Random walk with jumps in large-scale random geometric graphs☆
Introduction
One of the main challenges associated with large-scale, unstructured and dynamic networking environments is that of efficiently reaching out to all or a portion of the network nodes (i.e., disseminating information) in order to provide, e.g., software updates or announcements of new services or queries. The high dynamicity and the sheer size of such networking topologies ask for the adoption of decentralized approaches to information dissemination [1], [2], [3], [4]. In this paper, the problem of efficiently disseminating information (or queries) across a large-scale, resource-limited, ad hoc-structured wireless network, such as a wireless sensor network, is considered. One of the simplest approaches employed for disseminating information in such environments, is the traditional flooding approach. Under flooding [5], [6], [7], [8], each time a node receives a message for the first time from some node, it forwards it to all its neighbors except from that node. Despite its simplicity and speed (typically achieving the shortest cover time, upper bounded by the network diameter), the associated large message overhead is a major drawback.
As flooding is considered not to be an option for large-scale, wireless sensor networks (WSNs) due to strict energy limitations of individual sensor nodes, solutions based on variations of the random walk based information dissemination paradigm are viewed as reasonable choices for searching and/or routing in WSNs [9], [10], [11]. Furthermore, there has been a significant body of work in adopting random walks for search or information dissemination in large peer-to-peer (P2P) networks [12], [13], [14], [15]. The random walk based information dissemination paradigm possesses several good characteristics such as simplicity, robustness against dynamic failures or changes to the network topology, and lack of need for knowledge of the network physical and topological characteristics. A Random Walk agent (RW-agent) doing a simple random walk within a network of wireless sensors moves from neighbor node to neighbor node in a purely random manner, frequently revisiting previously covered nodes in a circular manner. Even when backtracking (returning to the node it just came from) is not allowed, some circular movement in the topology can not be eliminated; these revisits constitute overhead and impact negatively on the cover time [16] of the process. Such a poor behavior of the RW-agent is attributed to the random manner of its movement, combined with some problematic topological characteristics of large-scale wireless ad hoc networks, such as cliques and bottlenecks. To make sure that there is consistency in terminology in this paper, we note here that the term RW-agent denotes always an agent performing a simple random walk without backtracking in the network,unless otherwise stated.
Large-scale, random geometric graphs (RGGs) have been studied in the past in relationship with percolation theory, statistical physics and hypothesis testing [17]. Recently, the RGG has received significant attention due to its applicability in modelling wireless ad hoc and sensor networks, where N is the number of network nodes and is the connectivity radius of individual nodes in the field. The network connectivity of the RGG depends on (a) the connectivity radius ; and (b) the geometric position of nodes. In particular, any nodes having geometric (euclidean) distance below the connectivity radius are considered to be bi-directionally connected. Naturally, connectivity radius should be large enough, such that the network is connected, i.e., there are no isolated nodes within the network. Such a network connectivity model is substantially different from the well-known scale-free network model arising in many natural and man-made systems, like the Internet, the World Wide Web, citation networks and some social networks. Many such networks fall into the class of scale-free networks, meaning they have power-law (or scale-free) degree distributions. The Barabasi-Albert model is one of several proposed models that generates scale-free networks [18]. The rules of growth and preferential attachment used for creating such networks, allow for any two nodes (no matter how far away from each other) to be connected with non-zero probability. Existence of long-haul links (links connecting nodes residing far apart in physical distance), although perfectly valid in power-law graphs, do not appear in RGGs, due to physical limitations associated with the connectivity radius .
In this paper the Jumping Random Walk (J-RW) mechanism, originally proposed in [19], is shown both analytically and experimentally to be an efficient random walk based information dissemination/ retrieval mechanism for large-scale RGGs. The contributions in this paper, in addition to presenting the J-RW mechanism, include analytical results about the coverage of the RW-agent and J-RW-agent and a new set of simulation experiments to validate the analytical findings on coverage. The proposed J-RW scheme entails the well-known, key benefits of random walk based mechanisms, like simplicity, lack of need for centralized control and robustness to topology changes, while providing a “boost” in performance, i.e. accelerating the coverage process within a RGG (modelling a large-scale WSN). The latter is achieved by introducing a second state of operation of the J-RW-agent, in which the random walk based movement paradigm is replaced by a non-random “directional” movement paradigm. Note that the other state of operation of the J-RW-agent is similar with that of the RW-agent (i.e., it executes a random walk without backtracking). It turns out that this modification in the agent’s movement improves significantly the cover time in RGGs by allowing the J-RW-agent to traverse some “virtual” long links in the topology, which are otherwise absent in RGGs. A similar phenomenon has been described in [20] in the context of social networks. Individuals in that pioneering work were assumed to live on the vertices of a grid and thus they know their neighbors for some number of steps in all directions (links to local neighbor nodes are established). They also have a number of acquaintances distributed more broadly across the grid, representing a few, “long range” links of nodes. The author shows that networks constructed under such criterion can accurately model a social network as formed by a group of human beings through their relationships or by computers communicating in the World Wide Web. There can be also a decentralized algorithm (individuals may use only local information) for such network models that is capable of finding short paths between two individuals with high probability.
Section snippets
The RW-agent
A credible alternative to flooding for disseminating information in an unstructured, large-scale networking environment, is the random walk based information dissemination paradigm. In random walk based approaches, initiator nodes (representing an end user in P2P networks or a sink in WSNs) employ the random walk agent that will move randomly in the network, one hop/node per time slot, informing (or querying) all the nodes in its path. Random walks in large-scale P2P networks have been shown to
Motivation
Fig. 1a illustrates a random walk based agent movement initiated from the initiator node depicted inside the dotted ellipsis. The random walker spends some time revisiting nodes in the depicted “upper-left” network part, while nodes in other network parts are left unvisited. Suppose now that after a few time units – long enough to “cover” a certain network part – the random walker moves to a “new” (most likely uncovered) network part (“bottom right” network part in Fig. 1b). It is more likely
Coverage analysis
This section analyzes coverage under the random walk without backtracking mechanism in order to extract useful information regarding the performance of the RW-agent and consequently to use these results for further understanding of the particulars of the J-RW mechanism. The analysis followed in this section is different than any other previous analysis in the best of the authors’ knowledge.
Simulation results and evaluation
A simulation program exploiting the capabilities of the Omnet++ simulation platform, [25], was created for the simulation purposes. The aim of the simulation results presented in this section is twofold: to confirm the analytical findings of the previous section and to shed more light on the behavior of the J-RW mechanism (mostly in comparison to the RW mechanism) for cases not covered by the analysis.
There are multiple simulation runs executed under specific sets of parameters for the network
Conclusions
Random walk based solutions have been proposed for information dissemination in large-scale wireless sensor and ad hoc networks. However, such random walk based agents are prone to inefficiencies due to frequent revisits to already covered nodes, resulting in relatively large network cover time. In this paper a new class of random walk based agents, referred to as the Jumping Random Walk agent has been introduced. As this new class contains the (classical) random walk without backtracking agent
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This work has been supported in part by the project ANA (Autonomic Network Architecture) (IST-27489), the PENED 2003 program of the General Secretariat for Research and Technology (GSRT) co-financed by the European Social Funds (75%) and by National Sources (25%) and the NoE CONTENT (IST-384239).