many alternative mechanisms have been discovered since.<ref>{{Cite doi|10.1080/00018730110112519}}</ref>
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| The history of scale-free networks also includes some disagreement. On an empirical level, the scale-free nature of several networks has been called into question. For instance, the three brothers Faloutsos believed that the [[Internet]] had a power law degree distribution on the basis of [[traceroute]] data; however, it has been suggested that this is a [[Network Layer|layer 3]] illusion created by routers, which appear as high-degree nodes while concealing the internal [[Data Link Layer|layer 2]] structure of the [[Autonomous system (Internet)|ASes]] they interconnect.
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| <ref name="Willinger">{{Cite journal | |
| | last = Willinger
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| | first = Walter
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| | authorlink = Walter Willinger
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| | coauthors = David Alderson, and John C. Doyle
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| | title = Mathematics and the Internet: A Source of Enormous Confusion and Great Potential
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| | journal = Notices of the AMS
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| | volume = 56
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| | issue = 5
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| | pages = 586–599
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| | publisher = American Mathematical Society
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| | date = May 2009
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| | url = http://authors.library.caltech.edu/15631/1/Willinger2009p5466Notices_Amer._Math._Soc.pdf
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| | accessdate = 2011-02-03}}
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| </ref> On a theoretical level, refinements to the abstract definition of scale-free have been proposed. For example, Li et al. (2005) recently offered a potentially more precise "scale-free metric". Briefly, let ''G'' be a graph with edge set ''E'', and denote the degree of a vertex <math>v</math> (that is, the number of edges incident to <math>v</math>) by <math>\deg(v)</math>. Define | |
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| : <math>s(G) = \sum_{(u,v) \in E} \deg(u) \cdot \deg(v).</math>
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| This is maximized when high-degree nodes are connected to other high-degree nodes. Now define
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| : <math>S(G) = \frac{s(G)}{s_\mathrm{max}},</math>
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| where ''s''<sub>max</sub> is the maximum value of ''s''(''H'') for ''H'' in the set of all graphs with degree distribution identical to ''G''. This gives a metric between 0 and 1, where a graph ''G'' with small ''S''(''G'') is "scale-rich", and a graph ''G'' with ''S''(''G'') close to 1 is "scale-free". This definition captures the notion of [[self-similarity]] implied in the name "scale-free".
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| ==Characteristics==
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| [[Image:Scale-free network sample.png|thumb|Example of a random network and a scale-free network|400px|right|Random network (a) and scale-free network (b). In the scale-free network, the larger hubs are highlighted.]]
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| The most notable characteristic in a scale-free network is the relative commonness of vertices with a degree that greatly exceeds the average. The highest-degree nodes are often called "hubs", and are thought to serve specific purposes in their networks, although this depends greatly on the domain.
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| The scale-free property strongly correlates with the network's robustness to failure. It turns out that the major hubs are closely followed by smaller ones. These ones, in turn, are followed by other nodes with an even smaller degree and so on. This hierarchy allows for a [[fault-tolerance|fault tolerant]] behavior. If failures occur at random and the vast majority of nodes are those with small degree, the likelihood that a hub would be affected is almost negligible. Even if a hub-failure occurs, the network will generally not lose its [[connectedness]], due to the remaining hubs. On the other hand, if we choose a few major hubs and take them out of the network, the network is turned into a set of rather isolated graphs. Thus, hubs are both a strength and a weakness of scale-free networks. These properties have been studied analytically using [[percolation theory]] by Cohen et al.<ref>{{cite journal|title=Resilience of the Internet to Random Breakdowns|
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| journal=Phys. Rev. Lett.|
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| year=2000|
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| first=Reoven|
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| last=Cohen|
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| coauthors=K. Erez, D. ben-Avraham and [[Shlomo Havlin|S. Havlin]]|
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| volume=85|
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| pages=4626–8|
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| doi= 10.1103/PhysRevLett.85.4626|
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| url=http://link.aps.org/doi/10.1103/PhysRevLett.85.4626
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| |bibcode=2000PhRvL..85.4626C|arxiv = cond-mat/0007048 }}
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| </ref><ref>{{cite journal|title=Breakdown of the Internet under Intentional Attack| | |
| journal=Phys. Rev. Lett.|
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| year=2001|
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| first=Reoven|
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| last=Cohen|
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| coauthors=K. Erez, D. ben-Avraham and [[Shlomo Havlin|S. Havlin]]|
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| volume=86|
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| pages=3682–5|
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| doi= 10.1103/PhysRevLett.86.3682|
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| url=http://link.aps.org/doi/10.1103/PhysRevLett.86.3682
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| |bibcode=2001PhRvL..86.3682C
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| |pmid=11328053|arxiv = cond-mat/0010251 }}
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| </ref> and by Callaway et al.<ref>{{cite journal|title=Network Robustness and Fragility: Percolation on Random Graphs|
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| journal=Phys. Rev. Lett.|
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| year=2000|
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| first=Duncan S.|
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| last=Callaway|
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| coauthors=M. E. J. Newman, S. H. Strogatz and D. J. Watts|
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| volume=85|
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| pages=5468–71|
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| doi= 10.1103/PhysRevLett.85.5468|
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| url=http://link.aps.org/doi/10.1103/PhysRevLett.85.5468
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| |bibcode=2000PhRvL..85.5468C|arxiv = cond-mat/0007300 }}</ref>
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| Another important characteristic of scale-free networks is the [[clustering coefficient]] distribution, which decreases as the node degree increases. This distribution also follows a power law. This implies that the low-degree nodes belong to very dense sub-graphs and those sub-graphs are connected to each other through hubs. Consider a social network in which nodes are people and links are acquaintance relationships between people. It is easy to see that people tend to form communities, i.e., small groups in which everyone knows everyone (one can think of such community as a [[complete graph]]). In addition, the members of a community also have a few acquaintance relationships to people outside that community. Some people, however, are connected to a large number of communities (e.g., celebrities, politicians). Those people may be considered the hubs responsible for the [[small-world phenomenon]].
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| At present, the more specific characteristics of scale-free networks vary with the generative mechanism used to create them. For instance, networks generated by preferential attachment typically place the high-degree vertices in the middle of the network, connecting them together to form a core, with progressively lower-degree nodes making up the regions between the core and the periphery. The random removal of even a large fraction of vertices impacts the overall connectedness of the network very little, suggesting that such topologies could be useful for [[network security|security]], while targeted attacks destroys the connectedness very quickly. Other scale-free networks, which place the high-degree vertices at the periphery, do not exhibit these properties. Similarly, the clustering coefficient of scale-free networks can vary significantly depending on other topological details.
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| A final characteristic concerns the average distance between two vertices in a network. As with most disordered networks, such as the [[small world network]] model, this distance is very small relative to a highly ordered network such as a [[lattice graph]]. Notably, an uncorrelated power-law graph having 2 < γ < 3 will have ultrasmall diameter ''d'' ~ ln ln ''N'' where ''N'' is the number of nodes in the network, as proved by Cohen and Havlin. The diameter of a growing scale-free network might be considered almost constant in practice.
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| ==Examples==
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| Although many real-world networks are thought to be scale-free, the evidence often remains inconclusive, primarily due to the developing awareness of more rigorous data analysis techniques.<ref name="Clauset"/> As such, the scale-free nature of many networks is still being debated by the scientific community. A few examples of networks claimed to be scale-free include:
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| *[[Social network]]s, including collaboration networks. Two examples that have been studied extensively are [[Six Degrees of Kevin Bacon|the collaboration of movie actors in films]] and [[Erdős number|the co-authorship by mathematicians of papers]].
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| *Many kinds of [[computer network]]s, including the [[internet]] and the [[webgraph]] of the [[World Wide Web]].
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| *Some financial networks such as interbank payment networks <ref>{{cite journal|title=The topology of interbank payment flows|journal=Physica A: Statistical Mechanics and its Applications|year=2007|first=Kimmo|last=Soramäki|coauthors=et. al|volume=379|issue=1|pages=317–333|doi= 10.1016/j.physa.2006.11.093|url=http://www.sciencedirect.com/science/article/pii/S0378437106013124|bibcode = 2007PhyA..379..317S }}</ref>
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| *[[Protein-protein interaction]] networks.
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| *[[Semantic network]]s.<ref>{{cite journal|title=The Large-Scale Structure of Semantic Networks: Statistical Analyses and a Model of Semantic Growth|journal=Cognitive Science|year=2005|first=Mark|last=Steyvers|coauthors=Joshua B. Tenenbaum|volume=29|issue=1|pages=41–78|doi= 10.1207/s15516709cog2901_3|url=http://www.leaonline.com/doi/abs/10.1207/s15516709cog2901_3}}</ref>
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| *Airline networks.
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| Scale free topology has been also found in high temperature superconductors.<ref>{{cite journal |doi=10.1038/nature09260 |author=Fratini, Michela, Poccia, Nicola, Ricci, Alessandro, Campi, Gaetano, Burghammer, Manfred, Aeppli, Gabriel Bianconi, Antonio |title=Scale-free structural organization of oxygen interstitials in La2CuO4+y |journal=Nature |volume=466 |issue=7308 |pages=841–4 |year=2010 |url=http://www.nature.com/nature/journal/v466/n7308/full/nature09260.html |pmid=20703301|arxiv = 1008.2015 |bibcode = 2010Natur.466..841F }}</ref> The qualities of a high-temperature superconductor — a compound in which electrons obey the laws of quantum physics, and flow in perfect synchrony, without friction — appear linked to the fractal arrangements of seemingly random oxygen atoms and lattice distorsion.<ref>{{cite journal |doi=10.1073/pnas.1208492109 |author=Poccia, Nicola, Ricci, Alessandro, Campi, Gaetano, Fratini, Michela, Puri, Alessandro, Di Gioacchino, Daniele, Marcelli, Augusto, Reynolds, Michael, Burghammer, Manfred, Saini, Naurang L., Aeppli, Gabriel Bianconi, Antonio,|title=Optimum inhomogeneity of local lattice distortions in La2CuO4+y |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=109 |issue=39 |pages=15685–15690 |year=2012 |url=http://www.pnas.org/content/109/39/15685 |pmid=|arxiv = 1208.0101 |bibcode = }}</ref>
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| ==Generative models==
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| {{Merge from|Generalized scale-free model|discuss=Talk:Scale-free network#Merger with generalized scale-free model|date=August 2011}}
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| These scale-free networks do not arise by chance alone. [[Paul Erdős|Erdős]] and Rényi (1960) studied a model of growth for graphs in which, at each step, two nodes are chosen uniformly at random and a link is inserted between them. The properties of these [[random graph]]s are different from the properties found in scale-free networks, and therefore a model for this growth process is needed.
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| The mostly widely known generative model for a subset of scale-free networks is Barabási and Albert's (1999) [[rich get richer]] generative model in which each new Web page creates links to existing Web pages with a probability distribution which is not uniform, but
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| proportional to the current in-degree of Web pages. This model was originally discovered by [[Derek J. de Solla Price]] in 1965 under the term '''cumulative advantage''', but did not reach popularity until Barabási rediscovered the results under its current name ([[BA Model]]). According to this process, a page with many in-links will attract more in-links than a regular page. This generates a power-law but the resulting graph differs
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| from the actual Web graph in other properties such as the presence of small
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| tightly connected communities. More general models and networks characteristics have been proposed and studied (for a review see the book by Dorogovtsev and [[José Fernando Ferreira Mendes|Mendes]]).
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| A somewhat different generative model for Web links has been suggested by Pennock et al. (2002). They examined communities with interests in a specific topic such as the home pages of universities, public companies, newspapers or scientists, and discarded the major hubs of the Web. In this case, the distribution of links was no longer a power law but resembled a [[normal distribution]]. Based on these observations, the authors proposed a generative model that mixes preferential attachment with a baseline probability of gaining a link.
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| Another generative model is the '''copy''' model studied by Kumar et al. (2000),
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| in which new nodes choose an existent node at random and copy a fraction of the links of the existent node. This also generates a power law.
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| Interestingly, the ''growth'' of the networks (adding new nodes) is not a necessary condition for creating a scale-free network. Dangalchev (2004) gives examples of generating static scale-free networks. Another possibility (Caldarelli et al. 2002) is to consider the structure as static and draw a link between vertices according to a particular property of the two vertices involved. Once specified the statistical distribution for these vertices properties (fitnesses), it turns out that in some circumstances also static networks develop scale-free properties.
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| == Scale-free ideal network ==
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| In the context of [[network theory]] a '''scale-free ideal network''' is a [[random network]] with a [[degree distribution]] following the [[Scale-Free Ideal Gas|scale-free ideal gas]] [[Probability density function|density distribution]]. These networks have the special property of reproducing the city-size distribution and electoral results unravelling the size distribution of social groups with information theory on complex networks,<ref>{{cite arXiv |author=A. Hernando, D. Villuendas, C. Vesperinas, M. Abad, A. Plastino |eprint=0905.3704 |class=physics.soc-ph |title=Unravelling the size distribution of social groups with information theory on complex networks |year=2009 }}, submitted to ''European Physics Journal B''</ref>
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| when a competitive cluster growth process<ref>
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| {{cite arXiv |author=André A. Moreira, Demétrius R. Paula, Raimundo N. Costa Filho, José S. Andrade, Jr. |eprint=cond-mat/0603272 |title=Competitive cluster growth in complex networks |class=cond-mat.dis-nn |year=2006 }}</ref> is applied to the network. In models of scale-free ideal networks it is possible to demonstrate that [[Dunbar's number]] is the cause of the phenomenon known as the '[[six degrees of separation]]' .
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| == See also ==
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| * [[Random graph]]
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| * [[Erdős–Rényi model]]
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| * [[Bose-Einstein condensation: a network theory approach]]
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| * [[Scale invariance]]
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| * [[Complex network]]
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| * [[Webgraph]]
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| == References ==
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| <div class='references-small'>
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| {{Reflist}}
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| </div>
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| *{{cite journal |doi=10.1103/RevModPhys.74.47 |author=Albert R., Barabási A.-L. |title=Statistical mechanics of complex networks |journal=Rev. Mod. Phys. |volume=74 |pages=47–97 |year=2002 |url=http://www.nd.edu/~networks/Publication%20Categories/publications.htm#anchor-allpub0001 |bibcode=2002RvMP...74...47A|arxiv = cond-mat/0106096 }}
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| *{{cite journal |doi=10.1073/pnas.200327197 |author=Amaral, LAN, Scala, A., Barthelemy, M., Stanley, HE. |title=Classes of behavior of small-world networks |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=97 |issue=21 |pages=11149–52 |year=2000 |pmid=11005838 |pmc=17168 |arxiv=cond-mat/0001458|bibcode = 2000PNAS...9711149A }}
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| *{{cite book |author=Barabási, Albert-László |title=Linked: How Everything is Connected to Everything Else |year=2004 |isbn=0-452-28439-2 }}
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| *{{cite journal |doi=10.1038/scientificamerican0503-60 |author=Barabási, Albert-László |last2=Bonabeau |first2=Eric |title=Scale-Free Networks |journal=Scientific American |volume=288 |pages=60–9 |date=May 2003 |url=http://www.nd.edu/~networks/Publication%20Categories/01%20Review%20Articles/ScaleFree_Scientific%20Ameri%20288,%2060-69%20(2003).pdf |format=PDF |issue=5}}
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| *{{cite journal |doi=10.1103/PhysRevE.69.016113 |author=Dan Braha, Yaneer Bar-Yam |title=Topology of Large-Scale Engineering Problem-Solving Networks |journal=Phys. Rev. E |volume=69 |pages=016113 |year=2004 |url=http://necsi.edu/affiliates/braha/Topology--of--Large--Scale--Design--PRE69.pdf |format=PDF|bibcode = 2004PhRvE..69a6113B }}
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| * Caldarelli G. "[http://www.oup.com/us/catalog/general/subject/Physics/Mathematicalphysics/~~/dmlldz11c2EmY2k9OTc4MDE5OTIxMTUxNw== Scale-Free Networks"] Oxford University Press, Oxford (2007).
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| *{{cite journal |doi=10.1103/PhysRevLett.89.258702 |author=Caldarelli G., Capocci A., De Los Rios P., Muñoz M.A. |title=Scale-free networks from varying vertex intrinsic fitness |journal=Physical Review Letters |volume=89 |issue=25 |pages=258702 |year=2002 |pmid=12484927 |arxiv=cond-mat/0207366 |bibcode=2002PhRvL..89y8702C}}
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| *{{cite journal|title=Resilience of the Internet to Random Breakdowns|journal=Phys. Rev. Lett.|year=2000|authors=R. Cohen, K. Erez, D. ben-Avraham and [[Shlomo Havlin|S. Havlin]]|volume=85|pages=4626–8|doi= 10.1103/PhysRevLett.85.4626|url=http://link.aps.org/doi/10.1103/PhysRevLett.85.4626|bibcode=2000PhRvL..85.4626C|arxiv = cond-mat/0007048 }}
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| *{{cite journal|title=Breakdown of the Internet under Intentional Attack|journal=Phys. Rev. Lett.|year=2001|authors=R. Cohen, K. Erez, D. ben-Avraham and [[Shlomo Havlin|S. Havlin]]|volume=86|pages=3682–5|doi= 10.1103/PhysRevLett.86.3682|url=http://link.aps.org/doi/10.1103/PhysRevLett.86.3682|bibcode=2001PhRvL..86.3682C|pmid=11328053|arxiv = cond-mat/0010251 }}
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| *{{cite journal|title=Scale-free networks on lattices|journal=Phys. Rev. Lett.|year=2002|authors=A.F. Rozenfeld, R. Cohen, D. ben-Avraham, [[Shlomo Havlin|S. Havlin]]|volume=89|url=http://havlin.biu.ac.il/Publications.php?keyword=Scale-free+networks+on+lattices&year=*&match=all}}
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| *{{cite journal |author=Dangalchev, Ch. |title=Generation models for scale-free networks |journal=Physica A |volume=338 |year=2004 }}
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| *{{cite journal |author=Dorogovtsev, Mendes, J.F.F. , Samukhin, A.N. |title=Structure of Growing Networks: Exact Solution of the Barabási—Albert's Model |journal=Phys. Rev. Lett. |volume=85 |issue=21 |pages=4633–6 |year=2000 |pmid=11082614 |doi=10.1103/PhysRevLett.85.4633 |bibcode=2000PhRvL..85.4633D|arxiv = cond-mat/0004434 }}
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| *{{cite book |author=Dorogovtsev, S.N., Mendes, J.F.F. |title=Evolution of Networks: from biological networks to the Internet and WWW |publisher=Oxford University Press |year=2003 |isbn=0-19-851590-1 }}
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| *{{cite journal |author=Dorogovtsev, S.N., Goltsev A. V., Mendes, J.F.F. |title= Critical phenomena in complex networks |journal= Rev. Mod. Phys. |volume=80 |pages=1275 |year=2008 | doi = 10.1103/RevModPhys.80.1275 |bibcode=2008RvMP...80.1275D|arxiv = 0705.0010 }}
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| *{{cite journal |doi=10.1080/00018730110112519 |author=Dorogovtsev, S.N., Mendes, J.F.F. |title=Evolution of networks |journal=Advances in Physics |volume=51 |pages=1079–1187 |year=2002 |arxiv = cond-mat/0106144 |bibcode = 2002AdPhy..51.1079D }}
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| *{{cite book |last1=Erdős |first1=P. |authorlink1=Paul Erdős |last2=Rényi |first2=A. |authorlink2=Alfréd Rényi |title=On the Evolution of Random Graphs |publisher=Publication of the Mathematical Institute of the Hungarian Academy of Science |year=1960 |volume=5 |pages=17–61 |url=http://www.math-inst.hu/~p_erdos/1960-10.pdf |format=PDF}}
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| *{{cite journal |doi=10.1145/316194.316229 |author=Faloutsos, M., Faloutsos, P., Faloutsos, C. |title=On power-law relationships of the internet topology |journal=Comp. Comm. Rev. |volume=29 |pages=251 |year=1999 }}
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| *{{cite arXiv |author=Li, L., Alderson, D., Tanaka, R., Doyle, J.C., Willinger, W. |eprint=cond-mat/0501169 |title=Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version) |year=2005 |class=cond-mat.dis-nn }}
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| *{{cite conference |url=http://www.cs.brown.edu/research/webagent/focs-2000.pdf |title=Stochastic models for the web graph |author=Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A., Upfal, E. |year=2000 |publisher=IEEE CS Press |booktitle=Proceedings of the 41st Annual Symposium on Foundations of Computer Science (FOCS) |pages=57–65 |location=Redondo Beach, CA }}
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| *{{cite journal |author=Manev R., Manev H. |title=The meaning of mammalian adult neurogenesis and the function of newly added neurons: the "small-world" network |journal=Med. Hypotheses |volume=64 |issue=1 |pages=114–7 |year=2005 |pmid=15533625 |doi=10.1016/j.mehy.2004.05.013 |url=http://linkinghub.elsevier.com/retrieve/pii/S0306987704003524}}
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| *{{cite news |author=Matlis, Jan |title=Scale-Free Networks |url=http://www.computerworld.com/networkingtopics/networking/story/0,10801,75539,00.html |date=November 4, 2002 }}
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| *{{cite arXiv |author=Newman, Mark E.J. |eprint=cond-mat/0303516 |title=The structure and function of complex networks |year=2003 |class=cond-mat.stat-mech }}
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| *{{cite book |author=Pastor-Satorras, R., Vespignani, A. |title=Evolution and Structure of the Internet: A Statistical Physics Approach |publisher=Cambridge University Press |year=2004 |isbn=0-521-82698-5 }}
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| *{{cite journal |doi=10.1073/pnas.032085699 |author=Pennock, D.M., Flake, G.W., Lawrence, S., Glover, E.J., Giles, C.L. |title=Winners don't take all: Characterizing the competition for links on the web |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=99 |issue=8 |pages=5207–11 |year=2002 |url=http://www.modelingtheweb.com/ |pmid=16578867 |pmc=122747|bibcode = 2002PNAS...99.5207P }}
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| * Robb, John. [http://globalguerrillas.typepad.com/globalguerrillas/2004/05/scalefree_terro.html Scale-Free Networks and Terrorism], 2004.
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| *{{cite journal |doi=10.1002/bies.20294 |author=Keller, E.F. |title=Revisiting "scale-free" networks |journal=BioEssays |volume=27 |issue=10 |pages=1060–8 |year=2005 |url=http://www3.interscience.wiley.com/cgi-bin/abstract/112092785/ABSTRACT |pmid=16163729}}
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| *{{cite journal |doi=10.1103/PhysRevE.70.037103 |author=Onody, R.N., de Castro, P.A. |title=Complex Network Study of Brazilian Soccer Player |journal=Phys. Rev. E |volume=70 |pages=037103 |year=2004 |arxiv=cond-mat/0409609|bibcode = 2004PhRvE..70c7103O }}
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| *{{cite journal |doi=10.1103/PhysRevLett.90.058701 |author=Reuven Cohen, Shlomo Havlin |title=Scale-Free Networks are Ultrasmall |journal=Phys. Rev. Lett. |volume=90 |issue=5 |pages=058701 |year=2003 |url=http://havlin.biu.ac.il/Publications.php?keyword=Scale-Free+Networks+are+Ultrasmall&year=*&match=all|pmid=12633404 |arxiv=cond-mat/0205476 |bibcode=2003PhRvL..90e8701C}}
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| ==External links==
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| * [http://digital.csic.es/handle/10261/27556 snGraph] Optimal software to manage scale-free networks.
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| * [http://web-graph.org The Erdős Webgraph Server] describing the [[hyperlink]] structure of a weekly updated, constantly increasing portion of the [[WWW]].
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| {{DEFAULTSORT:Scale-Free Network}}
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| [[Category:Graph families]]
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| {{Link GA|zh}}
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