Performance Analysis and Evaluation of Random Walk Algorithms on Wireless Networks

Abstract—We propose a model of dynamically evolving random networks and give an analytical result of the cover time of the simple random walk algorithm on a dynamic random symmetric planar point graph. Our dynamic network model considers random node distribution and random node mobility. We analyze the cover time of the parallel random walk algorithm on a complete network and show by numerical data that k parallel random walks reduce the cover time by almost a factor of k. We present simulation results for four random walk algorithms on random asymmetric planar point graphs. These algorithms include the simple random walk algorithm, the intelligent random walk algorithm, the parallel random walk algorithm, and the parallel intelligent random walk algorithm. Our random network model considers random node distribution and random battery transmission power.
I. I NTRODUCTION The technique of random walks has been extensively used in wireless networks, mobile networks, ad-hoc networks, packet radio networks, sensor networks, peer-to-peer networks, and general networking and distributed systems for a wide range of applications, including query processing [2], data aggregation [4], membership service [5], group communication [8], searching [12], topology construction and maintenance [12], index quality measuring [13], routing [21], and token management [22]. The power and advantages of random walks are two fold. First, a random walk algorithm does not need any information of the size and the topology of a network. Thus, a random walk algorithm can be applied to any network, even a dynamic network with such evolutions as structural and topological change due to user mobility, node addition due to new users arriving to the network, node removal due to sleep modes and device deterioration and failure, reduction and loss of connectivity due to battery consumption, link removal due to weather condition, physical obstacles, noise, electrical and magnetic disturbance and interference, and other uncontrollable factors of channel fluctuations. Second, the performance of a random walk algorithm is practically acceptable. For instance, we will show in this paper that the normalized cover time of the simple random walk algorithm on a wireless and mobile network represented by a dynamic random symmetric planar point graph is very close to the
•
any network. We present simulation results for four random walk algorithms on random asymmetric planar point graphsic graphs in [4] as special cases. These algorithms include the simple random walk algorithm, the intelligent random walk algorithm, the parallel random walk algorithm, and the parallel intelligent random walk algorithm. Our random network model considers random node distribution and random battery transmission power. II. T HE DYNAMIC AND R ANDOM G RAPH M ODELS
978-1-4244-6534-7/10/$26.00 ©2010 IEEE
normalized cover time of the same algorithm on a complete network of the same size. The main performance measure of a random walk algorithm on a network is the cover time, i.e., the expected time to visit all nodes in the network. There are extensive literature on the investigation of the cover time of various graphs. It has been known that the cover time of an arbitrary graph with n vertices is in the range (1 + o(1))n ln n and 4 3 3 27 n + o(n ), where both the lower and upper bounds are tight [9], [10]. For random graphs, it was shown that if the radius is at least 8c log n/n with c > 1, a random geometric graph has optimal over time of Θ(n log n) with high probability [3]. Several models of dynamic graphs have been proposed in the literature, e.g., evolving graphs [6] and temporal networks [14]. However, there has been little work on the analysis of random walk algorithms on dynamic graphs. The contributions of the present paper are as follows. • Although the cover time of the simple random walk algorithm on a complete network with n nodes has been mentioned as Θ(n log n) [4], or more accurately, nHn , where Hn is the nth harmonic number [12], the exact result is not available. We derive a formula of the cover time of the simple random walk algorithm on a complete network. • We propose a model of dynamically evolving random networks and give an analytical result of the cover time of the simple random walk algorithm on a dynamic random symmetric planar point graph. Our dynamic network model considers random node distribution and random node mobility. To the best of our knowledge, this is the first analytical result on the cover time of the simple random walk algorithm on dynamic random networks. • We analyze the cover time of the parallel random walk algorithm on a complete network and show by numerical data that k parallel random walks reduce the cover time by almost a factor of k . Our analysis considers several different methods to initialize the starting nodes. To the best of our knowledge, this is the first analytical result on the calculation of the exact cover time of the parallel random walk algorithm on
Performance Analysis and Evaluation of Random Walk Algorithms on Wireless Networks
Keqin Li Department of Computer Science State University of New York New Paltz, New York 12561, USA Email: lik@