Overpayment Distribution in Scale-free Networks

We investigate the distribution of vertex overpayments in a kind of power-law networks when the degree distribution exponent is varied from 2 to infin . The overpayment of one vertex is the sum of the increase of shortest path lengths among all pairs of vertices in the network if the vertex were deleted. We find the overpayment increases with degree in the power-law form not only in heterogeneous networks but also in homogeneous networks, which discloses the fact that those high-connected vertices obtain more benefit, and also take on more traffic. Hence, we define the bonus per packet to describe how much the benefit of a vertex is for carrying one packet. Our investigation shows a vertex with a larger degree will charge more than that with a smaller degree for transmitting per packet in heterogeneous networks, and the difference between those vertices with different degrees becomes minor when the network becomes more homogeneous.