Scalefree

Scale-free refers to a property of certain networks whose degree distribution follows a power law over a range of scales. In these networks, most nodes have a small number of connections, while a few hubs accumulate a disproportionately large number of links. This heavy-tailed distribution contrasts with Erdős–Rényi random graphs, whose degree distribution is more uniform and characterized by a typical scale.

Scale-free structure often emerges in growing networks through mechanisms such as preferential attachment, where new nodes are more likely to connect to already well-connected nodes. The Barabási–Albert model is the canonical example illustrating how growth and preferential attachment can generate a power-law degree distribution. Other models incorporate factors like node fitness or aging. Key features of scale-free networks include the presence of hubs, high degree heterogeneity, and, in many cases, relatively short path lengths between nodes, though clustering behavior can vary across systems.

Implications of scale-free topology include robustness to random failures, since most random removals affect low-degree nodes that do not disrupt global connectivity. However, such networks can be particularly vulnerable to targeted attacks on hubs, influencing strategies for immunization, network design, and resilience planning. Real-world examples often cited include portions of the Internet, various social networks, and some biological networks; nevertheless, not all empirical networks exhibit a true power-law tail, and evidence for strict scale-free behavior is debated.

Criticism and limitations focus on the challenges of reliably estimating power-law exponents from data, finite-size effects, and the possibility that other heavy-tailed distributions (such as lognormal or stretched exponential) may provide comparable fits. Consequently, while scale-free is a useful descriptive label for networks with hub-dominated structure, it is not universally applicable to all real-world systems.