In this paper, we study network coding capacity for random wireless networks. Previous work on network coding capacity for wired and wireless networks have focused on the case where the capacities of links in the network are independent. In this paper, we consider a more realistic model, where wireless networks are modeled by random geometric graphs with interference and noise. In this model, the capacities of links are not independent. We consider two scenarios, single source multiple destinations and multiple sources multiple destinations. In the first scenario, employing coupling and martingale methods, we show that the network coding capacity for random wireless networks still exhibits a concentration behavior around the mean value of the minimum cut under some mild conditions. Furthermore, we establish upper and lower bounds on the network coding capacity for dependent and independent nodes. In the second one, we also show that the network coding capacity still follows a concentration behavior. Our simulation results confirm our theoretical predictions.
Deep Dive into Network Coding Capacity of Random Wireless Networks under a Signal-to-Interference-and-Noise Model.
In this paper, we study network coding capacity for random wireless networks. Previous work on network coding capacity for wired and wireless networks have focused on the case where the capacities of links in the network are independent. In this paper, we consider a more realistic model, where wireless networks are modeled by random geometric graphs with interference and noise. In this model, the capacities of links are not independent. We consider two scenarios, single source multiple destinations and multiple sources multiple destinations. In the first scenario, employing coupling and martingale methods, we show that the network coding capacity for random wireless networks still exhibits a concentration behavior around the mean value of the minimum cut under some mild conditions. Furthermore, we establish upper and lower bounds on the network coding capacity for dependent and independent nodes. In the second one, we also show that the network coding capacity still follows a concentra
Network coding was originally proposed by Ahlswede et al. in [1]. Unlike traditional store-and-forward routing algorithms, in network coding schemes, intermediate nodes encode their received messages and forward the coded messages to their next-hop neighbors. It has been shown that network coding can improve the network capacity, even by using simple linear or random codes [8], [9], [11], [12]. In most studies of network coding, network topologies are assumed to be known.
In [17], [18], the authors studied network coding capacity for weighted random graphs and random geometric graphs.
In the random graph model, each pair of nodes are connected by a bidirectional link with probability p < 1 independently [4], [10]. The capacity of each link is assumed to be i.i.d. according to some probability distribution. In the random geometric graph model, two nodes are connected to each other by a bidirectional link only when their distance is less than a predefined positive value r, the characteristic radius [15]. Each link has a unit capacity. For these two types of random networks, the authors showed that the network coding capacity is concentrated at the (weighted) mean degree of the graph, i.e., the (weighted) mean number of neighbors of each node. Essentially, the results reveal a concentration behavior of the size of the minimum cut between two nodes in random graphs or random geometric graphs. Similar problems have been studied in the literature, e.g., [6] and references there. In [3], the authors studied a generalized random geometric graph model, where two nodes are connected by a bidirectional link with probability 1 if their distance d is less than r 0 > 0 and with probability p < 1 if r 0 < d ≤ r 1 . They obtained similar concentration results there.
The geometric models in [3], [17], [18] assume that a link exists (possibly with a probability) between two nodes when the nodes are within each other’s transmission range.
Although each link has a direction, as all links are bidirectional (i.e., the link (i, j) implies the existence of the link (j, i)), the model in fact leads to an undirected graph and considerably simplifies the resulting analysis. In addition, interferences among wireless terminals were not considered in [3], [17], [18]. Nevertheless, in wireless networks, due to noise, interference, and heterogeneity of transmission power, significantly more sophisticated models for link connectivity are needed. For instance, a widely-used model for wireless communication channels is the Signal-to-Interference-plus-Noise-Ratio (SINR) model [16], [19]. In this paper, we study the capacity, i.e., the size of the minimum cut, of random wireless networks under the SINR model.
Since how to apply the network coding with noisy links is still an open problem, we assume that as long as the SINR of a link (i, j), β ij is greater than or equal to a predefined threshold β, then node i can transmit data at rate R packets/sec to node j without any error. That is links are noise-free once the SINR condition is met. In other words, we view the network coding as operation on a higher layer in the network communication stack, and assume there is an error correcting code at the lower layer which corrects errors on the links once the SINR threshold is met. Then, in this model, each link is indeed directional (not necessarily bidirectional), and the capacities of different links are not independent. We will show that the capacity still has a sharp concentration when the scale of the network is large enough.
This paper is organized as follows. In Section II, we describe the random wireless network model. In Section III, we study the network coding capacity for a single source and multiple destinations transmissions. Specifically, we investigate two cases. In the first one, all nodes have the same transmission power, and in the second one, the transmission powers are heterogeneous. We use different techniques for these two cases and show that the network coding capacity has a concentration behavior in both cases. In Section IV, we extend our result to multiple sources and multiple destinations transmission problem. In Section V, we present some simulation results, and finally, we conclude this paper in Section VI.
We use the following model for random wireless networks. Assume
random variables according to a homogeneous Poisson point process in the two-dimensional unit torus, where X i denotes the random location of node i, and n is the total number of nodes. (ii) Each node i has a transmission power P i , which follows a probability distribution f P (p), p ∈ [p min , p max ], where 0 < p min ≤ p max < ∞.
Here, the existence of a link from node i to node j depends on the ability to decode the transmitted signal from i to j, which is determined by the Signal-to-interference-plus-noiseratio (SINR) given by
where P i is the transmission power of node i, d ij is the distance between nodes i and j, and N 0 is the power of b
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