Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

: In wireless data networks each transmitter's power needs to be high enough to reach the intended receivers, while generating minimum interference on other receivers sharing the same channel. In particular, if the nodes in the network are assumed to cooperate in routing each others ' packets, as is the case in ad hoc wireless networks, each node should transmit with just enough power to guarantee connectivity in the network. Towards this end, we derive the critical power a node in the network needs to transmit in order to ensure that the network is connected with probability one as the number of nodes in the network goes to infinity. It is shown that if n nodes are placed in a disc of unit area in ! 2 and each node transmits at a power level so as to cover an area of ßr 2 = (log n + c(n))=n, then the resulting network is asymptotically connected with probability one if and only if c(n) ! +1. 1 Introduction Wireless communication systems consist of nodes which share a common commu...

Abstract—Many ad hoc routing protocols are based on some variant of flooding. Despite various optimizations, many routing messa ges are propagated unnecessarily. We propose a gossiping-based approa ch, where each node forwards a message with some probability, to reduce the ov erhead of the routing protocols. Gossiping exhibits bimodal behavio r in sufficiently large networks: in some executions, the gossip dies out quic kly and hardly any node gets the message; in the remaining executions, a sub stantial fraction of the nodes gets the message. The fraction of execution s in which most nodes get the message depends on the gossiping probability a nd the topology of the network. In the networks we have considered, using g ossiping probability between 0.6 and 0.8 suffices to ensure that almost every node gets the message in almost every execution. For large networ ks, this simple gossiping protocol uses up to 35 % fewer messages than flood ing, with improved performance. Gossiping can also be combined with va rious optimizations of flooding to yield further benefits. Simulations show that adding gossiping to AODV results in significant performance improv ement, even in networks as small as 150 nodes. We expect that the improvemen t should be even more significant in larger networks. I.

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We consider an unreliable wireless sensor gridnetwork with n nodes placed in a square of unit area. We are interested in the coverage of the region and the connectivity of the network. We first show that the necessary and sufficient conditions for the random grid network to cover the unit square region as well as ensure that the active nodes are connected are of the form p(n)r ,wherer(n) is the transmission radius of each node and p(n) is the probability that a node is "active" (not failed). This result indicates that, when n is large, even if each node is highly unreliable and the transmission power is small, we can still maintain connectivity with coverage.

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

We study the impact of interferences on the connectivity of large-scale ad-hoc networks, using percolation theory. We assume that a bi-directional connection can be set up between two nodes if the signal to noise ratio at the receiver is larger than some threshold. The noise is the sum of the contribution of interferences from all other nodes, weighted by a coefficient gamma, and of a background noise. We find that there is a critical value of gamma above which the network is made of disconnected clusters of nodes. We also prove that if gamma is non zero but small enough, there exist node spatial densities for which the network contains a large (theoretically infinite) cluster of nodes, enabling distant nodes to communicate in multiple hops. Since small values of gamma cannot be achieved without efficient CDMA codes, we investigate the use of a very simple TDMA scheme, where nodes can emit only every n-th time slot. We show qualitatively that it even achieves a better connectivity than the previous system with a parameter gamma/n.

In this paper, we analyze the critical transmitting range for connectivity in wireless ad hoc networks. More specifically, we consider the following problem: assume n nodes, each capable of communicating with nodes within a radius of r, are randomly and uniformly distributed in a d-dimensional region with a side of length l; how large must the transmitting range r be to ensure that the resulting network is connected with high probability? First, we consider this problem for stationary networks, and we provide tight upper and lower bounds on the critical transmitting range for one-dimensional networks, and non-tight bounds for two and three-dimensional networks. Due to the presence of the geometric parameter l in the model, our results can be applied to dense as well as sparse ad hoc networks, contrary to existing theoretical results that apply only to dense networks. We also investigate several related questions through extensive simulations. First, we evaluate the relationship between the critical transmitting range and the minimum transmitting range that ensures formation of a connected component containing a large fraction (e.g. 90%) of the nodes. Then, we consider the mobile version of the

Abstract—An achievable bit rate per source–destination pair in a wireless network of � 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 Ia � � transmission rate of arbitrarily located nodes. This contrasts with previous results suggesting that a Ia � � �� � � 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 by performing pairwise coding and decoding at each hop along the paths, and using a time division multiple access scheme. Index Terms—Ad-hoc networks, capacity, percolation theory, scaling laws, throughput, wireless networks.

In wireless data networks the range of each transmitter, and thus its power level, needs to be high enough to reach the intended receivers, while being low enough to avoid generating interference for other receivers on the same channel. If the nodes in the network are assumed to cooperate, perhaps in a distributed and decentralized fashion, in routing each others' packets, as is the case in ad hoc wireless networks, ([5]), and [7]), then each node should transmit with just enough power to guarantee connectivity of the overall network. Towards this end, we determine the critical power a node in the network needs to transmit in order to ensure that the network is connected with probability one as the number of nodes in the network goes to innity. Our main result is this: If n nodes are located randomly, uniformly i.i.d., in a disc of unit area in < 2 and each node transmits at a power level so as to cover an area of r 2 = (log n + c(n))=n, then the resulting network is asymptotical...