In mathematics, physics and sociology a small-world network is a type of 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:
In the context of a social network, this results in the 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 networks all exhibit small-world network characteristics.
A certain category of small-world networks were identified as a class of random graphs by Duncan Watts and Steven Strogatz in 1998. They noted that graphs could be classified according to two independent structural features, namely the clustering coefficient and average node-to-node distance, the latter 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 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 first description of the crossover in the Watts-Strogatz model between a "large world" (such as a lattice) and a small-world was described by Barthelemy and Amaral in 1999. This work was followed by a large number of studies including exact results (Barrat and Weigt, 1999; Dorogovtsev and Mendes; Barmpoutis and Murray, 2010). Braunstein et al[3] found that for weighted ER networks where the weights have a very broad distribution the optimal path scales becomes significantly longer as L~N1/3.
Construction of small-world networks
Several procedures generate small-world networks from scratch. One of these methods is known as preferential attachment. In this model, new nodes are added to a pre-existing network, and connected to each of the original nodes with a probability proportional to the number of connections each of the original nodes already had. I.e., new nodes are more likely to attach to hubs than peripheral nodes. Statistically, this method will generate a power-law distributed small-world network (that is, a scale-free network).
Elements of this mechanism can be seen to contribute to the small-worldness of the World Wide Web. A new site is more likely to have links to major pre-existing sites, such as Google or Wikipedia than arbitrary small obscure sites. This observation is known colloquially as a rich get richer model.
Small-world networks can also be introduced with time-delay, which will not only produces fractals but also chaos under the right conditions, or transition to chaos in dynamics networks.
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 small-world 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., 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
Properties of small-world networks
Small-world networks tend to contain 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). 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 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 fit with a 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 networks. Graphs of very different topology qualify as small-world networks as long as they satisfy the two definitional requirements above.
Cohen and Havlin showed analytically that scale-free networks are ultra-small worlds. In this case, due to hubs, the shortest paths become significantly smaller and scale as
L~LogLog N.
Examples of small-world networks
Small-world properties are found in many real-world phenomena, including road maps, food chains, electric power grids, metabolite processing networks, networks of brain neurons, voter networks, telephone call graphs, and social influence networks.
Networks of connected proteins have small world properties such as power-law obeying degree distributions. Similarly transcriptional networks, in which the nodes are genes, and they are linked if one gene has an up or down-regulatory genetic influence on the other, have small world network properties.
Examples of non-small-world networks
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.
For 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.
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.
Tacit assumptions, if not examined, can cause a bias in the literature on graphs in favour of finding small-world networks.
Network robustness
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 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 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 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. 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 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 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.
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.
