On the Degree Sequence and its Critical Phenomenon of an Evolving Random Graph Process

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Abstract

In this paper we focus on the problem of the degree sequence for the following random graph process. At any time-step $t$, one of the following three substeps is executed: with probability $\alpha_1$, a new vertex $x_t$ and $m$ edges incident with $x_t$ are added; or, with probability $\alpha-\alpha_1$, $m$ edges are added; or finally, with probability $1-\a$, $m$ random edges are deleted. Note that in any case edges are added in the manner of preferential attachment. we prove that there exists a critical point $\alpha_c$ satisfying: 1) if $\alpha_1<\alpha_c$, then the model has power law degree sequence; 2) if $\alpha_1>\alpha_c$, then the model has exponential degree sequence; and 3) if $\alpha_1=\alpha_c$, then the model has a degree sequence lying between the above two cases.