Error Scaling Laws for Linear Optimal Estimation from Relative Measurements
Abstract
We study the problem of estimating vectorvalued variables from noisy “relative” measurements. This problem arises in several sensor network applications. The measurement model can be expressed in terms of a graph, whose nodes correspond to the variables and edges to noisy measurements of the difference between two variables. We take an arbitrary variable as the reference and consider the optimal (minimum variance) linear unbiased estimate of the remaining variables.
We investigate how the error in the optimal linear unbiased estimate of a node variable grows with the distance of the node to the reference node. We establish a classification of graphs, namely, dense or sparse in , that determines how the linear unbiased optimal estimation error of a node grows with its distance from the reference node. In particular, if a graph is dense in ,, or D, then a node variable’s estimation error is upper bounded by a linear, logarithmic, or bounded function of distance from the reference, respectively. Corresponding lower bounds are obtained if the graph is sparse in , and D. Our results also show that naive measures of graph density, such as node degree, are inadequate predictors of the estimation error. Being true for the optimal linear unbiased estimate, these scaling laws determine algorithmindependent limits on the estimation accuracy achievable in large graphs.
I Introduction
Several applications in sensor and actuator networks lead to estimation problems where a number of variables are to be estimated from noisy measurements of the difference between certain pairs of them. Consider the problem of localization, where a sensor does not know its position in a global coordinate system, but can measure its position relative to a set of nearby nodes. These measurements can be obtained, for example, from range and angle data but are typically subjected to large noise (see Figure 1). In particular, two nearby sensors and located in a plane at positions and , respectively, have access to the measurement
(1) 
where denotes measurement error. The problem of interest is to use the ’s to estimate the positions of all the nodes in a common coordinate system whose origin is fixed arbitrarily at one of the nodes.
Similar estimation problems arise in time synchronization [1, 2, 3] and motion consensus in sensoractuator networks [4]; see [5, 4] for an overview of these applications. Motivated by these applications, we study the problem of estimating vector valued variables from noisy measurements of the difference between them. In particular, denoting the variables of interest by where , we consider problems for which noisy relative measurements of the form (1) are available. The ordered pairs of indices for which we have relative measurements form a set that is a (typically strict) subset of the set of all pairs of indices. Just with relative measurements, the ’s can be determined only up to an additive constant. To avoid this ambiguity, we assume that a particular variable (say ) is used as the reference, which is therefore assumed known. The problem of interest is to estimate the remaining node variables from all the available measurements.
The measurement equations (1) can be naturally associated with a directed graph with an edge from node to if the measurement is available. The graph is called the measurement graph, and each vector , is called the th node variable. The measurement noise is assumed zero mean and spatially uncorrelated, i.e., and if .
In this paper we investigate how the structure of the graph affects the quality of the optimal linear unbiased estimate of , measured in terms of the covariance of the estimation error . The optimal linear unbiased estimate refers to the one obtained with the classical best linear unbiased estimator (BLUE), which achieves the minimum variance among all linear unbiased estimators [6]. We examine the growth of the BLUE error variance of a node as a function of its distance to the reference node.
We are interested in the growth of error with distance in large graphs, for which infinite graphs (with a countably infinite number of nodes and edges) serve as proxies. This paper is focused on infinite graphs because the absence of boundary conditions in infinite graphs allows for more complete and simpler results. Using infinite graphs as proxies for large finite graphs is theoretically justified by the fact that the BLUE error variance of a node variable in a large but finite subgraph of an infinite graph is arbitrarily close to the BLUE estimation error in the infinite graph, as long as the finite graph is sufficiently large. This convergence result was established in [7].
When the measurement graph is a tree, there is a single path between the node and the reference node and one can show that the covariance matrix of the estimation error is the sum of the covariance matrices associated with this path. Thus, for trees, the variance of the BLUE estimation error of grows linearly with the distance from node to the reference node. It turns out that for graphs “denser” than trees, with multiple paths between pairs of nodes, the variance of the optimal linear unbiased estimation error can grow slower than linearly with distance.
In this paper, we introduce a novel notion of denseness for graphs that is needed to characterize how the estimation error grows with distance. In classical graphtheoretic terminology, a graph with vertices is called dense if its average node degree is of order , and is called sparse if its average node degree is a constant independent of [8]. We recall that the degree of a node is the number of edges incident on it (an edge is said to be incident on the nodes and ). Other notions of denseness include geodenseness introduced by [9], which requires uniform node density (nodes per unit area) but does not consider the role of edges. Accuracy of localization from distanceonly measurements have been extensively studied in the sensor networks literature, typically by evaluating the CramérRao lower bound (see [14, 10, 11, 12, 13] and references therein). In many of these studies, graph density (as measured by node degree or node density) is recognized to affect estimation accuracy [10, 11, 13]. However, we will see through examples in Remark IIB that for the estimation problem considered in this paper, such notions of denseness are not sufficient to characterize how the estimation error grows with distance.
A key contribution of this paper is the development of suitable notions of graph denseness and sparseness that are useful in determining BLUE error scaling laws. These notions exploit the relationship between the measurement graph and a lattice. We recall that the dimensional square lattice is defined as a graph with a node in every point in with integer coordinates and an edge between every pair of nodes at an Euclidean distance of (see Figure 4 for examples). The error scaling laws for a lattice measurement graph can be determined analytically by exploiting symmetry. It turns out that when the graph is not a lattice, it can still be compared to a lattice. Intuitively, if after some bounded perturbation in its node and edge set, the graph looks approximately like a dimensional lattice, then the graph inherits the denseness properties of the lattice. In that case the error covariance for the lattice can still be used to bound the error covariance in the original graph.
Our classification of dense and sparse graphs in , , characterizes BLUE error scaling laws. For dense graphs, they provide upper bounds on the growth rate of the error, while for sparse graphs, they provide lower bounds. The precise growth rates depend on which dimension the graph is dense or sparse in. When a graph is dense in D, D, or D, respectively, the error covariance of a node is upper bounded by a linear, logarithmic, or bounded function, respectively, of its distance from the reference. On the other hand, when a graph is sparse in D, D, or D, the error covariance of a node is lower bounded by a linear, logarithmic, and bounded function, respectively, of its distance from the reference. Our sparse graphs are also known as “graphs that can be drawn in a civilized manner” according to the terminology introduced by Doyle and Snell [15] in connection with random walks.
The BLUE error scaling laws derived in this paper provide an algorithmindependent limit to the estimation accuracy achievable in large networks, since no linear unbiased estimation algorithm can achieve higher accuracy than the BLUE estimator. For example, when a graph is sparse in D, the BLUE estimation error covariance grows at least linearly with the distance from the reference. Therefore the estimation accuracy will be necessarily poor in large 1D sparse graphs. On the other hand, when a graph is dense in D, the BLUE estimation error of every node variable remains below a constant, even for nodes that are arbitrarily far away from the reference. So accurate estimation is possible in very large 3D dense graphs.
The results in this paper are useful for the design and deployment of adhoc and sensor networks. Since we now know what structural properties are beneficial for accurate estimation, we can strive to achieve those structures when deploying a network. Specifically, we should try to achieve a densein structure, with as large as possible, for high accuracy estimation. Since the scaling laws are true for the optimal linear unbiased estimator, they can also help designers determine if design requirements are achievable. For example, if the requirement is that the estimation accuracy should not decrease with size, no matter how large a network is, the network must be dense in , for such a requirement to be satisfied.
Our results also expose certain misconceptions that exist in the sensor network literature about the relationship between graph structure and estimation error. In Section IIB, we provide examples that expose the inadequacy of the usual measures of graph denseness, such as node degree, in determining scaling laws of the estimation error.
In practice, more than one reference node (commonly referred to as anchors) may be used. We only consider the case of a single reference node since scaling laws with a single reference provide information on how many reference nodes may be needed. For example, since the estimation error in a 3D dense graph is bounded by a constant, one reference node may be enough for such a graph.
While we do not discuss the computation of the optimal linear unbiased estimates in this paper, we have developed distributed algorithms to compute these estimates with arbitrary precision (see [5] and references therein). These algorithms are distributed in the sense that every node computes its own estimate and the information needed to carry out this computation is obtained by communication with its neighbors.
A preliminary version of some of the results in this paper was presented in [16]. However, [16] used stricter assumptions to establish the upper bounds on error growth rates. Moreover, only sufficient conditions were obtained in [16] for some of the error scaling laws to hold; whereas here we derive necessary and sufficient conditions.
Organization: The rest of the paper is organized as follows. Section II describes the problem and summarizes the main results of the paper. Section III describes key properties of dense and sparse graphs. Section IV briefly describes the analogy between BLUE and generalized electrical networks from [7] that is needed to prove the main results. Section V contains the proof of the main result of the paper. Section VI deals with the question of how to check if a graph possesses the denseness/sparseness properties. The paper ends with a a few final conclusions and directions for future research in Section VII.
Ii Problem Statement and Main Results
Recall that we are interested in estimating vectorvalued variables , , from noisy relative measurements of the form:
(2) 
where denotes a zeromean measurement noise and is the set of ordered pairs for which relative measurements are available. The node set is either finite, or infinite but countable. We assume that the value of a particular reference variable is known and without loss of generality we take . The node set and the edge set together define a directed measurement graph .
The accuracy of a node variable’s estimate, measured in terms of the covariance of the estimation error, depends on the graph as well as the measurement errors. The covariance matrix of the error in the measurement is denoted by , i.e., . We assume that the measurement errors on different edges are uncorrelated, i.e., for every pair of distinct edges , . The estimation problem is now formulated in terms of a network where is a function that assigns to each edge the covariance matrix of the measurement error associated with the edge in the measurement graph . The symbol denotes the set of symmetric positivedefinite matrices.
As discussed in Section I, our results are stated for infinite networks. The following conditions are needed to make sure that the estimation problem is well posed and that the estimates satisfy appropriate convergence properties to be discussed shortly:
[measurement network] The measurement network satisfies the following properties:

The graph is weakly connected, i.e., it is possible to go from every node to every other node traversing the graph edges without regard to edge direction.

The graph has a finite maximum node degree^{1}^{1}1The degree of a node is the number of edges that are incident on the node. An edge is said to be incident on the nodes and ..

The edgecovariance function is uniformly bounded, i.e., there exists constant symmetric positive matrices such that , . \frqed
In the above, for two matrices , () means is positive definite (semidefinite). We write () if ().
We also assume throughout the paper that measurement graphs do not have parallel edges. A number of edges are said to be parallel if all of them are incident on the same pair of nodes. The condition of not having parallel edges is not restrictive since parallel measurements can be combined into a single measurement with an appropriate covariance, while preserving the BLUE error covariances (see Remark IVA).
Given a finite measurement network , where contains the nodes and , it is straightforward to compute the BLUE estimate of the unknown node variable in the network , as described in [7], and the covariance matrix of the estimation error exists as long as is weakly connected [7]. Due to the optimality of the BLU estimator, is the minimum possible estimation error covariance that is achievable by any linear unbiased estimator using all the measurements in the graph .
When the measurement graph is infinite, the BLUE error covariance for a node variable is defined as
(3) 
where the infimum is taken over all finite subgraphs of that contain the nodes and . We define a matrix to be the infimum of the matrix set , and denote it by
(4) 
if for every matrix , and for every positive real , there exists a matrix such that . Under Assumption II, it was shown in [7] that the infimum in (3) always exists. In this case, (3) means that the BLUE covariance is the the lowest error covariance that can be achieved by using all the available measurements.
In the sequel, we determine how the BLUE covariance grows as a function of the distance of node to the reference , and how this scaling law depends on the structure of the measurement graph . To this effect we start by providing a classification of graphs that is needed to characterize the error scaling laws.
Iia Graph Denseness and Sparseness
We start by introducing graph drawings, which will later allow us to define dense and sparse graphs.
IiA1 Graph Drawings
The drawing of a graph in a dimensional Euclidean space is obtained by mapping the nodes into points in by a drawing function . A drawing is also called a representation [17] or an embedding [8]. For a particular drawing , given two nodes the Euclidean distance between and induced by the drawing is defined by
where denoted the usual Euclidean norm in . It is important to emphasize that the definition of drawing allows edges to intersect and therefore every graph has a drawing in every Euclidean space. In fact, every graph has an infinite number of drawings in every Euclidean space. However, a particular drawing is useful only if it clarifies the relationship between the graph and the Euclidean space in which it is drawn. In what follows, given two nodes and , denotes the graphical distance between and , i.e., the number of edges in the shortest path between and . The graphical distance is evaluated without regards to edge directions, which are immaterial in determining BLUE error covariances (see Remark IVB).
For a particular drawing and induced Euclidean distance of a graph , four parameters are needed to characterize graph denseness and sparseness. The minimum node distance, denoted by , is defined as the minimum Euclidean distance between the drawing of two nodes
The maximum connected range, denoted by , is defined as the Euclidean length of the drawing of the longest edge
The maximum uncovered diameter, denoted by , is defined as the diameter of the largest open ball that can be placed in such that it does not enclose the drawing of any node
where the existential quantification spans over the balls in with diameter and centered at arbitrary points. Finally, the asymptotic distance ratio, denoted by , is defined as
Essentially provides a lower bound for the ratio between the Euclidean and the graphical distance for nodes that are far apart. The asymptotic distance ratio can be thought of as an inverse of the stretch for geometric graphs, which is a wellstudied concept for finite graphs [18].
The two parameters and defined above are especially useful to compare graphical and Euclidean distances, as stated in the following result.
[Euclidean vs. graphical distances] The following two statements are equivalent:

The asymptotic distance ratio is strictly positive.

There exist constants for which
(5)
Similarly, the following statements are equivalent:

The maximum connected range is finite.

There exist constants , for which
\frqed
The proof of this lemma is provided in Appendix A.
IiA2 Dense and Sparse Graphs
We call the drawing of a graph with finite maximum uncovered diameter () and positive asymptotic distance ratio () a dense drawing. We say that a graph is dense in if there exists a dense drawing of the graph in . Graph drawings for which the minimum node distance is positive () and the maximum connected range is finite () are called civilized drawings [15]. A graph is said to be sparse in if there exists a civilized drawing in .
It follows from these definitions and Lemma 2 that if a graph is dense in , then it has enough nodes and edges so that it is possible to draw it in in such a way that its nodes cover without leaving large holes (finite ), and yet a small Euclidean distance between two nodes in the drawing guarantees a small graphical distance between them (positive , which implies (5)). On the other hand, if a graph that is sparse in , then one can draw it in so as to keep a certain minimum separation between nodes (positive ) without making the edges arbitrarily long (finite ). It also follows from the definitions that a graph must be infinite to be dense in any dimension, and a finite graph is sparse in every dimension.
A graph can be both dense and sparse in the same dimension. For example, the dimensional lattice is both sparse and dense in . However, there is no civilized drawing of the dimensional lattice in for any . Moreover, there is no dense drawing of the dimensional lattice in for every . This means, for example, that the 3D lattice in not sparse in 2D and is not dense in 4D. In general, a graph being dense in a particular dimension puts a restriction on which dimensions it can be sparse in. The next result, proved in Section VI, states this precisely.
A graph that is dense in for some , cannot be sparse in for every . \frqed
Euclidean space and graph example 
Covariance matrix of the estimation error of in a sparse graph with a sparse drawing 
Covariance matrix of the estimation error of in a dense graph with a dense drawing 

[historical note] In the terminology of Doyle and Snell [15], sparse graphs (as defined here) are said to be graphs “that can be drawn in a civilized manner”. In this paper we refer to such graphs as sparse graphs since they are the antitheses of dense graphs. \frqed
IiB Error Scaling Laws
The concepts of dense and sparse graphs allow one to characterize precisely how the BLUE error covariance grows with the distance from the node to the reference . The next theorem, which establishes the BLUE error scaling laws for dense and sparse graphs, is the main result of the paper. The proof of the theorem is provided in Section V.
Before we present the theorem, we need to introduce some notation. The asymptotic notations and are used for matrix valued functions in the following way. For a matrixvalued function and a scalarvalued function , the notation means that there exists a positive constant and a constant matrix such that for all . Similarly, means there exists a positive constant and a constant matrix such that for all . Recall that is the set of all symmetric positive definite matrices.
[Error Scaling Laws] Consider a measurement graph that satisfies Assumption IIB, with a reference node . The BLUE error covariance for a node obeys the scaling laws shown in Table I. \frqed
A graph can be both sparse and dense in a particular dimension, in which case the asymptotic upper and lower bounds are the same. For a graph that is both sparse and dense in , the error covariance grows with distance in the same rate as it does in the corresponding lattice .
[Counterexamples to conventional wisdom] As noted in Section I, the average node degree of a graph or the number of nodes and edges per unit area of a deployed network are often used as measures of graph denseness. However, these measures do not predict error scaling laws. The three graphs in Figure 3 offer an example of the inadequacy of node degree as a measure of denseness. This figure shows a fuzz of the 1D lattice (see Section III for the formal definition of a fuzz), a triangular lattice, and a dimensional lattice. It can be verified from the definitions in Section IIA2 that the fuzz of the 1D lattice is both dense and sparse in , the triangular lattice is dense and sparse in , and the 3D lattice is dense and sparse in . Thus, it follows from Theorem IIB that the BLU estimation error scales linearly with distance in the fuzz of the D lattice, logarithmically with distance in the triangular lattice, and is uniformly bounded with respect to distance in the D lattice, even though every node in each of these graphs has the same degree, namely six. \frqed\draftnoteJoao wanted to remove the “Remark” heading, but is kept since it is referred to in the introduction.
We note that the notion of geodenseness introduced in [9] is also not useful for characterizing error scaling laws since geodenseness considers node density alone without regard to the edges.
Iii Dense and Sparse Graphs
This section establishes an embedding relationship between dense and sparse graphs and lattices, which is needed to prove Theorem IIB. Roughly speaking, a graph can be embedded in another graph if contains all the nodes and edges of , and perhaps a few more. The usefulness of embedding in answering the error scaling question is that when can be embedded in , the BLUE error covariances in are larger than the corresponding ones in (this statement will be made precise in Theorem IVB of Section IV).
The fuzz of a graph , introduced by Doyle and Snell [15], is a graph with the same set of nodes as but with a larger set of edges. Specifically, given a graph and a positive integer , a fuzz of , denoted by , is a graph that has an edge between two nodes and whenever the graphical distance between these nodes in is less than or equal to .
We say that a graph can be embedded in another graph if , and, whenever there is an edge between two nodes in , there is an edge between them in . More precisely, can be embedded in if there exists an injective map such that for every , either or . In the sequel, we use to denote that can be embedded in .
Iiia Relationship with lattices and Euclidean spaces
The next theorems (Theorem IIIA and IIIA) show that sparse graphs can be embedded in fuzzes of Lattices, and fuzzes of dense graphs can embed lattices. In these two theorems we use to denote the graphical distance in the lattice and to denote the Euclidean distance in induced by the drawing .
[Sparse Embedding] A graph is sparse in if and only if there exists a positive integer such that . Moreover, if is a civilized drawing of in , then there exists an embedding so that ,
(6) 
where is the minimum node distance in the drawing of . \frqed
In words, the theorem states that is sparse in if and only if can be embedded in an fuzz of a dimensional lattice. The significance of the additional condition (6) is that if the Euclidean distance between a pair of nodes and in a civilized drawing of the graph is large, the graphical distance in the lattice between the nodes that correspond to and must also be large.
The first statement of Theorem IIIA is essentially taken from [15], where it was proved that if a graph can be drawn in a civilized manner in , then it can be embedded in a fuzz of a lattice, where depends only on and . A careful examination of the proof in [15] reveals that it is not only sufficient but also a necessary condition for embedding in lattice fuzzes. The proof of this theorem is therefore omitted.
[Dense Embedding] A graph is dense in if and only if there exists finite, positive integers and such that the following conditions are satisfied

, and,

if is an embedding of into , then, such that .
Moreover, if is a dense drawing of in , then the embedding function in (ii) can be chosen so that , we can find satisfying
(7) 
where is the maximum uncovered diameter of the drawing of . \frqed
In words, the two conditions state that is dense in if and only if (i) the dimensional lattice can be embedded in an fuzz of for some positive integer and (ii) every node of that is not the image of a node in is at a uniformly bounded graphical distance from a node that is the image of a node in . The significance of (7) is that not only we can find for every node in a closeby node that has a preimage in the lattice, but also these closeby nodes can be so chosen so that if the Euclidean distance between a pair of nodes and in the drawing is small, then the graphical distance in the lattice between the preimages of their closeby nodes is small as well.
Iv Electrical Analogy
A crucial step in proving the main results of this paper is the analogy introduced in [7] between the BLU estimation problem and an abstract electrical network, where currents, potentials and resistances are matrix valued.
A generalized electrical network consists of a graph (finite or infinite) together with a function that assigns to each edge a symmetric positive definite matrix called the generalized resistance of the edge.
A generalized flow from node to node with intensity is an edgefunction such that
(8) 
A flow is said to have finite support if it is zero on all but a finite number of edges. We say that a flow is a generalized current when there exists a nodefunction for which
(9) 
The nodefunction is called a generalized potential associated with the current . Eq. (8) should be viewed as a generalized version of Kirchhoff’s current law and can be interpreted as: the net flow out of each node other than and is equal to zero, whereas the net flow out of is equal to the net flow into and both are equal to the flow intensity . Eq. (9) provides in a combined manner, a generalized version of Kirchhoff’s loop law, which states that the net potential drop along a circuit must be zero, and Ohm’s law, which states that the potential drop across an edge must be equal to the product of its resistance and the current flowing through it. A circuit is an undirected path that starts and ends at the same node. For , generalized electrical networks are the usual electrical networks with scalar currents, potentials, and resistors.
Iva Effective Resistance and BLUE Error Covariance
It was shown in [7] that when a current of intensity flows from node to node , the resulting generalized current is a linear function of the intensity and there exists a matrix such that
(10) 
We call the matrix the generalized effective resistance between and . In view of this definition, the effective resistance between two nodes is the generalized potential difference between them when a current with intensity equal to the identity matrix is injected at one node and extracted at the other, which is analogous to the definition of effective resistance in scalar networks [15]. Note that the effective resistance between two arbitrary nodes in a generalized network is a symmetric positive definite matrix as long as the network satisfies Assumption II, whether the network is finite or infinite [7].
Generalized electrical networks are useful in studying the BLU estimation error in large networks because of the following analogy between the BLU estimation error covariance and the generalized effective resistance.
[Electrical Analogy, from [7]] Consider a measurement network satisfying Assumption II with and a single reference node . Then, for every node , the BLUE error covariance defined in (3) is a symmetric positive definite matrix equal to the generalized effective resistance between and in the generalized electrical network :
\frqed 
In an electrical network, parallel resistors can be combined into one resistor by using the parallel resistance formula so that the effective resistance between every pair of nodes in the network remain unchanged. The same can be done in generalized electrical networks [19]. The analogy between BLUE covariance and effective resistance means that parallel measurement edges with possibly distinct measurement error covariances can be replaced by a single edge with an equivalent error covariance, so that the BLUE error covariances of all nodes remain unchanged. This explains why the assumption of not having parallel edges made at the beginning is not restrictive in any way.
IvB Graph Embedding and Partial Ordering of BLUE Covariances
Effective resistance in scalar electrical networks satisfies Rayleigh’s Monotonicity Law, which states that the effective resistance between any two nodes can only increase if the resistance on any edge is increased, and vice versa [15]. The next result (proved in [7]), states that the same is true for generalized networks, whether finite or infinite.
[Rayleigh’s Monotonicity Law [7]] Consider two generalized electrical networks and with graphs and , respectively, such that both the networks satisfy Assumption II. Assume that

can be embedded in , i.e., , and

for every edge .
Then, for every pair of nodes of ,
where and are the effective resistance between and in the networks and , respectively. \frqed
The usefulness of Rayleigh’s Monotonicity Law in answering the error scaling question becomes apparent when combined with the Electrical Analogy. It shows that when can be embedded in , the BLUE error covariances in are lower bounded by the error covariances in . Intuitively, since has only a subset of the measurements in , the estimates in are less accurate than those in .
Although the graph that defines the electrical network is directed, the edge directions are irrelevant in determining effective resistances. This is why Rayleigh’s Monotonicity Law holds with graph embedding, which is insensitive to edge directions. The electrical analogy also explains why the edge directions are irrelevant in determining error covariances.\frqed
IvC Triangle Inequality
Matrixvalued effective resistances satisfy a triangle inequality, which will be useful in proving the error scaling laws in Section V. It is known that scalar effective resistance obeys triangle inequality, and is therefore also referred to as the “resistance distance” [20]. Although the result in [20] was proved only for finite networks, it is not hard to extend it to infinite networks. The following simple extension of the triangle inequality to generalized networks with constant resistances on every edge was derived in [19]:
[Triangle Inequality] Let be a generalized electrical network satisfying Assumption II with a constant resistance on every edge of . Then, for every triple of nodes in the network,
\frqed 
IvD Effective Resistances in Lattices and Fuzzes
Recall that given a graph and a positive integer , the fuzz of , denoted by , is a graph that has an edge between two nodes and whenever the graphical distance between them in is less than or equal to .
An fuzz will clearly have lower effective resistance than the original graph because of Rayleigh’s Monotonicity Law, but it is lower only by a constant factor as stated in the following result, which is a straightforward extension to the generalized case of a result about scalar effective resistance established by Doyle and Snell (see the Theorem on page , as well as Exercise , in [15]). The interested reader can find a proof in [19].
Let be a generalized electrical network satisfying Assumption II with a constant generalized resistance on its every edge. Let be the electrical network similarly constructed on , the fuzz of . For every pair of nodes and in ,
where is the effective resistance in the network and is a positive constant that does not depend on and . \frqed
The following lemma establishes effective resistances in dimensional lattices and their fuzzes.
For a given positive integer , consider the electrical network with a constant generalized resistance at every edge of the fuzz of the dimensional square lattice . The generalized effective resistance between two nodes and in the electrical network satisfies


,

. \frqed
IVD The scalar effective resistance in D , D, and D lattices follow linear, logarithmic and bounded growth rates, respectively [21, 22]. Using these results, it was established in [7] that the matrix effective resistances in these lattices have the same scaling laws (see Lemma in [7]). Thus, 1D, 2D, and 3D lattices with matrixvalued resistances have linear, logarithmic, and bounded scaling laws for the effective resistance, which is the result with . The case follows from the application of Lemma IVD.
The slowing down of the growth of the effective resistance as the dimension increases can be attributed to the fact that the number of paths between each pair of nodes is larger in higher dimensional lattices. The scaling laws for effective resistance in lattices and their fuzzes also have intimate connections to the change from recurrence to transience of random walks in lattices as the dimension changes from to [15].
V Proof of Theorem IiB
We now prove Theorem IIB by using the tools that have been developed so far. The following terminology is needed for the proofs. For functions and , the notation means that and . The notations and are described in Section II.
IIB [Upper bounds:] We start by establishing the upper bounds on the effective resistance for graphs that are dense in . Throughout the proof of the upper bounds, we will use , for any graph , to denote the effective resistance between nodes and in the electrical network with every edge of having a generalized resistance of . From the Electrical Analogy theorem and Monotonicity Law (Theorems IVA and IVB), we get
To establish an upper bound on , we will now establish an upper bound on the resistance . To this effect, suppose that is a dense drawing of in . From dense embedding Theorem IIIA, we conclude that there exists a positive integer such that the D lattice can be embedded in the fuzz of . Moreover, Theorem IIIA tells us that there exists , a positive constant , and an embedding of into , such that
(11)  
(12) 
where is the maximum uncovered diameter of the drawing of . Note that . Consider the electrical network formed by assigning to every edge of a resistance of . From the triangle inequality for effective resistances (Lemma IVC),
(13) 
For any two nodes , application of the triangle inequality Lemma IVC to successive nodes on the shortest path joining and gives us . Using this bound in (V), and by using (11), we conclude that
(14) 
Since , from Rayleigh’s Monotonicity Law (Theorem IVB), we obtain
When is dense in, say, in , we have from Lemma IVD that
which implies
Combining this with (12) and (14), we get
From Lemma IVD we know that the effective resistance in and its fuzz is of the same order, so that
from which the desired result follows:
The statements of the upper bounds for and dimensions can be proved similarly. This concludes the proof of the upper bounds in Theorem IIB.
[Lower bounds:] Now we establish the lower bounds on the BLUE error covariance in a sparse graph. Throughout the proof of the lower bounds, for a graph , we will use to denote the effective resistance between nodes and in the electrical network with every edge of having a generalized resistance of . From the Electrical Analogy and Rayleigh’s Monotonicity Law (Theorems IVA and IVB), we get
(15) 
Therefore, to establish a lower bound on , we proceed by establishing a lower bound on the resistance . Since is sparse in , it follows from Theorem IIIA that there exists a positive integer , such that . Let be the embedding of into . Consider the generalized electrical network formed by assigning a generalized resistance of to every edge of . From Rayleigh’s monotonicity law, we get
(16) 
where refer to the nodes in that correspond to the nodes in . When the graph is sparse in, say, , it follows from (16) and Lemma IVD that
where the second statement follows from (6) in Theorem IIIA. Combining the above with (15), we get , which proves the lower bound for graph that are sparse in . The statements for the lower bounds graphs that are sparse in or can be proved in an analogous manner. This concludes the proof of the theorem.
Vi Checking Denseness and Sparseness
To show that a graph is dense (or sparse) in a particular dimension, one has to find a drawing in that dimension with the appropriate properties. For sensor networks, sometimes the natural drawing of a deployed network is sufficient for this purpose. By the natural drawing of a sensor network we mean the mapping from the nodes to their physical locations in the Euclidean space in which they are deployed. We can use this natural drawing to construct the following examples of dense and sparse graphs.

Deploy a countable number of nodes in so that the maximum uncovered diameter of its natural drawing is finite, and allow every pair of nodes whose Euclidean distance is no larger than to have an edge between them. The resulting graph is weakly connected and dense in . Such a graph is also sparse in if the nodes are placed such that every finite volume in contains a finite number of nodes.

Consider an initial deployment of nodes on a square lattice in , for which a fraction of the nodes has subsequently failed. Suppose that the number of nodes that failed in any given region is bounded by a linear function of the area of the region, i.e., that there exist constants and such that, for every region of area the number of nodes that failed in that region is no larger than . Assuming that , there will be an infinite connected component among the remaining nodes, which is dense and sparse in D. \frqed
The proof of the proposition above is provided in Appendix A.
The first example in the proposition is that of a geometric graph that is obtained by placing a number of nodes in a region and specifying a range such that a pair of nodes have an edge between them if and only if the Euclidean distance between them is no more than the given range. The second example refers to a network in which some of the initially deployed nodes have failed, with the stipulation that in large areas, no more than a certain fraction of the node may fail. For example, and satisfies the stated conditions. It can be shown that and means that in areas larger than , at most of the nodes may fail.
To show that a graph is not dense (or not sparse) in a particular dimension is harder since one has to show that no drawing with the required properties exists. Typically, this can be done by showing that the existence of a dense (or sparse) drawing leads to a contradiction. An application of this technique leads to the following result.

The dimensional lattice is not sparse in for every , and it is not dense in for every .

A regulardegree^{2}^{2}2A graph is called regulardegree if the degree of every node in the graph is the same. infinite tree is not dense or sparse in any dimension. \frqed
The first statement of the lemma is provided in Appendix A. The proof of the second statement is not provided since the method of the proof is similar.
We are now ready to prove Lemma IIA2.
IIA2 To prove the result by contradiction, suppose that a graph is dense in as well as sparse in , where . It follows from Theorems IIIA and IIIA that there exist positive integers such that and . It is straightforward to verify the following facts:

for every pair of graphs that do not have any parallel edges, for every positive integer .

for an arbitrary graph without parallel edges, and two positive integers , we have .
It follows that , which means, from sparse embedding Theorem IIIA, that a dimensional lattice is sparse in . This is a contradiction because of Lemma VI, which completes the proof.
Vii Summary and Future Work
In a large number of sensor and adhoc network applications, a number of node variables need to be estimated from measurements of the noisy differences between them. This estimation problem is naturally posed in terms of a graph.
We established a classification of graphs, namely, dense or sparse in , that determines how the optimal linear unbiased estimation error of a node grows with its distance from the reference node. The notion of denseness/sparseness introduced in this paper is distinct from the usual notion based on the average degree. In fact, we illustrated through examples that node degree is a poor measure of how the estimation error scales with distance.
The bounds and the associated graph classification derived here can be used in performance analysis, design and deployment of large networks. For example, if a sensor network is sparse in , then we know that the estimation error of a node will grow linearly with its distance from a reference. A large number of reference nodes will thus be needed for large networks that are sparse in . On the other hand, if one has control over the network deployment, then one should strive to obtain a network that is dense in with as large as possible. In the ideal case of , with a single reference node one can get bounded estimation error regardless of how large the network is.
There are several avenues for future research. The scaling laws described in this paper were derived for infinite measurement graphs. This is justified by the fact that the BLUE covariance of a node in an infinite graph is very close to the obtained in a large finite subgraph that contains the nodes and sufficiently inside it [7]. However, to gain a better understanding of the “boundary” effects that can occur in finite graphs, an interesting research direction is to determine how large the BLUE error covariance can be as a function of the size of the graph, for nodes that are close to the edge of the graph. A connection between the notions introduced in this paper and those in coarse geometry might be useful in this regard. It can be shown that a graph that is both sparse and dense in is coarsely equivalent to , which intuitively means that and are the same in their large scale structure (see [23] for a precise definition of coarse equivalence). Certain coarse geometric notions that were originally defined for infinite graphs have been extended to finite graphs (see [24]). This connection between coarse geometry and denseness/sparseness might provide a way to extend the techniques used in this paper to finite graphs.
Although the dense and sparse classification does allow randomness in the structure of the graph, the effect of such randomness on the scaling laws for the error is not explicitly accounted for in the present work. A useful research direction would be to characterize the estimation error covariances in graphs with random structure, such as random geometric graphs [25]. Another interesting avenue for future research is the investigation of estimation error growth in scalefree networks that do not satisfy the bounded degree assumption.
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