An achievable bit rate per source-destination pair in a wireless network of n randomly located nodes is determined adopting the scaling limit approach of statistical physics. It is shown that randomly scattered nodes can achieve, with high probability, the same 1= p n transmission rate of arbitrarily located nodes. This contrasts with previous results suggesting that a 1= p n log n reduced rate is the price to pay for the randomness due to the location of the nodes. The network operation strategy to achieve the result corresponds to the transition region between order and disorder of an underlying percolation model. If nodes are allowed to transmit over large distances, then paths of connected nodes that cross the entire network area can be easily found, but these generate excessive interference. If nodes transmit over short distances, then such crossing paths do not exist. Percolation theory ensures that crossing paths form in the transition region between these two extreme scenarios. Nodes along these paths are used as a backbone, relaying data for other nodes, and can transport the total amount of information generated by all the sources. A lower bound on the achievable bit rate is then obtained

Understanding the large-scale structural network formed by neurons is a major challenge in system neuroscience. A detailed connectivity map covering the entire brain would therefore be of great value. Based on diffusion MRI, we propose an efficient methodology to generate large, comprehensive and individual white matter connectional datasets of the living or dead, human or animal brain. This non-invasive tool enables us to study the basic and potentially complex network properties of the entire brain. For two human subjects we find that their individual brain networks have an exponential node degree distribution and that their global organization is in the form of a small world.

How can we localize the source of diffusion in a complex network? Because of the tremendous size of many real networks-such as the internet or the human social graph-it is usually unfeasible to observe the state of all nodes in a network. We show that it is fundamentally possible to estimate the location of the source from measurements collected by sparsely placed observers. We present a strategy that is optimal for arbitrary trees, achieving maximum probability of correct localization. We describe efficient implementations with complexity O(N(α)), where α=1 for arbitrary trees and α=3 for arbitrary graphs. In the context of several case studies, we determine how localization accuracy is affected by various system parameters, including the structure of the network, the density of observers, and the number of observed cascades.

Many complex networks are only a part of larger systems, where a number of coexisting topologies interact and depend on each other. We introduce a layered model to facilitate the description and analysis of such systems. As an example of its application, we study the load distribution in three transportation systems, where the lower layer is the physical infrastructure and the upper layer represents the traffic flows. This layered view allows us to capture the fundamental differences between the real load and commonly used load estimators, which explains why these estimators fail to approximate the real load. DOI: 10.1103/PhysRevLett.96.138701 PACS numbers: 89.75.Hc, 89.20.Hh, 89.40.Bb, 89.75.Fb In recent years, studies of biological, social, infrastructure, or technological networks have drawn a substantial amount of attention in the physics community. Although these networks are usually considered as distinct objects, they are often a part of larger complex systems, where a number of coexisting topologies interact and depend on each other. For instance, the topologies of the Internet at the IP layer [1], of the World Wide Web (WWW) [2], or of the networks formed by peer to peer (P2P) applications [3], although studied separately, are closely related: Each WWW or P2P link virtually connects two IP nodes. These two IP nodes are usually distant in the underlying IP topology, and the virtual connection is realized as a path found by IP routers. In other words, the graph formed by an application is mapped on the underlying IP network. Moreover, the IP links are in turn mapped on the physical layer [4] that consists of a mesh of optical fibers usually buried in the ground along roads, rails, or power lines. The resulting topologies at the three layers are very different from each other.Another important class of real-life systems is transportation networks. Graphs derived from the physical infrastructure of such networks were analyzed on the examples of a power grid [5], a railway network [6], road networks [7], or urban mass transportation systems [8]. This approach often gives a valuable insight into the studied topology, but it ignores the real-life traffic pattern. Interestingly, the networks of traffic flows were studied separately, for instance, the flows of people within a city [9] and commuting traffic flows between different cities [10]. These studies, in turn, neglect the underlying physical topology. A comprehensive view of the system often requires one to analyze both layers (physical and traffic) together. Only in some particular cases is one layer sufficient. This is the case, e.g., in airport networks [11], where all traffic flows are one-hop long and the full knowledge of the traffic pattern is introduced into the physical graph by setting the edge weights equal to the amount of traffic they carry. However, in the presence of traffic flows longer than one hop, a weighted physical graph is not sufficient.

Abstract-We consider a large-scale wireless network, but with a low density of nodes per unit area. Interferences are then less critical, contrary to connectivity. This paper studies the latter property for both a purely ad-hoc network and a hybrid network, where fixed base stations can be reached in multiple hops. We assume here that power constraints are modeled by a maximal distance above which two nodes are not (directly) connected.We find that the introduction of a sparse network of base stations does significantly help in increasing the connectivity, but only when the node density is much larger in one dimension than in the other. We explain the results by percolation theory. We obtain analytical expressions of the probability of connectivity in the 1-dim. case. We also show that at a low spatial density of nodes, bottlenecks are unavoidable. Results obtained on actual population data confirm our findings.