Degree Distribution

Degree distribution is the probability distribution of the number of connections — the degree — held by nodes in a network. It is the most fundamental macroscopic property of a network's topology, and it determines whether the network resembles a regular lattice, a random graph, or a hub-dominated system. Networks with heavy-tailed degree distributions, where a small number of nodes hold a large fraction of all connections, behave qualitatively differently from networks with concentrated or uniform distributions: they are robust to random failure but vulnerable to targeted attack, and they amplify cascades rather than damping them.

The study of degree distributions became central to network science after the claim that many real-world networks follow power laws. Subsequent reanalysis has shown that this claim was often premature — many claimed power-law networks are better described by lognormal or stretched-exponential distributions. The field's early confidence outran its statistical rigor, and the degree distribution remains a test case for how easily beautiful stories displace careful measurement.