Distributed State Estimation over Time-Varying Graphs: Exploiting the Age-of-Information

Given a discrete-time LTI system $\mathbf{x}[k+1]=\mathbf{Ax}[k]$, and a linear measurement model $\mathbf{y}[k]=\mathbf{Cx}[k]$, a classical result in control theory states that one can design an observer that generates an asymptotically correct estimate $\hat{\mathbf{x}}[k]$ of the state $\mathbf{x}[k]$, if and only if the pair $(\mathbf{A},\mathbf{C})$ is detectable [1]. Additionally, if the pair $(\mathbf{A},\mathbf{C})$ is observable, then one can achieve exponential convergence at any desired convergence rate. Over the last couple of decades, significant effort has been directed towards studying the distributed counterpart of the above problem, wherein observations of the process are distributed among a set of sensors modeled as nodes of a communication graph [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. A fundamental question that arises in this context is as follows: What are the minimal requirements on the measurement structure of the nodes and the underlying communication graph that guarantee the existence of a distributed observer? Here, by a distributed observer, we imply a set of state estimate update and information exchange rules that enable each node to track the entire state asymptotically. The question posed above was answered only recently in [5, 6, 7, 8, 9, 10] for static graphs. However, when the underlying network changes with time, very little is known about the distributed state estimation problem - a gap that we intend to bridge in this paper. Our study is motivated by applications in environmental monitoring tasks using mobile robot teams [13, 14], where time-varying communication graphs arise naturally either as a consequence of robot-mobility, or due to packet drops and link failures typical in wireless sensor networks.

Related Work: The existing literature on the design of distributed observers can be broadly classified in terms of the assumptions made on the system model, the observation model, and the network structure. Recent works on this topic [5, 6, 7, 8, 9, 10] that make minimal system- and graph-theoretic assumptions can be further classified based on a finer set of attributes, as follows. (i) Does the approach require multiple consensus iterations between two consecutive time-steps of the dynamics?¹¹1Such approaches, referred to as two-time-scale approaches, incur high communication cost, and might hence be prohibitive for real-time applications. (ii) What is the dimension of the estimator maintained by each node? (iii) Is the convergence asymptotic, or in finite-time? (iv) In case the convergence is asymptotic, can the convergence rate be controlled? The techniques proposed in [5, 6, 7, 8, 9, 10] operate on a single-time-scale, and those in [6, 8, 9, 10] require observers of dimension no more than that of the state of the system. The notion of convergence in each of the papers [5, 6, 7, 8, 9, 10] is asymptotic; the ones in [7, 8, 10] can achieve exponential convergence at any desired rate. For dynamic networks, there is much less work. Two recent papers [11] and [15] provide theoretical guarantees for certain specific classes of time-varying graphs; while the former proposes a two-time-scale approach, the latter develops a single-time-scale algorithm. However, the extent to which such results can be generalized has remained an open question.

Contributions: The main contribution of this paper is the development of a single-time-scale distributed state estimation algorithm in Section IV that requires each node to maintain an estimator of dimension equal to that of the state (along with some simple counters), and works under assumptions that are remarkably mild in comparison with those in the existing literature. Specifically, in terms of the observation model, we only require joint observability, i.e., the state is observable w.r.t. the collective observations of the nodes. This assumption is quite standard, and can be relaxed to joint detectability, with appropriate implications for the convergence rate.

However, the key departure from existing literature comes from the generality of our communication model, introduced formally in Section II. Based on this model, we require strong-connectivity to be preserved over non-overlapping, contiguous intervals that can even grow linearly with time at a certain rate. In other words, we allow the inter-communication intervals between the nodes to potentially grow unbounded. Even under the regime of such sparse communications, we establish that our proposed approach can track the state of any discrete-time LTI system exponentially fast, at any desired convergence rate. While all the works on distributed state estimation that we are aware of provide an asymptotic analysis, estimating the state of the system in finite-time might be crucial in certain safety-critical applications. To this end, we show that under a suitable selection of the observer gains, one can achieve finite-time convergence based on our approach. To put our results into context, it is instructive to compare them with the work closest to ours, namely [11]. In [11], the authors study a continuous-time analog of the problem under consideration, and develop a solution that leverages an elegant connection to the problem of distributed linear-equation solving [16]. In contrast to our technique, the one in [11] is inherently a two-time-scale approach, requires each node to maintain and update auxiliary state estimates, and works under the assumption that the communication graph is strongly-connected at every instant.

Our work is also related to the vast literature on consensus [17] and distributed optimization [18] over time-varying graphs, where one typically assumes strong-connectivity to be preserved over uniformly bounded intervals - a special case of our communication model. It is important to recognize that, in contrast to these settings, the problem at hand requires tracking potentially unstable external dynamics, making the stability analysis considerably more challenging. In particular, one can no longer directly leverage convergence properties of products of stochastic matrices. From a system-theoretic point of view, the error dynamics under our communication model evolves based on a switched linear system, where certain modes are potentially unstable. Thus, one way to analyze such dynamics is by drawing on techniques from the switched system stability literature [19]. However, such an analysis can potentially obscure the role played by the network structure. Instead, we directly exploit the interplay between the structure of the changing graph patterns on the one hand, and the evolution of the estimation error dynamics on the other. Doing so, we provide a comprehensive stability analysis of our estimation algorithm employing simple, self-contained arguments from linear system theory and graph theory.

Finally, in Section VI, we extend our study of distributed state estimation over dynamic graphs to settings where certain nodes are under attack, and can act arbitrarily. For the problem under consideration, accounting for adversarial behavior is highly non-trivial even for static graphs, with only a few recent results that provide any theoretical guarantees [20, 21, 22, 23]. Accounting for worst-case adversarial behavior over time-varying networks is considerably harder, since the compromised nodes can not only transmit incorrect, inconsistent state estimates, but also lie about the time-stamps of their data. Nonetheless, we develop a novel algorithm to handle such scenarios, and establish its correctness when the graph-sequence is sufficiently connected over time.

The key idea behind our approach is the use of a suitably defined “freshness-index” that keeps track of the age-of-information being diffused across the network. Loosely speaking, the “freshness-index” of a node quantifies the extent to which its estimate of the state is outdated. Exchanging such indices enables a node to assess, in real-time, the quality of the estimate of a neighbor. Accordingly, it can reject estimates that are relatively stale - a fact that contributes to the stability of the error dynamics. We point out that while this is perhaps the first use of the notion of age-of-information (AoI) in a networked control/estimation setting, such a concept has been widely employed in the study of various queuing-theoretic problems arising in wireless networks [24, 25, 26].²²2The notion of age-of-information (AoI) was first introduced in [24] as a performance metric to keep track of real-time status updates in a communication system. In a wireless network, it measures the time elapsed since the generation of the packet most recently delivered to the destination. In Section IV, we will see how such a concept applies to the present setting.

To sum up, we propose the first single-time-scale distributed state estimation algorithm that provides both finite-time and exponentially fast convergence guarantees, under significantly milder assumptions on the time-varying graph sequences than those in the existing literature. In addition, we show how our algorithm can be extended to allow for worst-case adversarial attacks as well.

A preliminary version of this paper appeared as [27]. Here, we significantly expand upon the content in [27] in two main directions. First, we generalize the results in [27] to allow for unbounded inter-communication intervals. Second, the extension to adversarial scenarios in Section VI is a new addition entirely. We also provide full proofs of all our results.

System and Measurement Model: We are interested in collaborative state estimation of a discrete-time LTI system of the form:

where $k\in\mathbb{N}$ is the discrete-time index, $\mathbf{A}\in\mathbb{R}^{n\times n}$ is the system matrix, and $\mathbf{x}[k]\in\mathbb{R}^{n}$ is the state of the system.³³3We use $\mathbb{N}$ and $\mathbb{N}_{+}$ to denote the set of non-negative integers and the set of positive integers, respectively. A network of sensors, modeled as nodes of a communication graph, obtain partial measurements of the state of the above process as follows:

where $\mathbf{y}_{i}[k]\in{\mathbb{R}}^{r_{i}}$ represents the measurement vector of the $i$-th node at time-step $k$, and $\mathbf{C}_{i}\in{\mathbb{R}}^{r_{i}\times n}$ represents the corresponding observation matrix. Let $\mathbf{y}[k]={\begin{bmatrix}\mathbf{y}^{T}_{1}[k]&\cdots&\mathbf{y}^{T}_{N}[k]\end{bmatrix}}^{T}$ and $\mathbf{C}={\begin{bmatrix}\mathbf{C}^{T}_{1}&\cdots&\mathbf{C}^{T}_{N}\end{bmatrix}}^{T}$ represent the collective measurement vector at time-step $k$, and the collective observation matrix, respectively. The goal of each node $i$ in the network is to generate an asymptotically correct estimate $\hat{\mathbf{x}}_{i}[k]$ of the true dynamics $\mathbf{x}[k]$. It may not be possible for any node $i$ in the network to accomplish this task in isolation, since the pair $(\mathbf{A},\mathbf{C}_{i})$ may not be detectable in general. Throughout the paper, we will only assume that the pair $(\mathbf{A},\mathbf{C})$ is observable; the subsequent developments can be readily generalized to the case when $(\mathbf{A},\mathbf{C})$ is detectable.

Communication Network Model: As is evident from the above discussion, information exchange among nodes is necessary for all nodes to estimate the full state. At each time-step $k\in\mathbb{N}$, the available communication channels are modeled by a directed communication graph $\mathcal{G}[k]=(\mathcal{V},\mathcal{E}[k])$, where $\mathcal{V}=\{1,\ldots,N\}$ represents the set of nodes, and $\mathcal{E}[k]$ represents the edge set of $\mathcal{G}[k]$ at time-step $k$. Specifically, if $(i,j)\in\mathcal{E}[k]$, then node $i$ can send information directly to node $j$ at time-step $k$; in such a case, node $i$ will be called a neighbor of node $j$ at time-step $k$. We will use $\mathcal{N}_{i}[k]$ to represent the set of all neighbors (excluding node $i$) of node $i$ at time-step $k$. When $\mathcal{G}[k]=\mathcal{G}\hskip 5.69054pt\forall k\in\mathbb{N}$, where $\mathcal{G}$ is a static, directed communication graph, the necessary and sufficient condition (on the system and network) to solve the distributed state estimation problem is the joint detectability of each source component of $\mathcal{G}$ [5].⁴⁴4A source component of a static, directed graph is a strongly connected component with no incoming edges. Our goal in this paper is to extend the above result to scenarios where the underlying communication graph is allowed to change over time. To this end, let the union graph over an interval $[k_{1},k_{2}],0\leq k_{1}\leq k_{2}$, denoted $\bigcup\limits_{\tau=k_{1}}^{k_{2}}\mathcal{G}[\tau]$, indicate a graph with vertex set equal to $\mathcal{V}$, and edge set equal to the union of the edge sets of the individual graphs appearing over the interval $[k_{1},k_{2}]$. Based on this convention, we now formally describe the communication patterns (induced by the sequence $\{G[k]\}^{\infty}_{k=0}$) that are considered in this paper. We assume that there exists a sequence $\mathbb{I}=\{t_{0},t_{1},\ldots\}$ of increasing time-steps with $t_{0}=0$ and each $t_{i}\in\mathbb{N}$, satisfying the following conditions.

1. (C1)

   Define the mapping $f:\mathbb{I}\rightarrow\mathbb{N}_{+}$ as $f(t_{q})=t_{q+1}-t_{q},\forall t_{q}\in\mathbb{I}.$ We assume that $f(t_{q})$ is a non-decreasing function of its argument.

2. (C2)

   For each $k\in\mathbb{N}$, let $m(k)\triangleq\max\{t_{q}\in\mathbb{I}:t_{q}\leq k\}$, and $M(k)\triangleq\min\{t_{q}\in\mathbb{I}:t_{q}>k\}$. Define $g:\mathbb{N}\rightarrow\mathbb{N}_{+}$ as $g(k)=M(k)-m(k)=f(m(k)).$ Then, we assume that the following holds:

   $$\limsup_{k\to\infty}\frac{2(N-1)g(k)}{k}=\delta<1.$$

   (3)

3. (C3)

   For each $t_{q}\in\mathbb{I}$, we assume that $\bigcup\limits_{\tau=t_{q}}^{t_{q+1}-1}\mathcal{G}[\tau]$ is strongly-connected.

Let us discuss what the above conditions mean. Condition (C1) in conjunction with condition (C3) tells us that the intervals over which strong-connectivity is preserved are non-decreasing in length (evident from the monotonicity of the function $f(\cdot)$).⁵⁵5Our results can be generalized to account for the case when $f(\cdot)$ is non-monotonic by suitably modifying condition (C2). Essentially, our aim here is to come up with distributed estimators that function correctly despite sparse communication; hence, we allow for potentially growing inter-communication intervals. Since we place no assumptions at all on the spectrum of the $\mathbf{A}$ matrix, stability of the estimation error process imposes natural restrictions on how fast the inter-communication intervals can be allowed to grow. Condition (C2) formalizes this intuition by constraining such intervals to grow at most linearly at a certain rate. Notably, conditions (C1)-(C3) cover a very general class of time-varying graph sequences. In particular, we are unaware of any other work that allows the inter-communication intervals to grow unbounded for the problem under consideration.

Consider the following two examples. (i) The mapping $f$ satisfies $f(t_{q})=T,\forall t_{q}\in\mathbb{I}$, where $T$ is some positive integer. (ii) The mapping $f$ satisfies $f(t_{q})=\lfloor\sqrt{t_{q}+1}\rfloor,\forall t_{q}\in\mathbb{I}$. It is easy to verify that (C2) is satisfied in each case. While example (i) represents the standard “joint strong-connectivity” setting where the inter-communication intervals remain bounded, example (ii) deviates from existing literature by allowing the inter-communication intervals to grow unbounded.

Background: For communication graphs satisfying conditions (C1)-(C3) as described above, our objective will be to design a distributed algorithm that ensures $\lim_{k\to\infty}\left\|\hat{\mathbf{x}}_{i}[k]-\mathbf{x}[k]\right\|=0,\forall i\in\mathcal{V}$, with $\hat{\mathbf{x}}_{i}[k]$ representing the estimate of the state $\mathbf{x}[k]$ maintained by node $i\in\mathcal{V}$. To this end, we recall the following result from [6].

We use the matrix $\mathbf{T}$ given by Lemma 1 to perform the coordinate transformation $\mathbf{z}[k]=\mathbf{T}^{-1}\mathbf{x}[k]$, yielding:

where $\bar{\mathbf{A}}={\mathbf{T}}^{-1}\mathbf{A}\mathbf{T}$ and $\bar{\mathbf{C}}_{i}=\mathbf{C}_{i}\mathbf{T}$ are given by (4). Commensurate with the structure of $\bar{\mathbf{A}}$, the vector $\mathbf{z}[k]$ is of the following form:

where $\mathbf{z}^{(j)}[k]$ will be referred to as the $j$-th sub-state. By construction, since the pair $(\mathbf{A}_{jj},\mathbf{C}_{jj})$ is locally observable w.r.t. the measurements of node $j$, node $j$ will be viewed as the unique source node for sub-state $j$. In this sense, the role of node $j$ will be to ensure that each non-source node $i\in\mathcal{V}\setminus\{j\}$ maintains an asymptotically correct estimate of sub-state $j$. For a time-invariant strongly-connected graph, this is achieved in [6] by first constructing a spanning tree rooted at node $j$, and then requiring nodes to only listen to their parents in such a tree for estimating sub-state $j$. The resulting unidirectional flow of information from the source $j$ to the rest of the network guarantees stability of the error process for sub-state $j$ [6].

However, the above strategy is no longer applicable when the underlying communication graph is time-varying, for the following reasons. (i) For a given sub-state $j$, there may not exist a common spanning tree rooted at node $j$ in each graph $\mathcal{G}[k],k\in\mathbb{N}$. (ii) Assuming that a specific spanning tree rooted at node $j$ is guaranteed to repeat at various points in time (not necessarily periodically), is restrictive, and qualifies as only a special case of conditions (C1)-(C3). (iii) Suppose for simplicity that $\mathcal{G}[k]$ is strongly-connected at each time-step (as in [11]), and hence, there exists a spanning tree $\mathcal{T}_{j}[k]$ rooted at node $j$ in each such graph. For estimating sub-state $j$, suppose consensus at time-step $k$ is performed along the spanning tree $\mathcal{T}_{j}[k]$. As we demonstrate in the next section, switching between such spanning trees can lead to unstable error processes over time. Thus, if one makes no further assumptions on the system model beyond joint observability, or on the sequence of communication graphs beyond conditions (C1)-(C3), ensuring stability of the estimation error dynamics becomes a challenging proposition. Nonetheless, we develop a simple algorithm in Section IV that bypasses the above problems. In the next section, we provide an example that helps to build the intuition behind this algorithm.

Consider a network of 3 nodes monitoring a scalar unstable process $x[k+1]=ax[k]$, as shown in Figure 1. The communication graph $\mathcal{G}[k]$ switches between the two topologies shown in Figure 1. Specifically, $\mathcal{G}[k]$ is the graph in Figure 1(a) at all even time-steps, and the one in 1(b) at all odd time-steps. Node 1 is the only node with non-zero measurements, and thus acts as the source node for this network. Suppose for simplicity that it perfectly measures the state at all time-steps, i.e., its state estimate is $\hat{x}_{1}[k]=x[k],\forall k\in\mathbb{N}$. Given this setup, a standard consensus based state estimate update rule would take the form (see for example [5, 6, 11]):

where the weights $w_{ij}[k]$ are non-negative, and satisfy $\sum_{j\in\mathcal{N}_{i}[k]\cup\{i\}}w_{ij}[k]=1,\forall k\in\mathbb{N}$. The key question is: how should the consensus weights be chosen to guarantee stability of the estimation errors of nodes 2 and 3? Even for this simple example, if such weights are chosen naively, then the errors may grow unbounded over time. To see this, consider the following two choices: (1) consensus weights are distributed evenly over the set $\mathcal{N}_{i}[k]\cup\{i\}$, and (2) consensus weights are placed along the tree rooted at node $1$. In each case, the error dynamics are unstable, as depicted in Figure 2. To overcome this problem, suppose nodes 2 and 3 are aware of the fact that node 1 has perfect information of the state. Since nodes 2 and 3 have no measurements of their own, intuitively, it makes sense that they should place their consensus weights entirely on node 1 whenever possible. The trickier question for node 2 (resp., node 3) is to decide when it should listen to node 3 (resp., node 2). Let us consider the situation from the perspective of node 2. At time-step 0, it adopts the information of node 1, and hence, the error of node 2 is zero at time-step 1. However, the error of node 3 is not necessarily zero at time-step 1. Consequently, if node 2 places a non-zero consensus weight on the estimate of node 3 at time-step 1, its error at time-step 2 might assume a non-zero value. Clearly, at time-step 1, node 2 is better off rejecting the information from node 3, and simply running open-loop. The main take-away point here is that adoption or rejection of information from a neighbor should be based on the quality of information that such a neighbor has to offer. In particular, a node that has come in contact with node 1 more recently is expected to have better information about the state than the other. Thus, to dynamically evaluate the quality of an estimate, the above reasoning suggests the need to introduce a metric that keeps track of how recent that estimate is with respect to (w.r.t.) the estimate of the source node 1. In the following section, we formalize this idea by introducing such a metric.

Building on the intuition developed in the previous section, we introduce a new approach to designing distributed observers for a general class of time-varying networks. The main idea is the use of a “freshness-index” that keeps track of how delayed the estimates of a node are w.r.t. the estimates of a source node. Specifically, for updating its estimate of $\mathbf{z}^{(j)}[k]$, each node $i\in\mathcal{V}$ maintains and updates at every time-step a freshness-index $\tau^{(j)}_{i}[k]$. At each time-step $k\in\mathbb{N}$, the index $\tau^{(j)}_{i}[k]$ plays the following role: it determines whether node $i$ should adopt the information received from one of its neighbors in $\mathcal{N}_{i}[k]$, or run open-loop, for updating $\hat{\mathbf{z}}^{(j)}_{i}[k]$, where $\hat{\mathbf{z}}^{(j)}_{i}[k]$ represents the estimate of $\mathbf{z}^{(j)}[k]$ maintained by node $i$. In case it is the former, it also indicates which specific neighbor in $\mathcal{N}_{i}[k]$ node $i$ should listen to at time-step $k$; this piece of information is particularly important for the problem under consideration, and ensures stability of the error process. The rules that govern the updates of the freshness indices $\tau^{(j)}_{i}[k]$, and the estimates of the $j$-th sub-state $\mathbf{z}^{(j)}[k]$, are formally stated in Algorithm 1. In what follows, we describe each of these rules.

Discussion of Algorithm 1: Consider any sub-state $j\in\{1,\ldots,N\}.$ Each node $i\in\mathcal{V}$ maintains an index $\tau^{(j)}_{i}[k]\in\{\omega\}\cup\mathbb{N}$, where $\omega$ is a dummy value. Specifically, $\tau^{(j)}_{i}[k]=\omega$ represents an “infinite-delay” w.r.t. the estimate of the source node for sub-state $j$, namely node $j$, i.e., it represents that node $i$ has not received any information (either directly or indirectly) from node $j$ regarding sub-state $j$ up to time-step $k$. For estimation of sub-state $j$, since delays are measured w.r.t. the source node $j$, node $j$ maintains its freshness-index $\tau^{(j)}_{j}[k]$ at zero for all time, to indicate a zero delay w.r.t. itself. For updating its estimate of $\mathbf{z}^{(j)}[k]$, it uses only its own information, as is evident from (8).

Every other node starts out with an “infinite-delay” $\omega$ w.r.t. the source (line 1 of Algo. 1). The freshness-index of a node $i\in\mathcal{V}\setminus\{j\}$ changes from $\omega$ to a finite value when it comes in contact with a neighbor with a finite delay, i.e., with a freshness-index that is not $\omega$ (line 4 of Algo. 1). At this point, we say that $\tau^{(j)}_{i}[k]$ has been “triggered”. Once triggered, at each time-step $k$, a non-source node $i$ will adopt the information of a neighbor $l\in\mathcal{N}_{i}[k]$ only if node $l$’s estimate of $\mathbf{z}^{(j)}[k]$ is “more fresh” relative to its own, i.e., only if $\tau^{(j)}_{l}[k]<\tau^{(j)}_{i}[k]$.⁶⁶6Under Case 1 or Case 2 in Algo. 1, when a node $i\in\mathcal{V}\setminus\{j\}$ updates $\tau^{(j)}_{i}[k]$ via (9), and $\hat{\mathbf{z}}^{(j)}_{i}[k]$ via (10), we say that “$i$ adopts the information of $u$ at time $k$ for sub-state $j$”; else, if it runs open-loop, we say it adopts its own information. Among the set of neighbors in $\mathcal{M}^{(j)}_{i}[k]$ (if $\tau^{(j)}_{i}[k]$ has not yet been triggered), or in $\mathcal{F}^{(j)}_{i}[k]$ (if $\tau^{(j)}_{i}[k]$ has been triggered), node $i$ only adopts the information (based on (10)) of the neighbor $u$ with the least delay. At this point, the delay of node $i$ matches that of node $u$, and this fact is captured by the update rule (9). In case node $i$ has no neighbor that has fresher information than itself w.r.t. sub-state $j$ (where informativeness is quantified by $\tau^{(j)}_{i}[k])$, it increments its own freshness-index by $1$ (as per (13)) to capture the effect of its own information getting older, and runs open-loop based on (12). Based on the above rules, at any given time-step $k$, $\tau^{(j)}_{i}[k]$ measures the age-of-information of $\hat{\mathbf{z}}^{(j)}_{i}[k]$, relative to the source node $j$. This fact is established later in Lemma 23. Finally, note that Algorithm 1 describes an approach for estimating $\mathbf{z}[k]$, and hence $\mathbf{x}[k]$, since $\mathbf{x}[k]=\mathbf{Tz}[k]$.

We now consider a scenario where a subset of agents in the network is adversarial, and can deviate from the prescribed algorithm. We will show how one can employ the notion of freshness-indices to account for such adversarial agents over a time-varying network. To avoid cumbersome notation and to present the key ideas in a clear way, we will consider a scalar LTI system of the form $x[k+1]=ax[k]$. Later, we will discuss how our approach can be naturally extended to more general system models. We consider a worst-case Byzantine adversary model, where an adversarial node is assumed to be omniscient, and can essentially act arbitrarily: it can transmit incorrect, potentially inconsistent information to its instantaneous out-neighbors, or choose not to transmit anything at all, while colluding with other adversaries in the process [28, 29, 30]. We will focus on an $f$-total adversarial model where the total number of adversaries in the network is bounded above by $f$, with $f\in\mathbb{N}$. The adversarial set will be denoted by $\mathcal{A}$, and the set of regular nodes by $\mathcal{R}=\mathcal{V}\setminus\mathcal{A}$. Finally, for the scalar model under consideration, we will define the set of source nodes as follows: $\mathcal{S}=\{i\in\mathcal{V}:c_{i}\neq 0,\,\textrm{where}\,y_{i}[k]=c_{i}x[k]\}$, i.e., $i\in\mathcal{S}$ if and only if it can measure the state on its own. Note that we will allow $\mathcal{S}\cap\mathcal{A}\neq\emptyset$.