Percolation and Loop Statistics in Complex Networks

Complex networks display various types of percolation transitions. We show that the degree distribution and the degree-degree correlation alone are not sufficient to describe diverse percolation critical phenomena. This suggests that a genuine structural correlation is an essential ingredient in characterizing networks. As a signature of the correlation we investigate a scaling behavior in $M_N(h)$, the number of finite loops of size $h$, with respect to a network size $N$. We find that networks, whose degree distributions are not too broad, fall into two classes exhibiting $M_N(h)\sim ({constant})$ and $M_N(h) \sim (\ln N)^\psi$, respectively. This classification coincides with the one according to the percolation critical phenomena.Comment: 4 pages and 2 figures; A major revision has been mad

Critical behavior of the contact process in annealed scale-free networks

Critical behavior of the contact process is studied in annealed scale-free networks by mapping it on the random walk problem. We obtain the analytic results for the critical scaling, using the event-driven dynamics approach. These results are confirmed by numerical simulations. The disorder fluctuation induced by the sampling disorder in annealed networks is also explored. Finally, we discuss over the discrepancy of the finite-size-scaling theory in annealed and quenched networks in spirit of the droplet size scale and the linking disorder fluctuation.Comment: 8 pages, 5 figure