A greedy topology design to accelerate consensus in broadcast wireless sensor networks☆
Introduction
Average consensus, in the general framework of networks of agents, means reaching an agreement on the average state of all agents. Recently, much effort has been directed to the study of the average consensus problem in Wireless Sensor Networks (WSNs) (see [1] and the references therein), since distributed consensus algorithms only require iterative local information exchanges among neighboring nodes and the computation of weighted sums at each node. Potential applications include detection, estimation, reputation management, load balancing, control of autonomous agents, etc. [2].
One important issue regarding distributed average consensus algorithms in WSNs is convergence speed: reducing the convergence time results in fewer transmissions and therefore in energy savings. Approaches from the literature to speed up convergence can be classified into two groups. If the topology of the network is fixed, one can design the weights intervening in the consensus scheme in order to minimize convergence time [3], [4], [5]. On the other hand, if the network topology can somehow be altered, then additional flexibility is available, and the optimization can be performed over the topology as well as the weights [6], [7], [8], [9], [10]. Generally speaking, topology optimization is a very difficult combinatorial problem and different suboptimal approaches can be adopted. In [6] the convergence properties of different topology classes are theoretically analyzed on average, given the number of nodes of a general network. In [8] it is shown that, starting from a given topology, removing certain links can be beneficial in terms of convergence speed; this approach was later refined in [9] in order to judiciously remove and add links with the goal of speeding up the consensus algorithm while keeping energy consumption at bay. In WSNs with static nodes, topology control can be achieved by varying the transmit power, as considered in [7], [10].
All of these approaches implicitly assume unicast pairwise communication1: each node can independently set the transmit power it allocates to communicate with each of its neighbors, e.g., by using orthogonal signaling. However, in distributed consensus schemes the information that nodes need to send at a given iteration is the same for all of its neighbors, so that in WSNs it is possible to exploit the broadcast nature of the wireless channel, also known as Wireless Multicast Advantage (WMA) [11]: at each consensus iteration, each node may broadcast its state while its neighbors simultaneously listen, thus reducing the number of required transmissions. On the other hand, exploiting WMA while varying the transmit power of a given node affects the links to all of its neighbors, so that these cannot be independently controlled now.2 This motivates specific topology optimization strategies that take this fact into account, since previous topology control schemes designed under the unicast assumption cannot be applied under these “broadcast communication” constraints.
This problem is related to the so-called range assignment (RA) problem in broadcast WSNs, usually oriented to other network-related goals (e.g. maintaining global connectivity [13]) and known to be difficult in general [14]. Our goal is to determine the transmit power for each node in a broadcast WSN in order to minimize the convergence time of a given distributed average consensus scheme. One issue featuring in such setting is the fact that, if the transmit powers of nodes i and j are different (non-homogeneous RA [13]), it may well happen that node i is out of the coverage range of node j whereas node j can listen to node i's transmissions; in other words, the underlying graph becomes directed. This has implications for consensus algorithms. Although reaching an agreement over a directed graph is easily achieved, the agreement value will be a weighted average of the agents' states, and the weights will depend on the topology. When the unweighted average is of interest, certain stringent requirements on the directed graph must be imposed (such as some sort of graph balancing [15]), which are generally difficult to enforce in practice. Hence, we focus on undirected graphs, for which reaching an agreement on the unweighted average by consensus algorithms is not a problem. To this end, we adopt a simple strategy by which nodes just ignore transmissions received from neighbors which are not within their own transmit range (in the previous example, node j would simply ignore packets received from node i), thus obtaining an undirected topology. With this framework, we start from a maximally connected setting (all nodes transmit at full power), and then proceed to iteratively reduce the power of one node at a time in a centralized greedy fashion in order to maximize the convergence rate.
As previous approaches to topology control [6], [7], [8], [9], [10], ours is a centralized scheme which can be run by a central entity after deployment and previously to the network becoming operative; after such step, network operation may become decentralized. Fully distributed topology control methods are desirable and should be the target of future research.
In Section 2 the network model and the basics of consensus schemes are presented. The proposed greedy algorithm for non-homogeneous RA is presented in Section 3. Simulation results and conclusions are provided in Sections 4 and 5.
Section snippets
Graph model
Consider a set of randomly deployed nodes with indices . Let be the distance between nodes i and j, and let be a set of connectivity radii (i.e. transmit ranges), with the maximum allowable range. We adopt a simple model by which a link between two nodes exists iff their distance does not exceed the transmit range of the transmitter, which can be controlled by setting the transmit power [10], [13]. As discussed in Section 1, the edge set is
A greedy approach to fast consensus
An important observation is that the maximally connected topology3 obtained by setting for all i is not necessarily optimal in terms of η. For example, with a constant weight assignment with optimum stepsize, minimizing η amounts to maximizing the eigenvalue ratio . It is known that the eigenvalues of the Laplacian matrix cannot increase (resp. decrease) by removing links from (resp. adding links to) a
Simulation results
The performance of the greedy schemes was checked in a number of randomly generated deployments in . The maximum transmit range is obtained as a function of the number of nodes n as ; thus, the larger the value of c, the more connected the topology. For nodes and , we averaged the results over 100 random deployments for each pair. A constant-weight assignment is assumed.
Fig. 2(a) shows the relative convergence time , where
Conclusions
New methods to optimize a topology-dependent cost function in the context of broadcast WSNs have been introduced. They start with a dense topology and successively remove one link at a time in a greedy fashion; the best configuration among those obtained is then picked. The methods effectively improve convergence speed for average consensus algorithms, with reduced energy consumption as an important side benefit. The full greedy version requires n EVDs of an matrix per step, with n the
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Research supported by the European Regional Development Fund (ERDF) and the Spanish Government (TEC2013-47020-C2-1-R COMPASS, CONSOLIDER-INGENIO 2010 CSD2008-00010 COMONSENS, TACTICA), and the Galician Regional Government (R2014/037, GRC2013/009 and AtlantTIC).