## Degree and clustering coefficient in sparse random intersection graphs.(English)Zbl 1273.05197

Summary: We establish asymptotic vertex degree distribution and examine its relation to the clustering coefficient in two popular random intersection graph models of E. Godehardt and J. Jaworski [Electron. Notes Discrete Math. 10, 129–132 (2001; Zbl 1182.05108)]. For sparse graphs with a positive clustering coefficient, we examine statistical dependence between the (local) clustering coefficient and the degree. Our results are mathematically rigorous. They are consistent with the empirical observation of I. Foudalis et al. [Lect. Notes Comp. Sci. 6732, 85–102 (2011; Zbl 1328.91262)] that, “clustering correlates negatively with degree.” Moreover, they explain empirical results on $$k^{-1}$$ scaling of the local clustering coefficient of a vertex of degree $$k$$ reported in E. Ravasz and A.L. Barabási [Phys. Rev. E67, 026112 (2003)].

### MSC:

 05C80 Random graphs (graph-theoretic aspects) 91D30 Social networks; opinion dynamics 05C07 Vertex degrees

### Citations:

Zbl 1182.05108; Zbl 1328.91262
Full Text:

### References:

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