spatiotemporal information coupling in network navigation · network localization are derived in...

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 64, NO. 12, DECEMBER 2018 7759

Spatiotemporal Information Couplingin Network Navigation

Santiago Mazuelas , Senior Member, IEEE, Yuan Shen , Member, IEEE, and Moe Z. Win , Fellow, IEEE

Abstract— Network navigation, encompassing both spatial andtemporal cooperation to locate mobile agents, is a key enablerfor numerous emerging location-based applications. In suchcooperative networks, the positional information obtained byeach agent is a complex compound due to the interaction amongits neighbors. This information coupling may result in poor per-formance: algorithms that discard information coupling are ofteninaccurate, and algorithms that keep track of all the neighbors’interactions are often inefficient. In this paper, we develop aprincipled framework to characterize the information couplingpresent in network navigation. Specifically, we derive the equiv-alent Fisher information matrix for individual agents as the sumof effective information from each neighbor and the coupledinformation induced by the neighbors’ interaction. We furthercharacterize how coupled information decays with the networkdistance in representative case studies. The results of this papercan offer guidelines for the development of distributed techniquesthat adequately account for information coupling, and henceenable accurate and efficient network navigation.

Index Terms— Fisher information, localization, navigation,spatiotemporal cooperation, information coupling, inference.

I. INTRODUCTION

NETWORK NAVIGATION is an emerging paradigm forproviding location awareness of mobile nodes in a net-

work with unprecedented accuracy and reliability [1]–[3].This new paradigm will enable numerous future location-based applications such as healthcare monitoring, personnel/asset tracking, emergency evacuation, search/rescue opera-tions, autonomous vehicles, and military operations [4]–[12].

Manuscript received December 21, 2016; revised April 5, 2018; acceptedJuly 3, 2018. Date of publication July 24, 2018; date of current versionNovember 20, 2018. This work was supported in part by the Office of NavalResearch under Grant N00014-16-1-2141, in part by the MIT Institute forSoldier Nanotechnologies, in part by the Spanish Ministry of Economy andCompetitiveness through Ramon y Cajal under Grant RYC-2016-19383 andin part by the National Natural Science Foundation of China under Grant61501279. This paper was presented at the 2012 IEEE Global Communica-tions Conference.

S. Mazuelas was with the Wireless Information and Network Sciences Lab-oratory, Massachusetts Institute of Technology, Cambridge, MA 02139 USA.He is now with the Basque Center for Applied Mathematics (BCAM),IKERBASQUE–Basque Foundation for Science, 48009 Bilbao, Spain (e-mail:[email protected]).

Y. Shen was with the Wireless Information and Network Sciences Labo-ratory, Massachusetts Institute of Technology, Cambridge, MA 02139 USA.He is now with the Department of Electronic Engineering, Tsinghua Uni-versity, and Beijing National Research Center for Information Science andTechnology, Beijing 100084, China (e-mail: [email protected]).

M. Z. Win is with the Laboratory for Information and DecisionSystems Laboratory, Massachusetts Institute of Technology, Cambridge,MA 02139 USA (e-mail: [email protected]).

Communicated by R. La, Associate Editor for Communication Networks.Color versions of one or more of the figures in this paper are available

online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIT.2018.2859330

Fig. 1. A network navigation scenario where blue dots represent differentstates, while edges represent pairs of states that share information. Positionalinformation for state ι is obtained from neighbors i1, i2, i3, and i4 thatlikewise obtain information from states in components C1, C2, C3, and C4,respectively. The double slash denotes an interaction set, as will be definedin Section III.

These potential applications have motivated increasingresearch interest in localization and tracking technologies inthe past decade [13]–[26].

In classic localization techniques, the states of themobile agents are inferred from the measurements betweenanchors (with known positions) and agents [27]–[30]. Thestate usually includes the agent position and possibly othermobility parameters such as velocity and acceleration, whiletypical measurements for localization are inter-node distancesand angles-of-arrival. However, in harsh propagation environ-ments without sufficient line-of-sight measurements betweenthe anchors and agents, these techniques yield unsatisfactoryperformance or even completely fail [31]–[34]. For exam-ple, the global positioning system (GPS)-based navigationhardly works in indoor environments [35]–[37]. Other local-ization techniques use intra-node inertial measurements of anagent to determine its moving trajectory [38]–[42], whichis commonly known as dead-reckoning. These techniquessuffer from cumulative errors (positional drift) and thus loselocalization accuracy over time. The above limitations haveresulted in remarkable ongoing research efforts to overcomethe drawbacks of harsh propagation environments for anchor-based techniques [43]–[46] and cumulative errors for inertial-based techniques [38]–[40]. However, the performance of suchtechniques still relies on high-density anchor deployment,high-power anchors, or high-grade inertial measurementunits—none of which are cost-effective solutions—to achievereliable and accurate ubiquitous localization.

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https://orcid.org/0000-0002-6608-8581

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7760 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 64, NO. 12, DECEMBER 2018

Driven by the success in wireless communica-tions [47]–[50], cooperative techniques have been introducedto improve the localization accuracy by sharing informationand performing measurements among spatial neigh-bors [51]–[63]. In particular, network navigation providesa new framework in which agents exploit both spatial andtemporal cooperation to infer their states [1]–[3]. Under thisframework, each agent obtains information about its statefrom prior knowledge, inter- and intra-node measurements,and messages from neighbors (see Fig. 1). It was shown thatthe spatiotemporal cooperation can significantly improve thelocalization accuracy by providing additional information.

In terms of implementation, distributed algorithms that donot rely on a central processor are often preferred for networknavigation since they can reduce communication requirementsand increase system robustness [64]–[66]. The main difficultyfor the distributed schemes lies in that the information foreach state depends not only on its interaction with neigh-boring states but also on the interaction of those neigh-bors among themselves [67]–[71].1 The interaction amongneighbors causes information coupling, that is, the positionalinformation of a state is not the sum of effective informa-tion obtained from each neighbor. Such information couplinghinders the development of efficient and accurate distributedalgorithms since monitoring completely the interaction amongneighbors requires a difficult network coordination.

Most current distributed algorithms either simply discard theinformation coupling or use ad-hoc methods to keep track ofneighbors’ interaction, resulting in sub-optimal performances.For instance, techniques based on belief propagation (BP)may fail to obtain accurate solutions due to the presence ofcycles in the associated inference graph [51]–[53] (see alsoSection VI-A). In fact, loopy BP algorithms only obtainapproximate beliefs since the underlying independenceassumptions do not hold in general and iterative algorithmsmay use the same measurements repeatedly. The informationcoupling phenomenon has also been observed using otheralgorithms and some mitigation techniques have been pro-posed. Specifically, it was observed in [72]–[74] that ignoringthe cross-correlations among positional estimates can havea negative effect, often leading to biased position estimates.Those studies explore mitigation methods that either preventcertain cooperations [53] or perform a careful bookkeepingof cross-correlations and the origins of the used data [74].To fully unleash the potential of network navigation, the designof distributed algorithms requires a principled characterizationof the information coupling due to the complex informationdynamics induced by spatiotemporal cooperation.

Based on Fisher information analysis, the fundamentallimits or the performance bounds of localization accuracy fornetwork localization are derived in [55] and then the resultsare extended to network navigation in [2].2 In particular, since

1A neighboring state refers to the state of either a spatial neighbor or atemporal neighbor, where a temporal neighbor means the same agent at aconsecutive time step.

2The Fisher information is the most common metric to characterize theperformance limit of inference problems in terms of the information inequal-ity or Cramér-Rao lower bound (CRLB) [75].

the Fisher information matrix (FIM) for network navigationis non-diagonal [55], the equivalent FIM (EFIM) for a stateis not the sum of effective information obtained from eachneighbor, and depends also on the coupled information due tothe interaction among neighbors. Recent studies analyzed theposition error propagation in spatial and temporal domainsfor cooperative localization [68], [69], where the recursiveexpressions of the EFIM are obtained by approximating part ofthe non-diagonal terms in the FIM. To the best of the authors’knowledge, the only works that have studied informationcoupling are [70] and [71], which focus on simple networkswith three or four cooperating agents.

In this paper, we characterize the effect of informationcoupling arising from neighbors’ interaction in general net-works, and derive the closed-form expressions for the coupledinformation. Specifically, the main contributions of the paperare as follows.

• We derive the EFIM for individual states in navigationnetworks and the effective information obtained fromeach neighbor.

• We characterize the coupled information induced bysets of pair-wise information links and determine howinformation coupling decays with the network distance.

• We derive the closed-form expression of the coupledinformation arising from neighbors’ interaction throughone information link.

• We determine the effect of the information coupling inrepresentative case studies through analytical expressionsand numerical results.

• We show the impact of information coupling in theperformance of distributed algorithms, and outline designguidelines for coupling-aware algorithms inspired by thetheoretical results.

The rest of the paper is organized as follows. Section IIdescribes the FIM for network navigation and the graphstructure induced by the FIM in the set of states. Section IIIderives the EFIMs for individual states and characterizes theinformation coupling. Section IV specifies the expressions forcoupled information in representative case studies. Section Vprovides numerical results for coupled information in severalscenarios, and Section VI shows the relevance of the resultspresented for distributed algorithms. Finally, conclusions aredrawn in Section VII.

Notations: The notation [ · ]T denotes the transpose of itsargument. Random variables are shown in sans serif, uprightfonts; their realizations in serif, italic fonts. Vectors aredenoted by bold lowercase letters. For example, a randomvariable (RV) and its realization are denoted by x and x; arandom vector and its realization are denoted by x and x. Thefunction fx(x) and, for brevity when possible, f(x) denotethe probability density function (PDF) of a continuous RV x;fx|y(x|y) and, for brevity when possible, f(x|y) denote thePDF of x conditional on y = y. eS

i denotes the S-dimensionalvector with all zeros except a one at ith element. I denotesa generic identity matrix and IS denotes a S × S identitymatrix. ⊗ denotes the Kronecker product. ‖A‖∗ denotesthe nuclear norm of a matrix A. diag{M1, M2, . . . , Mk}denotes the block diagonal matrix formed by concatenating the

MAZUELAS et al.: SPATIOTEMPORAL INFORMATION COUPLING IN NETWORK NAVIGATION 7761

matrices Mi, i = 1, 2, . . . , k. SD+ and S

D++ denote the set of

D×D real positive semi-definite and positive definite matrices,respectively. For A, B ∈ S

D+ , A ≺ B, A � B, A � B, and

A � B denote, respectively the fact that B − A ∈ SD++,

B − A ∈ SD+ , A − B ∈ S

D++, and A − B ∈ S

D+ .

II. FISHER INFORMATION IN NETWORK NAVIGATION

This section describes the FIM for network navigationand the graph structure that such FIM induces in the set ofstates. These matrices and graph structure are used in thesubsequent sections to characterize the information couplingvia the EFIMs.

A. Preliminaries

Consider a cooperative network consisting of K nodes.Let x

(n)k ∈ R

D be the state of node k at time tn fork = 1, 2, . . . , K and n = 1, 2, . . . , N . The goal of networknavigation is to infer the states of the nodes from measure-ments and prior knowledge.3 Let S = {1, 2, . . . , S} withS = KN be an index set of all the states. For notationalconvenience, we refer to the state of node k at time step tneither by x

(n)k or by its index i ∈ S.

It has been proven [2] that the FIM for the S states can bedecomposed as

J =∑

(i,j)∈S2,i�j

GSi,j ⊗ Ki,j (1)

where

GSi,j =

{(eS

i − eSj

)(eS

i − eSj

)T, i �= j

eSi

(eS

i

)T, i = j.

(2)

The matrix Ki,j ∈ SD+ accounts for the pair-wise positional

information from the measurements or prior knowledge relatedjointly to the states i and j. For example, if yi,j are themeasurements related to states xi and xj , i, j ∈ S,

Ki,j = E

{−∂2 ln f(yi,j |xi, xj)

∂xi∂xTi

}. (3)

As special cases, Ki,j is the zero matrix in the absence ofsuch measurements or prior knowledge, and Ki,i accountsfor the self positional information related only with state i,e.g., prior knowledge or information from anchors. Moreover,we assume that measurements and prior knowledge are relatedto the states pair-wise differences, and hence Ki,j = Kj,i.

Proposition 1: Each term GSi,j ⊗ Ki,j in (1) is a positive

semidefinite matrix that has rank equal to the rank of Ki,j .In addition, if Ki,i � 0 for all i ∈ S then J � 0.4

Proof: See Appendix A. �Let A be a subset of the natural numbers, M be a matrix

of size |A|D × |A|D, so that we can associate each element

3The state includes position and possibly other mobility parameters suchas velocity, acceleration, orientation, and angular velocity; examples ofmeasurement sensors include RF radios and inertial devices; and examplesof prior knowledge include positions of certain nodes and mobility models.

4In the following, we assume Ki,i � 0 for all i ∈ S , while Ki,j for i �= jare not necessarily full rank.

in A with a D× |A|D block-row or |A|D×D block-columnof M . We denote by MA1,A2 the sub-matrix of M formedby the blocks corresponding to rows associated with A1 andcolumns associated with A2 for A1,A2 ⊆ A. For instance,in a navigation network with three states, J{1},{2,3} denotesthe upper right D × 2D block of the FIM.

In the example of Fig. 1, the FIM for all the states can bewritten as

J =

⎡

⎢⎢⎢⎢⎣

Jι,ι Jι,C1 Jι,C2 Jι,C3 Jι,C4

JC1,ι JC1,C1 JC1,C2 0 0JC2,ι JC2,C1 JC2,C2 0 0JC3,ι 0 0 JC3,C3 0JC4,ι 0 0 0 JC4,C4

⎤

⎥⎥⎥⎥⎦(4)

where

Jι,ι = Kι,ι +4∑

k=1

Kι,ik(5)

Jι,Ck= JT

Ck,ι = −[ Kι,ik0 · · · 0 ] (6)

for k = 1, 2, 3, 4, and

[JC1,C2 ]j1,j2= [JC2,C1 ]j2,j1

= −Kj1,j2

[JC1,C2 ]i,j = [JC2,C1 ]j,i = 0 (7)

for (i, j) �= (j1, j2).Based on the structure of the FIM, we next introduce a

graph structure for the set of states, which will be used todetermine the interrelationship among the different informa-tion components.

B. Structure of the States from FIM

The FIM induces a graph-structure for the set of statesthat we refer to as the navigation information graph (NIG).Its vertices correspond to the states, and its edges (links)correspond to the pairs of states (i, j) ∈ S2 for which Ki,j �= 0(see Fig. 1).5 Based on the terminologies in graph theory,we next give several definitions for the NIG.

Definition 1: The matrix GSi,j ⊗Ki,j is the information link

related to the states i and j for (i, j) ∈ S2, and the matrixKi,j ∈ S

D+ is the capacity of the link. Two states i �= j are

neighbors if Ki,j �= 0, and Ni denotes the set of states thatare neighbors of state i.

Definition 2: Two states i and j are connected in a set ofstates S̄ ⊂ S if there exists a path between i and j in S̄,i.e., a sequence of states {k1, k2, . . . , kr+1} ⊂ S̄ with k1 = iand kr+1 = j such that Kks,ks+1 �= 0 for s = 1, 2, . . . , r.The network distance between two connected states i and j,denoted by ND(i, j), is the length of the shortest path betweeni and j. A connected component of S̄ is a maximal subset ofstates connected in S̄.

In this paper, without loss of generality we focus on aspecific state ι ∈ S. The set S̄ = S \ {ι} can be uniquelypartitioned as the disjoint union of connected components.In the network example shown in Fig. 1, S̄ has three connectedcomponents (i.e., (C1∪C2), C3, and C4) if Kj1,j2 �= 0 and four

5Notice that the FIM in (1) corresponds to the Laplacian of the undirectedgraph with vertices corresponding to the states and edges weighted by thesemidefinite matrices Ki,j [76].


connected components (i.e., C1, C2, C3, and C4) if Kj1,j2 = 0.In the former case the component (C1 ∪ C2) contains twoneighbors of ι, and in the latter case each component containsonly one neighbor of ι.

In the next section, we characterize the effective informationobtained from each neighbor as well as the coupled informa-tion due to neighbors’ interaction.

III. INFORMATION COUPLING IN NETWORK NAVIGATION

This section first derives the EFIM for individual states andcharacterizes the effect of the information coupling inducedby sets of information links. Then, we determine the decayof information coupling with the network distance and theclosed-form EFIM for networks coupled by one informationlink.

A. EFIM for Individual States

The FIM J in (1) captures the information for all the statesin S, and thus determines the performance bound for jointestimation of the states. To further study information coupling,we will next adopt the notion of the EFIM [32]. Such matrixrepresents the equivalent information for states’ subsets andshows how coupled information arises as a consequence ofother states’ interaction.

Definition 3 (EFIM [32]): Given a parameter vectorθ = [ θT

1 θT2 ]T and the FIM Jθ of the form

Jθ =[

A B

BT C

](8)

where θ ∈ RN , θ1 ∈ R

n, A ∈ Rn×n, B ∈ R

n×(N−n), andC ∈ R

(N−n)×(N−n) with 1 � n < N , the EFIM for θ1 isgiven by

Je(θ1) := A − BC−1BT. (9)

Note that the right-hand side of (9) is known as the Schurcomplement of block-matrix A in matrix Jθ [77]. It isclear that the EFIM retains all the necessary information toderive the information inequality for the parameter θ1. In otherwords, the CRLB for θ1 can be obtained by the inverseof Je(θ1) rather than directly inverting a high-dimensionalmatrix Jθ . More importantly, the use of the EFIM allows tocharacterize all the information components of each individualstate, which is essential for studying the effect of informationcoupling.

The following theorem shows that the EFIM Je for a genericstate ι ∈ S can be decomposed as a sum of terms, eachcorresponding to a connected component of S̄ that containsneighbors of ι.

Theorem 1: Let S̄ = C1∪C2∪. . .∪Cn be a disjoint partitionof S̄ as a union of connected components, where only the firstm components have nonempty intersection with Nι, i.e., Ck ∩Nι = ∅ for m < k � n. Then the EFIM for state ι is

Je = Kι, ι +m∑

k=1

[ ∑

i∈Nι∩Ck

Kι, i − Jι,Nι∩CkΛk JT

ι,Nι∩Ck

]

(10)

where Λk =[J−1Ck,Ck

]Nι∩Ck,Nι∩Ck

.

Proof: From the definition of the EFIM, we have that

Je = Kι,ι +∑

i∈Nι

Kι,i − [ Jι,C1 Jι,C2 · · · Jι,Cn ]

· J−1S̄,S̄ [ Jι,C1 Jι,C2 · · · Jι,Cn ]T

= Kι,ι +∑

i∈Nι

Kι,i − [ Jι,C1 Jι,C2 · · · Jι,Cm ]

· diag{J−1C1,C1

, J−1C2,C2

, . . . , J−1Cm,Cm

}

· [ Jι,C1 Jι,C2 · · · Jι,Cm ]T

= Kι,ι +∑

i∈Nι

Kι,i −m∑

k=1

Jι,CkJ−1Ck,Ck

JTι,Ck

(11)

where the second equality is due to

JS̄,S̄ = diag{JC1,C1 , JC2,C2 , . . . , JCn,Cn

}(12)

and Jι,Ci = 0 for i > m. The result (10) is obtained byobserving that

Jι,CkJ−1Ck,Ck

JTι,Ck

= Jι,Nι∩CkΛk JT

ι,Nι∩Ck(13)

and that the non-zero blocks of Jι,Ckare those corresponding

to Jι,Nι∩Ck. �

The theorem shows that the information obtained by state ιis the sum of the self positional information and that obtainedfrom each connected component, where the latter is thedifference between the capacity of the links to the neighbors insuch a component and a term that accounts for the uncertaintyof neighbors in the component.

Definition 4: The neighbors of a state ι ∈ S in Nι areisolated if each connected component of S̄ contains at mostone neighbor of ι, i.e., for each connected component Ck, |Ck∩Nι| � 1.

In the network example in Fig. 1, the neighbors of state ι areisolated if Kj1,j2 = 0. The following result shows that whenthe neighbors are isolated, the information obtained throughcooperation can be decoupled as the sum of the informationobtained from each neighbor.

Corollary 1: If the neighbors of state ι are isolated, thenthe EFIM for state ι is

Je = Kι,ι +∑

i∈Nι

(I − Kι,iΛki)Kι,i (14)

where ki is the index of the component corresponding toneighbor i, i.e., Cki ∩Nι = {i}.

Proof: The proof is a direct consequence of Theorem 1since Jι,Nι∩Cki

= Jι,i = −Kι,i if i ∈ Nι and Nι ∩ Cki = ∅

in other case. �This result shows that when the neighbors are isolated,

the information obtained through cooperation is the sum of theinformation obtained from each neighbor. In addition, each ofthose terms depends on the capacity of the link to the neighbor,i.e., Kι,i, and on the neighbor’s uncertainty induced by itscomponent, i.e., Λki . Such amenable situation characterizesuncoupled neighbors.

Definition 5: The neighbors of state ι ∈ S in Nι areuncoupled if

Je = Kι,ι +∑

i∈Nι

Ki→ι (15)


with

Ki→ι � (I − Kι,iΛki) Kι,i (16)

representing the effective information that can be obtainedfrom neighbor i taking into account its own uncertainty. Inaddition, we say that a state is subject to information couplingwhen its neighbors are not uncoupled.

B. Information Coupling and Effective Information

The results below further characterize the terms correspond-ing to the information that can be effectively obtained fromeach neighbor as well as the information coupling.

Definition 6: Let B be a set of links between the states, theB-reduced FIM is defined as

JB =∑

(i,j)∈S2\B,i�j

GSi,j ⊗ Ki,j

= J −∑

(i,j)∈B,i�j

GSi,j ⊗ Ki,j . (17)

Then, the B-effective information from i ∈ Nι to ι is definedas

KBi→ι =

(I − Kι,i ΛB

ki

)Kι,i (18)

where

ΛBki

=[(

JBCki

,Cki

)−1]

i,i(19)

and i ∈ Cki ∩ Nι. Moreover, Ki→ι = KBi→ι when B = ∅.

Definition 7: A set of links between the states in S̄,B ⊂ S̄2, is an interaction set for state ι if the neighbors ofstate ι are isolated for the FIM JB but not for the FIM J .

Remark 1: This definition characterizes the sets of linkssuch that their removal isolates the neighbors of a state,since the matrix JB corresponds with the FIM for S whenKj1,j2 = 0 for all (j1, j2) ∈ B [see (17)]. For instance, in thenetwork shown in Fig. 1, the set (j1, j2) is an interaction setcontaining a single link.

Proposition 2: Let B be an interaction set for state ι. TheEFIM for state ι is bounded as

Kι,ι +∑

i∈Nι

KBi→ι � Je � Kι,ι +

∑

i∈Nι

Kι,i. (20)

Proof: The first inequality is a consequence of Corollary 1since J � JB implies that Λk � ΛB

k for any k. The secondinequality is a direct consequence of Theorem 1 since thematrices Λk are positive definite. �

Proposition 2 shows that the information obtained fromcooperation by each state is upper bounded by the sum ofthe capacities of the neighboring links and lower bounded bythe sum of the B-effective information from neighbors for anyinteraction set B. Moreover, such lower bound becomes anequality for B = ∅ if the neighbors of state ι are isolated. Notethat the effective information for cooperative localization wasfirst introduced in [55], B-effective information proposed inthis paper is a generalization of that in [55], which correspondsto the specific interaction set B = {(i, j) ∈ (S \ {ι})2 fori > j}.

Remark 2: Proposition 2 has the following operationalmeaning: discarding several measurements or prior knowl-edges, i.e., letting several link capacities Ki,j = 0, decouplesthe neighbors of state ι. Although such an operation reducesthe total information for state ι, it decomposes the informationfor state ι as a sum of that from each neighbor, which isamenable for distributed algorithms.

The next proposition shows that the effective informationis always smaller than or equal to the link capacity and thatthe effective information increases when the interaction set Bdecreases.

Proposition 3:

1) If B and B′ are two sets of links between the states withB ⊆ B′, then for any i ∈ Nι

0 � KB′i→ι � KB

i→ι � Ki→ι � Kι,i. (21)

2) If Ki,i = ξiI , for any set of links Blim

ξi→∞KB

i→ι = Kι,i. (22)

Proof: See Appendix C. �Proposition 4: If Kι,i is rank-one, i.e., Kι,i = vι,iv

Tι,i for

vι,i ∈ RD, the B-effective information KB

i→ι is proportionalto Kι,i, KB

i→ι = ξBι,i Kι,i with 0 < ξBι,i < 1 for any set B, andgiven by

ξBι,i = 1 − vTι,iΛ

Bki

vι,i. (23)

Proof: Using the definition of effective information,we have that

KBi→ι =

(I − vι,iv

Tι,iΛ

Bki

)vι,iv

Tι,i

=(1 − vT

ι,i ΛBki

vι,i

)Kι,i. (24)

The remaining part of the corollary, i.e., ξBι,i > 0, is provenusing the Lemma 1 in Appendix B with C = JB

Cki,Cki

and

A = vι,ivTι,i + Ki,i. Then, the matrix [(JB

Cki,Cki

)−1]i,i can bewritten as (vι,iv

Tι,i+Ki,i+R)−1 with Ki,i +R � 0 for some

matrix R. Hence, the result is obtained since

ξBι,i =(1 + vT

ι,i(Ki,i + R)−1vι,i

)−1(25)

by using the matrix inversion lemma. �Remark 3: The effective information a state obtains from a

neighbor is a positive semidefinite matrix smaller than or equalto the link capacity, and it is proportional to such capacity ifit is a rank-one matrix. In addition the B-effective informationincreases with smaller sets B. Finally, the effective informationequals the link capacity when the self positional informationof the neighbor tends to infinity.6

The next definition quantifies the information couplinginduced by an interaction set as the difference between theactual EFIM and the sum of effective information from eachneighbor.

6We refer to the matrices Ki,j as link capacities since they play a similarrole to the capacities of network flows as upper bounds of flows throughlinks [78].


Definition 8: Let B be an interaction set for state ι, the cou-pled information for state ι induced by the links B is thepositive semidefinite matrix

ΞB = Je − JBe = Je − Kι,ι −

∑

i∈Nι

KBi→ι � 0. (26)

where JBe is the EFIM for state ι based on the B-reduced

FIM JB .

C. Information Coupling Decay with Network Distance

This section shows how the information induced by a linkchanges with the network distance. These results will be usedin Section IV-D to show the exponential decay of informationcoupling with the network distance in a simple network. Here,the first result shows the difference in the EFIM, caused by thepresence of a specific link, for states with increasing networkdistance to such link. The second result shows how the EFIMdifference decreases with the network distance.

We consider subsets of the states with increasing networkdistance to a link B = {(j1, j2)}. Specifically, Sk is the set ofthe states at network distance at least k to j1 and j2, and Ek isthe set of the states in Sk at network distance k to j1 or j2, i.e.,

Sk ={i ∈ S : ND(i, j1) � k, ND(i, j2) � k

}

Ek ={i ∈ S : min{ND(i, j1), ND(i, j2)} = k

}. (27)

The EFIM difference due to information link B is given in thefollowing theorem.

Theorem 2: Let Jk and JBk be the EFIMs of states Sk for

J and JB, respectively. Then

[Jk]i,j =

{[JB

k

]i,j

if (i, j) /∈ E2k[

JBk

]i,j

+ [Dk]i,j if (i, j) ∈ E2k

(28)

with

Dk = ΓkUj1,j2

(I + UT

j1,j2ΔkUj1,j2

)−1UT

j1,j2ΓTk

� ΓkKj1,j2ΓTk (29)

where Uj1,j2 is a square root of Kj1,j2 , i.e., Kj1,j2 =Uj1,j2U

Tj1,j2 , Γ0 = [I,−I]T, Δ0 = 0, and

Γk = JEk,Ek−1

([JB

k−1

]Ek−1,Ek−1

)−1

Γk−1

Δk = Δk−1 + Γ Tk−1

([JB

k−1

]Ek−1,Ek−1

)−1

Γk−1 (30)

for k = 1, 2, . . . , n, with n being the largest integer for whichSn �= ∅.

Proof: See Appendix D. �Remark 4: Theorem 2 shows that the difference between

the EFIMs with and without a specific link (j1, j2) dependson the link capacity Kj1,j2 and the network distance k to thelink (j1, j2) through the recursively defined matrices Γk.

The following result shows that the difference betweenEFIMs for a network distance k + 1 is smaller than thedifference between EFIMs for a network distance k multipliedby the matrices that define the recurrence for Γk.

Corollary 2: Under the conditions of Theorem 2, we havethat for k = 0, 1, . . . , n

Dk+1 � JEk+1,Ek

([JB

k

]Ek,Ek

)−1

Dk

([JB

k

]Ek,Ek

)−1

JEk,Ek+1 .

(31)

Proof: The result is a straightforward consequence of theprevious result taking into account that(I + UT

j1,j2Δk+1Uj1,j2

)−1 �(I + UT

j1,j2ΔkUj1,j2

)−1(32)

since Δk � Δk+1. �These two results illustrate how the information induced by

sets of links decay with the network distance in the network.The next section further explores information coupling fornetworks where there is one coupling link.

D. Information Coupling for One-Link Coupled Neighbors

The detailed closed-form expression of the EFIM for gen-eral networks is too complex. We present such EFIM for thecase where there is an interaction set formed by one link.

Theorem 3: Let B = {(j1, j2)} be an interaction set forstate ι and C1, C2, . . . , Cn be a partition of S̄ into connectedcomponents for the structure induced by the B-reduced FIMJB. If {i1, i2} ⊆ Nι with i1, j1 ∈ C1 and i2, j2 ∈ C2, then theEFIM for state ι is

Je = Kι,ι +∑

i∈Nι

KBi→ι + Ξ

(j1,j2)i1,i2

(33)

where

Ξ(j1,j2)i1,i2

= Γ[K

(Φ−1 + K

)−1Φ−1

]Γ T (34)

with

K = Kj1,j2

Φ =[(

JBC1,C1

)−1]

j1,j1+

[(JBC2,C2

)−1]

j2,j2

Γ = Kι,i1

[(JBC1,C1

)−1]

i1,j1− Kι,i2

[(JBC2,C2

)−1]

i2,j2. (35)

Proof: See Appendix E. �Note that the coupled information Ξ

(j1,j2)i1,i2

in (34) is a func-

tion of K, Φ and Γ , and we rewrite it as Ξ(j1,j2)i1,i2

(K, Φ, Γ )in the following corollary to show the monotonicity of thecoupled information with the variables.

Corollary 3: Under the conditions of Theorem 3,1) if K′ � K,

Ξ(j1,j2)i1,i2

(K ′, Φ, Γ

)� Ξ

(j1,j2)i1,i2

(K, Φ, Γ

); (36)

2) if Φ′ � Φ,

Ξ(j1,j2)i1,i2

(K, Φ′, Γ

)� Ξ

(j1,j2)i1,i2

(K, Φ, Γ

); (37)

3) if Γ ′ = λΓ with λ � 1,

Ξ(j1,j2)i1,i2

(K, Φ, Γ ′) � Ξ

(j1,j2)i1,i2

(K, Φ, Γ

). (38)

Proof: For the first two cases, notice that(I − Kj1,j2

(Φ−1 + Kj1,j2)−1)Kj1,j2 is the Schur complement of the

matrix

M =[

Kj1,j2 Kj1,j2

Kj1,j2 Φ−1 + Kj1,j2

].

Then the results are obtained by observing that K′j1,j2 �

Kj1,j2 or Φ′ � Φ implies that the corresponding matrixM ′ satisfies M ′ � M . The last case is straightforwardsince λ2 � 1. �

Remark 5: Theorem 3 and Corollary 3 show that the infor-mation obtained from cooperation is the sum of effective


Fig. 2. Three representative simple scenarios of network navigation.(a): Three nodes cooperate via inter-node measurements; (b): Two nodescooperate at two different times via inter- and intra-node measurements;(c): Three nodes cooperate at two different times via inter- and intra-nodemeasurements. The double slash denotes the interaction set.

information from each neighbor plus the coupled informationby the link in the interaction set. Such coupled informationdepends on the link capacities between state ι and the coupledneighbors, as well as on the link capacity of the interac-tion set. In addition, the coupled information increases when1) the capacity of the link in the interaction set increases,2) the uncertainty of the interaction set decreases, and 3) thecapacities of the links to the coupled neighbors increaseproportionally with the same ratio.

Corollary 4: Under the conditions of Theorem 3, if Kι,i1 =vι,i1v

Tι,i1

, Kι,i2 = vι,i2vTι,i2

, and Kj1,j2 = vj1,j2vTj1,j2

, then

Ξ(j1,j2)i1,i2

= δj1,j2(μi1vι,i1 − μi2vι,i2)(μi1vι,i1 − μi2vι,i2)T

(39)

where

δj1,j2 =(1 + vT

j1,j2 Φ vj1,j2

)−1

μi1 = vTι,i1

[(JBC1,C1

)−1]

i1,j1vj1,j2

μi2 = vTι,i2

[(JBC2,C2

)−1]

i2,j2vj1,j2 . (40)

Proof: See Appendix F. �Remark 6: This result shows that when the information

capacities of the coupled neighbors and interaction set arerank-one matrices, the coupled information is also a rank-one matrix. In addition, such matrix is the outer product ofa combination of the vectors forming the link capacity fromthe neighbors weighted by the coefficients μi1/

√δj1,j2 and

−μi2/√

δj1,j2 .

The next section focuses on four representative cases andderives the corresponding EFIMs. In Section V these scenariosare further examined and the coupled information is numeri-cally quantified.

IV. CASE STUDIES

This section presents the EFIMs for a state in the scenariosdepicted in Fig. 2 and Fig. 3 to provide insights into thespatiotemporal information coupling arising in network navi-gation. These scenarios are composed by small-scale networkswith 3, 4, and 5 states as well as by a simple network with2n + 1 states.

In the following, the link capacities Ki,j for i �= jaccount for the positional information obtained from the inter-or intra-node measurements. When inter-node measurements

Fig. 3. A simple scenario of network navigation in which all the nodes arelinked by a single loop. The double slash denotes a potential interaction set.

are related to the distances between pairs of nodes andintra-node measurements are related to the speed of nodes,each Ki,j for i �= j is a rank-one matrix. Specifically,Ki,j = vi,jv

Ti,j , and vi,j is a vector with direction joining

states i and j and norm√

λi,j ∈ R+, where λi,j is called theranging information intensity (RII) [32] and speed informationintensity (SII) [71] for inter- and intra-node measurements,respectively.

A. Scenario A

Three nodes cooperate to determine their positions via inter-node measurements, e.g., time-of-arrival (TOA) or angle-of-arrival (AOA), as described in Fig. 2(a). In this case, the statesare x1, x2, and x3 (denoted by states 1, 2, and 3, respectively),which represent the positions of the three nodes, and the FIMbecomes7

J =

⎡

⎣J{1},{1} −K1,2 −K1,3

−K1,2 J{2},{2} −K2,3

−K1,3 −K2,3 J{3},{3}

⎤

⎦ (41)

where

J{1},{1} = K1,1 + K1,2 + K1,3

J{2},{2} = K2,2 + K1,2 + K2,3

J{3},{3} = K3,3 + K1,3 + K2,3. (42)

As a special case of Theorem 3 with interaction set B ={(2, 3)}, the EFIM for x1 in Scenario A is (see Appendix G)

Je = K1,1 + K(2,3)2→1 + K

(2,3)3→1 + Ξ

(2,3)2,3 (43)

where

Ξ(2,3)2,3 = Γ

(K2,3

(Φ−1 + K2,3

)−1Φ−1

)Γ T

Γ = K1,2(K2,2 + K1,2)−1 − K1,3(K3,3 + K1,3)−1

Φ = (K2,2 + K1,2)−1 + (K3,3 + K1,3)−1 (44)

and

K(2,3)i→1 =

(I − K1,i(Ki,i + K1,i)−1

)K1,i

= K1,i(Ki,i + K1,i)−1Ki,i. (45)

The EFIM in (43) is the sum of four terms. The first termcorresponds to the information that depends only on node 1,and the remaining three terms correspond to the informa-tion obtained through spatial cooperation. The second and

7Notice that in this scenario for notational convenience we do not specifythe time index because there is only one time step involved.


third terms correspond to the decoupled effective informationobtained from neighbor nodes 2 and 3, respectively, whilethe last term represents the coupled information due to thecooperation between neighbor nodes 2 and 3. The coupledinformation depends on the capacity of the link betweennodes 2 and 3 as well as the uncertainties of nodes 2 and 3.

If the link capacities are rank-one matrices, i.e., Ki,j =vi,jv

Ti,j with vi,j ∈ R

D for i, j ∈ {1, 2, 3}, i �= j, then theEFIM Je can be simplified as (see Appendix G)

Je = K1,1 + ξ1,2K1,2 + ξ1,3K1,3

+ δ2,3(μ2v1,2 − μ3v1,3)(μ2v1,2 − μ3v1,3)T (46)

where

ξ1,i =(1 + vT

1,iK−1i,i v1,i

)−1

μi = ξ1,i

(vT

1,iK−1i,i v2,3

)

δ2,3 =(1+vT

2,3

((K2,2+K1,2)−1+(K3,3 + K1,3)−1

)v2,3

)−1.

(47)

In this special case, each effective information for node 1 isproportional to the corresponding link capacity. The coupledinformation is also a rank-one matrix with positive eigen-value δ2,3‖μ2v1,2 − μ3v1,3‖2 and corresponding eigenvectorμ2v1,2−μ3v1,3, both depending on the RIIs, the uncertaintiesof nodes 2 and 3, and the nodes’ spatial topology.

B. Scenario B

Two nodes cooperate at two consecutive time steps to deter-mine their positions via inter- and intra-node measurements,as described in Fig. 2(b). In this case, the states are x

(2)1 , x

(2)2 ,

x(1)1 , and x

(1)2 (denoted by states 1, 2, 3, and 4, respectively),

which represent the states of the two nodes at two time steps.Note that K1,4 = K2,3 = 0, and hence the FIM becomes

J =

⎡

⎢⎢⎣

J{1},{1} −K1,2 −K1,3 0−K1,2 J{2},{2} 0 −K2,4

−K1,3 0 J{3},{3} −K3,4

0 −K2,4 −K3,4 J{4},{4}

⎤

⎥⎥⎦ (48)

where

J{1},{1} = K1,1 + K1,2 + K1,3

J{2},{2} = K2,2 + K1,2 + K2,4

J{3},{3} = K3,3 + K1,3 + K3,4

J{4},{4} = K4,4 + K2,4 + K3,4. (49)

As a special case of Theorem 3 with interaction set B ={(4, 3)},8 the EFIM for x

(2)1 in Scenario B is (see Appendix G)

Je = K1,1 + K(4,3)2→1 + K

(4,3)3→1 + Ξ

(4,3)2,3 (50)

where

Ξ(4,3)2,3 = Γ

(K4,3

(Φ−1 + K4,3

)−1Φ−1

)Γ T

Γ = K1,2Ω−12 K2,4(K4,4 + K2,4)−1

−K1,3(K3,3 + K1,3)−1

Φ = Ω−14 + (K3,3 + K1,3)−1

8Note that one can also consider the interaction set to be B = {(4, 2)}which leads to analogous results.

K(4,3)2→1 =

(I − K1,2Ω

−12

)K1,2

K(4,3)3→1 =

(I − K1,3(K3,3 + K1,3)−1

)K1,3 (51)

andΩ2 = K2,2 + K1,2 +

(I − K2,4(K4,4 + K2,4)−1

)K2,4

Ω4 = K4,4 +(I − K2,4(K2,2 + K1,2 + K2,4)−1

)K2,4.

(52)The EFIM in (50) is the sum of four terms. The first term

corresponds to the information that depends only on node 1at time step t2, and the remaining three terms correspond tothe information obtained through spatiotemporal cooperation.The second and third terms correspond to the effective infor-mation obtained from spatial neighbor (node 2 at time step t2)and temporal neighbor (node 1 at time step t1), respectively,while the last term represents the coupled information dueto the cooperation between neighbors (nodes 1 and 2 shareinformation at time step t1). The coupled information dependson the capacity of the link between nodes 1 and 2 at timestep t1 as well as the uncertainties of node 2 at time t2 andnode 1 at time t1.

If the link capacities are all rank-one matrices, i.e., K1,2 =v1,2v

T1,2, K1,3 = v1,3v

T1,3, and K3,4 = v3,4v

T3,4, then the

EFIM Je can be simplified as (see Appendix G)

Je = K1,1 + ξ1,2K1,2 + ξ1,3K1,3

+ δ3,4(μ2v1,2 − μ3v1,3)(μ2v1,2 − μ3v1,3)T (53)

where

ξ1,2 =(1+vT

1,2

(K2,2+

(1 + vT

2,4K−14,4 v2,4

)−1K2,4

)−1

v1,2

)−1

ξ1,3 =(1 + vT

1,3K−13,3 v1,3

)−1

μ2 = ξ1,2

(1 + vT

2,4K−14,4 v2,4

)−1(vT

2,4K−14,4 v3,4

)

·vT1,2

(K2,2 +

(1 + vT

2,4K−14,4 v2,4

)−1K2,4

)−1

v2,4

μ3 = ξ1,3

(vT

1,3K−13,3 v3,4

)

δ3,4 =(1 + vT

3,4

(Ω−1

4 + (K3,3 + K1,3)−1)v3,4

)−1

. (54)

In this special case, each effective information for node1 is proportional to the corresponding link capacity. Thecoupled information is a rank-one matrix with positive eigen-value δ3,4‖μ2v1,2 − μ3v1,3‖2 and corresponding eigenvec-tor μ2v1,2 − μ3v1,3 both depending on the RIIs, the SIIs,the uncertainties of nodes 1 and 2 at different time steps, andthe nodes’ spatial topology.

C. Scenario C

Node 1 cooperates with nodes 2 and 3 at time step t2 whilenodes 2 and 3 have cooperated at time step t1, as describedin Fig. 2(c). In this case, the states are x

(2)1 , x

(2)2 , x

(2)3 , x

(1)2 ,

and x(1)3 (denoted by states 1, 2, 3, 4, and 5, respectively),

which represent the states of three nodes at time step t2 andtwo nodes at time step t1. Then, the FIM becomes

J =

⎡

⎢⎢⎢⎢⎣

J{1},{1} −K1,2 −K1,3 0 0−K1,2 J{2},{2} 0 −K2,4 0−K1,3 0 J{3},{3} 0 −K3,5

0 −K2,4 0 J{4},{4} −K4,5

0 0 −K3,5 −K4,5 J{5},{5}

⎤

⎥⎥⎥⎥⎦(55)


where

J{1},{1} = K1,1 + K1,2 + K1,3

J{2},{2} = K2,2 + K1,2 + K2,4

J{3},{3} = K3,3 + K1,3 + K3,5

J{4},{4} = K4,4 + K2,4 + K4,5

J{5},{5} = K5,5 + K3,5 + K4,5. (56)

As a special case of Theorem 3 with interaction set B ={(4, 5)},9 the EFIM for x

(2)1 in Scenario C is (see Appendix G)

Je = K1,1 + K(4,5)2→1 + K

(4,5)3→1 + Ξ

(4,5)2,3 (57)

where

Ξ(4,5)2,3 = Γ

(I − K4,5

(Φ−1 + K4,5

)−1)K4,5Γ

T

Γ = K1,2Ω−12 K2,4(K4,4 + K2,4)−1

−K1,3Ω−13 K3,5(K5,5 + K3,5)−1

Φ = Ω−14 + Ω−1

5 (58)

K(4,5)2→1 =

(I − K1,2Ω

−12

)K1,2

K(4,5)3→1 =

(I − K1,3Ω

−13

)K1,3

Ωi = Ki,i + K1,i

+(I − Ki,i+2(Ki+2,i+2 + Ki,i+2)−1

)Ki,i+2 (59)

for i = 2, 3, and

Ωi = Ki,i +(I − Ki−2,i(Ki−2,i−2 + K1,i−2 + Ki−2,i)−1

)

·Ki−2,i (60)

for i = 4, 5.The EFIM in (57) is the sum of four terms. The first term

corresponds to the information that depends only on node 1 attime step t2, and the remaining 3 terms correspond to the infor-mation obtained through spatiotemporal cooperation. The sec-ond and third terms correspond to the decoupled effectiveinformation obtained from spatial neighbors (nodes 2 and 3 attime step t2), respectively, while the last term represents thecoupled information due to the cooperation between neighbors(nodes 2 and 3 share information at time step t1). The coupledinformation depends on the capacity of the link betweennodes 2 and 3 at time step t1 as well as the uncertaintiesof nodes 2 and 3 at time t2.

If the link capacities are all rank-one matrices, i.e., K1,i =v1,iv

T1,i, Ki,i+2 = vi,i+2v

Ti,i+2 for i = 2, 3, and K4,5 =

v4,5vT4,5, then the EFIM Je can be simplified as (see

Appendix G)

Je = K1,1 + ξ1,2K1,2 + ξ1,3K1,3

+ δ4,5(μ2v1,2 − μ3v1,3)(μ2v1,2 − μ3v1,3)T (61)

where

ξ1,i

=1

1+vT1,i

(Ki,i+

(1+vT

i,i+2K−1i+2,i+2vi,i+2

)−1Ki,i+2

)−1

v1,i

μi

9Note that one can also consider the interaction set to be B = {(4, 2)} andB = {(3, 5)}, which lead to analogous results.

= ξ1,i

(1+vT

i,i+2K−1i+2,i+2vi,i+2

)−1(vT

i,i+2K−1i+2,i+2v4,5

)

·vT1,i

(Ki,i+

(1+vT

i,i+2K−1i+2,i+2vi,i+2

)−1Ki,i+2

)−1

vi,i+2

(62)

for i = 2, 3, and

δ4,5 =(1 + vT

4,5

(Ω−1

4 + Ω−15

)v4,5

)−1

. (63)

In this case, each effective information for node 1is proportional to the corresponding link capacity. Thecoupled information is a rank-one matrix with positive eigen-value δ4,5‖μ2v1,2 − μ3v1,3‖2 and corresponding eigenvec-tor μ2v1,2 − μ3v1,3 both depending on the RIIs, the SIIs,the uncertainties of nodes 1, 2 and 3 at different time steps,and the nodes’ spatial topology.

D. Scenario D

Finally, we characterize the exponential decay of informa-tion coupling with respect to the network distance for thesimple network shown in Fig. 3. This behavior is also observedin Section V for more general networks.

In Fig. 3, the subset of the states with network distance atleast k to B = {i0, j0} is

Sk = {ι} ∪ {(ir, jr)}r�k, k = 0, 1, . . . , n (64)

and the subset of the states with network distance k toB = {(i0, j0)} is

Ek =

{{(ik, jk)}, k = 0, 1, . . . , n − 1{ι}, k = n.

(65)

Then, as a consequence of Theorem 2 and Corollary 2 wehave the following result.

Corollary 5: Let S be a set of the states with NIG describedby Fig. 3, (64), and (65), with Ks,r = Kr,s = λr,sI forr, s ∈ {i0, i1, . . . , in−1, j0, j1, . . . , jn−1}. Then, the coupledinformation for state ι induced by B = {(i0, j0)} is upperbounded as

Ξ(i0,j0) � 4λi0,j0εnI (66)

where ε = (λ2/(λ1 + λ2))2 < 1, λ1 = min{λr,r}, and λ2 =max{λr,s} for r, s ∈ {i0, i1, . . . , in−1, j0, j1, . . . , jn−1}.

Proof: See Appendix H. �Remark 7: This result shows that when two neighbors are

linked by a single loop {in−1, in−2, . . . , i0, j0, j1, . . . , jn−1}and the link capacities are “isotropic,” the information cou-pling induced by the interaction set {i0, j0} decays exponen-tially with the network distance of the interaction set. Thisexponential decay as a function of the distance of the linksforming the interaction set is further examined in Section Vfor more general networks through numerical results.

The exponential decay of coupled information implies thatonly close links induce a significant information coupling.This locality of the information coupling can be exploited bydistributed algorithms since the effect of information couplingcan be safely discarded when the neighbors are coupled bydistant links. Hence, distributed algorithms that track theinformation coupling induced only by close links can resultin near-optimal implementations.


Fig. 4. Quantification of the coupled information for Scenario A. (a): Vectors v2→1, v3→1, and vc representing the effective information and the coupledinformation from cooperation with nodes 2 and 3; (b): Intensity of information coupling for different values x1, measured by the norm of vector vc

(nuclear norm of Ξ(2,3)2,3 ).

Fig. 5. Quantification of the coupled information for Scenario B. (a): Vectors v2→1, v3→1, and vc representing the effective information and the coupledinformation from cooperation with node 2 at time step t2 and node 1 at time step t1 ; (b): Intensity of information coupling for different values x1, measuredby the norm of vector vc (nuclear norm of Ξ

(4,3)2,3 ).

V. NUMERICAL RESULTS

This section presents numerical results quantifying the cou-pled information. We first assess the effect of informationcoupling for scenarios A, B, and C studied in the previoussection, then we describe the effect of information coupling forlarge networks and how it decreases with the network distanceof the links inducing the coupling.

A. Representative Scenarios

We simulate the three specific scenarios in the square[−5 m, 5 m]× [−5 m, 5 m] and quantify the terms correspond-

ing to the effective information from neighbors as well as thecoupled information.

1) Scenario A: In this scenario, the positions of nodes2 and 3 are (−1.66 m, 0 m) and (1.66 m, 0 m), the matricesKi,i = I , for i = 1, 2, 3, and the RIIs λi,j = 50/d2

i,j

for i �= j [32].10 Fig. 4(a) shows the vectors v2→1, v3→1,and vc where K

(2,3)2→1 = v2→1v

T2→1, K

(2,3)3→1 = v3→1v

T3→1,

and Ξ(2,3)2,3 = vcv

Tc for a grid of 36 different values of x1

10For simplicity, we illustrate the numerical results of the informationcoupling using the free-space path-loss model for the wireless ranging signals.Similar observations can be made with other distributions for fading models.


Fig. 6. Quantification of the coupled information for Scenario C. (a): Vectors v2→1, v3→1, and vc representing the effective information and the coupledinformation from cooperation with nodes 2 and 3 at time step t2; (b): Intensity of information coupling for different values x

(2)1 , measured by the norm of

vector vc (nuclear norm of Ξ(4,5)2,3 ).

in the square area. Fig. 4(b) shows the norm of vector vc,which corresponds to the square root of the nuclear norm ofthe matrix Ξ

(2,3)2,3 , for different positions of x1 in the square

area.From these figures we can observe that the coupled informa-

tion increases when node 1 is far from neighbors. Moreover,the direction corresponding to the coupled information alignswith one of the directions of neighbors’ cooperation when v1,i

and v2,3 are orthogonal for i = 2 or i = 3.

2) Scenario B: In this scenario, the position of node 2 attime step t2 is (1.66 m, 0 m), the position of node 1 at timestep t1 is (−1.66 m, 0 m), and the position of node 2 attime step t1 is (0 m,−5 m). The matrices Ki,i = I fori ∈ {1, 2, 3, 4}, and the RIIs and SIIs λi,j = 50/d2

i,j for i, j ∈{1, 2, 3, 4}, where di,j is the Euclidean distance between statesi and j. Fig. 5(a) shows the vectors v2→1, v3→1, and vc

where K(4,3)2→1 = v2→1v

T2→1, K

(4,3)3→1 = v3→1v

T3→1, and

Ξ(4,3)2,3 = vcv

Tc for a grid of 36 different positions of x1 in

the square area. Fig. 5(b) shows the norm of vector vc, whichcorresponds to the squared root of the nuclear norm of thematrix Ξ

(4,3)2,3 , for different positions of x

(2)1 in the square

area.From these figures, we can observe that the coupled

information decreases when the node 1 at time step t2 isfar from neighbors. Moreover, the coupled information alsodecreases when the direction formed by node 1 at times t1 andt2 is orthogonal to the direction formed by node 1 at time t1and node 2 at time t1, that is, when the direction with neighborx

(1)1 is orthogonal to the direction formed by the interaction

set {x(1)1 , x

(1)2 }. Conversely, the information coupling is more

severe when the direction with neighbor x(1)1 is the same as

the interaction set and when node 1 at time t2 is closer to suchneighbor. Finally, the direction corresponding to the coupled

information is closer to the direction of cooperation withnode 1 at time t1. These results agree with the intuition that thecoupling directly affects neighbor x

(1)1 since this state is in the

interaction set, and this effect comes mainly in the directionwith the other state in the interaction set (node 2 at time t1)while its effect is negligible in the orthogonal direction.

3) Scenario C: In this scenario, the position ofnodes 2 and 3 at time step t2 are (−1.66 m, 0 m) and(1.66 m, 0 m), respectively, and the positions of nodes 2 and 3at time step t1 are (1.66 m,−5 m) and (−1.66 m,−5 m),respectively. The matrices Ki,i = I for i ∈ {1, 2, 3, 4, 5},and the RIIs and SIIs λi,j = 50/d2

i,j for i, j ∈ {1, 2, 3, 4, 5},where di,j is the Euclidean distance between statesi and j. Fig. 5(a) shows the vectors v2→1, v3→1, andvc where K

(4,5)2→1 = v2→1v

T2→1, K

(4,5)3→1 = v3→1v

T3→1, and

Ξ(4,5)2,3 = vcv

Tc for a grid of 36 different values of x

(2)1 in the

square area. Fig. 6(b) shows the norm of vector vc, whichcorresponds to the square root of the nuclear norm of thematrix Ξ

(4,5)2,3 , for different positions of x

(2)1 in the square

area.From these figures we can observe that the information

coupling is much less severe than in previous scenarios anddecreases when the node 1 at time step t2 is far from neigh-bors. Moreover, the coupled information also decreases whenthe direction formed by node 1 at time t1 and each neighboris orthogonal to the direction formed by each neighboringnode with itself at previous time step. Conversely, the coupledinformation increases when the node 1 at time step t2 is inthe same direction as the direction formed by each neighboringnode with itself, and hence the information coupling is moresevere when node 1 at time step t2 is in the intersection of bothdirections. These results agree with the intuition that firstly,the information coupling is much less severe because the


Fig. 7. Localization network formed by 100 nodes. Magenta, light blue, anddark green edges represent different connected components while red dashededges represent an interaction set.

coupling affects both neighbors indirectly (none of them arein the interaction set). In addition, the information couplingaffects the neighbors in the direction they form with theirneighbors in the interaction set, and hence, the informationcoupling affects node 1 at time step t2 mainly in such direc-tions while its effect is negligible in the orthogonal directions.

B. Large Networks

This section evaluates the intensity of information couplingover the total EFIM for large networks in terms of the nuclearnorm of the corresponding matrices. In particular, we studyhow the closeness of the interaction set affects the coupledinformation. In addition, we show that close information linksin terms of network distance are the truly relevant ones.

In this set of numerical simulations, we carried10,000 Monte Carlo network emulations where 100 and200 nodes are deployed uniformly at random in the square[−50 m, 50 m] × [−50 m, 50 m] and the node ι is located inthe origin (see Fig. 7). Each pair of nodes obtains rangemeasurements with probability 0.5 if their distance is smallerthan 16m and the prior knowledge of nodes is set to Ki,i = I,Ki,i = 4I, or Ki,i = 10I.

In the first simulations, we study the relationship betweeninformation coupling and network distance of the interactionset. In these simulations the interaction set is formed by onelink with different network distances to state ι. In Fig. 8 weshow the ratio between the nuclear norms of the coupledinformation Ξ and the total EFIM Je as a function of thenetwork distance between the interaction set and state ι fornetworks formed by 100 and 200 nodes. In both figures,it can be observed that coupled information sharply decayswith the network distance of the interaction set and also thatinformation coupling is more severe in cases with small priorknowledge.

In the second simulations, we study how the informationinduced by a set of links decays with the network distance.In these simulations we compare the EFIM for state ι using

all the information links Je with that obtained using the infor-mation links between nodes with network distances at mostk, JBk

e , where Bk = {(i, j) ∈ S2: max{ND(ι, i), ND(ι, j)} >k}. Then, Fig. 9 shows the information intensities obtainedfrom the nodes with network distances at most k normalizedby the total information intensity and the number of links tonodes with network distances at most k for k = 1, 2, . . . , 10.From this figure we can observe that the information intensityprovided by each link in a network with network distances atmost k decays sharply with the network distance k.

VI. INFORMATION COUPLING AND ALGORITHMS

In network navigation, nodes obtain position estimates fromspatial and temporal cooperation with neighbors. In previoussections, we have shown that the information an node obtainsfrom its neighbors is often coupled since it depends not onlyon the information obtained from the links to each neighbor(e.g., links (ι, i1) and (ι, i2) in NIG shown in Fig. 1) butalso on how those neighbors share information among them-selves (e.g., link (j1, j2) in NIG shown in Fig. 1). In thissection, we show how such information coupling affects theperformance of distributed algorithms and how our theoreticalresults can aid the design of such algorithms.

A. Impact of Information Coupling on Algorithms

In centralized algorithms, one processor collects all theinformation and determines jointly the positional estimatesfor all nodes. Such implementations can take full accountof the information coupling since the centralized processorcan have access to all pair-wise states’ cooperations. That is,the processor can have complete access to the NIG formed bythe pair-wise information links, and thus it can determine thecoupled information for each state. For instance, in cases withlinear and Gaussian models for measurements, a centralizedprocessor can determine the posterior distribution of the statesgiven all measurements as a joint Gaussian distribution. Themean of such distribution μ represents the minimum mean-squared error (MMSE) positional estimates for all states andthe covariance matrix Σ represents the joint uncertainty for allstate estimates. In particular, the diagonal blocks in such jointcovariance matrix represent the uncertainty of each state, whileeach non-diagonal block represents the correlation betweenpairs of states estimates. Note that a non-diagonal block Σi,j

corresponding to states i and j is non-zero when there is pathconnecting states i and j in the NIG [79]. Such non-diagonalblocks quantify the interrelation between the estimates ofstates i and j.

In distributed algorithms, several processors collect differentsubsets of information and determine in parallel positionalestimates of subsets of nodes. In such implementations, it isdifficult to account for information coupling since each proces-sor has access to only a subset of pair-wise states cooperations.That is, each processor has access to a subgraph of theNIG formed by pair-wise information links. The most com-mon current distributed implementations are based on beliefpropagation [51]–[53]. In these approaches, each processorkeeps track of the positional estimate and uncertainty for eachnode separately, and individual estimates and uncertainties


Fig. 8. Coupled information sharply decays with the network distance of the interaction set and is more severe in cases with small prior knowledge. They-axis represents averaged ratios between the nuclear norms of coupled information and the total equivalent information, while the x-axis represents differentnetwork distances of the interaction link. (a) Networks with 100 nodes. (b) Networks with 200 nodes.

Fig. 9. The impact of the links in the total information decays sharply with the network distance and increases with the amount of prior knowledge. They-axis represents averaged nuclear norms of the EFIMs J

Bke obtained from nodes with network distances at most k normalized by the total EFIM intensity

and the number of links with network distances at most k, while the x-axis represents network distance k. (a) Networks with 100 nodes. (b) Networks with200 nodes.

are updated by means of messages among different proces-sors. For instance, in cases with linear and Gaussian modelsfor measurements, each processor approximates the marginalposterior of each individual state given all measurements asa Gaussian distribution. Each mean μi of such distributionsrepresents the positional estimate of each individual state andthe corresponding covariance Σi,i represents the individualuncertainty of each state. Both means and covariances areupdated through messages communicated with other proces-sors through an iterative process. Note that those techniquesare unaware of the interrelations among nodes’ uncertaintiesand each processor does not keep track of how neighbors shareinformation among themselves.

In the following we show the impact of informationcoupling in the algorithms performance via both analyticalexpressions and numerical results using linear and Gaussian

models for measurements for the ease of closed-formexpressions. In particular, we consider a localization systemthat uses inter-node measurements related to the differencesof positions and velocities, i.e., an inter-node measurementrelated to nodes i and j is yi,j given by

yi,j =[pi − pj

vi − vj

]+ εi,j (67)

with instantiation of pi ∈ Rd and vi ∈ R

d being positionand velocity of node i, and εi,j being additive white noisefollowing a zero-mean Gaussian distribution with covariance

Qi,j = diag{σ2

p Id, σ2v Id

}(68)

The state of node k at time tn is x(n)k =

[(p(n)

k )T, (v(n)k )T

]T ∈R

2d. Let x(n) =[(x(n)

1 )T, (x(n)2 )T, . . . , (x(n)

K )T]T ∈


R2dK be the concatenation of all states, and y(n) =[(y(n)

i1,j1)T, (y(n)

i2,j2)T, . . . , (y(n)

im,jm)T]T ∈ R

2dM be a set ofinter-node measurements. Then, we have that

y(n) = H(n)x(n) + ε(n) (69)

where

H(n) =[eK

i1 − eKj1 , e

Ki2 − eK

j2 , . . . , eKim

− eKjm

]T ⊗ I2d (70)

with ε(n) additive white noise that follows a zero-meanGaussian distribution with covariance

Q(n) = diag{Q

(n)i1,j1

, Q(n)i2,j2

, . . . , Q(n)im,jm

}. (71)

We also assume that the nodes move with constantvelocities, that is x

(n)k = Ex

(n−1)k with

E =[

1 Δ0 1

]⊗ I2d (72)

and Δ = tn − tn−1 is a constant observation interval.Therefore, x(n) = F x(n−1) with F = IK ⊗ E.

Under such assumptions, if μ(n) and Σ(n) denotethe mean and covariance of x(n) given measurements{y(1), y(2), . . . , y(n)}, the usual Kalman filtering recursion(see e.g., [80]–[86]) give us that

μ(n) = μ̃ + Σ̃(H(n)

)T(Q(n) + H(n)Σ̃

(H(n)

)T)−1

z (73)

Σ(n) =(Σ̃

−1+

(H(n)

)T(Q(n)

)−1H(n)

)−1

(74)

where

μ̃ = Fμ(n−1)

Σ̃ = FΣ(n−1)F T

z = y(n) − H(n)μ̃. (75)

In the specific case where only one measurement is obtained

at time tn (i.e., y(n) = y(n)i,j ), (73) and (74) lead to

(see Appendix I)

μ(n)i = W1(W1 + W2)−1Eμ

(n−1)i

+W2(W1 + W2)−1(y

(n)i,j + Eμ

(n−1)j

)(76)

[Σ(n)

]i,i

= W1(W1 + W2)−1E[Σ(n−1)

]i,i

ET

+W2(W1 + W2)−1E[Σ(n−1)

]j,i

ET (77)

with

W1 = Q(n)i,j + E

([Σ(n−1)

]j,j

− [Σ(n−1)]j,i)ET

W2 = E([

Σ(n−1)]i,i

− [Σ(n−1)

]i,j

)ET. (78)

In this simple case, at time tn node i cooperates withtwo neighbors: node i at time tn−1 (temporal cooperation)and node j at time tn (spatial cooperation). Both the stateestimates and its covariance in (76) and (77) become theweighted sum of estimates and covariances due to the temporalcooperation and spatial cooperation. Note that both weightedsums depend on the uncertainty of the two neighbors but alsoon how those uncertainties are interrelated. If the neighborshave interacted, then the non-diagonal term [Σ(n−1)]i,j in thecovariance matrix is non-zero. Discarding such information

Fig. 10. The RMSE per node in a navigation network for centralizedalgorithm that estimates the states and joint nodes uncertainties versus adistributed algorithm based on BP, and a centralized algorithm that estimatethe states and individual nodes’ uncertainties. Position errors can rapidlyincrease if the information coupling among nodes’ estimates are discarded.

coupling leads to an inaccurate positional estimate in (76) anda wrong covariance in (77). Such covariances lead to a harmfulcascade effect since they are used by other neighbors in nexttime steps.

We next quantify the localization performance loss due todiscarding information coupling. In particular, we comparethe position error of algorithms that keep only the individualuncertainties of each state with that of algorithms that keep thejoint uncertainty of all the states. We simulate a navigation net-work composed of 100 nodes and three anchors deployed uni-formly at random in the square [−50 m, 50 m]×[−50 m, 50 m].Nodes move with constant velocities with components sam-pled uniformly in the interval [−1 m/s, 1 m/s]. The statesare estimated using inter-node measurements for positionsand velocities differences as described in (67). In particular,in every second each pair of nodes i, j obtain measurementswith probability 0.5 if their distance di,j is smaller than 20 m,and σ2

p = σ2v = d2

i,j/50 m2. Positions are estimated usingthree algorithms:

1) a centralized algorithm that obtains MMSE estimates forthe states using (73) and (74);

2) a distributed algorithm that estimates the states using BP(see e.g., [67]);11

3) a centralized algorithm that estimates the statesusing (73) and (74) but discard the non-diagonal blocksin the covariance matrix.

These three algorithms are the most relevant for this evaluationsince the first one is optimal in the mean-squared error (MSE)sense, the second one is the most common distributed algo-rithm, and the third one is the same as the optimal algorithmexcept that it discards the information coupling.

Fig. 10 shows the RMSE per node for the position estimatesobtained by the three algorithms used during 30 seconds. It can

11We use four iterations for message passing as this is a common practiceused for cooperative localization [51].


Fig. 11. Ratio between RMSEs of BP-based and MMSE algorithms for nodeswith different number of neighbors. The performance gap of algorithms thatdiscard information coupling increases with the number of neighbors.

be observed that the three algorithms obtain accurate positionestimates for the first time steps, but that the algorithmsthat are not aware of the interrelations among estimates fordifferent nodes (non-diagonal blocks in covariance matrix)quickly become highly inaccurate. In addition, Fig. 11 showsthe RMSE ratio between the BP-based and MMSE algorithmsfor nodes with different number of neighbors. It can beobserved that the performance gap between the BP-based andMMSE algorithms increases with the number of neighbors.As shown by these results, algorithms that discard the infor-mation coupling among different nodes can result in poorperformances, especially in sequential implementations wherethe positions and uncertainties estimates are based on thoseobtained in previous steps.

B. Design Insights Toward Coupling-Aware Algorithms

The paper characterizes the information coupling in networknavigation via Fisher information analysis, and shows howthe accuracy bound for each state depends on the effectiveinformation obtained from neighbors and also on the coupledinformation by neighbors’ interaction. Specifically,

• Theorem 1 shows that the information obtained by eachnode through cooperation is the sum of that obtainedfrom each connected component in the NIG. In addi-tion, it shows that the information obtained from eachcomponent depends on the capacity of the links to theneighbors in such component and a term that accountsfor the joint uncertainty of neighbors in the component.Corollary 1 shows that when neighbors are isolated,i.e., belong to different components, the informationobtained through cooperation is the sum of the effectiveinformation obtained from each neighbor. Such effectiveinformation accounts for the individual uncertainty of theneighbor in its component.

• Neighbors in the same connected component can beuncoupled by removing the information links in aninteraction set. Proposition 2 shows that the informationobtained through cooperation can be decoupled as the

sum of the effective information from each neighbor bydiscarding the information links in an interaction set.Propositions 3 and 4 further characterize the effectiveinformation that can be obtained from neighbors in termsof the interaction set, neighbors uncertainty, and informa-tion links to neighbors.

• Theorem 2 with Corollaries 2 and 5 describes howcoupled information decreases with the network distanceto the interaction set. In particular Corollary 5 and thenumerical results in Section V-B show the sharp decay ofcoupled information with network distance. Such decayof coupled information implies that only close linksinduce a significant information coupling.

• Theorem 3 with Corollary 3 shows that the coupledinformation increases with the capacity of the link inthe interaction set, and decreases with the uncertaintyof the interaction set. In addition, the analytical expres-sions for specific simple representative networks shownin Section IV and the corresponding numerical resultsshown in Section V-A describe how information couplingalso depends on the spatial topology among the states.

Our results show how interrelations among neighbors resultin information coupling, how such coupling affects positionalaccuracy bounds, and the main aspects influencing informationcoupling. Several insights derived from such results can yieldspecially useful guidelines for the design of coupling-awaredistributed algorithms.

• Algorithms that discard the information provided byinteraction sets can directly lead to efficient distributedimplementations. Such techniques can exploit a trade-offbetween positional accuracy and implementation effi-ciency where the information provided by interaction setscan be sacrificed in order to enable simple distributedimplementations (see Propositions 2, 3, and 4).

• Algorithms that keep track of the most relevant infor-mation coupling can result in near optimal implemen-tations. Our results show in what circumstances theinformation coupling is negligible depending on 1) thenetwork distance to the interaction set (see Corollary 5and Section V-B); 2) the interaction set uncertainties andlink capacities (see Theorem 3 and Corollary 3); and3) the spatial topology of neighbors and the interactionset (see Section V-A).

Efficient distributed estimation is critical not only for net-work navigation but also for a diverse set of applicationsin fields such as digital communications and sensor net-works [87], [88]. Dependencies among different informationsources hinder the development of effective distributed estima-tion techniques for general problems. For instance, BP-baseddistributed algorithms fail to take into account the dependencesbetween messages from neighbors by assuming conditionalindependences, which are valid for tree-structured inferencegraphs but fail for graphs with loops. Capturing the dependen-cies among different information sources in distributed settingsis challenging. Generalized BP techniques assemble variablesin groups, and hence can keep the intra-group dependences butnot inter-group dependences [88]. Other works suggest to keeptrack of the path followed by messages exchanged in oder to


capture dependences between messages [87]. The theoreticalresults and design insights presented in this paper for networknavigation can also be useful for general distributed estimationproblems.

VII. CONCLUSION

In this paper, we characterized the information couplingarising in network navigation via Fisher information analysis.In particular, we derived the EFIM for individual states,characterizing the effective information obtained from eachneighbor and the coupled information induced by neighbors’interaction. In addition, we determined how the coupled infor-mation decays with network distance, analyzed informationcoupling in representative case studies, and showed the impactof information coupling in the performance of current distrib-uted algorithms.

The results show that the information coupling depends onthe links capacities, the node uncertainties, the spatiotemporaltopology of the nodes, and the network distance of neighborsinteraction. Information coupling can highly impact the per-formance of network navigation algorithms. The theoreticalfindings in the paper characterize how information couplingarises and which factors determine the intensity of informa-tion coupling. Our results can serve as guidelines for thedesign of efficient and accurate coupling-aware algorithms fornetwork navigation that enable numerous emerging location-based applications.

APPENDIX APROOF OF PROPOSITION 1

Proof: The matrix GSi,j ⊗ Ki,j is positive semidefinite

because GSi,j ⊗Ki,j = Ci,jKi,jC

Ti,j where Ci,j is a SD ×D

matrix given by

Ci,j =

{eS

i ⊗ ID, i = j

eSi ⊗ ID − eS

j ⊗ ID, i �= j.(79)

Then, the rank of Ci,jKi,jCTi,j is equal to that of Ki,j because

Ci,j has rank D which is larger than or equal to the rankof Ki,j .

The second part of the proposition follows since using (1)we have that

J �∑

i∈SGS

i,i ⊗ Ki,i (80)

where the right-hand side is full rank since it is block diagonaland Ki,i ∈ S

D++ for all i ∈ S. �

APPENDIX BLEMMA

Lemma 1: Let

C =

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

A +∑n

i=1 Ai −A1−A2 . . . −An

−A1

−A2

. . . diag{A1, A2, . . . , An} + B

−An

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

(81)

where A1, A2, . . . , An ∈ SD+ , A ∈ S

D++, and B ∈ S

Dn++.

Then, the matrix C is positive definite and the Schur comple-ment of A +

∑ni=1 Ai in C is A + R with R � 0.

Proof: The matrix C can be rewritten as

C =n∑

i=1

(en

1 ⊗ Ui − eni ⊗ Ui

)(en


)T

+[

A 00 B

](82)

where Ai = UiUTi . Then, C is positive definite because

the first term in that decomposition is the sum of positivesemidefinite matrices and the last term is a positive definitematrix.

Finally, the Schur complement of A +∑n

i=1 Ai in C isA + R where R is the Schur complement of

n∑

i=1

(en


)(en


)T +[

0 00 B

]

(83)

and hence, the result is obtained by observing that R is theSchur complement of a positive semidefinite matrix. �

APPENDIX CPROOF OF PROPOSITION 3

Proof:1) The inequality KB

i→ι � Kι,i is trivial since ΛBki

is apositive semidefinite matrix. The inequality KB′

i→ι � 0 isobtained by observing that it is the Schur complement of

[Kι,i Kι,i

Kι,i

([(JB′C′

ki,C′

ki

)−1]

i,i

)−1

](84)

which is a positive semidefinite matrix since([(

JB′C′

ki,C′

ki

)−1]

i,i

)−1

� Kι,i (85)

using the Lemma 1 with C = JB′C′

ki,C′

ki

and A = Kι,i +

Ki,i. Finally, the inequality KB′i→ι � KB

i→ι is obtainedby observing that JB � JB′

.2) The result is obtained by using the Lemma 1 with C =

JB′C′

ki,C′

ki

and A = Ki,i. Notice that the result for B canbe obtained analogously.

�APPENDIX D

PROOF OF THEOREM 2

Proof: The fact that [Jk]i,j =[JB

k

]i,j

= Ji,j for(i, j) /∈ E2

k is straightforward by the definition of Sk and Ek.The remaining of the theorem can be proven by induction overk as follows.

The result for k = 0 is straightforward. Suppose theresult holds for k = r � 0. Since [Jr+1]Er+1,Er+1 and[JB

r+1

]Er+1,Er+1

are the Schur complement of the upper leftblock of [

JEr+1,Er+1 JEr+1,Er

JTEr+1,Er

[Jr]Er ,Er

]


[(JS̄,S̄

)−1]

i1,j1=

[(JBC1,C1

)−1]

i1,i1−

[(JBC1,C1

)−1]

i1,j1Uj1,j2ΔUT

j1,j2

[(JBC1,C1

)−1]

i1,j1(91)

[(JS̄,S̄

)−1]

i1,j2=

([(JS̄,S̄

)−1]

i2,j1

)T=

[(JBC1,C1

)−1]

i1,j1Uj1,j2ΔUT

j1,j2

[(JBC2,C2

)−1]

i2,j2(92)

[(JS̄,S̄

)−1]

i2,j2=

[(JBC2,C2

)−1]

i2,i2−

[(JBC2,C2

)−1]

i2,j2Uj1,j2ΔUT

j1,j2

[(JBC2,C2

)−1]

i2,j2(93)

and [JEr+1,Er+1 JEr+1,Er

JTEr+1,Er

[JB

r

]Er,Er

]

respectively, then

[Jr+1]Er+1,Er+1 −[JB

r+1

]Er+1,Er+1

= JEr+1,Er

[([JB

r

]Er ,Er

)−1

−([Jr]Er,Er

)−1]JTEr+1,Er

(86)

and the result is obtained since by the induction hypothesis

[Jr]Er ,Er =[JB

r

]Er ,Er

+ΓrUj1,j2

(I + UT

j1,j2ΔrUj1,j2

)−1UT

j1,j2ΓTr

(87)

and hence([

JBr

]Er,Er

)−1 − ([Jr]Er,Er)−1

=([

JBr

]Er ,Er

)−1ΓrUj1,j2

·[I + UT

j1,j2

(Δr + Γ T

r

([JB

r

]Er,Er

)−1

Γr

)Uj1,j2

]−1

·UTj1,j2Γ

Tr

([JB

r

]Er,Er

)−1

(88)

using the matrix inversion lemma. �

APPENDIX EPROOF OF THEOREM 3

Proof: Note that

JS̄,S̄ = J B̄S,S̄ + V V T (89)

where V =(eS−1j1

− eS−1j2

) ⊗ Uj1,j2 and Kj1,j2 =Uj1,j2U

Tj1,j2 . Then, using the matrix inversion lemma

(JS̄,S̄)−1 =(J B̄S,S̄

)−1 − (J B̄S,S̄

)−1V ΔV T(

J B̄S,S̄

)−1(90)

where Δ =(I + V T(

J B̄S,S̄

)−1V

)−1. Since J B̄

S,S̄ is blockdiagonal with each block corresponding to a different compo-nent of the partition of S̄, all the components of (JS̄,S̄)−1 and(J B̄S,S̄

)−1are equal except for those corresponding to (i1, j1),

(i1, j2), (i2, j1), and (i2, j2) shown by (91), (92), and (93)respectively, at the top of this page.

Then the result is obtained since

Uj1,j2ΔUTj1,j2 = Uj1,j2

(I + UT

j1,j2ΦUj1,j2

)−1UT

j1,j2

=(I − Kj1,j2

(Φ−1 + Kj1,j2

)−1)Kj1,j2

= Kj1,j2

(Φ−1 + Kj1,j2

)−1Φ−1 (94)

where the second equality is obtained using the matrix inver-sion lemma. �

APPENDIX FPROOF OF COROLLARY 4

Proof: Note that using (34) we have that

Ξ(j1,j2)i1,i2

= Γ(I − Kj1,j2

(Φ−1 + Kj1,j2

)−1)Kj1,j2Γ

T (95)

and by the hypothesis of this corollary we have that(I − Kj1,j2

(Φ−1 + Kj1,j2

)−1)Kj1,j2

=(1 −

(vT

j1,j2

(Φ−1 + vj1,j2v

Tj1,j2

)−1)vj1,j2

)vj1,j2v

Tj1,j2

=(1 + vT

j1,j2Φvj1,j2

)−1vj1,j2v

Tj1,j2

=(1 + vT

j1,j2Φvj1,j2

)−1Kj1,j2 (96)

where the second equality is a consequence of the matrixinversion lemma. Then, using the expression for Γ in (35),we have that

ΓKj1,j2 =(vT

ι,i1

[(JBC1,C1

)−1]

i1,j1vj1,j2vι,i1

)vT

j1,j2

−(vT

ι,i2

[(JBC2,C2

)−1]

i2,j2vj1,j2vι,i2

)vT

j1,j2

= (μi1vι,i1 − μi2vι,i2)vTj1,j2 (97)

and the result is obtained observing that

vTj1,j2Γ

T = (μi1vι,i1 − μi2vι,i2)T. (98)

�

APPENDIX GDERIVATIONS FOR SECTIONS IV-A, IV-B, AND IV-C

The EFIM in (43) is obtained from Theorem 3 since inScenario A, i1 = j1 = 2, i2 = j2 = 3, C1 = {2}, andC2 = {3}. Hence

(JBC1,C1

)−1 = (K2,2 + K1,2)−1

(JBC2,C2

)−1 = (K3,3 + K1,3)−1 (99)

which leads to (43) using the expressions inTheorem 3 and (18).

The EFIM for the case with rank-one link capacities in (46)is obtained from Corollary 4 since from (18)

ξ1,i = 1 − vT1,i(Ki,i + K1,i)−1v1,i

=(1 + vT

1,iK−1i,i v1,i

)−1(100)

using the matrix inversion lemma. The expression for μi isobtained since

vT1,i(Ki,i + K1,i)−1 = ξ1,iv

T1,iK

−1i,i (101)

using again the matrix inversion lemma. The expression forδ2,3 is a direct consequence of Corollary 4 and (43).


(JBC1,C1

)−1 =[

K2,2 + K1,2 + K2,4 −K2,4

−K2,4 K4,4 + K2,4

]−1

=[

Ω−12 Ω−1

2 K2,4(K4,4 + K2,4)−1

(K4,4 + K2,4)−1K2,4Ω−12 Ω−1

4

](108)

(JBC2,C2

)−1 =[

K3,3 + K1,3 + K3,5 −K3,5

−K3,5 K5,5 + K3,5

]−1

=[

Ω−13 Ω−1

3 K3,5(K5,5 + K3,5)−1

(K5,5 + K3,5)−1K3,5Ω−13 Ω−1

5

](109)

The EFIM in (50) is obtained from Theorem 3 since inScenario B, i1 = 2, j1 = 4, i2 = j2 = 3, C1 = {2, 4}, andC2 = {3}. Hence,

(JBC1,C1

)−1is given by (108), shown at the

top of this page, and

(JBC2,C2

)−1 = (K3,3 + K1,3)−1 (102)

which leads to (50) using the expressions in Theorem 3and (18).

The EFIM for the case with rank-one link capacities in (53)is obtained from Corollary 4. The derivations for ξ1,3 andμ2 are the same as those in Scenario A. The expression forδ3,4 is a direct consequence of Corollary 4 and (50), andξ1,2 is obtained since from (18) and the matrix inversionlemma

ξ1,2 = 1 − vT1,2Ω

−11,2 v1,2 =

(1 + vT

1,2(Ω2 − K1,2)v1,2

)−1

(103)

and

Ω2 − K1,2 = K2,2 + K2,4(K4,4 + K2,4)−1K2,4

= K2,2 +(1 + vT

2,4K−14,4 v2,4

)−1K2,4. (104)

The expression for μ2 is obtained since from Corollary 4and (43)

μ2 = vT1,2Ω

−12 K2,4(K4,4 + K2,4)−1v3,4 (105)

and using the matrix inversion lemma

vT1,2Ω

−12

= vT1,2(Ω2 − K1,2)−1

(1 + vT

1,2(Ω2 − K1,2)−1v1,2

)−1

= ξ1,2vT1,2(Ω2 − K1,2)−1 (106)

and similarly

K2,4(K4,4+K2,4)−1 =v2,4vT2,4K

−14,4

(1+vT

2,4K−14,4 v2,4

)−1.

(107)

The EFIM in (57) is obtained from Theorem 3 since inScenario C, i1 = 2, j1 = 4, i2 = 3, j2 = 5, C1 = {2, 4},and C2 = {3, 5}. Hence, (JB

C1,C1)−1 and (JB

C2,C2)−1 are given

by (108) and (109), respectively, shown at the top of thispage, which leads to (57) using the expressions in Theorem 3and (18).

The EFIM for the case with rank-one link capacities in (61)is obtained from Corollary 4. The derivations for ξ1,i and μi

are the same as those of ξ1,2 and μ2 in Scenario B. Theexpression for δ4,5 is a direct consequence of Corollary 4and (57).

APPENDIX HPROOF OF COROLLARY 5

Proof: The proof is divided into two steps: we first provethat for k = 0, 1, . . . , n − 1,

JBEk,Ek

� diag{(λik ,ik

+ λik,ik+1) I, (λjk,jk+ λjk,jk+1) I

}

(110)

where we let in = jn = ι for notational convenience.In the second step, we prove that for k = 0, 1, . . . , n − 1,Dk � 2λi0,j0ε

k I . Finally, the result is obtained by observingthat Ξ(i0,j0) = Dn � 4λi0,j0ε

n I .First step: notice that

JBE0,E0

= diag{(λi0,i0 + λi0,i1) I, (λj0,j0 + λj0,j1) I

}(111)

and then the result is obtained by induction since

JBEk+1,Ek+1

=[

(λik+1,ik+1 + λik+1,ik+2)I 00 (λjk+1,jk+1 + λjk+1,jk+2)I

]

+[

λik+1,ikI 0

0 λjk+1,jkI

]−

[λik+1,ik

I 00 λjk+1,jk

I

]

· (JBEk,Ek

)−1[

λik+1,ikI 0

0 λjk+1,jkI

]

(112)

and hence, if the inequality in (110) is true for k we have thatit is also true for k + 1 using (110) in (112) and since

λik+1,ik− λ2

ik+1,ik

λik,ik+ λik+1,ik

> 0

λjk+1,jk− λ2

jk+1,jk

λjk,jk+ λjk+1,jk

> 0. (113)

Second step: notice that

D0 = λi0,j0 [1,−1]T[1,−1]⊗ I � 2λi0,j0I (114)

and then Dk � 2λi0,j0εkI for k = 0, 1, . . . , n−1, by induction

using Corollary 2 and the fact that(JBEk,Ek

)−2 � diag{(λik,ik

+ λik ,ik+1)−2 I,

(λjk ,jk+ λjk,jk+1)

−2 I}

(115)

andλik+1,ik

λik,ik+ λik,ik+1

<λ2

λ1 + λ2

λjk+1,jk

λjk,jk+ λjk,jk+1

<λ2

λ1 + λ2. (116)

Finally, from Corollary 2

Dn � J1,En−1

(JBEn−1,En−1

)−1Dn−1

(JBEn−1,En−1

)−1JEn−1,1

� 4 λi0,j0εnI (117)


where the second inequality is obtained using (110) andJ1,En−1 = (λ1,in−1 , λ1,jn−1) ⊗ I = JT

En−1,1 �

APPENDIX IDERIVATIONS FOR EQUATIONS (76) AND (77)

Since y(n) = y(n)i,j , we have that H(n) =

[eK

i −eKj

]T ⊗ I2d

and hence

z = y(n)i,j −E

(μ

(n−1)i −μ

(n−1)j

)(118)

Q(n)+H(n)Σ̃(H(n)

)T = W1 + W2 (119)

with W1 and W2 given in (78). Therefore, using (73) weobtain

μ(n)i = Eμ

(n−1)i + W2(W1 + W2)−1

·(y

(n)i,j − E

(μ

(n−1)i − μ

(n−1)j

))(120)

that directly leads to (76) after rearranging. Finally, (77)is obtained similarly after using the matrix inversionlemma in (74).

ACKNOWLEDGMENT

The authors would like to thank Z. Liu, F. Meryer, andH. Shin for their valuable suggestions and careful reading ofthe manuscript.

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Santiago Mazuelas (M’10–SM’17) received the Ph.D. in Mathematics andPh.D in Telecommunications Engineering from the University of Valladolid,Spain, in 2009 and 2011, respectively.

Since 2017 he has been Ramon y Cajal Researcher at the Basque Centerfor Applied Mathematics (BCAM). Prior to joining BCAM, he was a StaffEngineer at Qualcomm Corporate Research and Development from 2014 to2017. He previously worked from 2009 to 2014 as Postdoctoral Fellow andAssociate in the Laboratory for Information and Decision Systems (LIDS) atthe Massachusetts Institute of Technology (MIT). His general research interestis the application of mathematics to solve engineering problems, currently hiswork is primarily focused on statistical signal processing, machine learning,and data science.

Dr. Mazuelas is Associate Editor for the IEEE COMMUNICATIONS LET-TERS and served as Co-chair for the Symposiums on Wireless Communi-cations at the 2014 IEEE Globecom and at the 2015 IEEE ICC. He hasreceived the Best Doctorate Thesis Award from University of Valladolidin 2011, the Young Scientist Prize from the Union Radio-Scientifique Interna-tionale (URSI) Symposium in 2007, and the Early Achievement Award fromthe IEEE ComSoc in 2018. His papers received the IEEE CommunicationsSociety Fred W. Ellersick Prize in 2012, and Best Paper Awards from theIEEE ICC in 2013, the IEEE ICUWB in 2011, and the IEEE Globecomin 2011.

Yuan Shen (S’05–M’14) received the Ph.D. degree and the S.M. degree inelectrical engineering and computer science from the Massachusetts Instituteof Technology (MIT), Cambridge, MA, USA, in 2014 and 2008, respectively,and the B.E. degree (with highest honor) in electronic engineering fromTsinghua University, Beijing, China, in 2005.

He is an Associate Professor with the Department of Electronic Engi-neering at Tsinghua University. Prior to that, he was a Research Assistantand then Postdoctoral Associate with the Wireless Information and NetworkSciences Laboratory at MIT in 2005-2014. He was with the Hewlett-PackardLabs in winter 2009 and the Corporate R&D at Qualcomm Inc. in summer2008. His research interests include statistical inference, network science,communication theory, information theory, and optimization. His currentresearch focuses on network localization and navigation, inference techniques,resource allocation, and intrinsic wireless secrecy.

Professor Shen was a recipient of the Qiu Shi Outstanding Young ScholarAward, the China’s Youth 1000-Talent Program, the Marconi SocietyPaul Baran Young Scholar Award, and the MIT Walter A. Rosenblith Presiden-tial Fellowship. His papers received the IEEE Communications Society FredW. Ellersick Prize and three Best Paper Awards from the IEEE internationalconferences. He is elected Vice Chair (2017-2018) and Secretary (2015-2016)for the IEEE ComSoc Radio Communications Committee. He serves as TPCsymposium Co-Chair for the IEEE Globecom (2018 and 2016), the Euro-pean Signal Processing Conference (EUSIPCO) (2016), and the IEEE ICCAdvanced Network Localization and Navigation (ANLN) Workshop (2016,2017, and 2018). He also serves as Editor for the IEEE TRANSACTIONS ON

WIRELESS COMMUNICATIONS since 2018, IEEE WIRLESS COMMUNICA-TIONS LETTERS since 2018, IEEE CHINA COMMUNICATIONS since 2017,IEEE COMMUNICATIONS LETTERS (2015-2018), and Guest-Editor for theInternational Journal of Distributed Sensor Networks (2015).

Moe Z. Win (S’85–M’87–SM’97–F’04) is a Professor at the MassachusettsInstitute of Technology (MIT) and the founding director of the WirelessInformation and Network Sciences Laboratory. Prior to joining MIT, he waswith AT&T Research Laboratories and NASA Jet Propulsion Laboratory.

His research encompasses fundamental theories, algorithm design, andnetwork experimentation for a broad range of real-world problems. Hiscurrent research topics include network localization and navigation, networkinterference exploitation, and quantum information science. He has served theIEEE Communications Society as an elected Member-at-Large on the Boardof Governors, as elected Chair of the Radio Communications Committee,and as an IEEE Distinguished Lecturer. Over the last two decades, heheld various editorial positions for IEEE journals and organized numerousinternational conferences. Currently, he is serving on the SIAM DiversityAdvisory Committee.

Dr. Win is an elected Fellow of the AAAS, the IEEE, and the IET.He was honored with two IEEE Technical Field Awards: the IEEE KiyoTomiyasu Award (2011) and the IEEE Eric E. Sumner Award (2006, jointlywith R. A. Scholtz). Together with students and colleagues, his papers havereceived several awards. Other recognitions include the IEEE CommunicationsSociety Edwin H. Armstrong Achievement Award (2016), the InternationalPrize for Communications Cristoforo Colombo (2013), the Copernicus Fel-lowship (2011) and the Laurea Honoris Causa (2008) from the Universitàdegli Studi di Ferrara, and the U.S. Presidential Early Career Award forScientists and Engineers (2004). He is an ISI Highly Cited Researcher.

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