For realistic scale-free networks, we investigate the traffic properties of stochastic routing inspired by a zero-range process known in statistical physics. By parameters @math and @math , this model controls degree-dependent hopping of packets and forwarding of packets with higher performance at more busy nodes. Through a theoretical analysis and numerical simulations, we derive the condition for the concentration of packets at a few hubs. In particular, we show that the optimal @math and @math are involved in the trade-off between a detour path for @math ; In the low-performance regime at a small @math , the wandering path for @math and @math is small, neither the wandering long path with short wait trapped at nodes ( @math ), nor the short hopping path with long wait trapped at hubs ( @math ) is advisable. A uniformly random walk ( @math ) yields slightly better performance. We also discuss the congestion phenomena in a more complicated situation with packet generation at each time step.
In the next two sections, based on a straightforward approach introduced in Refs. @cite_9 @cite_8 , we derive the phase transition in the ZRP on SF networks with the degree-dependent hopping rule for both @math and @math , inspired by preferential @cite_11 @cite_3 @cite_21 and congestion-aware walks. Although the rule is not identical to the congestion-aware routing scheme @cite_13 based on occupied queue length @math , @math corresponds to avoiding hubs with large degrees, where many packets tend to be concentrated. Furthermore, we study the traffic properties in the case with neighbor search into a terminal node at the last step.
@cite_8 @cite_9 @cite_21 @cite_3 @cite_13 @cite_11
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We present a study of transport on complex networks with routing based on local information. Particles hop from one node of the network to another according to a set of routing rules with different degrees of congestion awareness, ranging from random diffusion to rigid congestion-gradient driven flow. Each node can be either source or destination for particles and all nodes have the same routing capacity, which are features of ad-hoc wireless networks. It is shown that the transport capacity increases when a small amount of congestion awareness is present in the routing rules, and that it then decreases as the routing rules become too rigid when the flow becomes strictly congestion-gradient driven. Therefore, an optimum value of the congestion awareness exists in the routing rules. It is also shown that, in the limit of a large number of nodes, networks using routing based on local information jam at any nonzero load. Finally, we study the correlation between congestion at node level and a betweenness centrality measure. Compact routing addresses the tradeoff between table sizes and stretch, which is the worst-case ratio between the length of the path a packet is routed through by the scheme and the length of an actual shortest path from source to destination. We adapt the compact routing scheme by Thorup and Zwick [2001] to optimize it for power-law graphs. We analyze our adapted routing scheme based on the theory of unweighted random power-law graphs with fixed expected degree sequence by [2000]. Our result is the first analytical bound coupled to the parameter of the power-law graph model for a compact routing scheme. Let n denote the number of nodes in the network. We provide a labeled routing scheme that, after a stretch--5 handshaking step (similar to DNS lookup in TCP IP), routes messages along stretch--3 paths. We prove that, instead of routing tables with O(n1 2) bits (O suppresses factors logarithmic in n) as in the general scheme by Thorup and Zwick, expected sizes of O(nγ log n) bits are sufficient, and that all the routing tables can be constructed at once in expected time O(n1+γ log n), with γ = τ-22 τ-3 + e, where τ∈(2,3) is the power-law exponent and e 0 (which implies e With the same techniques as for the compact routing scheme, we also adapt the approximate distance oracle by Thorup and Zwick [2001, 2005] for stretch-3 and we obtain a new upper bound of expected O(n1+γ) for space and preprocessing for random power-law graphs. Our distance oracle is the first one optimized for power-law graphs. Furthermore, we provide a linear-space data structure that can answer 5--approximate distance queries in time at most O(n1 4+e) (similar to γ, the exponent actually depends on τ and lies between e and 1 4 + e). This paper considers two inter-related questions: (i) Given a wireless ad-hoc network and a collection of source-destination pairs (s i ,t i ) , what is the maximum throughput capacity of the network, i.e. the rate at which data from the sources to their corresponding destinations can be transferred in the network? (ii) Can network protocols be designed that jointly route the packets and schedule transmissions at rates close to the maximum throughput capacity? Much of the earlier work focused on random instances and proved analytical lower and upper bounds on the maximum throughput capacity. Here, in contrast, we consider arbitrary wireless networks. Further, we study the algorithmic aspects of the above questions: the goal is to design provably good algorithms for arbitrary instances. We develop analytical performance evaluation models and distributed algorithms for routing and scheduling which incorporate fairness, energy and dilation (path-length) requirements and provide a unified framework for utilizing the network close to its maximum throughput capacity.Motivated by certain popular wireless protocols used in practice, we also explore "shortest-path like" path selection strategies which maximize the network throughput. The theoretical results naturally suggest an interesting class of congestion aware link metrics which can be directly plugged into several existing routing protocols such as AODV, DSR, etc. We complement the theoretical analysis with extensive simulations. The results indicate that routes obtained using our congestion aware link metrics consistently yield higher throughput than hop-count based shortest path metrics. We study wireless ad hoc networks with a large number of nodes. We first focus on a network of n immobile nodes, each with a destination node chosen in random. We develop a scheme under which, in the absence of fading, the network can provide each node with a traffic rate spl lambda sub 1 (n)=K sub 1 (nlog n) sup -1 2 . This result was first shown in J. Hightower and G. Borriello (2001) under a similar setting, however the proof presented here is shorter and uses only basic probability tools. We then proceed to show that, under a general model of fading, each node can send data to its destination with a rate spl lambda sub 2 (n)=K sub 2 n sup -1 sup 2 (log n) sup -3 2 . Next, we extend our formulation to study the effects of node mobility. We first develop a simple scheme under which each of the a mobile nodes can send data to a randomly chosen destination node with a rate spl lambda sub 3 (n)=K sub 3 n sup -1 2 (log n) sup -3 2 , and with a fixed upper bound on the packet delay d sub max that does not depend on n. We subsequently develop a scheme under which each of the nodes can send data to its destination with a rate spl lambda sub 4 (n)=K sub 4 n sup (d-1) 2 (log n) sup -5 2 provided that nodes are willing to tolerate packet delays smaller than d sub max (n)<K sub 5 n sup d , where 0<d<1. With both schemes, a general model of fading is assumed. In addition, nodes require no global topology or routing information, and only need to coordinate locally. The above results hold for an appropriate choice of values for the constants K sub i , and with probability approaching 1 as the number of nodes n approaches infinity.