“The key to generating the small-world phenomenon is the presence of a small fraction of [...] edges, which contact otherwise distant parts of the graph” (Duncan J Watts, "Small Worlds: The Dynamics of Networks Between Order and Randomness", 1999)
"[...] the transition to a small world is essentially undetectable at a local level. If you were living through the morph, nothing about your immediate neighborhood would tell you that the world had become small." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"We investigate structure in the network by characterizing motifs that represent order. A simple motif is the existence of a triangle, three vertices connected one to the next. The ratio of the number of observed to expected triangles is synonymous with the standard definition of the clustering coefficient for a small world network. This statistic is sensitive to organization over short length scales. To investigate organization over longer length scales, we investigate the distribution of longer cycles. This distribution may be measured for an empirical network We introduce a simple mathematical model for a network organized to have one level of clustering and show that this model is sufficient to explain the observed cycle distribution. Thus, there is no need to invoke a continuous distribution of length scales. Moreover, the one-level model immediately yields a characteristic, testable scaling length for the network, which again stands in contrast to scale-free behavior." (J S Bader, "The Drosophila Protein Interaction Network May Be neither Power-Law nor Scale-Free", 2006)
"If a network is solely composed of neighborhood connections, information must traverse a large number of connections to get from place to place. In a small-world network, however, information can be transmitted between any two nodes using, typically, only a small number of connections. In fact, just a small percentage of random, long-distance connections is required to induce such connectivity. This type of network behavior allows the generation of 'six degrees of separation' type results, whereby any agent can connect to any other agent in the system via a path consisting of only a few intermediate nodes." (John H Miller & Scott E Page, "Complex Adaptive Systems", 2007)
"A side effect of computing in a small world web is vulnerability to viruses or other malicious software. A computer virus may be broadly defined as a program that can replicate itself while executing instructions on infected computers, without the user's knowledge. Modelling computer viruses may lead to new methods to combat them. How can we model computer viruses loose on scale-free networks?" (Anthony Bonato, "A Course on the Web Graph", 2008)
"These networks arise naturally in many diverse disciplines (such as biology, cOInputer sci- ence, and social science), but all share the common characteristics of be- ing Inassive, sparse graphs with a power law degree distribution and small world structure. Such networks have now been coined scale-free, complex, self-organizing, or heterogeneous. Our own preferred term is self-organizing, since this is suggestive of the generative mechanisms underlying the network: over time, vertices choose their neighbour according to their own predisposition, rather than necessarily followil1g global rules." (Anthony Bonato, "A Course on the Web Graph", 2008)
"To put it simply, a small world is characterized by the fact that every two people can reach each other through a chain of just a handful of messages. This phenomenon is also known as the 'six degrees of separation' […]" (Maarten van Steen, "Graph Theory and Complex Networks: An Introduction", 2010)
"With the birth of Internet-based social networks, information on personal interaction on a large-scale became available and the interest in developing mathematical models that are able to capture and explain certain stylized features of large and complex real-world networks has grown significantly. Among the most central phenomena observed in large-scale real-world networks is the so-called small-world phenomenon, which refers to the fact that despite the large network size most of the nodes have a surprisingly small graph (hop count) distance in between them, and the so-called scale-free nature of networks, meaning that the degree sequence has polynomial decay." (Francesca Biagini et al (Eds), "Network Science: An Aerial View", 2019)
"A small-world network is characterized by a high clustering coefficient and a short average path length, meaning that most nodes can be reached from any other node through a small number of intermediate connections. This structure often mirrors real-world social networks, where individuals are typically connected through a few mutual acquaintances, allowing for rapid information dissemination." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)
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