A scale-free network with limiting on vertices

We propose and analyze a random graph model which explains a phenomena in the economic company network in which company may not expand its business at some time due to the limiting of money and capacity The random graph process is defined as follows at any time-step t. (i) with probability alpha(k) and independently of other time-step, each vertex upsilon(1) < (t <= t - 1) is inactive which means it cannot be connected by more edges, where k is the degree of upsilon(t) at the time-step t. (ii) a new vertex upsilon(t) is added along with m edges incident with upsilon(t) at one time and its neighbors are chosen in the manner of preferential attachment We prove that the degree distribution P(k) of this random graph process satisfies P(k) proportional to C(1)k(-3-alpha 0/1-alpha 0) if alpha( ) is a constant alpha(0): and P(k) proportional to C(2)k(-3) if alpha(l) down arrow 0 as l up arrow infinity, where C-1, C-2 are two positive constants The analytical result is found to be in good agreement with that obtained by numerical simulations Furthermore, we get the degree distiibutions in this model with m-varying functions by simulation. (C) 2010 Elsevier B.V. All rights reserved