Growing network: Models following nonlinear preferential attachment rule

• We discuss a nonlinear preferential attachment rule for growing networks.
• For this rule the probability to attach to a node depends on degree kk as f(k)f(k).
• The node degree distribution can be calculated for any ff based on derived formulas.
• The inverse problem of the calculation ff for a given degree distribution is solved.
• So we can calibrate the graph model and explain growing networks features.

We investigate the preferential attachment graphs proceeding from the following two assumptions. The first one: the probability that a new vertex connects to a vertex ii is proportional to an arbitrary nonnegative function ff of a vertex degree kk. The second assumption: a new vertex can have a random number of edges. We derive formulas for any ff to determine the vertex degree distribution {Qk}{Qk} in generated graphs. The inverse problem is solved: we have obtained formulas, that allow from a given distribution {Qk}{Qk} to determine ff (the problem of a model calibration). The formulas allowing for any ff to calculate the joint distribution of vertex degrees at the ends of randomly selected edge are also obtained. Some other results are presented in the paper.