Rate Allocation

In this section, we focus on the numerical solutions of (3.30) and infer some intuition about the optimal distributions of the quantization rates in the MSE sense. For a 2-D regular graph of 64 nodes, connectivity radius 0.25 and initial observations uncorrelated (i.e., σi(0) = σj(0), ∀ {i, j}), the total number of bits spent over one thousand iterations versus the final MSE is given in Fig. 3.6. For each MSE-β pair, the algorithm is initialized with a quantization rate 5≤ R(0) ≤ 20and R(t) is calculated by (3.30) for a given 0.70≤ β ≤ 0.95. We note that such a family of β satisfies β > maxjω_j². We observe that given a total number of bits to be spent, one can achieve a lower MSE by utilizing a larger β. Fig. 3.7 shows quantization rates over iterations for, β = 0.95, β = 0.90 and β = 0.85 such that the total number of bits spent is fixed, i.e., ∑200

t=0R(t) ≃ 225. For β = 0.95, the algorithm starts with 20 bits per node and slowly converges to 0.0050 bits

0 20 40 60 80 100 0

5 10 15 20

iteration number

number of bits spent

β=0.95 → MSE=−76 dB β=0.90 → MSE=−74 dB β=0.85 → MSE=−70.4 dB

Figure 3.7: MSE (dB) vs sum of the quantization rates for different β.

per node. On the other hand, for β = 0.90, the algorithm initializes with 19.2 bits per node, stays under the β = 0.95 curve initially, then crosses the curve and follows a similar behavior converging to 0.047. The β = 0.85 curve starts at 18.2bits per node and converges to 0.091 slower than both 0.90 and 0.95 curves.

The MSE performances of these schemes are different, i.e., −76dB, −74dB and

−70.4dB respectively, since as we noted in Fig. 3.6, a larger β results in a better MSE performance for a fixed number of total bits spent.

Compiling the theoretical and numerical results, we note that a slower de-crease in the noise variances will result in a smaller MSE when the total number of quantization bit to be spent is fixed. In other words, transmitting more bits in the beginning and decreasing transmission cost gradually results in a better performance.

3.5 Discussions

In this study, we have studied rate constrained average consensus problem un-der both fixed and variable rate quantization rate scenarios. We have focused on simple predictive coding scenario where at time instant t, nodes transmit only the innovation with respect to previous state value. In the case of fixed rate quantization, we have presented a closed form MSE expression and correspond-ing upper bounds in terms of the network connectivity, quantization rate and initial node correlation. We have also derived asymptotic behavior of the quan-tization rate in a scenario where quanquan-tization rates are variable and consensus with finite MSE is achieved. In a special case where the quantization rates are chosen such that quantization noises decrease like a geometric series, we have determined the rate regions where asymptotic quantization rate vanishes such that sum-rate and MSE is bounded.

CHAPTER 4

BROADCAST GOSSIPING ALGORITHM

4.1 Motivation and Related Work

As it was discussed in Chapter 1, geographic-type gossiping improves upon the convergence speed of the standard gossip by increasing the diversity of pair-wise exchanges. However, the problem of packet loss is exacerbated by the requirement that messages must be sent on long routes, creating congestion is-sues. Moreover, it does not mitigate the major bottleneck associated with the fact that the messages between two peers need to be routed and exchanged to perform two updates. Finally, successfully setting up a two-way route exacer-bates the problem by requiring information about the location of the nodes in the network.

The wireless medium has the advantage of being inherently broadcast and, at the cost of one transmission, one can reach several terminals. Our objec-tive in this chapter is to analyze a broadcasting-based gossip algorithm that en-ables all nodes in range to perform an update by exploiting the wireless medium, and thereby avoiding the need of complex routing and problematic pairwise exchange operations.

4.1.1 Summary of Main Contributions

To overcome the drawbacks of the standard packet based gossip algorithms, we study a broadcast based gossiping algorithm for wireless sensor networks. In

the studied algorithm, a node in the network wakes up uniformly at random according to the asynchronous time model and broadcasts its value. This value is successfully received by the nodes in the predefined radius of the broadcast-ing node, i.e., connectivity radius. The nodes that have received the broadcasted value update their own state value and the remaining nodes sustain their value.

It is shown here that by iterating this procedure, this type of gossiping algo-rithm is capable of achieving consensus over the network with probability one.

We also show that the random consensus value is, in expectation, equal to the desired value, i.e., the average of initial node measurements. Because the sum of the node state values is not preserved at each iteration, the broadcast gossip-ing algorithm converges to a value that is in the neighborhood of the desired average.

The question that motivates this chapter is investigating if it is possible to avoid the partner selection process altogether, analyzing a broadcast communi-cation protocol where each random transmission triggers an update by all nodes within range, without a mechanism of reply in place to maintain the network average. We initiated this study in [9]. Fagnani and Zampieri have concur-rently studied the convergence to consensus characteristics of general random-ized algorithms which do not necessarily converge to the initial average (such as asymmetric gossip, broadcast gossip and packet-drop gossip) [38]. In par-ticular, the authors have shown that random consensus algorithms in general achieve probabilistic consensus, and discussed their mean squared error char-acteristics (Proposition 4.4 and Corollary 3.2). In this chapter, we provide an in depth study of broadcast gossip algorithms’ speed of convergence and mean squared error characteristics. Our results also address the choice of the mixing parameter and its effect on both the mean square error and the convergence rate,

which provides insight for implementation.

More specifically, we provide theoretical and simulation results on the mean square error and communication cost performance of the broadcast gossip al-gorithm. Moreover, we study the effect of the so called mixing parameter on the convergence rate and limiting mean square error through theoretical results and numerical experiments. In addition, we derive the optimal mixing param-eter when approached from the convergence rate perspective. Although the convergence time of our algorithm is commensurate with the standard pairwise gossip algorithms, we present simulations showing that for more modest net-work sizes our algorithm converges to consensus faster than other algorithms based on pairwise averages or routing.

4.1.2 Chapter Organization

The remainder of this chapter is organized as follows. Section 4.2 introduces the average consensus problem and the graph and time models adopted in this study. The studied broadcast gossip algorithm is introduced in Section 4.3 and its convergence characteristics are studied in Section 4.4. In Section 4.5 we de-rive the optimal mixing parameter considering the worst-case convergence rate and analyze the effects of various network parameters on the optimal value.

The MSE characterization and communication complexity analysis are given in Section 4.6 along with the convergence rate expression. Finally, we conclude with some discussion and future directions in Section 4.7.