Generalized Preferential Attachment: Tunable Power-Law Degree Distribution and Clustering Coefficient

Abstract

We propose a common framework for analysis of a wide class of preferential attachment models, which includes LCD, Buckley-Osthus, Holme-Kim and many others. The class is defined in terms of constraints that are sufficient for the study of the degree distribution and the clustering coefficient. We also consider a particular parameterized model from the class and illustrate the power of our approach as follows. Applying our general results to this model, we show that both the parameter of the power-law degree distribution and the clustering coefficient can be controlled via variation of the model parameters. In particular, the model turns out to be able to reflect realistically these two quantitative characteristics of a real network, thus performing better than previous preferential attachment models. All our theoretical results are illustrated empirically.

Limit theorems for the weights and the degrees in anN-interactions random graph model

“…This result was presented in [15] for the case of M D 2, N D 3 and also for the case of M D N for arbitrary N > 2. The general case can be obtained by using the ideas of [18]. To this end one has to consider a slight modification of the general model of [18] and to prove appropriate versions of Theorems 2 and 3 in [18].…”

“…The general case can be obtained by using the ideas of [18]. To this end one has to consider a slight modification of the general model of [18] and to prove appropriate versions of Theorems 2 and 3 in [18]. First, we study the weight of a fixed M -clique.…”

“…By the definition of ZOEn; M; 1; j , we see that bOEn; M; 1W OEn; M; j also converges almost surely on the event fW OEl; M; j > 0g. This fact, (26) and (18) imply that (8) is true with non-negative M;j . Now we will show that M;j is positive with probability 1.…”

“…We have already seen that, for a fixed k, the sequence bOEn; 1; k is decreasing. Therefore, applying also (18),…”

“…Above we applied that, by (18), Let QOEm; n D max mÄj Än W OEn; 1; j . Then 0 Ä W n M OEm; n Ä QOEm; n. The maximum of increasing numbers of submartingales is also a submartingale, so bOEn; 1; k QOEm; n C k 1 k !…”