Exploiting the Small-World Effect to Increase Connectivity ... - CiteSeerX

Exploiting the Small-World Effect to Increase Connectivity in Wireless Ad hoc Networks Dave Cavalcanti1, Dharma Agrawal1, Judith Kelner2 and Djamel Sadok2 1

ORB Center for Distributed and Mobile Computing, University of Cincinnati, Cincinnati, Ohio, USA {cavalcat, dpa}@ececs.uc.edu http://www.ececs.uc.edu/~cdmc 2 Centro de Informatica, Universidade Federal de Pernambuco, Recife, Brazil {jk, jamel}@cin.ufpe.br

Abstract. This paper investigates how the small world concept can be applied in the context of wireless ad hoc networks. Different from wireless ad hoc networks, small world networks have small characteristic path lengths and are highly clustered. This path length reduction is caused by long-range edges between randomly selected nodes. However, in a wireless ad hoc network there are no such long-range connections. Then, we propose to use a fraction of nodes in the network equipped with two radios with different transmission ranges in order to introduce the long-range shortcuts. We analyze the system from a percolation perspective and show that a small fraction of these “special nodes” can improve connectivity in a significant way. We also study the effects of the special nodes on the process of information diffusion and on network robustness.

1 Introduction Wireless ad hoc and sensor networks have been extensively studied in the recent years. The efficiency of multihop routing used in these networks depends on the network connectivity, i.e., the existence of a direct or multiple hops path between any two nodes in the network. However, connectivity depends on factors such as, interference, noise and energy constraints and cannot be always assured [5]. Connectivity also affects the network capacity, and as shown in [6], a fundamental condition for scalability is to keep the average distance between any source/destination pair small as the system grows. Recent results by Watts and Strogatz [8] have shown the existence Small World networks that have a small degree of separation between nodes while maintaining highly clustered neighborhoods. The small degree of separation in such networks is obtained through the introduction of some long-range edges that result in faster information propagation. The small word effect has been widely studied in the context of relation graphs, such as, the graph formed by hyperlinks in Web. However, these long-range edges are not possible in wireless ad hoc networks due to the limited the transmission range of the nodes.

In this paper, we propose to introduce some long-range edges in an ad hoc network by using special nodes equipped with two radios, one with a short transmission range and another with a longer transmission range. In fact, nodes with multiple interfaces are expected to be common in a near future. For instance, wireless devices can be equipped with cellular and 802.11 interfaces. As we will show, the introduction of the special nodes can improve network connectivity, reduce the route length and consequently, enhance the network capacity and scalability. The remainder of this work is organized as follows: In Section 2, we present the basic concepts and some related works. Next, in Section 3, we evaluate the effect of the special nodes with multiple radios in the network connectivity, average hop distance between nodes and network robustness. Finally, we present some concluding remarks and future work in Section 4.

2 Background and Related Work Recently, Watts and Strogatz [6] have studied a class of networks that present a small characteristic path length (L), and are highly clustered, known as Small World networks (or small world graphs). Given the network graph G, the characteristic path length can be calculated as the mean number of hops between any two nodes in G. The clustering coefficient Cu of a given node u in G, is the relation between number of edges connecting neighbors of u in G, and the total number of possible edges between the neighbors of u. The overall clustering coefficient C is the average over all Cu, for u in G. The reduction in L is important in various aspects, including the network capacity and the way information propagates through the network. The small world effect has been observed in real networks such as the WWW and overlay peer-to-peer networks. However, these networks form relational graphs in which there is no distance constraint in the edges lengths. Thus, they are not appropriate to model wireless networks, in which the connections are limited by physical distance. The fixed radius graph G=G(N, r) is a spatial model widely used in studies of wireless ad hoc networks. In this case, given N points placed randomly according to some distribution on the Euclidian plane, G is constructed connecting the nodes whose corresponding points are within a distance r of each other. Clearly, the constraint on the connection range r precludes the existence of long-range shortcuts in G. Nevertheless, Watts suggested in [7] that, if a graph is constructed following the spatial graph concept but using a probability distribution to connect nodes with slowdecay and infinite variance, one might expect that this distribution would generate small world spatial graphs. In terms of ad hoc and sensor networks, the shortcuts could be interpreted as the global edges serving as the links between the clusters heads which could gather information about cluster members using local connections. However, some practical questions have to be considered: How far a shortcut should be? How many shortcuts are needed to keep the network connected? An attempt is made to answer these questions in the remainder of this paper.

In [3], Helmy has the suggested to use long-range shortcuts to reduce the number of queries during the search for a given target node in a large-scale ad hoc network. In this case, a shortcut is defined as a logical link that a node maintains with a random selected node. Therefore, a logical link between a pair of nodes can eventually correspond to several physical hops, and this logical link is implemented by making each node in the path to keep a route to the long-range contact. In [2], the authors have used percolation theory to analyze the introduction of fixed base stations to increase connectivity in large-scale ad hoc networks. The main idea is that base stations would allow distant nodes to communicate through a fixed, wired infrastructure. The authors have assumed the base stations transmission ranges are the same as the ones of the wireless nodes. The connectivity level is measured as the fraction of nodes connected to a giant cluster, which is defined as percolation probability. Although analytical modeling and simulations [2] have shown that, in the onedimensional case, the fixed infrastructure does improve the network connectivity, in the 2-dimensional case, the base stations do not enhance connectivity significantly. The idea of using base stations is also proposed in [1], where the authors proved that if base stations can communicate at a distance larger than twice the maximum communication distance to the users, a giant connected cluster forms almost surely for large values of the density of users, regardless of the covering algorithm used to place the base stations in the covered region.

3 Long-range Shortcuts in Ad hoc Networks Transforming the network into a small world by simple increasing the transmission range of the nodes is an alternative to reduce the average separation between nodes without imposing restrictions to network traffic as suggested in [4]. However, this approach has practical restrictions, such as power consumption, and interference, and the impact of these constraints on the network performance need to be well understood. Helmy [3] has also pointed out the possibility of having all nodes equipped with several radios, one for short-range communications and another with a longer transmission range, in order to introduce long-range shortcuts in the network. Indeed, a more practical approach is to have a limited number of special nodes equipped with two different radios. A typical short-range radio (r) and a longer range one for introducing the long-range shortcuts (rf). We consider that the longer-range radios operate in a different spectrum from the typical short-range radios. Then, two nodes i and j are directly connected in the network graph if they satisfy one of the following conditions: the distance dij between i and j is smaller than their typical communication range r; or, if i and j are special nodes and dij < rf. In the remainder of this section we analyze the effect of the introduction of the special nodes on the network connectivity, on the broadcast efficiency and on the network robustness. We measure the connectivity level as the percolation probability,

as in [2]. We consider a network in which 1000 nodes are uniformly distributed over a 1kmx1km square area and all results plotted are averaged over 100 runs. Initially, we considered a disconnected scenario where r = 35 m. As shown in Figure 1(a), the percolation probability increases with f for different values of rf. For f < 0,01, the network remains disconnected regardless of the rf used. On the other hand, for f > 0,2 all values of rf result in an almost connected network. From this point on, there is not much connectivity gain by adding more special nodes. This is an important result, as it suggests that there is an upper bound on the number of special nodes required to transform a disconnected topology in an almost connected network. Figure 1(b) shows the percolation probability as a function of the rf/r relation. We can observe that for all values of f simulated there is a critical value of rf after which the connectivity has a drastic increase and that, further increase in rf has no effect on connectivity. This confirms that there is a maximal range rf after which there is no reduction on L. The reduction in L can be seen in Figure 1(c), for two values of rf (10r and 3r). Although we do not show, no significant change was observed in the clustering coefficient. In this case, we have assumed r=60, to make sure that the network was initially connected.

1 2r 0.8 0.6 0.4

Characteristic path length

Percolation probability (r=35)

Percolation probability (r=35)

3r

1

12

0.8

10 f=0,05

8

f=0,1

6

0.4

f=0,2

4

0.2

f=0,3

0.6

5r 8r

0.2

f=0,7

0 0.001

0

0.01

f

0.1

(a)

1

1

3

5 rf/r

7

(b)

9

rf =10r rf=3r

2 0 0.001

0.01

0.1 f

1

(c)

Fig. 1. (a) Percolation probability as a function of the fraction of nodes with two radios, (b) Percolation probability as a function of rf, (c) Characteristic path length as a function of f

We have also studied the effects of the special nodes on the propagation of a broadcast packet in the network. We randomly select a node to start a broadcast and at each time step, each node that received the packet forwards it to its neighbors with a transmission probability ptr. Then, we computed the fraction of nodes at the end of the simulation that did not receive the message S(tmax). We call S(tmax) as the susceptible nodes as an analogy to an epidemic model in which the susceptible nodes represent the nodes that were not infected by the information at the end of the simulation. The diamonds in Figure 2(a) represent S(tmax) for the fixed radius model (spatial graph) with r=60m. Note that in general, the models with special nodes (f=0,1) result in more efficient information diffusion than the simple spatial model for 0.1