- Open Access
An integrated diversity and fusion framework for binary consensus over fading channels
© Malmirchegini and Mostofi; licensee Springer. 2013
- Received: 22 April 2012
- Accepted: 19 June 2013
- Published: 4 September 2013
In this paper, we consider a cooperative network that is trying to reach consensus on the occurrence of an event by communicating over not fully connected and time-invariant network topologies with fading channels. We first discuss the fusion and diversity decision-making strategies over time-invariant network topologies and shed light on the underlying trade-offs. We then propose an integrated diversity and fusion framework. Our approach properly takes advantage of both fusion to enable information flow and diversity to increase robustness to link errors. We mathematically analyze the proposed framework and show how the network achieves accurate consensus asymptotically. To show an example, we then utilize the proposed framework over regular ring lattice networks. Our theoretical and simulation results indicate that the proposed technique improves the consensus performance considerably.
- Fading Channel
- Fading Coefficient
- Link Quality
- Fusion Level
- Fusion Approach
Cooperative decision-making over sensor networks has gotten considerable interest in recent years. These networks have a variety of applications such as environmental monitoring, target tracking, and surveillance. Consider the scenario where a network of nodes, distributed in a harsh environment, aims to cooperatively accomplish a task. Each node has limited local capabilities and can therefore only accomplish the task in a cooperative manner. In this paper, we are interested in group agreement problem, where a group of agents need to reach consensus on the value of a parameter of interest. The consensus problems can be categorized into two main groups: estimation consensus and detection consensus. Estimation consensus refers to problems in which each agent has an estimate of the parameter of interest, where the parameter of interest can take values over an infinite set or an unknown finite set. For instance, it may be of interest that all the mobile agents that started in different directions reach an agreement on their asymptotic headings in a cooperative multi-agent network . Recently, there has been considerable interest in estimation consensus problems from the signal processing and communication community, with more emphasis on link uncertainties [2–4].
Detection consensus, on the other hand, refers to problems in which the parameter of interest takes values from a finite known set. Then the update protocol, which each agent will utilize, becomes nonlinear. By binary consensus, we refer to a subset of detection consensus problems where the network is trying to reach consensus over a binary parameter. For instance, in a cooperative fire detection scenario, each node has an initial opinion as to if there is a fire or not. However, as a network, they may act based only on the majority vote. Therefore, the goal of the network is for each node to reach consensus over the majority of initial votes. Another application of binary consensus is in cooperative spectrum sensing in cognitive radio networks. In this scenario, the secondary users communicate with each other in order to reach consensus on busy or idle status of the primary user, which is a binary value .
1.1 Related work and our contribution
While there exists a rich literature on estimation consensus, detection consensus problems only recently started to receive attention. In , the authors consider convergence in a detection consensus setup over perfect channels with repeated sensing and known probabilistic sensing models. In , the authors consider a distributed hypothesis testing problem over perfect communication channels to which they refer to as belief consensus. They consider the case where each node transmits its belief (conditional probability) to other nodes. As a result, their problem immediately takes the form of the traditional average estimation consensus for which a rich literature exists.
In [5, 9], the binary consensus scenario is considered. In this scenario, each node in the network has an opinion regarding the occurrence of an event. The nodes will then communicate over the network topology. The goal for every node is to reach the majority of the initial votes without any prior knowledge on the sensing qualities. Authors in  characterized phase transition of such a binary consensus problem in the presence of a uniformly distributed communication noise. In most applications, however, the agents will communicate their values wirelessly and will experience Gaussian receiver noise as opposed to a uniformly distributed noise. In , we considered reaching binary consensus over regular network topologies (all nodes have the same number of neighbors) with additive white Gaussian noise channels. We characterized the transient behavior of the network probabilistically. We showed that in the presence of noise, the network state is asymptotically memoryless, i.e., independent of the initial state. This is undesirable since the group agreement is not related to the initial state of the system and is merely a function of channel errors.
In , we studied the binary consensus over a fully connected network topology with fading channels. We proposed a novel consensus-seeking protocol that utilizes information of link qualities. We showed that by incorporating the information of link qualities, the network will be in consensus with a higher probability but still holds the undesirable asymptotic behavior. In , we considered binary consensus over rapidly changing network topologies with fading channels. We mathematically characterized the impact of fading, noise, network connectivity and time-varying topology on consensus performance.
In this paper, we consider the binary consensus problem over the general time-invariant network topologies (not necessarily fully connected) with fading channels. We study two decision-making strategies that differ in terms of using the available transmissions: fusion and diversity. In the first strategy, the given resources are used to increase the flow of information in the network, whereas the second strategy aims to increase robustness to link error by channel coding . We characterize the underlying trade-offs between these two strategies for binary consensus over a not fully connected and time-invariant network topology. In particular, we show that fusion-based scheme results in an asymptotic memoryless behavior, which is not desirable. Furthermore, diversity-based scheme only outperforms fusion-based scheme, when the main bottleneck is link quality and the network has a good connectivity. The main contribution of this paper is to propose a framework that keeps the benefits of both fusion and diversity strategies, in terms of the network information flow and link error robustness, for binary consensus over time-invariant network topologies with fading channels. We mathematically analyze the proposed framework and show that it achieves accurate consensus asymptotically. The proposed framework solves the undesirable memoryless behavior of the network consensus and results in a drastic performance improvement.
In this section, we propose our diversity-fusion framework for binary consensus over fading channels. In this strategy, each agent sends a vector to its neighbors. This vector consists of the estimations of the votes corresponding to the different fusion levels. Throughout the repeated communications (diversity), each node tries to refine its assessments of different fusion levels in order to reach consensus.
Time progression of transmitted vector by node i
t = 0
t = 1
t = u G −1
t = k
for k ≥ u, 0 ≤ u ≤ u G − 1 and ∀i,k. Next we show that the decision-making function of Equation 12 achieves accurate consensus asymptotically and overcomes the asymptotic memoryless behavior of the traditional fusion approaches.
Consider binary consensus over a time-invariant network topology with i.i.d. Rayleigh fading channels. Then, the decision-making function of Equation 12 asymptotically converges (in probability) to accurate consensus if .
We prove the theorem by induction. Define . For u = 0, we have . From Lemma 1, it can be easily confirmed that . Assume . We next prove that .
Since is a non-negative random variable, it goes to 0 as k → ∞. Therefore, we have, . Furthermore, similar to Equation 10, we can show that . By substituting these values in Equation 13, it can be easily confirmed that . Using induction, we have and as a result for 1 ≤ i ≤ M. Therefore, if , then is an accurate consensus state with the probability of one, which proves the theorem. □
for k ≥ u, 0 ≤ u ≤ u G −1, and ∀i,k. This receiver can be considered as a special case of the decision-making function of Equation 12, where σ j,i,u = 0. We can similarly show that it achieves accurate asymptotic consensus. Furthermore, we have and .
In the subsequent sections, we utilize the integrated framework for a special class of undirected graphs and show the performance of the proposed framework.
In Section 4, we introduced our proposed framework, which asymptotically achieves accurate consensus. In this approach, node i will send a vector B i (k) to all its neighbors over fading channels. The length of this vector is u G for t ≥ u G −1, which is a function of graph connectivity. Intuitively, networks with higher connectivity require smaller values of u G . For instance, for fully connected networks, we have u G = 1. In general, u G is a function of network topology and independent of communication quality. This parameter needs to be determined before running the algorithm. In this section, we mainly focus on L-regular ring lattice topologies with ideal communication links in order to characterize u G . Let denote the vertex set. Without loss of generality, we assume that the vertices are ordered clockwise on the ring (see Figure 1). Furthermore, we assume that M is odd. Under the above assumptions, the adjacency matrix of an L-regular ring lattice, i.e., A L , can be represented by a circulant matrix with the first row of . For this class of graphs, L is a notion of connectivity. Therefore, we try to show how u G changes as a function of L. We then have the following lemma, which will be used in Theorem 2.
For an L-regular ring lattice, if L = M − 1, then . Moreover, if L ≤ M − 3, then . The equality is achieved if and only if denotes a set of consecutive nodes on the corresponding ring.
The proof is straightforward and we skip it. □
Let . We then have the following theorem.
Assume , such that represents a set of consecutive nodes on the ring that vote the same and . Then the corresponding network state is an absorbing state.
For , the accurate consensus is achievable after one level of fusion.
For and , accurate consensus is achievable at most after two fusion steps.
Let denote the i th partition, where represents the index of j th node in the i th partition. Therefore, we have, for all . Since all the nodes in vote the same, we have for 1 ≤ i ≤ c and .
We next prove the second part. Consider the case where the majority of the initial votes is 1. If , we then have , which results in . For the case where the majority of the initial votes is 0, we have , which results in . Therefore, for accurate consensus is achievable in one iteration.
Next, we show the third statement. First we show that for and , we have . Let denote an ordered set of the nodes, which vote to the minority of the initial votes. Lemma 2 says that at most nodes can have in their neighbor set if and only if is a set of consecutive vertices. Therefore, is achievable if and only if the initial state is an absorbing state (see Theorem 2-1). Therefore, if , then , which reaches accurate consensus in u = 2 (see Theorem 2-2). □
Figure 4 shows the performance of the proposed framework for L = 8,10, and 14. For these simulations, we assume that node i does not have the knowledge of for . So it simply assumes in the decision-making function of Equation 12. As can be seen, the the integrated approach, independent of network connectivity, achieves accurate consensus asymptotically. Therefore, the proposed approach overcomes the memoryless asymptotic behavior of traditional binary consensus approaches. Furthermore, networks with higher connectivity, i.e., larger values of L, reach their steady state in fewer transmissions. For the case of L = 8, the performance of both fusion and diversity approaches of Section 3 are also shown for comparison. As can be seen, the proposed approach keeps the benefits of both fusion and diversity in terms of the transient and asymptotic behaviors respectively. Furthermore, the performance of the integrated approach for the case where L = 8 and knowledge of σ j,i,u is not available (Remark 1) is also shown in Figure 4. As can be seen, the integrated approach with known link quality slightly outperforms the case of unknown link quality. However, both cases provide a similar performance asymptotically as mentioned earlier in Remark 1.
In this paper we considered a cooperative network that is trying to reach binary consensus over not fully connected and time-invariant network topologies with fading channels. We first characterized the underlying trade-offs between fusion and diversity strategies. We then proposed a framework that keeps the benefits of both strategies. We mathematically analyzed the proposed algorithm and showed how it achieves accurate consensus asymptotically. Our results indicated that the proposed technique improves the consensus performance considerably and solves the undesirable memoryless asymptotic behavior of the original problem.
a We also use the term ‘agent’ to refer to each node.
b Note that in a binary consensus scenario, each node only exchanges 1 bit of information with its neighbors in each time step.
c An L-regular ring lattice is an L-regular graph with M vertices in a ring in which each vertex is connected to its L neighbors ( on each side for an even value of L).
d Note that without loss of generality, we assumed that the modulation is on-off keying.
e We assume an FDMA- or TDMA-based MAC approach. For instance, in FDMA-base approach, to each node we assign different frequency subbands, corresponding to different fusion levels.
f Note that we chose u G as the maximum length based on the fact that, in the case of no noise, this is the number of fusion levels needed for convergence. However, in this case that we have noise, this may not be the optimum length and choosing longer lengths can possibly result in a better performance.
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