A scale-free network is a 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
A preferential attachment process is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects according to how much they already have, so that those who are already wealthy receive more than those who are not. "Preferential attachment" is only the most recent of many names that have been given to such processes. They are also referred to under the names "Yule process", "cumulative advantage", "the rich get richer", and, less correctly, the "Matthew effect". They are also related to Gibrat's law. The principal reason for scientific interest in preferential attachment is that it can, under suitable circumstances, generate power law distributions. If preferential attachment is non-linear, measured distributions may deviate from a power law. These mechanisms may generate distributions which are approximately power law over transient periods.
The Barabási–Albert (BA) model is an algorithm for generating random scale-free networks using a preferential attachment mechanism. Several natural and human-made systems, including the Internet, the world wide web, citation networks, and some social networks are thought to be approximately scale-free and certainly contain few nodes with unusually high degree as compared to the other nodes of the network. The BA model tries to explain the existence of such nodes in real networks. The algorithm is named for its inventors Albert-László Barabási and Réka Albert and is a special case of an earlier and more general model called Price's model.
The Watts–Strogatz model is a random graph generation model that produces graphs with small-world properties, including short average path lengths and high clustering. It was proposed by Duncan J. Watts and Steven Strogatz in their article published in 1998 in the Nature scientific journal. The model also became known as the (Watts) beta model after Watts used to formulate it in his popular science book Six Degrees.
In complex network theory, the fitness model is a model of the evolution of a network: how the links between nodes change over time depends on the fitness of nodes. Fitter nodes attract more links at the expense of less fit nodes.
In applied probability theory, the Simon model is a class of stochastic models that results in a power-law distribution function. It was proposed by Herbert A. Simon to account for the wide range of empirical distributions following a power-law. It models the dynamics of a system of elements with associated counters. In this model the dynamics of the system is based on constant growth via addition of new elements as well as incrementing the counters at a rate proportional to their current values.
In the study of scale-free networks, a copying mechanism is a process by which such a network can form and grow, by means of repeated steps in which nodes are duplicated with mutations from existing nodes. Several variations of copying mechanisms have been studied. In the general copying model, a growing network starts as a small initial graph and, at each time step, a new vertex is added with a given number k of new outgoing edges. As a result of a stochastic selection, the neighbors of the new vertex are either chosen randomly among the existing vertices, or one existing vertex is randomly selected and k of its neighbors are ‘copied’ as heads of the new edges.
Assortativity, or assortative mixing is a preference for a network's nodes to attach to others that are similar in some way. Though the specific measure of similarity may vary, network theorists often examine assortativity in terms of a node's degree. The addition of this characteristic to network models more closely approximates the behaviors of many real world networks.
Network science is an academic field which studies complex networks such as telecommunication networks, computer networks, biological networks, cognitive and semantic networks, and social networks, considering distinct elements or actors represented by nodes and the connections between the elements or actors as links. The field draws on theories and methods including graph theory from mathematics, statistical mechanics from physics, data mining and information visualization from computer science, inferential modeling from statistics, and social structure from sociology. The United States National Research Council defines network science as "the study of network representations of physical, biological, and social phenomena leading to predictive models of these phenomena."
Evolving networks are networks that change as a function of time. They are a natural extension of network science since almost all real world networks evolve over time, either by adding or removing nodes or links over time. Often all of these processes occur simultaneously, such as in social networks where people make and lose friends over time, thereby creating and destroying edges, and some people become part of new social networks or leave their networks, changing the nodes in the network. Evolving network concepts build on established network theory and are now being introduced into studying networks in many diverse fields.
Evolving networks are dynamic networks that change through time. In each period there are new nodes and edges that join the network while the old ones disappear. Such dynamic behaviour is characteristic for most real-world networks, regardless of their range - global or local. However, networks differ not only in their range but also in their topological structure. It is possible to distinguish:
Price's model is a mathematical model for the growth of citation networks. It was the first model which generalized the Simon model to be used for networks, especially for growing networks. Price's model belongs to the broader class of network growing models whose primary target is to explain the origination of networks with strongly skewed degree distributions. The model picked up the ideas of the Simon model reflecting the concept of rich get richer, also known as the Matthew effect. Price took the example of a network of citations between scientific papers and expressed its properties. His idea was that the way an old vertex gets new edges should be proportional to the number of existing edges the vertex already has. This was referred to as cumulative advantage, now also known as preferential attachment. Price's work is also significant in providing the first known example of a scale-free network. His ideas were used to describe many real-world networks such as the Web.
A hyperbolic geometric graph (HGG) or hyperbolic geometric network (HGN) is a special type of spatial network where (1) latent coordinates of nodes are sprinkled according to a probability density function into a hyperbolic space of constant negative curvature and (2) an edge between two nodes is present if they are close according to a function of the metric. A HGG generalizes a random geometric graph (RGG) whose embedding space is Euclidean.
In mathematical modeling of social networks, link-centric preferential attachment is a node's propensity to re-establish links to nodes it has previously been in contact with in time-varying networks. This preferential attachment model relies on nodes keeping memory of previous neighbors up to the current time.
The Bianconi–Barabási model is a model in network science that explains the growth of complex evolving networks. This model can explain that nodes with different characteristics acquire links at different rates. It predicts that a node's growth depends on its fitness and can calculate the degree distribution. The Bianconi–Barabási model is named after its inventors Ginestra Bianconi and Albert-László Barabási. This model is a variant of the Barabási–Albert model. The model can be mapped to a Bose gas and this mapping can predict a topological phase transition between a "rich-get-richer" phase and a "winner-takes-all" phase.
In network science, a hub is a node with a number of links that greatly exceeds the average. Emergence of hubs is a consequence of a scale-free property of networks. While hubs cannot be observed in a random network, they are expected to emerge in scale-free networks. The uprise of hubs in scale-free networks is associated with power-law distribution. Hubs have a significant impact on the network topology. Hubs can be found in many real networks, such as the brain or the Internet.
In a scale-free network the degree distribution follows a power law function. In some empirical examples this power-law fits the degree distribution well only in the high degree region, however for small degree nodes the empirical degree-distribution deviates from it. See for example the network of scientific citations. This deviation of the observed degree-distribution from the theoretical prediction at the low-degree region is often referred as low-degree saturation.
The initial attractiveness is a possible extension of the Barabási–Albert model. The Barabási–Albert model generates scale-free networks where the degree distribution can be described by a pure power law. However, the degree distribution of most real life networks cannot be described by a power law solely. The most common discrepancies regarding the degree distribution found in real networks are the high degree cut-off and the low degree cut-off. The inclusion of initial attractiveness in the Barabási–Albert model addresses the low-degree cut-off phenomenon.
In the scale-free network theory, a mediation-driven attachment (MDA) model appears to embody a preferential attachment rule tacitly rather than explicitly. According to MDA rule, a new node first picks a node from the existing network at random and connect itself not with that but with one of the neighbors also picked at random.
In the analysis of social networks, the Uniform-Preferential-Attachment model, or UPA model is a variation of the Barabási–Albert model in which the preferential attachment is perceived as having a double nature. New nodes joining the network may either attach themselves with high-degree nodes or with most recently added nodes. This behaviour can be noticed in some examples of social networks, such as the citation network of scientific publications.
