We propose a new concept, two-step degree. Defining it as the capacity of a node of complex networks, we establish a novel capacity--load model of cascading failures of complex networks where the capacity of nodes decreases during the process of cascading failures. For scale-free networks, we find that the average two-step degree increases with the increase of the heterogeneity of the degree distribution, showing that the average two-step degree can be used for measuring the heterogeneity of the degree distribution of complex networks. In addition, under the condition that the average degree of a node is given, we can design a scale-free network with the optimal robustness to random failures by maximizing the average two-step degree.
We propose a new concept, two-step degree. Defining it as the capacity of a node of complex networks, we establish a novel capacity--load model of cascading failures of complex networks where the capacity of nodes decreases during the process of cascading failures. For scale-free networks, we find that the average two-step degree increases with the increase of the heterogeneity of the degree distribution, showing that the average two-step degree can be used for measuring the heterogeneity of the degree distribution of complex networks. In addition, under the condition that the average degree of a node is given, we can design a scale-free network with the optimal robustness to random failures by maximizing the average two-step degree.
[1] Albert R, Jeong H and Barab\'{asi A L 2000 Nature 406 378 [2] Cohen R, Erez K, ben-Avraham D and Havlin S 2000 Phys. Rev. Lett. 85 4626 [3] Cohen R, Erez K, ben-Avraham D and Havlin S 2001 Phys. Rev. Lett. 86 3682 [4] Callaway D S, Newman M E J, Strogatz S H and Watts D J2000 Phys. Rev. Lett. 85 5468 [5] Gallos L K, Cohen R, Argyrak is P and Havlin S 2005 Phys. Rev. Lett. 94 188701 [6] Shargel B, Sayama H, Epstein I R and Bar-Yam Y 2003 Phys. Rev. Lett. 90 068701 [7] Paul G, Tanizaw A T, Havlin S and Stanley H E 2004 Eur. Phys. J. B 38 187 [8] Valente A X C N, Sarkar A and Stone H A 2004 Phys.Rev. Lett. 92 118702 [9] Tanizawa T, Paul G, Cohen R, Havlin S and Stanley H E 2005 Phys. Rev. E 71 047101 [10] Wang B, Tang H W, Guo C H and Xiu Z L 2006 PhysicaA 363 591 [11] Moreno Y, Gomez J B and Pacheco A. F 2002 Europhys.Lett. 58 630 [12] Motter A E, Nishikawa T and Lai Y C 2002 Phys. Rev.E 66 065102(R) [13] Newnan M E J, Strogatz S H and Watts D J 2001 Phys.Rev. E 64 026118 [14] Home P and Kim B J 2002 Phys. Rev. E 65056109 [15] Crucitti P,Latora V and Marchiori m 2004 Phys. Rev.E 69 045104 [16] Barab\'{asi A L and Albert R 1999 Science 286509
2008, Vol. 25 
