We study the optimal distance in networks, ℓ opt , defined as the length of the path minimizing the total weight, in the presence of disorder. Disorder is introduced by assigning random weights to the links or nodes. For strong disorder, where the maximal weight along the path dominates the sum, we find that ℓ opt ∼ N 1/3 in both Erdős-Rényi (ER) and Watts-Strogatz (WS) networks. For scale free (SF) networks, with degree distribution P (k) ∼ k −λ , we find that ℓ opt scales as N (λ−3)/(λ−1) for 3 < λ < 4 and as N 1/3 for λ ≥ 4. Thus, for these networks, the small-world nature is destroyed. For 2 < λ < 3, our numerical results suggest that ℓ opt scales as ln λ−1 N . We also find numerically that for weak disorder ℓ opt ∼ ln N for both the ER and WS models as well as for SF networks. Recently much attention has been focused on the topic of complex networks which characterize many biological, social, and communication systems [1,2,3]. The networks can be visualized by nodes representing individuals, organizations, or computers and by links between them representing their interactions.The classical model for random networks is the Erdős-Rényi (ER) model [4,5]. An important quantity characterizing networks is the average distance (minimal hopping) ℓ min between two nodes in the network of total N nodes. For the Erdős-Rényi network, and the related, more realistic Watts-Strogatz (WS) network [6] ℓ min scales as ln N [7], which leads to the concept of "six degrees of separation".In most studies, all links in the network are regarded as identical and thus the relevant parameter for information flow including efficient routing, searching, and transport is ℓ min . In practice, however, the weights (e.g., the quality or cost) of links are usually not equal, and thus the length of the optimal path minimizing the sum of weights is usually longer than the distance. In many cases, the selection of the path is controlled by the sum of weights (e.g., total cost) and this case corresponds to regular or weak disorder. However, in other cases, for example, when a transmission at a constant high rate is needed (e.g., in broadcasting video records over the Internet) the narrowest band link in the path between the transmitter and receiver controls the rate of transmission. This situation-in which one link controls the selection of the path-is called the strong disorder limit. In this Letter we show that disorder or inhomogeneity in the weight of links may increase the distance dramatically, destroying the "small-world" nature of the networks.To implement the disorder, we assign a weight or "cost" to each link or node. For example, the weight could be the time τ i required to transit the link i. The optimal path connecting nodes A and B is the one for which i τ i is a minimum. While in weak disorder all links contribute to the sum, in strong disorder one term dominates it. The strong disorder limit may be naturally realized in the vicinity of the absolute zero temperature if passing through a link is an activation process with a random...

Models of epidemic spreading on complex networks have attracted great attention among researchers in physics, mathematics, and epidemiology due to their success in predicting and controlling scenarios of epidemic spreading in real-world scenarios. To understand the interplay between epidemic spreading and the topology of a contact network, several outstanding theoretical approaches have been developed. An accurate theoretical approach describing the spreading dynamics must take both the network topology and dynamical correlations into consideration at the expense of increasing the complexity of the equations. In this short survey we unify the most widely used theoretical approaches for epidemic spreading on complex networks in terms of increasing complexity, including the mean-field, the heterogeneous mean-field, the quench mean-field, dynamical messagepassing, link percolation, and pairwise approximation. We build connections among these approaches to provide new insights into developing an accurate theoretical approach to spreading dynamics on complex networks.