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Distributed Average Consensus With Dithered Quantization Tuncer Can Aysal, Member, IEEE, Mark J. Coates, Member, IEEE, and Michael G. Rabbat, Member, IEEE
Abstract—In this paper, we develop algorithms for distributed computation of averages of the node data over networks with bandwidth/power constraints or large volumes of data. Distributed averaging algorithms fail to achieve consensus when deterministic uniform quantization is adopted. We propose a distributed algorithm in which the nodes utilize probabilistically quantized information, i.e., dithered quantization, to communicate with each other. The algorithm we develop is a dynamical system that generates sequences achieving a consensus at one of the quantization values almost surely. In addition, we show that the expected value of the consensus is equal to the average of the original sensor data. We derive an upper bound on the mean-square-error performance of the probabilistically quantized distributed averaging (PQDA). Moreover, we show that the convergence of the PQDA is monotonic by studying the evolution of the minimum-length interval containing the node values. We reveal that the length of this interval is a monotonically nonincreasing function with limit zero. We also demonstrate that all the node values, in the worst case, converge to the final two quantization bins at the same rate as standard unquantized consensus. Finally, we report the results of simulations conducted to evaluate the behavior and the effectiveness of the proposed algorithm in various scenarios. Index Terms—Average consensus, distributed algorithms, dithering, probabilistic quantization, sensor networks.
I. INTRODUCTION D HOC networks of autonomous sensors and actuators are attractive solutions for a broad range of applications. Such networks find use in civilian and military applications, including target tracking and surveillance for robot navigation, source localization, weather forecasting, medical monitoring and imaging. In general, the networks envisioned for many of these applications involve large numbers of possibly randomly distributed inexpensive sensors, with limited sensing, processing and communication power on board. In many of the applications, limitations in bandwidth, sensor battery power and computing resources place tight constraints in the rate
Manuscript received August 20, 2007; revised April 17, 2008. First published June 13, 2008; current version published September 17, 2008. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Mounir Ghogho. Portions of this paper were presented at the IEEE Statistical Signal Processing WorkshopMadison, WI, August 2007, and the Allerton Conference on Communications, Control, and Computing, Monticello, IL, September 2007. T. C. Aysal is with the Communications Research in Signal Processing Group, School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853 USA (e-mail:
[email protected]
). M. J. Coates and M. G. Rabbat are with the Telecommunications and Signal Processing-Computer Networks Laboratory, Department of Electrical and Computer Engineering, McGill University, Montreal, QC H3A 2A7, Canada (e-mail:
[email protected]
[email protected]
). Digital Object Identifier 10.1109/TSP.2008.927071
and form of information that can be exchanged [3]–[5]. Other applications such as camera networks and distributed tracking demand communication of large volumes of data. When the power and bandwidth constraints, or large volume data sets are considered, communication with unquantized values is impractical. Distributed average consensus—the task of ensuring that all nodes in a network are aware of the average of a set of network-wide measurements—is a fundamental problem in ad hoc network applications, including distributed agreement and synchronization problems [6], distributed coordination of mobile autonomous agents [7], [8], and distributed data fusion in sensor networks [3], [9], [10]. It is also a central topic for load balancing (with divisible tasks) in parallel computers [11], [12]. Our previous work has illustrated how distributed average consensus can be used for two distributed signal processing tasks: source localization [13], and data compression [14]. Decentralized data compression, in particular, requires the computation of many consensus values in parallel (one for each compression coefficient). By appropriately quantizing each coefficient, multiple coefficients can be transmitted in a single packet, leading to a significantly more efficient implementation. Distributed averaging algorithms are extremely attractive for applications in wirelessly networked systems because nodes only exchange information and maintain state for their immediate neighbors. Consequently, there is no need to establish or maintain complicated routing structures. Also, there is no single bottleneck link (as in a tree) where the result of in-network computation can be compromised or lost or jammed by an adversary. Finally, consensus algorithms have the attractive property that, at termination, the computed value is available throughout the network, so a network user can query any node and immediately receive a response, rather than waiting for the query and response to propagate to and from a fusion center. In both wireless sensor and peer-to-peer networks, there is interest in simple protocols for computing aggregate statistics [15]–[18]. In this paper, we focus on a particular class of iterative algorithms for average consensus. Each node updates its state with a weighted sum of values from neighboring nodes, i.e. (1) for and . Here is a weight assoand is the total number of nodes. ciated with the edge These weights are algorithm parameters [3], [9]. Furthermore, denotes the set of nodes that have a direct (bidirectional) communication link with node . The state at each node in the
iteration consists of a single real number, which overwrites the are time-indepenprevious value. The algorithm parameters dent, i.e., do not depend on . Under easily verified conditions on it is easy to show that the value at each node conasymptotically as . verges to Xiao, Boyd, and Kim extended the distributed consensus algorithm to admit noisy communication links where each node updates its local variable with a weighted average of its neighbors' values, and each new value is corrupted by an additive noise with zero mean [19] (2) where is the additive zero-mean noise with fixed variance. that They pose and solve the problem of designing weights lead to optimal steady-state behavior, based on the assumption are independent. that the noise terms A. Related Work While there exists a substantial body of work on average consensus protocols with infinite precision and noise-free peer-to-peer communications, little research has been done introducing distortions in the message exchange. Recently, Yildiz and Scaglione, in [20], explored the impact of quantization noise through modification of the consensus algorithm proposed by Xiao, Boyd, and Kim [19]. They note that the noise component in (2) can be considered as the quantization noise and they develop an algorithm for predicting neighbors' unquantized values in order to correct errors introduced by quantization [20]. Simulation studies for small indicate that if the increasing correlation among the node states is taken into account, the variance of the quantization noise diminishes and nodes converge to a consensus. Kashyap et al. examine the effects of quantization in consensus algorithms from a different point of view [21]. They require that the network average be preserved at every iteration. To do this using quantized transmissions, nodes must carefully account for round-off errors. Suppose we have a network of nodes and let denote the "quantization resolution" or distance between two quantization lattice points. If is not a multiple of , then it is not possible for the network to reach a strict consensus (i.e., ) while also preserving the network average, , since nodes only ever exchange units of . Instead, Kashyap et al. define the notion of a "quantized consensus" to be such that all take on one of two neighboring quantization values while preserving the network average; i.e., for all and some , and . They show that, under reasonable conditions, their algorithm will converge to a quantized consensus. However, the quantized consensus is clearly not a strict consensus, i.e., all nodes do not have the same value. Even when the algorithm has converged, as many as half the nodes in the network may have different values. If nodes are strategizing and/or performing actions based on these values (e.g., flight formation), then differing values may lead to inconsistent behavior.
Of note is that both related works discussed above utilize standard deterministic uniform quantization schemes to quantize the data. In contrast to [20], where quantization noise terms are modeled as independent zero-mean random variables, we explicitly introduce randomization in our quantization procedure, i.e., "dithering." Careful analysis of this randomization allows us to provide concrete theoretical rates of convergence in addition to empirical results. Moreover, the algorithm proposed in this paper converges to a strict consensus, as opposed to the approximate "quantized consensus" achieved in [21]. In addition to proving that our algorithm converges, we show that the network average is preserved in expectation, and we characterize the limiting mean squared error (MSE) between the consensus value and the network average. B. Summary of Contributions Constraints on sensor cost, bandwidth, and energy budget dictate that information transmitted between nodes has to be quantized in practice [3], [4]. In this paper, we propose a simple distributed and iterative scheme to compute the average at each sensor node utilizing only quantized information communication. Standard, deterministic uniform quantization does not lead to the desired result. Although the standard distributed averaging algorithm converges to a fixed point when deterministic uniform quantization is used, it fails to converge to a consensus as illustrated in Fig. 1(a). Instead, we adopt the probabilistic quantization (PQ) scheme described in [4]. PQ has been shown to be very effective for estimation with quantized data since the noise introduced by PQ is zero-mean [4]. This makes PQ suitable for average-based algorithms. As shown in Section II, the PQ algorithm is a form dithered quantization method. Dithering has been widely recognized as a method to render the quantization noise independent of the quantized data, reducing some artifacts created by deterministic quantization and there is a vast literature on the topic, see [22] and references therein. In the scheme considered here, each node exchanges quantized state information with its neighbors in a simple and bidirectional manner. This scheme does not involve routing messages in the network; instead, it diffuses information across network by updating each node's data with a weighted average of its neighbors' quantized ones. We do not burden the nodes with extensive computations, and we provide theoretical results, i.e., we show here that the distributed average computation utilizing probabilistic consensus indeed achieves a consensus almost surely (Fig. 1), and the consensus is one of the quantization levels. Furthermore, the expected value of the achieved consensus is equal to the desired value, i.e., the average of the initial analog node measurements. We also give an upper bound on the MSE performance of the probabilistically quantized distributed averaging (PQDA) algorithm. We also investigate the evolution with time of the interval occupied by the node values. Specifically, we show that the size of this interval is a monotonically nonincreasing function, with limit zero. These results indicate that the convergence of the PQDA algorithm is monotonic in the sense that the global trend of the node values is towards the consensus. Moreover, we show here that all the node values, in the worst case, arrive in the final two quantization bins at the same rate as standard unquantized
AYSAL et al.: DISTRIBUTED AVERAGE CONSENSUS
Fig. 1. Individual node trajectories (i.e., x (t); 8i) taken by the distributed average consensus using (a) deterministic uniform quantization and (b) probabilistic quantization. The number of nodes is N = 50, the nodes' initial average is x(0) = 0:85, and the quantization resolution is set to 1 = 0:1. The consensus value, in this case, is 0.8.
consensus. Of note is that there is always a nonzero probability of achieving consensus when all the node values are in the final two bins. Finally, we present simulation results evaluating the proposed algorithm in varying scenarios and showing the effectiveness of the PQDA algorithm.
We assume that transmissions are always successful and that the topology is fixed. We assume connected network topologies and adthe connectivity pattern of the graph is given by the , where jacency matrix if otherwise.
C. Paper Organization The remainder of this paper is organized as follows. Section II reviews the graph theoretical concepts, introduces the distributed average consensus problem along with the probabilistic quantization scheme, and, reveals the connections between probabilistic quantization and dithering theory. The proposed algorithm, along with its properties, is detailed in Section III. Section IV presents results regarding the convergence characteristics of the proposed PQDA algorithm. Numerical examples evaluating the performance of the proposed algorithm in varying scenarios are provided in Section V. Some extensions of the proposed algorithm along with additional practical considerations are detailed in Section VI. Finally, we conclude with Section VII. II. PRELIMINARIES In the following, the distributed average consensus problem is formulated utilizing the probabilistic quantization concept. We first review some basic concepts from graph theory and then formulate the consensus problem in which the nodes communicate with quantized data. Finally, we provide a brief review of probabilistic quantization and reveal the connections between probabilistic quantization and dithering theory. A. Review of Graph Theoretical Concepts be a graph consisting of a set of vertices, , Let and a set of edges . Let denote the number of verdenotes the cardinality. We denote an edge betices, where . The prestween vertices and as an unordered pair ence of an edge between two vertices indicates that they can establish bidirectional noise-free communication with each other.
Moreover, we denote the neighborhood of the node by . Also, the degree of the node is . given by B. Distributed Average Consensus We consider a set of nodes of a network (vertices of the graph), each with an initial real valued scalar , where . Let denote the vector of ones. Our goal is to develop a distributed iterative algorithm that computes at every node in the network, while the value using quantized communication. We hence aim to design a system such that the states at all nodes converge to a consensus and the expectation of the consensus achieved, in the limit, is the average of the initial states. The average of node measurements is a sufficient statistic for many problems of interest. The following two remarks briefly discusses two examples. where Remark 1: When the observations follow and is the scalar to be estimated, and the , are independent and identinoises, cally distributed (i.i.d.) zero-mean Gaussian with variance , the maximum-likelihood estimate is given by the average, with the associated MSE . Remark 2: Suppose node measurements are i.i.d. conditioned on some hypothesis , with , where . Let, , where . Then, the optimal decision is to perform the following detection rule: where
C. Probabilistic Quantization and Dithering In the following, we present a brief review of the quantization scheme adopted in this paper. Suppose that the scalar value is bounded to a finite interval . Furthermore, suppose that we wish to obtain a quantized message with length bits, where is application dependent. We, therefore, have quantization points given by the set where and . The points are uniformly spaced such for . It follows that that . Now suppose and let where denotes the PQ operation. Then is quantized in a probabilistic manner and
. Of note is that when the variable to where quantize is exactly equal to a quantization centroid, there is zero probability of choosing another centroid. The following lemma, adopted from [4], discusses two important properties of PQ. and let be an -bit Lemma 1: [4] Suppose quantization of . The message is an unbiased representation of , i.e., and
As noted in the following lemma, a careful observation shows that probabilistic quantization is equivalent to a "dithered quantization" method. and let . Lemma 2: Suppose Probabilistic quantization is equivalent to the following dithered quantization scheme: (6) where
is a uniform random variable with support on . Proof: Without loss of generality, suppose . Moreover, suppose we are utilizing a deterministic uniform quantizer. Then (7) (8) (9) , so the proof is Note that the last line is equivalent to complete. Thus, before we perform any quantization, we add uniform random variable with support defined on and . Now, performing standard deterministic we form , yields uniform quantization, i.e., letting quantized values, s that are statistically identical to the ones of the probabilistic quantization. Thus, probabilistic quantization is a form of dithering where one, before performing standard deterministic uniform quantization, adds a uniform random variable with support equal to the quantization bin size. This is a substractively dithered system [22]. It has been shown by
Schuchman that the substractive dithering process utilizing uniyields error form random variable with support on signal values that are statistically independent from each other and the input [23]. III. DISTRIBUTED AVERAGE CONSENSUS WITH PQ COMMUNICATION In the following, we propose a quantized distributed average consensus algorithm and incorporate PQ into the consensus framework for networks. Furthermore, we analyze the effect of PQ on the consensus algorithm. Specifically, we present theorems revealing the limiting consensus, expectation and MSE of the proposed PQDA algorithm. (after all sensors have taken the measurement), each At , i.e., , where node initializes its state as denotes the initial states at the nodes. It then quantizes . At each following step, its state to generate each node updates its state with a linear combination of its own quantized state and the quantized states at its neighbors (10) for , where , and denotes the is the weight on at node . Moreover, time step. Also, whenever , the distributed iterative setting process reduces to the following recursion: (11) where
denotes the quantized state vector, followed by (12)
The PQDA algorithm hence refers to the iterative algorithm defined by (11) and (12). In the sequel, we assume that , the weight matrix, is symmetric, nonnegative, and satisfies the conditions required for asymptotic average consensus without quantization [19] and
where denotes the spectral radius of a matrix (i.e., the largest eigenvalue of in absolute value), and , projects onto the -dimensional "diagonal" subwhere corresponding to a strict space (i.e., the set of vectors in consensus). Weight matrices satisfying the required convergence conditions are easy to find if the underlying graph is connected and nonbipartite, e.g., Maximum-degree and Metropolis weights [19]. The following theorem considers the convergence of the probabilistically quantized distributed average computation. Theorem 1: The probabilistically quantized distributed iterative process achieves a consensus, almost surely, i.e., (14) where . Proof: Without loss of generality, we focus on integer . Define as the discrete quantization in the range Markov chain with initial state and transition matrix
AYSAL et al.: DISTRIBUTED AVERAGE CONSENSUS
defined by the combination of the deterministic transforand the probabilistic quantizer mation . Let be the set of quantization points that can be represented for some integer and denote by the set of in the form quantization points with Manhattan distance from . Morebe the open hypercube centered at and defined over, let as . Here denotes the th coefficient of . Note that any point in has a nonzero probability of being quantized to . Let (15) The consensus operator has the important property that for , denotes the projection of its argument onto where . The latter propthe -vector. Moreover, erty implies that is an absorbing state, since . The former property implies that there are no other absorbing states, since cannot equal (it must be closer to the -vector). This implies, from the properties of the quantizer , that there . is a nonzero probability that is an absorbing Markov chain, it In order to prove that remains to show that it is possible to reach an absorbing state from any other state. We prove this by induction, demonstrating first that (16)
The theorem reveals that the probabilistically quantized distributed process indeed achieves a strict consensus at one of the quantization values. It is of interest to note that the stationary points of the PQDA algorithm are in the form of where . We, hence, construct an absorbing Markov chain where the absorbing states are given by the stationarity points and show that for any starting state, there exists a sequence of transitions with nonzero probability whose application results in absorption. The following theorem discusses the expectation as of the limiting random vector, i.e., the expected value of tends to infinity. Theorem 2: The expectation of the limiting random vector is given by (21) Proof: Note that , for , and, is bounded for all . Moreover, from Theorem converges in 1, we know that the random vector sequence for some . Thus, by the the limit, i.e., Lebesgue dominated convergence theorem [24], we have (22) In the following, we derive and utilize the above relationship to arrive at the desired result. , we can write In terms of quantization noise . The distributed iterative process reduces to the fol. Repeatedly lowing recursion: utilizing the state recursion gives
and subsequently that (23) (17) Define the open set
as (18)
. The distance To commence, observe that for . Hence, if , and Similarly, the set
Taking the statistical expectation of that the only random variables are yields
as for
and noting ,
(24) (25)
since for 1. Furthermore, noting that
; a corollary of Lemma gives (26)
is contained in the union of the first hypercubes, , . The maximum distance for any point is . This implies that (20) There is, thus, some
and some such that . This argument implies that for such that for some , there any starting state exists a sequence of transitions with nonzero probability whose application results in absorption.
Recalling (22) gives the desired result. This result indicates that the expectation of the limiting random vector is indeed equal to the initial analog node measurements' average. Furthermore, this theorem, combined with the previous one, indicates that the consensus value is a discrete random variable with support defined by , and whose expectation is equal to the average of the initial states. After establishing that the consensus value is a random variable with the desired expectation in the limit, the next natural quantity of interest is the limiting MSE, i.e., the limiting average squared distance of the consensus random variable from
the desired initial states' average value. The following theorem, thus, considers the expectation of the error norm of probabilistically quantized consensus as and tend to infinity. Theorem 3: Let us define . The expectation of the error norm of the probabilistically quantized distributed average consensus is asymptotically bounded by
(27) where denotes the spectral radius of its argument. Proof: See Appendix A. The proof exploits averaging characteristics of , properties of norm operators, and uses a Law of Large Numbers argument to bound the error contributed by quantization noise. Note that the upper bound decreases with decreasing spectral , where a smaller (larger) can be, in radius of a loose manner, interpreted as better (worse) "averaging ability" of the weight matrix. Furthermore, as expected, the upper bound on the error norm increases with decreasing quantization resolution (i.e., increasing ). IV. CONVERGENCE CHARACTERISTICS OF PQDA CONSENSUS The convergence characteristics of the PQDA are essential for further understanding of the algorithm. In the following, we consider the evolution of the intervals occupied by the quantized and unquantized state values. Interestingly, we reveal that the length of the smallest interval containing all of the quantized state values, (i.e., the range of the quantized state values) is nonincreasing with a limit of zero as the time step tends to infinity. Moreover, we show that size of the minimum length interval, with boundaries constrained to the quantization points, that contains all of the unquantized node state values, is also nonincreasing. This also has limit zero as the time step tends to infinity. Let us denote the smallest and largest order statistics of any as and , vector denote the interval of the respectively. Furthermore, let node state values at time , i.e., the interval in which values lie (28) and denote the domain of the quantized node state values at time , i.e.,
range quantization bin that encloses the node state values. The theorem reveals that both intervals are nonexpanding. , suppose that , Theorem 4: For some and, , for . By construction (32) Then, for , the following holds. i) The interval of the quantized state vector is nonexpanding, i.e., (33) ii) The minimum length interval with boundaries defined by quantization points that encloses the state vector values is nonexpanding, i.e., (34) , for Proof: Consider (i) first. Suppose that , and recall that the state recursion follows as . Let denote the row vector formed as the th row of the weight matrix . Now, we can write the node specific update equation as (35) Note that is a linear combination of quantized local for , where denotes node values and , since . Thus, the th entry of . Moreover, is a convex combination of the quantized node state values and its own quantized state. The node state value is then in the convex hull of quantized state values . The convex hull of the quantized state values at , indicating that time is given by (36) for
, and subsequently (37)
Hence, we see that (38) for some
(29) Moreover, let
(39) for some
(30) and (31) along with . The following theorem discusses the evolution of the interval of the quantized node state values, and the minimum
. It follows that (40)
Repeatedly utilizing the above steps completes the proof. for Now consider (ii). Suppose that . Then, by construction, . Furthermore, since
AYSAL et al.: DISTRIBUTED AVERAGE CONSENSUS
quantized node state values at time step , with the initial values and , respectively. Then (42)
it follows that (43) for . The convex combination property, similar for to the previous case, indicates that, , and subsequently, . Moreover, since, , it follows that (44) indicating that results, i.e.,
. Finally combining all the (45)
and repeatedly utilizing the above steps completes the proof. The proof of this theorem indicates that each iteration is indeed a convex combination of the previous set of quantized node state values, and uses of the properties of convex functions to arrive at the stated results. as the Lebesgue measure of the domain Let us define of the quantized state vector at time step , i.e., the range of (46) where . Similar to the quantized state vector case, we define as the length of the interval . The following corollary (the proof of which is omitted since it follows directly from Theorem 1 and Theorem 4), compiled from Theorem 1 and Theorem 4, discusses properties of interest and . of and , with initial conCorollary 1: The functions and , tend to zero as tends to ditions infinity, i.e., (47) and (48) Moreover, and are monotonically nonincreasing functions. The presented theorem and corollary indicate that the convergence of the PQDA is monotonic in the sense that the global trend of both the quantized and unquantized node state values is towards the consensus and that the minimum-length intervals containing all values do not expand, and in fact, converge to zero-length monotonically. The following theorem investigates the rate of convergence of the PQDA to a state where there is a first time nonzero probability of converging to the consensus (all values are contained within two quantization bins). and Theorem 5: Let denote the range of the quantized and un-
denotes the spectral radius of its argument. where Proof: See Appendix B. In the Appendix, we compile an upper and lower bound on the largest and smallest order statistics of the quantized node state vector using results from [25], [26]. Then, the task reduces to deriving a bound on the convergence rate of the normed difference of any row and with time, and combining this bound with the bounds on the order statistics gives the desired result. Theorem 5 reveals that the PQDA converges to the final two bins with the same rate as standard consensus. Theorem 5 also relates the convergence of the quantized node state values range to the range of initial node measurements. After all the node state values are in the final two bins, there is always a nonzero probability to immediately converge to consensus. Note that, in the absence of knowledge of the norm of the initial node states or the initial state range, the bound given above reduces to (50) where we used the facts that and . To understand the convergence of the PQDA algorithm after all the quantized states converged to the final two bins, first, let us discuss the behavior of the PQDA algorithm in the final . Suppose bin, i.e., , for some . In this case, all the nodes state values or to achieve a consensus at need to be quantized to time step . Hence, the effect of the weight matrix on the convergence rate significantly decreases and the convergence rate is mainly dominated by the probabilistic quantization. Moreover, we hence believe that the time interval, where all the node state , is a transition period between the values are in and dominating effect of the weight matrix, i.e., the spectral radius of , and the dominating effect of probabilistic quantization. Obtaining analytical expressions of convergence rate for these transition and final bin regions appears to be a challenging task. Although our current research efforts focus on this challenge, we assess the convergence performance of the PQDA algorithm with extensive simulations in the following section. V. NUMERICAL EXAMPLES This section details numerical examples evaluating the performance of the distributed average computation using probabilistic quantization. Throughout the simulations we utilized the [3]. Metropolis weight matrix defined for a graph The Metropolis weights on a graph are defined as follows: , and otherwise.
Fig. 4 shows the behavior of the average MSE per iteration defined as (52)
Fig. 2. The plotted are is the interval in which the quantized state vector is. The , the nodes' initial average is x , and the number of nodes is quantization resolution is set to .
1 = 0:1
(0) = 0:85
This method of choosing weights is adapted from the Metropolis algorithm in the literature of Markov chain Monte Carlo [3], [19]. The Metropolis weights are very simple to compute and are well suited for distributed implementation. In particular, each node only needs to know the degrees of its neighbors to determine the weights on its adjacent edges. Furthermore, the nodes do not need any global knowledge of the communication graph or even the total number of nodes. nodes randomly disWe simulate a network with persed on the unit square [0,1] [0,1], connecting two nodes by an edge if the distance between them is less than the connec. Thus, a link exists between tivity radius, i.e., any two nodes that are at a range less than . Throughout this section, the initial states are drawn i.i.d. from a uniform distri, where is i.i.d. uniformly bution as following: interval. The initial states distributed with support in the . The quantiare then regularized such that zation resolution is taken as . Plotted in Fig. 2 is and at every time step (corresponding to node trajectories given in Fig. 1). The figure indicates that the proposed algorithm does indeed achieve consensus as the interval in which the quantized state vector converges to zero and is monotonically nonexpanding, corroborating the theoretical results. In this case, the , which is in agreement with consensus is the theoretical results indicating that the consensus is at one of the quantization levels. We next investigate the effect of the quantization resolution and the location of the initial state average on the consensus standard deviation. Fig. 3 plots the error norm of the consensus when and for varying for varying when . Also plotted is the derived upper bound on the PQDA. Note that each data point in the plots is an ensemble average of 1000 trials. The variance, as expected, increases and exhibits a harmonic betends to increase as havior as the location of the average changes. This is due to the effect induced by the distance of the average to the quantization levels.
. In other words, is the for average mean squared distance of the states at iteration from the initial mean. Each curve is an ensemble average of 1000 ex, periments and the network parameters are: , and . The plots suggest that smaller quantization bins yield a smaller steady state MSE and that as quantization bin size increases, the number of iterations taken by PQDA to reach the final quantization bin decreases. The quasi-convex shape of the MSE curves are due to the fact that the algorithm, after all the state values converge into a quantifor some , drifts to zation range a quantization value. Considered next is the consensus value of the PQDA algorithm. Fig. 5 plots the histograms of the consensus value for varying initial state average, i.e., for . The number of nodes in the network is . Note that the consensus values shift as the initial average value shifts from 0.80 to 1.00. This is directly related to the fact that the consensus, in expectation, is equal to the average of initial states as provided by the theoretical results. We investigate the average convergence time of the distributed average consensus using probabilistic quantization for against the number of nodes in the varying network, Fig. 6(a) and (b). We also show the average number of iterations taken to achieve the final quantization bin. Moreover, Fig. 6(c) and (d) plots the average normalized distance to the closest absorbing state at the first time step when all the quantized node state values are in the final quantization bin. The and , and the initial state averages are . Each data point is connectivity radius is an ensemble average of 10 000 trials. Note that the convergence time increases with the number of nodes in the network. The plots suggest that the number of iterations taken by the PQDA algorithm to converge to final quantization bin decreases as increases. This can be seen by noting that the algorithm has to go through less "averaging" (multiplication with the weight matrix) before arriving at the final bin. It is, hence, clear that the algorithm needs to run for a smaller number of iterations to arrive at a larger final bin size. On the other hand, as discussed in more detail below, the expected number of iterations taken to achieve consensus is dominated by the number of iterations taken to converge to an absorbing state after all the node values are in the final bin. The probabilistic quantization is the dominant effect in the final bin. The time taken to converge to an absorbing state is heavily dependent on the distance to that absorbing state at the first time step when all values enter the final bin. This distance is affected by two factors. First, if more averaging operations occur prior to the entry step, then there is more uniformity in the values, decreasing the distance. Second, if the initial data average is close to a quantization value, then, on average, the entry point will be
AYSAL et al.: DISTRIBUTED AVERAGE CONSENSUS
Fig. 3. The error norm of the consensus with respect to (a) the quantization resolution, i.e., . The network parameter are: N N =N . and d : with
1 = 0 25
= 50
= log( )
Fig. 4. The average MSE of the probabilistically quantized distributed average consensus for varying quantization resolution where 2 , f : ; : ; : ; : g. The remaining network parameters are: N x : , and d N =N .
0 05 0 1 0 15 0 2 (0) = 0 85 = log( )
1 = 50
closer to an absorbing state (note that ). These observations explain the results of Fig. 6. Note that the and cases convergence time order for flip for and . That is due to the fact that the average distance to an absorbing when, at the first time step, all the when node values enter the final bin is smaller for compared to , and is smaller for when compared to . Moreover, note that yields the smallest distance to an absorbing state for both initial conditions. Although, it takes more iterations to converge to final bin, in both cases, PQDA algorithm with yields the smallest average distance to an absorbing state when all the node values enter to the final bin for the first time step, hence, the smallest average number of iterations to achieve the consensus.
1 2 [0 05 0 5] with x(0) = 0 85 and (b) the initial state average :
Fig. 5. Histograms of the consensus value achieved by the probabilistically quantized consensus for varying initial state average where x 2 f : ; : ; ; : g and : . The number of nodes in the network is N .
0 80 0 825 . . . 1 00 = 50
1 = 02
We consider next the effect of the connectivity radius on the average number of iterations taken to achieve the consensus. Fig. 7 depicts the average number of iterations to achieve the consensus for the cases where the initial state average is and . As expected, the average number of iterations taken to achieve consensus decreases with increasing connectivity radius. This is related to the fact that higher connectivity radius, implies a lower second largest eigenvalue for the weight matrix. Moreover, as in the previous case, the convergence time is related to the distance of the initial state average to a quantization value for a given quantization resolution. Of , and 0.90 note is that 0.85 is a quantization point for and . The is a quantization point for both combined results of the presented experiments indicate that the
Fig. 6. Average number of iterations taken by the probabilistically quantized distribute average computation to achieve final quantization bin (dashed) and consensus (solid) for 2 f : ; : ; : g and for varying N with (a) x : and (b) x : , along with the corresponding average distance to the : and (d) x : . closest absorbing state at the first time step when all the quantized node state value are in the final quantization bin for (c) x
0 05 0 1 0 2
(0) = 0 85
expected number of iterations required to reach a consensus depends on the following factors: 1) quantization resolution; 2) initial node measurements; 3) number of nodes in the network; 4) connectivity radius. Note that (1), (3), and (4) are system design choices, affected by energy budgets and bandwidth constraints, but (2) is datadependent. This implies that the quantization resolution, given the bandwidth and power constraints of the application, should be chosen to minimize the expected (or worst-case) convergence time over the range of possible initial averages. VI. FURTHER CONSIDERATIONS The analysis presented in this paper makes two main simplifying assumptions: 1) the network topology does not change over time, and 2) communication between neighboring nodes
(0) = 0 90
(0) = 0 85
(0) = 0 90
is always successful. The simplifying assumptions essentially , allow us to focus on the case where the weight matrix, does not change with time. However, time-varying topologies and unreliable communications are important practical issues which have been addressed for unquantized consensus algohas the same rithms (see, e.g., [3], [27], and [28]). Since support as the adjacency matrix of the underlying communication graph, when the topology changes with time, the averaging weights must also vary. Likewise, an unsuccessful transmission between two nodes is equivalent to the link between those nodes vanishing for one iteration. In either case, we can as random process. Typical results for this now think of scenario roughly state that average consensus is still accomplished when the weight matrix varies with time, so long as , is connected. This conthe expected weight matrix, dition ensures that there is always nonzero probability that information will diffuse throughout the network. We expect that the same techniques employed in [3], [27], [28] can be used to
AYSAL et al.: DISTRIBUTED AVERAGE CONSENSUS
0 05 0 1 0 2
(0) = 0 85
Fig. 7. Average number of iterations taken by the probabilistically quantized distribute average computation for 2 f : ; : ; : g with (a) x : . The number of nodes in the network is N and (b) x : . Connectivity radius factor is defined as the modulating constant k , in the expression k N =N . for the connectivity radius d
(0) = 0 90
log( )
= 50
show convergence of our average consensus with probabilistic . quantization with time-varying In this paper, we also restricted ourselves to the scenario where the quantization step size remains fixed over all time. Recall that when the algorithm has converged to a consensus, all are at the same quantization point, so . Letting denote Euclidean distance to convergence, we know that when the algorithm is far from large), quantization errors have less of converging (i.e., an effect on convergence of the algorithm. This is because are multiplicative and thus have a the averaging effects of is large, whereas the quantization stronger influence when error is bounded by a constant which only depends on and not on . When is of the same order as the quantization noise variance, quantization essentially wipes away the effects of averaging and hampers the time to convergence. A natural extension of the algorithm proposed in this paper involves over time, e.g., setting shrinking the quantization step size once is established to be below the threshold where quantization effects outweigh averaging. We expect that this modification should improve the rate at which tends to zero without affecting statistical properties of the limiting consensus values (i.e., unbiased with respect to , and no increase in the limiting variance). Solidifying this procedure is a topic of current investigation. VII. CONCLUDING REMARKS We have described PQDA, a framework for distributed computation of averages of the node data over networks with bandwidth/power constraints or large volumes of data. The proposed method unites the distributed average consensus algorithm and probabilistic quantization, which is a form of "dithered quantization." The proposed PQDA algorithm achieves a consensus, and the consensus is a discrete random variable whose support
is the quantization values and expectation is equal to the average of the initial states. We have derived an upper bound on the MSE performance of the PQDA algorithm. Our analysis demonstrates that the minimum-length intervals (with boundaries constrained to quantization points) containing the quantized and unquantized state values are nonexpanding. Moreover, the lengths of these intervals are nonincreasing functions with limit zero, indicating that convergence is monotonic. In addition, we have shown that, all the node state values, in the worst case, arrive in the final two quantization bins at the same rate as standard, unquantized consensus algorithms. Finally, we have provided numerical examples illustrating the effectiveness of the proposed algorithm and highlighting the factors that impact the convergence rate. APPENDIX A PROOF OF THEOREM 3—ERROR NORM OF PQDA CONSENSUS Consider the following set of equalities:
(53) (54)
where we use the facts that , for , for . Now the eigendecomposition of and yields
where values
denotes the eigenvector associated with the eigen. Eigendecomposition further indicates that
Proof: It follows from Lemma 1 that
and (65)
Since and , the eigenvector associated is given by . Subwith the eigenvalue stituting this information into the error norm equation gives
(67) (68) Now using Chebyshev's Inequality, we obtain
(69) (70)
, we see that the RHS of the Now, taking the limit as . Thus, the probability of above goes to zero for all being greater than zero is equal to zero for , and this im, since convergence in probplies that ability implies convergence in expectation for bounded random variables. The error norm equation, after taking the expectation and , since the limit of each exists and limit as equals zero (from Lemma 3), reduces to
Moreover, applying the Triangle inequality and using the facts and that that (61)
after multiplying both sides with
(71) Furthermore, utilizing the Norm inequality gives
(72) (62) We need to following lemma to continue with the proof. Lemma 3: The average quantization error, at a fixed time step , converges in probability as the network size tends to infinity, i.e.,
In the following, we derive an upper bound of for to bound . Consider the expectation of the quantization noise (73)
(63) for
Note that is a concave function. The concavity indicates that utilizing Jensen's inequality gives
and all . Thus, (64)
(74) for all .
AYSAL et al.: DISTRIBUTED AVERAGE CONSENSUS
Now using the upper bound for the expectation of the quantization noise variance term, i.e., Lemma 1, indicates that the expectation of the quantization noise norm is bounded by
values at time step . To prove the proposed theorem, we make use of the following bounds for the maximum and minimum samples [25], order statistics of (possibly dependent) [26]:
(75) Now, substituting this result into the error norm equation, after some manipulations, gives
Recall that equality, i.e.,
(82) and
(83) respectively. Using these bounds, in our setup, for the largest order statistics gives
, hence, applying the Geometric Series
(84) (85)
to be -th row of the weight matrix taken to where we define for the power and and used the properties of probabilistic quantization and the fact almost surely. Similarly, we have shown that that
further yields (86)
where yielding
Now, taking the limit as tends to infinity yields
Utilizing the Cauchy–Schwartz inequality reduces the above expression to (88) (79) Note that the limit of each term exists. Also consider the following:
, we need to upper bound Clearly, to upper bound . Hence, we derive an upper bound for for any pair such that , in the following: (89)
(80) since
, and, subsequently (81)
Combining these findings and substituting them into (79) yields the desired result. APPENDIX B PROOF OF THEOREM 5—CONVERGENCE RATE TO THE FINAL TWO BINS Note that . In order to bound the expected range, we will upper and lower bound the largest and smallest order statistics of the quantized node state
(91) (92) (93) (94) where (a) follows from the Triangle inequality, (b) from the fact that the norm of any row of a matrix is smaller that the norm of the matrix, (c) using the properties of the weight matrix, (d) by the Norm inequality, and (e) due to the symmetric assumption on the weight matrix. Finally, substituting (94) into (88) yields (95)
Moreover, using Thomson's sharp bound relating order statistics and sample standard deviation (for even, but a very similar result exists for odd ) [29]
(96) one can relate the bound given on the quantized node state values range to the initial states' range, i.e., the result stated in the theorem. REFERENCES [1] T. C. Aysal, M. J. Coates, and M. G. Rabbat, "Distributed average consensus using probabilistic quantization," presented at the EEE Statist. Signal Process. Workshop, Madison, WI, Aug. 2007. [2] T. C. Aysal, M. J. Coates, and M. G. Rabbat, "Rates of convergence of distributed average consensus with probabilistic quantization," presented at the Allerton Conf. Commun., Control, Comput., Monticello, IL, Sep. 2007. [3] L. Xiao, S. Boyd, and S. Lall, "A scheme for robust distributed sensor fusion based on average consensus," presented at the IEEE/ACM Int. Symp. Inf. Process. Sens. Netw., Los Angeles, CA, Apr. 2005. [4] J.-J. Xiao and Z.-Q. Luo, "Decentralized estimation in an inhomogeneous sensing environment," IEEE Trans. Inf. Theory, vol. 51, no. 10, pp. 3564–3575, Oct. 2005. [5] H. Papadopoulos, G. Wornell, and A. Oppenheim, "Sequential signal encoding from noisy measurements using quantizers with dynamic bias control," IEEE Trans. Inf. Theory, vol. 47, no. 3, pp. 978–1002, Mar. 2001. [6] N. Lynch, Distributed Algorithms. San Francisco, CA: Morgan Kaufmann, 1996. [7] W. Ren and R. Beard, "Consensus seeking in multiagent systems under dynamically changing interaction topologies," IEEE Trans. Autom. Control, vol. 50, no. 5, pp. 655–661, 2005. [8] D. S. Scherber and H. C. Papadopoulos, "Locally constructed algorithms for distributed computations in ad-hoc networks," presented at the 3rd Int. Symp. Inf. Process. Sens. Netw., Berkeley, CA, Apr. 2004. [9] C. C. Moallemi and B. V. Roy, "Consensus propagation," IEEE Trans. Inf. Theory, vol. 52, no. 11, pp. 4753–4766, Nov. 2006. [10] D. P. Spanos, R. Olfati-Saber, and R. M. Murray, "Distributed sensor fusion using dynamic consensus," presented at the 16th IFAC World Congr., Prague, Czech Republic, Jul. 2005. [11] C.-Z. Xu and F. Lau, Load Balancing in Parallel Computers: Theory and Practice. Dordrecht, Germany: Kluwer, 1997. [12] Y. Rabani, A. Sinclair, and R. Wanka, "Local divergence of Markov chains and the analysis of iterative load-balancing schemes," presented at the IEEE Symp. Found. Comp. Sci., Palo Alto, CA, Nov. 1998. [13] M. Rabbat, R. Nowak, and J. Bucklew, "Robust decentralized source localization via averaging," presented at the IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Philadelphia, PA, Mar. 2005. [14] M. Rabbat, J. Haupt, A. Singh, and R. Nowak, "Decentralized compression and predistribution via randomized gossiping," presented at the Inf. Process. Sens. Netw., Nashville, TN, Apr. 2006. [15] C. Intanagonwiwat, R. Govindan, and D. Estrin, "Directed diffusion: A scalable and robust communication paradigm for sensor networks," presented at the ACM/IEEE Int. Conf. Mobile Comput. Netw., Boston, MA, Aug. 2000. [16] J. Zhao, R. Govindan, and D. Estrin, "Computing aggregates for monitoring wireless sensor networks," presented at the Int. Workshop on Sens. Netw. Protocols and Appl., Anchorage, AL, May 2003. [17] S. R. Madden, R. Szewczyk, M. J. Franklin, and D. Culler, "Supporting aggregate queries over ad-hoc wireless sensor networks," presented at the Workshop on Mobile Comput. Syst. Appl., Callicoon, NY, Jun. 2002. [18] A. Montresor, M. Jelasity, and O. Babaoglu, "Robust aggregation protocols for large-scale overlay networks," presented at the Int. Conf. Depend. Syst. Netw. , Florence, Italy, Jun. 2004.
[19] L. Xiao, S. Boyd, and S.-J. Kim, "Distributed average consensus with least-mean-square deviation," J. Parallel Distrib. Comput., vol. 67, no. 1, pp. 33–46, Jan. 2007. [20] M. E. Yildiz and A. Scaglione, "Differential nested lattice encoding for consensus problems," presented at the Inf. Process. Sens. Netw., Cambridge, MA, Apr. 2007. [21] A. Kashyap, T. Basar, and R. Srikant, "Quantized consensus," Automatica, vol. 43, pp. 1192–1203, Jul. 2007. [22] R. A. Wannamaker, S. P. Lipshitz, J. Vanderkooy, and J. N. Wright, "A theory of nonsubtractive dither," IEEE Trans. Signal Process., vol. 8, no. 2, pp. 499–516, Feb. 2000. [23] L. Schuchman, "A theory of nonsubtractive dither," IEEE Trans. Commun. Technol., vol. COMM-12, pp. 162–165, Dec. 1964. [24] O. Kallenberg, Foundations of Modern Probability, 2nd ed. New York: Springer-Verlag, 2002. [25] L. P. Devroye, "Inequalities for the completion times of PERT networks," Math. Operat. Res., vol. 4, no. 4, pp. 441–447, Nov. 1979. [26] T. Aven, "Upper (lower) bounds on the mean of the maximum (minimum) of a number of random variables," J. Appl. Probab., vol. 22, no. 3, pp. 723–728, Sep. 1985. [27] R. Olfati-Saber and R. M. Murray, "Consensus problems in networks of agents with switching topology and time-delays," IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 1520–1533, Sep. 2004. [28] M. G. Rabbat, R. D. Nowak, and J. A. Bucklew, "Generalized consensus algorithms in networked systems with erasure links," presented at the IEEE Workshop on Signal Process. Adv. Wireless Commun., New York, Jun. 2005. [29] G. W. Thomson, "Bounds for the ratio of range to standard deviation," Biometrika, vol. 42, no. 1/2, pp. 268–269, Jun. 1955. Tuncer Can Aysal (S'05–M'08) received the B.E. degree (high honors) from Istanbul Technical University, Istanbul, Turkey, in 2003, and the Ph.D. degree from the University of Delaware, Newark, in 2007, both in electrical and computer engineering. He is currently a Postdoctoral Research Fellow with the Electrical and Computer Engineering Department, Cornell University, Ithaca, NY. His research interests include distributed/decentralized signal processing, sensor networks, consensus algorithms, and robust, nonlinear, statistical signal, and image processing.
Mark J. Coates (M'99) received the B.E. degree (first class honors) in computer systems engineering from the University of Adelaide, Australia, in 1995, and the Ph.D. degree in information engineering from the University of Cambridge, Cambridge, U.K., in 1999. Currently, he is an Assistant Professor with McGill University, Montreal, Canada. He was awarded the Texas Instruments Postdoctoral Fellowship in 1999 and was a research associate and lecturer at Rice University, Houston, TX, from 1999 to 2001. His research interests include communication and sensor/actuator networks, statistical signal processing, causal analysis, and Bayesian and Monte Carlo inference.
Michael G. Rabbat (S'02–M'06) received the B.S. degree from the University of Illinois at Urbana-Champaign in 2001, the M.S. degree from Rice University, Houston, TX, in 2003, and the Ph.D. degree from the University of Wisconsin-Madison, in 2006, all in electrical engineering. Since January, 2007, he has been an Assistant Professor with McGill University. He was a visiting researcher with Applied Signal Technology Inc., Sunnyvale, CA, during the summer of 2003. His current research is focused on distributed information processing in sensor networks, network monitoring, and network inference.
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Shaded Panther: Wakanda Forever supplied comics icon Namor to the Shock Cinematic Universe
Actor Tenoch Huerta said he'd fancy to come lend a hand, however rights points indicate it's no longer that easy
Universal has reportedly blocked Disney from being ready to make a standalone Namor movie
It's such as the points that beget plagued the likes of Hulk and Spider-Man
The Shock Cinematic Universe certain loves a correct row over rights, and while it's wrangled the Amazing Four and the X-Males lend a hand from the clutches of 20th Century Fox, its most as much as the moment addition is caught in a sophisticated web of legalities.
At the centre of this court chaos is Tenoch Huerta's Namor — a character who made his comics debut in 1939 — who made his movie debut in Ryan Coogler's Shaded Panther: Wakanda Forever. As we dived into the mythology of Ample'uk'ulkan and the people of Talokan, Huerta made a splash as Namor (dropping the Sub-Mariner) bit of his comic ebook counterpart's name.
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As if truth be told likely the most MCU's oldest teases, Namor changed into once hinted at in Iron Man 2, earlier than Avengers: Endgame gave a nod to his impending arrival through the level out of underwater earthquakes.
Praised as if truth be told likely the most franchise's finest villains to this level — and with Shuri (Letitia Wright) sparing Namor's life — there are glaring questions about when he'll return.
Why can't Disney make a Namor standalone?
In step with Wakanda Forever producer Nate Moore, the rights to Namor as a character belong to Universal as a change of Disney. Talking to The Wrap, Moore confirmed 'he can return' however there are some caveats. Even in Wakanda Forever, Disney wasn't allowed to use Namor in advertising materials except it changed into once for a series of character posters.
Though this implies Disney presumably brokered a deal to use Namor in Wakanda Forever, the studio can't forge ahead with a full-blown Namor coast-off movie. This likely takes a Disney sequence feature in Talokan off the desk, that methodology Namor and his nautical friends are reduced to cameos in other initiatives.
Universal studios bought the rights to "The Sub-Mariner" from a disturbed Shock Studios round 2001 — planning a movie after Ang Lee's Hulk (2003). Harry Potter's Chris Columbus changed into once linked to declare this legend of a 'disturbed revolt with a instant temper', which changed into once feature to be a 'main franchise' for Universal. Though the mission fell into vogue hell, it changed into once quiet being talked about in 2009.
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The rights beget remained up in the air since, with MCU overlord Kevin Feige telling IGN as currently as 2018: "It's no longer as a sexy or constructive as the bulk of the other characters."
Doctor Irregular in the Multiverse of Madness creator Michael Waldron outlined to Diversity how Namor changed into once belief to be for its Illuminati lineup correct fancy the comics, nonetheless, it changed into once dropped due to the 'other plans'.
History is repeating itself
This isn't the main time the MCU has been left over a barrel with character conundrums. Louis Leterrier's The Amazing Hulk is the MCU's gloomy sheep, however irrespective of being fragment of the franchise's canon and Trace Ruffalo changing Edward Norton from The Avengers onward, it's easy to neglect Universal Studios technically owns the Emerald Extensive.
Feige has shared his pleasure at being ready to use Galactus and the Silver Surfer after the Dwelling of Mouse bought Fox, however there are quiet about a outliers who don't somewhat tumble beneath Mickey's umbrella. Following Artisan Leisure's Man-Thing in 2005 and its closure, it changed into once belief Lionsgate owned the rights to the character. In other areas, James Gunn tweeted about how obscure correct wranglings methodology he can't use Rom the Build of dwelling Knight and the Micronauts' Bug in the Guardians of the Galaxy movies.
The mountainous one is Sony's Spider-Man standalones. It looks fancy simplest the day earlier than at the present time the two aspects went to battle and it appeared fancy the wall-crawling hero changed into once out of the MCU.The Sony Photos Universe of Shock Characters is much more complex, as the Venom and Morbius movies beget a disturbed relationship with the mainline MCU.
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Sony's Amy Pascal has constantly maintained the SPUMC is fragment of the MCU, which ended in a now-terrifying watch from an ungainly Feige. There beget been whispers of a Spider-Man and Venom crossover, however even that looks fancy a logistical nightmare because of Disney successfully borrows Peter Parker.
What's subsequent for Hulk and Namor?
The distinction is, Sony owns the movie rights to Spider-Man whereas Universal simplest has Hulk and Namor's distribution rights. This puts these two juggernauts in limbo… for now. A Reddit submit from earlier in 2022 confirmed off an SEC submitting that suggests Universal's contract will roam out in June 2023. Even then, it's no longer assured it'll be full steam ahead with Hulk and Namor.
Whatever happens subsequent, all people is conscious of Huerta is ready to come lend a hand. Talking to Total Movie (through The Disclose), the star said, "The mythology round Namor is huge. You furthermore mght will be loopy with all this cultural component, and you would possibly maybe well well presumably also make a ramification of issues with Namor, because of they pick an fabulous source of reports and mythology and faith and the entire lot."
A long way from Namor, we would possibly maybe well ultimately salvage every other Hulk movie due to the the She-Hulk finale neatly constructing a World Battle Hulk time out. Noteworthy fancy that can use a couple of characters fancy Bruce Banner, Jennifer Walters, and Skaar, an ensemble would possibly maybe well very well be one methodology to salvage round the mess of Namor's rights.
We by no methodology belief we would watch Hugh Jackman's Wolverine or Tobey Maguire's Spider-Man in the MCU, so glimpse how a long way we've come in this kind of instant home of time when all people decides to half.
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The seas of Namor's rights seem a bit of simpler to lag after Wakanda Forever, however that doesn't salvage us nearer to giving Huerta the standalone his followers are cheering for. For now, we're pinning our hopes on the inevitable Shaded Panther 3 or assorted Wakanda Forever coast-offs which would be reportedly in the works.
If no longer, who wouldn't need to observe Huerta star alongside John Krasinski and Patrick Stewart for an Illuminati movie?
Shaded Panther: Wakanda Forever is in cinemas now.


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Maximizing High Net Worth to Fund Retirement Plans for Business Owners So why would you want to set up an Insured Retirement Plan as a business owner? To dividend out your retained earning subjects you pay taxes at your personal marginal tax rate. Depending upon how much you pay yourself (Salary, Bonuses, Dividends) in any given year can mean that … Read More
4 Key Things to Consider When Building Business Owner Wealth
The Keys To Building Business Owner Wealth You've built a thriving and successful business. You continue to be profitable and continue to reinvest into your business while building wealth for your eventual retirement. You have a succession plan in place and a plan to exit the business at some point over the next 15 years. The plan is to work … Read More


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Bark for A Cure Scarf in Pink
I See Spot
SKU: iss-36-01
Bark for A Cure Dog Scarf in PinkA scarf with a cause!
Profits from the sale of this super comfortable cotton/lycra scarf help fund breast cancer research. Thank you for your support!!!
Small fits neck sizes 8 to 14 inches, Large fits neck sizes 15 to 20 inches.
The Bark for A Cure Scarf in Pink destined to impress, and priced at only $14.99, for a limited time.
Inventory Last Updated: Feb 02, 2023
Inventory Last Updated: Feb 02, 2023


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Founded Year2005
StageAcquired | Acquired
About ALPS Funds Services
ALPS is a financial services firm focused on asset servicing and asset gathering. With more than 300 employees, nearly 200 clients, and an executive team that has been in place for over 16 years, ALPS continues to actively promote all of its various business segments, from asset servicing through ALPS Fund Services, to asset gathering through ALPS Distributors, and ALPS Advisors. In November 2011, ALPS was acquired by DST Systems. The valuation of ALPS was $250 million.
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ALPS Funds Services Patents
ALPS Funds Services has filed 3 patents.
Application Date
Grant Date
Related Topics
Valves, Milk, Brand name snack foods, Plumbing valves, Food preservation
Application Date
Grant Date
Related Topics
Valves, Milk, Brand name snack foods, Plumbing valves, Food preservation
ALPS Funds Services Frequently Asked Questions (FAQ)
When was ALPS Funds Services founded?
ALPS Funds Services was founded in 2005.
Where is ALPS Funds Services's headquarters?
ALPS Funds Services's headquarters is located at 1290 Broadway, Denver.
What is ALPS Funds Services's latest funding round?
ALPS Funds Services's latest funding round is Acquired.
Who are the investors of ALPS Funds Services?
Investors of ALPS Funds Services include DST Systems and Lovell Minnick Partners.
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An Open Door to Australia - 482 Visa
How can you finally reach the dream of moving to Australia? The 482 Visa might be precisely what you need. What is it? How can you get it? Learn about it here.
Australia is one of the most popular ex-pat destinations. It is also welcoming to skilled workers from a variety of sectors. It has great beaches, a stable economy and a good work/life balance. What's the catch I hear you say? Well, it's not always easy to secure a 482 Visa to be able to work in Australia.
What is a 482 work visa?
The 482 Temporary Skill Shortage Visa lets an employer sponsor a suitably skilled worker to fill a position they can't find a suitably skilled Australian to fill. There are three streams of the 482 visa. Short-term, medium-term and labour-agreement stream.
Short-term stream
You can stay for up to 2 years. This can be extended to 4 years if an international Trade Obligation applies (ITO). Hong Kong passport holders may stay for up to 5 years.
The 482 visa processing times according to the Australian Department of Home Affairs are anywhere between 44 days and 11 months. It costs from AUD1,330.
Medium-term stream
You can stay for up to 4 years unless you are a Hong Kong passport holder, then it's 5 years.
Processing times are between 30 days and 6 months and the cost is AUD2,770.
Labour-agreement stream
You can stay up to 4 years depending on the terms in the agreement, except for Hong Kong passport holders who can stay up to 5 years.
Processing times are between 36 days and 6 months and the cost is AUD2,770.
There is also a Subsequent Entrant Visa. This is for family members of subclass 457 and TSS visa holders who are applying separately for their TSS visa and want to join the primary subclass 457 or TSS visa holder in Australia.
You can stay as long as your visa is valid and it costs AUD1,330.
What is it like to work and live in Australia?
According to the current (Oct 2022) Mercer's 2022 Quality of Life Index, Canberra is in first place with Adelaide in third. Perth and Brisbane are both in the top 10 with Sydney coming in 21st place. Sydney has risen five places since the start of the year. The UN states Australia is the second-best country in the world to live.
Australia offers some of the highest graduate salaries in the world and only has 5% of unemployment. The services sector dominates the Australian economy, followed by Industry and agriculture. Australia is a significant exporter of energy, food and natural resources.
Working hours are 38 per week on average and workers are entitled to four weeks paid leave every year. Australia also offers one of the best average salaries in the world.
Other advantages are the mandatory employee benefits such as holiday pay, long service leave, maternity/paternity leave and sick pay.
Australia boasts a great climate in general with around 3000 hours of sunshine a year. Australians definitely get the work/life balance and love to spend time on the beach having a 'Barbie' or surfing. You'll find many workers go straight to the beach after work or chill out around their pools with friends and family.
Australia has a friendly and welcoming culture due to its long history of immigration. The big cities are diverse and multicultural. There are so many more advantages to Australian life that I would have to write another blog just to describe them.
482 visa eligibility requirements
So, have I sold the prospect of moving to Australia to you? Is it something you've been contemplating for a while? Before you bite the bullet, you need to know if you're eligible for the 482 visa. Here are the general requirements to apply for a Temporary Skill Shortage visa.
- Have the relevant skills and qualifications required for the position as well as having an occupation on the Consolidated Skilled Occupations list.
- Have a vocational level of English
- Have an offer of employment by an Australian employer
- Have at least 2 years of work experience in your occupationMeet the DHA health and character requirements
- Meet the DHA health and character requirements
- Meet relevant licensing, registration or professional membership requirements
- Have made arrangements for adequate health insurance in Australia to cover you for your stay
If you are an employer who wants to sponsor someone to work in Australia you must meet the following criteria:
- The job position for the foreign national worker must be full-time and genuine
- You must be an active and lawfully operating business
- Meet the required Australian employee training benchmarksMeet employment and salary condition-related requirements
- Meet employment and salary condition-related requirements
Applying for a 482 visa
The visa process in Australia is a complex and difficult one. Many wisely choose to hire the services of a registered migration agent. A migration agent can help you as a company with the process of sponsoring a foreign national to fill a position in your company. They can help with the relevant paperwork and work out the best sponsorship package for you and your potential employee.
Your potential employer must apply to become a standard sponsor before you apply for the 482 visa. They must apply online, submit the relevant documents and pay a sponsorship fee of AUD420. This doesn't apply if you are applying for the labour agreement stream. However, your employer must be part of the labour agreements.
Your sponsor is required to pay the skills levy fund once they begin the nomination process. The amount varies depending on the size of the business and the work period length.
Small businesses with less than AUD10million turnover have to pay AUD1200 yearly. Businesses with a turnover of over AUD10million have to pay AUD1800.
Once your sponsor's nomination has been approved, you can start the 482 visa application process. Migration agents can help with this process as it can be complicated and any mistakes will mean the process may fail or take longer.
Can a 482 visa lead to permanent residency?
There is a possibility of achieving permanent residency from a TSS visa if:
- You are under 45 years of age at the time of application
- Your application is on the medium-term listYou've worked for the same employer for 3 years on your TSS visa
- You've worked for the same employer for 3 years on your TSS visa
- Have an adequate level of English (IELTS 6 in each band or equivalent)
There are other pathways to permanent residency from a TSS 482 visa. If you would like to discuss them, contact a migration agent in Sydney who will be able to help you further.


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Signature Healthcare of Galion is one of just three nursing homes in Galion, Ohio. Featuring an overall grade of B-, this facility actually is the best nursing home in Galion. Based on our assessment, this nursing home ought to meet the needs of many people. This facility was better in some categories than others, but it didn't have any bad scores in any of our major categories discussed below. More information about these categories can be found below.
Signature Healthcare of Galion
- Nursing Home
Updated Sep 1, 2022 by Nick Reese
935 Rosewood Dr,
Galion OH 44833
89.35% estimated occupancy 1
Quick Details
- Accepts Medicare 1
- Accepts Medicaid 1
- No CCRC
- Has Resident Council
- No Family Council
- For profit - Corporation
- Offers Hospice
- Offers Post-Acute Care
- Offers Respite Care
- Offers Inpatient Rehab
- Offers Outpatient Rehab
Specialized Services
- Medication Management
- Transportation
- Speech Therapy
- Stroke Recovery
- IV Antibiotic Therapy
- Palliative Care
- Dentistry
- Wound Care
- Occupational Therapy
- Cardiac Therapy
- Tracheotomy Care
- Podiatry
- Mental Health
- Diabetes Management
- Physical Therapy
- Respiratory Therapy
- Nutritional Counseling
- Pharmacy
- Recreational Therapy
- Pain Management
- Optometry
- Oncology Care
- Orthopedic Rehabilitation
- Total Parenteral Nutrition
Registered Staff Hours
Facility Inspections
The primary reason this facility received a decent overall score is its impeccable inspections in recent years. We gave them one of our higher scores in that area, with an A. This is a much better score than the facility's overall score, which was decent but certainly not elite. Our inspection grades are based on items located in the a nursing home's recent inspection reports. Places that score well in this area tend to have few deficiencies on their reports. Most importantly, these facilities generally do not have any severe deficiencies which are associated with risks to patient safety. This place was assessed 5 deficiencies on its inspection report, but fortunately none were considered to be severe deficiencies. This means that CMS did not deem any of the deficiencies to create an immediate risk to patient health or safety. We should point out that deficiency-free inspection reports are rare in the industry.
Nurse Quality
Building on this facility's list of strengths, it also earned an excellent nursing rating. We awarded them an impressive grade of B- in this category. Nursing grades are primarily based on the facility's level of nurse staffing. This facility provided just 0.4 hours of nursing care per resident on a daily basis. This is a relatively low total compared to most nursing homes. Lastly, we also looked at some nursing quality measures. While this nursing home got docked a bit for its nursing hours per resident, it excelled in some of these quality-based measures. With fewer than five percent of its residents suffering from pressure ulcers, it performed very well in this statistic. This can be an indicator of quality nursing care. Pressure ulcers can frequently be avoided by offering better nursing care, such as by employing a policy of moving patients at least once a day.
Does this page answer your questions about Signature Healthcare of Galion?
Long-term Care Quality
This facilities third most favorable area was long-term care. We gave it a B- in this area. This is essentially a slightly below average grade in this category. For patients in need of a permanent residence rather than rehabilitation, long-term care grades are a key measure. On top of considering the quantity of care provided by nurses aids and other staff, we looked at the percentage of residents vaccinated against pneumonia. Fortunately, this nursing home gave the vaccine to 98.92473 percent of its patients, which is a very solid figure. This place was able to keep its residents out of the hospital. In fact, it had only 0.02 hospitalizations per 1,000 long-term resident days, which is a remarkably low figure.
Short-term Care Quality
Switching gears to our last category, this facility received a C for short-term care, which we consider to be a decent score. Short-term care scores are based on a facility's quantity of highly-skilled skilled healthcare professionals. This means a vast scope of services, spanning from registered nurses to physical and speech therapists, in addition to other forms of therapy. Fortunately, it looks like this facility has registered nurses on staff. Not every facility employs these skilled professionals. On the other hand, according to the information this nursing home provided, they do not appear to employ physical therapists. Lastly, we considered the percentage of residents who were able to return to the community from this facility. We found that 4.4 percent of this facility's residents returned home. At most facilities, closer to half of their residents are able to return home so this figure is below average.
Rating Over Time
Compared to national and state averages across all facilities.
Signature Healthcare of Galion Quality Metrics
Minimizes Pressure Ulcers
In Signature Healthcare of Galion, 7.61% of Patients had Pressure Ulcers
This statistic measures the percentage of long-term patients who have pressure ulcers. We have found that pressure ulcers are a great indicator of a facility's nursing care quality.
Minimizes Serious Falls
In Signature Healthcare of Galion, 3.74% of Patients had Serious Falls
This is the percent of patients that suffered from a major fall. Falls resulting in major injury are routinely associated with lower levels of patient supervision. Better nursing care can reduce the percentage of residents in a nursing home who sustain major falls.
Minimizes Urinary Tract Infections
In Signature Healthcare of Galion, 1.71% of Patients had UTIs
This tells you the percentage of patients who sustained a urinary tract infection. UTI's are considered to be an indicator of nursing care at a nursing home. UTI's are routinely linked to lower quality nursing care. However, this statistic could also be skewed for certain facilities due to inconsistent reporting standards for these infections.
Appropriately Uses Anti-Psychotic Medication
In Signature Healthcare of Galion, 13.23% of Patients use Anti-Psychotic Medication
This figure gauges the percentage of long-term residents taking antipsychotic drugs. While antipsychotic drugs play an important role in caring for many residents, it is important to make sure these drugs are being used appropriately. In limited cases, high levels of antipsychotic drug use may mean a facility is using these drugs to subdue residents.
Appropriately Uses Anti-Anxiety Medication
In Signature Healthcare of Galion, 22.46% of Patients use Anti-Anxiety Medication
This gauges the percentage of long-term residents taking antianxiety medications.
Managing Depression Among Residents
In Signature Healthcare of Galion, 20.26% of Patients
This indicates the percent of patients who are showing depressive symptoms. Increased levels of depression could imply lower quality care.
Appropriate Vaccine Usage
In Signature Healthcare of Galion, 90.12% of Patients
This metric is a measure of the percentage of long-term care residents who received the pneumonia and flu vaccines. Respiratory viruses can be deadly for seniors, making these types of vaccines critical to patient well-being.
Residents Maintain Autonomy
In Signature Healthcare of Galion, 14.88% Percentage of Patients
This metric measures the percent of long-term care patients who required more assistance with activities of daily living over time. ADL's include activities such as dressing and eating.
Ability to Keep Residents Mobile
In Signature Healthcare of Galion, 16.29% Percentage of Residents
This datapoint is a measure of the percentage of long-term care residents who were able to retain mobility. Some experts would argue that mobility is important for patients physical and mental well-being.
In Signature Healthcare of Galion, 1.36 Hospitalizations per 1,000 resident days
This metric measures the number of hospitalizations per 1,000 days of patient care. Avoiding the hospital is important to preserving the physical well-being of nursing home patients.
Short-term Care: Rehospitalizations
In Signature Healthcare of Galion, 16.29 Percentage of Residents Rehospitalized
This tells you the number of rehospitalizations per 1,000 days of short-term patient care.
Short-term Care: ER Visits
In Signature Healthcare of Galion, 12 Percentage of Patients
This metric measures the number of emergency room visits per thousand days of short-term patient care.
Short-term Care: Facilitates Functional Improvement
In Signature Healthcare of Galion, 75.25% Percentage of Resident
This tells you the percentage of short-term stay residents that experienced functional improvements, such as with activities of daily living.
Nearby Hospitals
- Bucyrus Community HospitalCritical Access Hospitals 10.52 miles away629 North Sandusky Avenue Bucyrus Ohio 44820Voluntary non-profit - Other(419) 562-4677
- Avita OntarioAcute Care Hospitals 11.72 miles away715 Richland Mall Mansfield Ohio 44906Voluntary non-profit - Private(567) 307-7666
- Morrow County HospitalCritical Access Hospitals 12.6 miles away651 West Marion Road Mount Gilead Ohio 43338Government - Local(419) 946-5015
- Ohiohealth Shelby HospitalCritical Access Hospitals 12.82 miles away199 West Main Street Shelby Ohio 44875Voluntary non-profit - Private(419) 342-5015
- Ohiohealth Mansfield HospitalAcute Care Hospitals 14.72 miles away335 Glessner Avenue Mansfield Ohio 44903Voluntary non-profit - Private(419) 526-8000
- Marion General HospitalAcute Care Hospitals 19.95 miles away1000 Mckinley Park Drive Marion Ohio 43302Voluntary non-profit - Private(740) 383-8400
Nearby Dialysis
- Davita - Galion Dialysis0.49 miles away856 Harding Way West Galion Ohio 44833Dialysis Stations: 17 (419) 462-0897
- Fmc - Crawford County Kidney Center10.72 miles away701 Tiffin Street Bucyrus Ohio 44820Dialysis Stations: 12 (419) 562-3000
- Davita - Mid Ohio Dialysis11.9 miles away2148 W 4th St Mansfield Ohio 44906Dialysis Stations: 13 (419) 747-4039
- Fmc - Richland County13.54 miles away680 Bally Row Mansfield Ohio 44906CMS Rating: 4 stars Dialysis Stations: 24 (419) 774-0180
- Fmc - Mansfield Kidney Center13.57 miles away647 Bally Row Mansfield Ohio 44906CMS Rating: 4 stars Dialysis Stations: 19 (419) 775-1730
- Davita - Heart of Marion Dialysis19.86 miles away1221 Delaware Avenue Marion Ohio 43302Dialysis Stations: 13 (740) 375-5058
- Fmc - Heart of Ohio Dialysis Ctr19.94 miles away1730 Marion-Waldo Rd Marion Ohio 43302CMS Rating: 3 stars Dialysis Stations: 24 (740) 389-4111


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Workers 50+ may make contributions to their qualified retirement plans above the limits imposed on younger workers.
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text: Overview Reviews Product Description Product Videos Custom Field Brand Atlantic Traders Size X-Large Decade 1990s Color Bronze Chest Width 26" Length 26" Vintage Condition Notes - Item is generally in good condition. Like all vintage clothing, it shows some signs of wear, but there are no outstanding flaws. final_sale true Product Reviews Write a Review Write a Review × Vintage Atlantic Traders 80s Sweater Rating * Select Rating 1 star (worst) 2 stars 3 stars (average) 4 stars 5 stars (best) Name Review Subject * Comments *


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2003 Honda Cbr 600rr Sportbike on 2040-motos
lancaster, Pennsylvania, US
Honda CBR tech info
Honda CBR description
2003 Honda Cbr 600RR, 2003 cbr600rr 2006 motor and front suspension Yosh exhaust New chain and sprockets New battery Aftermarket plastics Frame sliders $5,000.00 7172264301
Honda CBR for Sale
- 2005 honda cbr1000rr sportbike (US $5,999.00)
- 2004 honda cbr 600rr sportbike (US $4,999.00)
- 2013 honda cbr250r sportbike (US $0.00)
- 2009 honda cbr 600rr sportbike (US $8,999.00)
- 2005 honda cbr 1000rr sportbike (US $8,499.00)
- 2004 honda cbr 600rr sportbike (US $5,899.00)
Moto blog
Casey Stoner To Be Named MotoGP LegendFri, 11 Oct 2013
There's no question Casey Stoner was (and still is) one of the best riders ever to throw a leg over a motorcycle. His accomplishments on two wheels, especially his championships on both Ducati and Honda machinery, are proof. Next week at the MotoGP round at Phillip Island in Australia, Stoner will join a distinct group of riders as he will be officially inducted into the MotoGP Hall of Fame.
Helmet Law Upheld in OntarioTue, 16 Jun 2009
A Sikh in Ontario, Canada, has lost his challenge to the $110 ticket he received back in September 2005 for not wearing a helmet while riding his Honda Shadow. Baljinder Badesha, 39, claims the Ontario helmet law is "100 per cent" discrimination no matter what the safety issues are when riding without a helmet. We cannot put anything over the turban, it is against our religion. Mr. Badesha said outside court.
Electric Moto CorporationTue, 15 Dec 2009
After launching their website last week, the California based Electric Moto Corporation has now announced the IPG Group as their agency of record to help increase brand awareness and bring in some investor interest. According to their website, other then the EMAX 72 DS electric motorcycle, there is a scooter, ATV, bicycle and a personal watercraft on the drawing board. Some available specs after the jump. Weight: 210 lb Voltage: 72 V Capacity: 14 kW Top Speed: 60 mph HP (approx): 21 Torque: 40 Nm Range: 3 hrs Construction: Aluminum Wheels (front/back): 21/19 Hubs: Talon Brakes: Disc/F&R Type: Brembo Suspension: Ohlins Price (est): $9,995 Mitch Leonard is the newly appointed CEO of Electric Moto Corporation and has more than eighteen years in the motorcycle industry. Leonard, credentials include working with: American Honda Racing, Pro Circuit Racing, Suzuki Racing, Yoshimura Racing, Troy Lee Designs, Showa Suspension, Ohlins Suspension and Dunlop Tire.
Honda CBR by State
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Discussion in 'Northeastern US Wireless Forum' started by BriGuy, May 15, 2002.
They are taking away all the benefits of nextel
www.corporatewireless.com (sells nextel service and they said that they still have the 1 second billing on the national shared plans)
Did they start it back up again? I went in today and spoke with them and they told me that it is still 1 second billing, and will stay that way from now on. True?
I called Nextel CS today - to cancel my work phone, the one year was up. When I was sent to the retention department they asked why I am closeing the account, and I explained to her my bill has never been over $100.00 over the two years I have used your service, and that was because of the per second billing, for the last two bills they have both been over $300.00. I told her I would be happy to stay with them if the kept my plan on per second billing. She stated the only plans were the National shared plans that kept the per second billing the rest is per min. She was very short with me, I guess they are getting alot of these calls. I asked why they did not wait til the end of the contracts to change this, and she rudely said if I was dumb enough to sing a contract with out understanding they (NEXTEL)can change it when ever they want, she did not need to answer any more of my questions. There you go. Best of all after she gave me my comfirmation number she hung up on me.
How Rude!
I would have asked to talk to her boss.
Correct -- only National Shared plans retain per-second billing.