Exploring generic scale-free networks∗

where qk is the probability that a randomly chosen node has degree k, and cγ and γ are positive constants. As this formula makes sense only for k > 0, we will tacitly assume that the graph contains no isolated nodes, that is, q0 = 0. Graphs with a power-law distribution of degrees are often called scale-free networks. This phrase needs to be handled with care. Since qk > 0 for all k > 1, Equation (1) could be literally true only if there were infinitely many nodes in the graph.2 For graphs with finitely many nodes, (1) can be satisfied only approximately. If this is the case, we will write that the graph is approximately a scale-free network. A number of constructions of approximately scale-free networks have been proposed in the literature. The preferential attachment model of Barabási and Albert [2] was studied in some detail in our module [4]. Here we focus on some properties that are common to all types of approximately scale-free networks and on properties of generic scale-free networks that are constructed by the procedure outlined in our module [3].