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{{Network Science}}
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A '''small-world network''' is a type of [[Graph (mathematics)|mathematical graph]] in which most nodes are not neighbors of one another, but most nodes can be reached from every other by a small number of hops or steps.  Specifically, a small-world network is defined to be a network where the typical distance ''L'' between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodes ''N'' in the network, that is:<ref>http://www.nature.com/nature/journal/v393/n6684/full/393440a0.html</ref>


:<math>L \propto \log N</math>
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In the context of a social network, this results in the [[Small-world experiment|small world phenomenon]] of strangers being linked by a mutual acquaintance.  Many empirical graphs are well-modeled by small-world networks. [[Social networks]], the connectivity of the [[Internet]], wikis such as Wikipedia, and [[gene regulatory network|gene networks]] all exhibit small-world network characteristics.
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A certain category of small-world networks were identified as a class of [[random graph]]s by [[Duncan J. Watts|Duncan Watts]] and [[Steven Strogatz]] in 1998.<ref>{{cite journal | author=Watts, Duncan J.; [[Steven Strogatz|Strogatz, Steven H.]] | title=Collective dynamics of 'small-world' networks | journal=Nature |date=June 1998 | volume=393 | pages=440–442 | doi=10.1038/30918 | pmid=9623998 | issue=6684 | bibcode=1998Natur.393..440W}} [http://web.archive.org/web/20070418032327/http://www.tam.cornell.edu/SS_nature_smallworld.pdf]</ref> They noted that graphs could be classified according to two independent structural features, namely the [[clustering coefficient]], and average node-to-node [[distance (graph theory)|distance]] (also known as average [[shortest path]] length). Purely random graphs, built according to the [[Erdős–Rényi model|Erdős–Rényi (ER) model]], exhibit a small average shortest path length (varying typically as the logarithm of the number of nodes) along with a small clustering coefficient. Watts and Strogatz measured that in fact many real-world networks have a small average shortest path length, but also a clustering coefficient significantly higher than expected by random chance. Watts and Strogatz then proposed a novel graph model, currently named the [[Watts and Strogatz model]], with (i) a small average shortest path length, and (ii) a large clustering coefficient. The crossover in the Watts-Strogatz model between a "large world" (such as a lattice) and a small world was first described by Barthelemy and Amaral in 1999.<ref>{{cite journal | last1 = Barthelemy | first1 = M. | last2 = Amaral | first2 = LAN | year = 1999 | title = Small-world networks: Evidence for a crossover picture | url = | journal = Phys. Rev. Lett. | volume = 82 | issue = 15| page = 3180 | doi=10.1103/PhysRevLett.82.3180 | bibcode=1999PhRvL..82.3180B|arxiv = cond-mat/9903108 }}</ref> This work was followed by a large number of studies, including exact results (Barrat and Weigt, 1999; Dorogovtsev and [[José Fernando Ferreira Mendes|Mendes]]; Barmpoutis and Murray, 2010).
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==Properties of small-world networks==
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Small-world networks tend to contain [[clique (graph theory)|cliques]], and near-cliques, meaning sub-networks which have connections between almost any two nodes within them. This follows from the defining property of a high [[clustering coefficient]]. Secondly, most pairs of nodes will be connected by at least one short path. This follows from the defining property that the mean-shortest path length be small.
Several other properties are often associated with small-world networks. Typically there is an over-abundance of ''hubs'' - nodes in the network with a high number of connections (known as high [[degree (graph theory)|degree]] nodes). These hubs serve as the common connections mediating the short path lengths between other edges. By analogy, the small-world network of airline flights has a small mean-path length (i.e. between any two cities you are likely to have to take three or fewer flights) because many flights are routed through [[airline hub|hub]] cities.


This property is often analyzed by considering the fraction of nodes in the network that have a particular number of connections going into them (the degree distribution of the network). Networks with a greater than expected number of hubs will have a greater fraction of nodes with high degree, and consequently the degree distribution will be enriched at high degree values. This is known colloquially as a [[fat-tailed distribution]]. Specifically, if a network has a degree-distribution which can be [[goodness of fit|fit]] with a [[Power law#Power-law distributions|power law distribution]], it is taken as a sign that the network is small-world. Networks with power law degree distribution are also known as [[scale-free network]]s. Graphs of very different topology qualify as small-world networks as long as they satisfy the two definitional requirements above.
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Network small-worldness has been quantified by comparing clustering and path length of a given to network to an equivalent random network with same degree distribution.<ref name="D. Humphries, K 1639">Is there a brainstem substrate for action selection?
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M. D. Humphries, K. Gurney and T. J. Prescott, Phil. Trans. R. Soc. B 2007 362, 1627-1639, {{doi|10.1098/rstb.2007.2057}}</ref> Another method for quantifying network small-worldness utilizes the original definition of the small-world network comparing the clustering of a given network to an equivalent lattice network and its path length to an equivalent random network.<ref>The ubiquity of small-world networks Q.K. Telesford, K.E. Joyce, S. Hayasaka, J.H. Burdette, P.J. Laurienti, Brain Connect. 2011;1(5):367-75, {{doi|10.1089/brain.2011.0038}}</ref> The small-world measure (<math>\omega</math>) is defined as


:<math>\omega = \tfrac{L_r}{L} - \tfrac{C}{C_l}</math>
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:<math>L \propto \log \log N</math>
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==Examples of small-world networks==
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Small-world properties are found in many real-world phenomena, including road maps, food chains, electric power grids, metabolite processing networks, [[neural networks|networks of brain neurons]],
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Networks of [[protein-protein interaction|connected proteins]] have small world properties such as power-law obeying degree distributions.<ref>{{cite journal | last1 = Bork | first1 = P. | last2 = Jensen | first2 = LJ | last3 = von Mering | first3 = C. | last4 = Ramani | first4 = A. | last5 = Lee | first5 = I. | last6 = Marcotte | first6 = EM. | year = 2004 | title = Protein interaction networks from yeast to human | url = http://polaris.icmb.utexas.edu/paper-pdfs/cosb-review.pdf | format = PDF | journal = Current Opinion in Structural Biology | volume = 14 | issue = 3| pages = 292–299 | doi = 10.1016/j.sbi.2004.05.003 | pmid = 15193308 }}</ref> Similarly [[transcriptional regulation|transcriptional network]]s, in which the nodes are [[gene]]s, and they are linked if one gene has an up or down-regulatory genetic influence on the other, have small world network properties.<ref>{{cite journal | last1 = Van Noort | first1 = V | last2 = Snel | first2 = B | last3 = Huynen | first3 = MA. |date=Mar 2004 | title = The yeast coexpression network has a small-world, scale-free architecture and can be explained by a simple model | journal = EMBO Rep. | volume = 5 | issue = 3| pages = 280–4 | doi = 10.1038/sj.embor.7400090 | pmid = 14968131 | pmc = 1299002 }}</ref>
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==Examples of non-small-world networks==
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Networks are less likely to have the small-world properties if links between nodes arise mainly from spatial or temporal proximity, because there may be no short path between two "distant" nodes. Being constrained to physical space or time, as in a subway system or road network, tends to impede the formation of particularly long links that are conducive to hub formation.


In another example, the famous theory of "[[six degrees of separation]]" between people tacitly presumes that the [[domain of discourse]] is the set of people alive at any one time. The number of degrees of separation between [[Albert Einstein]] and [[Alexander the Great]] is almost certainly greater than 30 and this network does not have small-world properties. A similarly constrained network would be the "went to school with" network: if two people went to the same college ten years apart from one another, it is unlikely that they have acquaintances in common amongst the student body.
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Similarly, the number of relay stations through which a message must pass was not always small. In the days when the post was carried by hand or on horseback, the number of times a letter changed hands between its source and destination would have been much greater than it is today. The number of times a message changed hands in the days of the visual telegraph (circa 1800–1850) was determined by the requirement that two stations be connected by line-of-sight.
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Tacit assumptions, if not examined, can cause a bias in the literature on graphs in favor of finding small-world networks (an example of the [[File drawer problem#File drawer effect|file drawer effect resulting from the publication bias]]).
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==Network robustness==
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It is hypothesized by some researchers such as [[Albert-László Barabási|Barabási]] that the prevalence of small world networks in biological systems may reflect an evolutionary advantage of such an architecture. One possibility is that small-world networks are more robust to perturbations than other network architectures. If this were the case, it would provide an advantage to biological systems that are subject to damage by [[mutation]] or [[virus|viral infection]].


In a [[power law]] distributed small world network, deletion of a random node rarely causes a dramatic increase in [[mean-shortest path]] length (or a dramatic decrease in the [[clustering coefficient]]). This follows from the fact that most shortest paths between nodes flow through [[Spoke-hub distribution paradigm|hubs]], and if a peripheral node is deleted it is unlikely to interfere with passage between other peripheral nodes. As the fraction of peripheral nodes in a small world network is much higher than the fraction of [[Spoke-hub distribution paradigm|hubs]], the probability of deleting an important node is very low. For example, if the small airport in [[Sun Valley, Idaho]] was shut down, it would not increase the average number of flights that other passengers traveling in the United States would have to take to arrive at their respective destinations. That said, if random deletion of a node hits a hub by chance, the average path length can increase dramatically. This can be observed annually when northern hub airports, such as Chicago's [[O'Hare International Airport|O'Hare airport]], are shut down because of snow; many people have to take additional flights.
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By contrast, in a random network, in which all nodes have roughly the same number of connections, deleting a random node is likely to increase the mean-shortest path length slightly but significantly for almost any node deleted. In this sense, random networks are vulnerable to random perturbations, whereas small-world networks are robust. However, small-world networks are vulnerable to targeted attack of hubs, whereas random networks cannot be targeted for catastrophic failure.
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Appropriately, viruses have evolved to interfere with the activity of hub proteins such as [[p53]], thereby bringing about the massive changes in cellular behavior which are conducive to viral replication.
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==Construction of small-world networks==
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The main mechanism to construct small-world networks is the [[Watts and Strogatz Model|Watts-Strogatz mechanism]].
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Small-world networks can also be introduced with time-delay,<ref>X. S. Yang, Fractals in small-world networks with time-delay, Chaos, Solitons & Fractals, vol. 13, 215-219 (2002)</ref> which will not only produces fractals but also
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[[degree diameter|Degree-Diameter]] graphs are constructed such that the number of neighbors each vertex in the network has is bounded, while the distance from any given vertex in the network to any other vertex (the [[Distance (graph theory)|diameter]] of the network) is minimized. Constructing such small-world networks is done as part of the effort to find graphs of order close to the [[Moore graph|Moore bound]].
 
 
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Another way to construct a small world network from scratch is given in Barmpoutis ''et al.'',<ref name=BarmpoutisMurray2010>{{Cite arXiv
 
| author = [[D.Barmpoutis and R.M. Murray]]
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| title = Networks with the Smallest Average Distance and the Largest Average Clustering
 
| eprint = 1007.4031
</ul>
| year = 2010
| class = q-bio.MN
}}</ref> where a network with very small average distance and very large average clustering is constructed. A fast algorithm of constant complexity is given, along with measurements of the robustness of the resulting graphs. Depending on the application of each network, one can start with one such "ultra small-world" network, and then rewire some edges, or use several small such networks as subgraphs to a larger graph.
 
''See also:'' [[Diffusion-limited aggregation]], [[pattern formation]]
 
== Applications to sociology ==
The advantages to small world networking for [[social movement| social movement groups]] are their resistance to change due to the filtering apparatus of using highly connected nodes, and its better effectiveness in relaying information while keeping the number of links required to connect a network to a minimum.<ref name = "shirky">[[Clay Shirky|Shirky, Clay]]. 2008. ''[[Here Comes Everybody]]''</ref>
 
The small world network model is directly applicable to [[affinity group]] theory represented in sociological arguments by [[William Finnegan]]. Affinity groups are social movement groups that are small and semi-independent pledged to a larger goal or function. Though largely unaffiliated at the node level, a few members of high connectivity function as connectivity nodes, linking the different groups through networking. This small world model has proven an extremely effective protest organization tactic against police action.<ref name ="finnegan">Finnegan, William "Affinity Groups and the Movement Against Corporate Globalization"</ref> [[Clay Shirky]] argues that the larger the social network created through small world networking, the more valuable the nodes of high connectivity within the network.<ref name= "shirky"/> The same can be said for the affinity group model, where the few people within each group connected to outside groups allowed for a large amount of mobilization  and adaptation. A practical example of this is small world networking through affinity groups that William Finnegan outlines in reference to the [[1999 Seattle Protests|1999 Seattle WTO protests]].
 
== Applications to earth sciences ==
 
Many networks studied in geology and geophysics have been shown to have characteristics of small-world networks.  Networks defined in fracture systems and porous substances have demonstrated these characteristics.<ref>X. S. Yang, Small-world networks in geophysics,  Geophys. Res. Lett., 28(13), 2549–2552 (2001)</ref> The seismic network in the Southern California region may be a small-world network.<ref>A. Jimenez, K. F. Tiampo, and A. M. Posadas, Small-world in a seismic network: the California case, Nonlin. Processes Geophys., 15, 389-395 (2008)</ref>  The examples above occur on very different spatial scales, demonstrating the [[scale invariance]] of the phenomenon in the earth sciences.
 
== Applications to computing ==
 
Small-world networks have been used to estimate the usability of information stored in large databases.  The measure is termed the Small World Data Transformation Measure.<ref>http://mike2.openmethodology.org/wiki/Small_Worlds_Data_Transformation_Measure</ref><ref>{{cite book|last=Hillard|first=Robert|title=Information-Driven Business|year=2010|publisher=Wiley|isbn=978-0-470-62577-4}}</ref> The greater the database links align to a small-world network the more likely a user is going to be able to extract information in the future. This usability typically comes at the cost of the amount of information that can be stored in the same repository.
 
The [[Freenet]] peer-to-peer network has been shown to form a small-world network in simulation,<ref>{{cite web|last=Sandberg|first=Oskar|title=Searching in a Small World|url=https://freenetproject.org/papers/lic.pdf}}</ref> allowing information to be stored and retrieved in a manner that scales efficiency as the network grows.
 
== Small-world neural networks in the brain ==
Both anatomical connections in the brain<ref>{{cite journal|last=Sporns|first=Olaf|coauthors=Tononi G, Edelman GM|title=Organization, development and function of complex brain networks|journal=Trends Cogn Sci.|year=2004|volume=8|issue=9|pages=418–425.|doi=10.1016/j.tics.2004.07.008|pmid=15350243}}</ref> and the synchronization networks of cortical neurons<ref>{{cite journal|last=Yu|first=Shan|coauthors=D. Huang, W. Singer and D. Nikolić|title=A Small World of Neuronal Synchrony|journal=Cerebral Cortex|year=2008|volume=18|issue=12|pages=2891–2901|doi=10.1093/cercor/bhn047|pmid=18400792|pmc=2583154}}</ref> exhibit small-world topology.
 
A small-world network of neurons can exhibit [[short-term memory]]. A computer model developed by Solla ''et al.''<ref>Cohen, Philip. [http://www.newscientist.com/article.ns?id=dn5012 Small world networks key to memory]. ''New Scientist''. 26 May 2004.</ref><ref>Sara Solla's [http://online.itp.ucsb.edu/online/brain04/solla/ Lecture & Slides: Self-Sustained Activity in a Small-World Network of Excitable Neurons]</ref>
had two stable states, a property (called [[bistable|bistability]]) thought to be important in [[memory]] storage. An activating pulse generated self-sustaining loops of communication activity among the neurons. A second pulse ended this activity. The pulses switched the system between stable states: flow (recording a "memory"), and stasis (holding it).
 
On a more general level, many large-scale neural networks in the brain, such as the visual system and brain stem, exhibit small-world properties.<ref name="D. Humphries, K 1639"/>
 
==Small world with a distribution of link length==
The WS model includes a uniform distribution of long-range links. When the distribution of link lengths follows a power law distribution, the mean distance between two sites changes depending on the power of the distribution.<ref>{{cite journal|last=D. Li|first=K. Kosmidis, A. Bunde, [[Shlomo Havlin|S. Havlin]]|title=Dimension of spatially embedded networks|journal=Nature Physics|doi=10.1038/nphys1932|year=2011|url=http://havlin.biu.ac.il/Publications.php?keyword=Dimension+of+spatially+embedded+networks&year=*&match=all|bibcode=2011NatPh...7..481D}}</ref>
 
==See also==
* [[Erdős–Rényi model|Erdős–Rényi (ER) model]]
* [[Watts and Strogatz Model]]
* [[Barabási–Albert model]]
* [[Erdős number]]
* [[Scale-free network]]
* [[Six degrees of Kevin Bacon]]
* [[Small world experiment]]
* [[Social network]]
 
==References==
{{reflist}}
 
===Books===
*{{cite book | author=Buchanan, Mark | title=Nexus: Small Worlds and the Groundbreaking Theory of Networks | publisher=Norton, W. W. & Company, Inc | year=2003 | isbn=0-393-32442-7}}
*{{cite book | author=Dorogovtsev, S.N. and 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}}
*{{cite book | author=Watts, D. J. | year=1999 | title=Small Worlds: The Dynamics of Networks Between Order and Randomness | publisher=Princeton University Press | isbn=0-691-00541-9}}
*[[James H. Fowler|Fowler, JH]]. (2005) "Turnout in a Small World," in Alan Zuckerman, ed., ''Social Logic of Politics'', Temple University Press, 269-287
* {{cite book |title= Complex Networks: Structure, Robustness and Function |authors= Reuven Cohen and [[Shlomo Havlin]] |year= 2010 |url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_com_net.php |publisher= Cambridge University Press}}
 
===Journal articles===
*{{cite journal | author=Albert, R.; Barabási A.L. | title= Statistical mechanics of complex networks | journal=Rev. Mod. Phys. | year=2002 | volume=74 | pages=47–97 | doi= 10.1103/RevModPhys.74.47 | bibcode=2002RvMP...74...47A|arxiv = cond-mat/0106096 }}
*{{cite journal | author=Albert, R.; Barabási A.L. | title= Emergence of scaling in random networks | year=1999 | volume=286 | issue=5439 | journal=Science | arxiv=cond-mat/9910332 | doi = 10.1126/science.286.5439.509 | pmid=10521342|bibcode = 1999Sci...286..509B }}
*{{cite journal | author=Barthelemy, M.; Amaral, LAN. | title= Small-world networks: Evidence for a crossover picture| journal=Phys. Rev. Lett. | year=1999 | volume=82 | pages=3180 | doi= 10.1103/PhysRevLett.82.3180 | issue=15 | bibcode=1999PhRvL..82.3180B|arxiv = cond-mat/9903108 }}
*{{cite journal | author=Dorogovtsev, S.N.; Mendes, J.F.F. | title= Exactly solvable analogy of small-world networks| journal=Europhys. Lett. | year=2000 | volume=50 | pages=1–7 | doi= 10.1209/epl/i2000-00227-1|arxiv = cond-mat/9907445 |bibcode = 2000EL.....50....1D }}
*{{cite journal | author=[[Stanley Milgram|Milgram, Stanley]] | title=The Small World Problem | journal=Psychology Today | year=1967 | volume=1 | issue=1 |pages=60–67}}
*{{cite journal | author=Newman, Mark | title=The Structure and Function of Complex Networks | journal=SIAM Review | year=2003 | volume=45 | pages=167–256 | doi=10.1137/S003614450342480 | issue=2|arxiv = cond-mat/0303516 |bibcode = 2003SIAMR..45..167N }} [http://www-personal.umich.edu/~mejn/courses/2004/cscs535/review.pdf pdf]
*{{cite journal | author=Ravid, D.; [[Sheizaf Rafaeli|Rafaeli, S.]] | title=Asynchronous discussion groups as Small World and Scale Free Networks | journal=First Monday | year=2004 | volume=9 | issue=9 }} [http://firstmonday.org/issues/issue9_9/ravid/index.html]
*{{cite journal |author= R. Parshani, S.V. Buldyrev, [[Shlomo Havlin|S. Havlin]] |year= 2011 |title= Critical effect of dependency groups on the function of networks |journal= PNAS |volume= 108 |page= 1007 |url= http://havlin.biu.ac.il/Publications.php?keyword=Critical+effect+of+dependency+groups+on+the+function+of+networks++&year=*&match=all|arxiv = 1010.4498 |bibcode = 2011PNAS..108.1007P |doi = 10.1073/pnas.1008404108 }}
*{{cite journal |author= S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, [[Shlomo Havlin|S. Havlin]] |year= 2010 |title= Catastrophic cascade of failures in interdependent networks |journal= Nature|volume= 464 |pages= 1025–8 |url= http://havlin.biu.ac.il/Publications.php?keyword=Catastrophic+cascade+of+failures+in+interdependent+networks++&year=*&match=all | doi = 10.1038/nature08932 |issue=7291 |bibcode=2010Natur.464.1025B|arxiv = 0907.1182 }}
 
==External links==
* [http://demonstrations.wolfram.com/DynamicProximityNetworks/ Dynamic Proximity Networks] by Seth J. Chandler, [[The Wolfram Demonstrations Project]].
* [http://www.scholarpedia.org/article/Small-world_network Small-World Networks] entry on Scholarpedia (by Mason A. Porter)
 
{{Social networking}}
 
[[Category:Networks]]
[[Category:Graph families]]
 
[[fr:Petit monde]]
[[ko:작은세상네트워크]]

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