Available online at www.sciencedirect.com Physica A 330 (2003) 246–252 www.elsevier.com/locate/physa Length of optimal path in random networks with strong disorder Sergey V. Buldyrev a , Lidia A. Braunstein a; c , Reuven Cohen b , Shlomo Havlin a; b , H. Eugene Stanley a ; ∗ a Department of Physics, Center for Polymer Studies, Boston University, Boston, MA 02215, USA b Minerva Center and Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel c Departamento de F sica, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, 7600 Mar del Plata, Argentina Abstract We study the optimal distance ‘ opt in random networks in the presence of disorder implemented by assigning random weights to the links. The optimal distance between two nodes is the length of the path for which the sum of weights along the path (“cost”) is a minimum. We study the case of strong disorder for which the distribution of weights is so broad that its sum along any path is dominated by the largest link weight in the path. We nd that in random graphs, ‘ opt scales as N 1=3 , where N is the number of nodes in the network. Thus, ‘ opt increases dramatically compared to the known small-world result for the minimum distance ‘ min , which scales as log N . We also study, theoretically and by simulations, scale-free networks characterized by a power law distribution for the number of links, P(k ) ∼ k - , and nd that ‘ opt scales as N 1=3 for ¿ 4 and as N (-3)=(-1) for 3 ¡¡ 4. For 2 ¡¡ 3, our numerical results suggest that ‘ opt scales logarithmically with N . c  2003 Published by Elsevier B.V. PACS: 87.10.+e; 64.60.Cn; 75.10.Hk; 02.50.−r Keywords: Scale-free networks; Small-world networks; Percolation; Strong disorder; Optimal path Much attention has been focused on the topic of complex networks characterizing many biological, social, and communication systems [1–3]. The networks can be vi- sualized by nodes representing individuals, organizations, or computers and by links between them representing their interactions. The classical model for random networks * Corresponding author. Tel.: +1-617-3532617; fax: +1-617-3533783. E-mail address: hes@bu.edu (H.E. Stanley). 0378-4371/$ - see front matter c  2003 Published by Elsevier B.V. doi:10.1016/j.physa.2003.08.030