Smallworld network
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A smallworld network is a type of mathematical graph in which most nodes are not neighbors of one another, but the neighbors of any given node are likely to be neighbors of each other and most nodes can be reached from every other node by a small number of hops or steps. Specifically, a smallworld 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:^{[1]}
while the clustering coefficient is not small. In the context of a social network, this results in the small world phenomenon of strangers being linked by a short chain of acquaintances. Many empirical graphs show the smallworld effect, including social networks, wikis such as Wikipedia, gene networks, and even the underlying architecture of the Internet. It is the inspiration for many networkonchip architectures in contemporary computer hardware.^{[2]}
A certain category of smallworld networks were identified as a class of random graphs by Duncan Watts and Steven Strogatz in 1998.^{[3]} They noted that graphs could be classified according to two independent structural features, namely the clustering coefficient, and average nodetonode distance (also known as average shortest path length). Purely random graphs, built according to the 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 realworld 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.^{[4]} This work was followed by many studies, including exact results (Barrat and Weigt, 1999; Dorogovtsev and Mendes; Barmpoutis and Murray, 2010). Braunstein et al ^{[5]} found that for weighted ER networks where the weights have a very broad distribution, the optimal path scales becomes significantly longer and scales as N^{1/3}.
Properties of smallworld networks[edit]
Smallworld networks tend to contain cliques, and nearcliques, meaning subnetworks 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 meanshortest path length be small. Several other properties are often associated with smallworld networks. Typically there is an overabundance of hubs – nodes in the network with a high number of connections (known as high degree nodes). These hubs serve as the common connections mediating the short path lengths between other edges. By analogy, the smallworld network of airline flights has a small meanpath length (i.e. between any two cities you are likely to have to take three or fewer flights) because many flights are routed through 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 fattailed distribution. Graphs of very different topology qualify as smallworld networks as long as they satisfy the two definitional requirements above.
Network smallworldness has been quantified by a smallcoefficient, , calculated by comparing clustering and path length of a given network to an equivalent random network with same degree on average.^{[6]}^{[7]}
 if ( and ), network is smallworld. However, this metric is known to perform poorly because it is heavily influenced by the network's size.^{[8]}^{[9]}
Another method for quantifying network smallworldness utilizes the original definition of the smallworld network comparing the clustering of a given network to an equivalent lattice network and its path length to an equivalent random network. The smallworld measure () is defined as^{[8]}
Where the characteristic path length L and clustering coefficient C are calculated from the network you are testing, C_{ℓ} is the clustering coefficient for an equivalent lattice network and L_{r} is the characteristic path length for an equivalent random network.
Still another method for quantifying smallworldness normalizes both the network's clustering and path length relative to these characteristics in equivalent lattice and random networks. The Small World Index (SWI) is defined as^{[9]}
Both ω′ and SWI range between 0 and 1, and have been shown to capture aspects of smallworldness. However, they adopt slightly different conceptions of ideal smallworldness. For a given set of constraints (e.g. size, density, degree distribution), there exists a network for which ω′ = 1, and thus ω aims to capture the extent to which a network with given constraints as small worldly as possible. In contrast, there may not exist a network for which SWI = 1, the thus SWI aims to capture the extent to which a network with given constraints approaches the theoretical small world ideal of a network where C ≈ C_{ℓ} and L ≈ L_{r}.^{[9]}
R. Cohen and Havlin^{[10]}^{[11]} showed analytically that scalefree networks are ultrasmall worlds. In this case, due to hubs, the shortest paths become significantly smaller and scale as
Examples of smallworld networks[edit]
Smallworld properties are found in many realworld phenomena, including websites with navigation menus, food webs, electric power grids, metabolite processing networks, networks of brain neurons, voter networks, telephone call graphs, airport networks,^{[12]} and social influence networks.^{[13]} Cultural networks,^{[14]} semantic networks ^{[15]} and word cooccurrence networks^{[16]} have also been shown to be smallworld networks.
Networks of connected proteins have small world properties such as powerlaw obeying degree distributions.^{[17]} Similarly transcriptional networks, in which the nodes are genes, and they are linked if one gene has an up or downregulatory genetic influence on the other, have small world network properties.^{[18]}
Examples of nonsmallworld networks[edit]
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^{[19]} and this network does not have smallworld 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.
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 lineofsight.
Tacit assumptions, if not examined, can cause a bias in the literature on graphs in favor of finding smallworld networks (an example of the file drawer effect resulting from the publication bias).
Network robustness[edit]
It is hypothesized by some researchers, such as Barabási, that the prevalence of small world networks in biological systems may reflect an evolutionary advantage of such an architecture. One possibility is that smallworld 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 viral infection.
In a small world network with a degree distribution following a powerlaw, deletion of a random node rarely causes a dramatic increase in meanshortest path length (or a dramatic decrease in the clustering coefficient). This follows from the fact that most shortest paths between nodes flow through 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 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. However, 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 airport, are shut down because of snow; many people have to take additional flights.
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 meanshortest path length slightly but significantly for almost any node deleted. In this sense, random networks are vulnerable to random perturbations, whereas smallworld networks are robust. However, smallworld networks are vulnerable to targeted attack of hubs, whereas random networks cannot be targeted for catastrophic failure.
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. A useful method to analyze network robustness is the percolation theory.^{[20]}
Construction of smallworld networks[edit]
The main mechanism to construct smallworld networks is the Watts–Strogatz mechanism.
Smallworld networks can also be introduced with timedelay,^{[21]} which will not only produce fractals but also chaos^{[22]} under the right conditions, or transition to chaos in dynamics networks.^{[23]}
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 diameter of the network) is minimized. Constructing such smallworld networks is done as part of the effort to find graphs of order close to the Moore bound.
Another way to construct a small world network from scratch is given in Barmpoutis et al.,^{[24]} 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 smallworld" network, and then rewire some edges, or use several small such networks as subgraphs to a larger graph.
Smallworld properties can arise naturally in social networks and other realworld systems via the process of dualphase evolution. This is particularly common where time or spatial constraints limit the addition of connections between vertices The mechanism generally involves periodic shifts between phases, with connections being added during a "global" phase and being reinforced or removed during a "local" phase.
Smallworld networks can change from scalefree class to broadscale class whose connectivity distribution has a sharp cutoff following a power law regime due to constraints limiting the addition of new links.^{[25]} For strong enough constraints, scalefree networks can even become singlescale networks whose connectivity distribution characterized as a fast decaying.^{[25]}
See also: Diffusionlimited aggregation, Pattern formation
Applications[edit]
Applications to sociology[edit]
The advantages to small world networking for 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.^{[26]}
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 semiindependent 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.^{[27]} 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.^{[26]} 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 WTO protests.
Applications to earth sciences[edit]
Many networks studied in geology and geophysics have been shown to have characteristics of smallworld networks. Networks defined in fracture systems and porous substances have demonstrated these characteristics.^{[28]} The seismic network in the Southern California region may be a smallworld network.^{[29]} The examples above occur on very different spatial scales, demonstrating the scale invariance of the phenomenon in the earth sciences. Climate networks may be regarded as small world networks where the links are of different length scales.^{[30]}
Applications to computing[edit]
Smallworld networks have been used to estimate the usability of information stored in large databases. The measure is termed the Small World Data Transformation Measure.^{[31]}^{[32]} The greater the database links align to a smallworld 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 peertopeer network has been shown to form a smallworld network in simulation,^{[33]} allowing information to be stored and retrieved in a manner that scales efficiency as the network grows.
Smallworld neural networks in the brain[edit]
Both anatomical connections in the brain^{[34]} and the synchronization networks of cortical neurons^{[35]} exhibit smallworld topology.
A smallworld network of neurons can exhibit shortterm memory. A computer model developed by Solla et al.^{[36]}^{[37]} had two stable states, a property (called bistability) thought to be important in memory storage. An activating pulse generated selfsustaining 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). Small world neuronal networks have also been used as models to understand seizures.^{[38]}
On a more general level, many largescale neural networks in the brain, such as the visual system and brain stem, exhibit smallworld properties.^{[6]}
Small world with a distribution of link length[edit]
The SW model includes a uniform distribution of longrange 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.^{[39]}
See also[edit]
 Barabási–Albert model
 Climate as complex networks – Conceptual model to generate insight into climate science
 Dualphase evolution – A process that drives selforganization within complex adaptive systems
 Dunbar's number – suggested cognitive limit important in sociology and anthropology
 Erdős number – Closeness of someone's association with mathematician Paul Erdős
 Erdős–Rényi (ER) model – Two closely related models for generating random graphs
 Percolation theory – Mathematical theory on behavior of connected clusters in a random graph
 Network science  mathematical theory of networks
 Scalefree network – Network whose degree distribution follows a power law
 Six degrees of Kevin Bacon – Parlour game on degrees of separation
 Smallworld experiment – Experiments examining the average path length for social networks
 Social network – Social structure made up of a set of social actors
 Watts–Strogatz model
 Network on a chip – systems on chip modeled on smallworld networks
 Zachary's_karate_club
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Further reading[edit]
Books[edit]
 Buchanan M (2003). Nexus: Small Worlds and the Groundbreaking Theory of Networks. Norton, W. W. & Company, Inc. ISBN 9780393324426.
 Dorogovtsev SN, Mendes JF (2003). Evolution of Networks: from biological networks to the Internet and WWW. Oxford University Press. ISBN 9780198515906.
 Watts DJ (1999). Small Worlds: The Dynamics of Networks Between Order and Randomness. Princeton University Press. ISBN 9780691005416.
 Fowler JH (2005). "Turnout in a Small World". In Zuckerman A (ed.). Social Logic of Politics. Temple University Press. pp. 269–287.
 Cohen R, Havlin S (2010). Complex Networks: Structure, Robustness and Function. Cambridge University Press.
Journal articles[edit]
 Albert R, Barabási AL (2002). "Statistical mechanics of complex networks". Rev. Mod. Phys. 74 (1): 47–97. arXiv:condmat/0106096. Bibcode:2002RvMP...74...47A. doi:10.1103/RevModPhys.74.47. S2CID 60545.
 Barabasi AL, Albert R (October 1999). "Emergence of scaling in random networks". Science. 286 (5439): 509–12. arXiv:condmat/9910332. Bibcode:1999Sci...286..509B. doi:10.1126/science.286.5439.509. PMID 10521342. S2CID 524106.
 Barthelemy M, Amaral LA (1999). "Smallworld networks: Evidence for a crossover picture". Phys. Rev. Lett. 82 (15): 3180–3183. arXiv:condmat/9903108. Bibcode:1999PhRvL..82.3180B. doi:10.1103/PhysRevLett.82.3180. S2CID 119398712.
 Dorogovtsev SN, Mendes JF (2000). "Exactly solvable analogy of smallworld networks". Europhys. Lett. 50 (1): 1–7. arXiv:condmat/9907445. Bibcode:2000EL.....50....1D. doi:10.1209/epl/i2000002271. S2CID 11334862.
 Milgram S (1967). "The Small World Problem". Psychology Today. 1 (1): 60–67.
 Newman M (2003). "The Structure and Function of Complex Networks". SIAM Review. 45 (2): 167–256. arXiv:condmat/0303516. Bibcode:2003SIAMR..45..167N. doi:10.1137/S003614450342480. S2CID 65837. pdf
 Ravid D, Rafaeli S (2004). "Asynchronous discussion groups as Small World and Scale Free Networks". First Monday. 9 (9). doi:10.5210/fm.v9i9.1170. S2CID 6388295. [1]
 Parshani R, Buldyrev SV, Havlin S (January 2011). "Critical effect of dependency groups on the function of networks". Proceedings of the National Academy of Sciences of the United States of America. 108 (3): 1007–10. arXiv:1010.4498. Bibcode:2011PNAS..108.1007P. doi:10.1073/pnas.1008404108. PMC 3024657. PMID 21191103.
 Buldyrev SV, Parshani R, Paul G, Stanley HE, Havlin S (April 2010). "Catastrophic cascade of failures in interdependent networks". Nature. 464 (7291): 1025–8. arXiv:0907.1182. Bibcode:2010Natur.464.1025B. doi:10.1038/nature08932. PMID 20393559. S2CID 1836955.
External links[edit]
 Dynamic Proximity Networks by Seth J. Chandler, The Wolfram Demonstrations Project.
 SmallWorld Networks entry on Scholarpedia (by Mason A. Porter)