On the Degree Sequence and its Critical Phenomenon of an Evolving Random Graph Process
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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 $α_1$, a new vertex $x_t$ and $m$ edges incident with $x_t$ are added; or, with probability $α-α_1$, $m$ 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 $α_c$ satisfying: 1) if $α_1<α_c$, then the model has power law degree sequence; 2) if $α_1>α_c$, then the model has exponential degree sequence; and 3) if $α_1=α_c$, then the model has a degree sequence lying between the above two cases.
18 pages
18 pages