scalesfree

Scale-free networks are a class of complex networks that exhibit a power-law degree distribution, meaning that the probability P(k) that a node has k connections follows a power law, P(k) ~ k^(-γ), where γ is a constant typically in the range 2 < γ < 3. This property is in contrast to random networks, which have a Poisson degree distribution, and regular networks, which have a uniform degree distribution.

Scale-free networks are ubiquitous in nature and technology, including the World Wide Web, social networks, biological networks, and infrastructure networks. They are characterized by the presence of a few highly connected nodes, known as hubs, and a large number of nodes with only a few connections. This structure gives scale-free networks robustness and resilience to random failures, but vulnerability to targeted attacks on hubs.

The scale-free nature of a network can be quantified using the degree distribution, which can be estimated from empirical data or generated using models such as the Barabási-Albert model. The degree distribution can be plotted on a log-log scale to reveal the power-law relationship. The exponent γ of the power law can be estimated using linear regression on the log-log plot.

Scale-free networks have been the subject of extensive research in the fields of physics, computer science, and social sciences. They have been used to model a wide range of phenomena, including the spread of diseases, the propagation of information, and the evolution of social networks. The study of scale-free networks has led to the development of new algorithms and techniques for network analysis and optimization.