Robust distributed cooperative RSSbased localization for directed graphs in mixed LoS/NLoS environments
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Abstract
The accurate and lowcost localization of sensors using a wireless sensor network is critically required in a wide range of today’s applications. We propose a novel, robust maximum likelihoodtype method for distributed cooperative received signal strengthbased localization in wireless sensor networks. To cope with mixed LoS/NLoS conditions, we model the measurements using a twocomponent Gaussian mixture model. The relevant channel parameters, including the reference path loss, the path loss exponent, and the variance of the measurement error, for both LoS and NLoS conditions, are assumed to be unknown deterministic parameters and are adaptively estimated. Unlike existing algorithms, the proposed method naturally takes into account the (possible) asymmetry of links between nodes. The proposed approach has a communication overhead upperbounded by a quadratic function of the number of nodes and computational complexity scaling linearly with it. The convergence of the proposed method is guaranteed for compatible network graphs, and compatibility can be tested a priori by restating the problem as a graph coloring problem. Simulation results, carried out in comparison to a centralized benchmark algorithm, demonstrate the good overall performance and high robustness in mixed LoS/NLoS environments.
Keywords
Cooperative localization Received signal strength (RSS) Maximum likelihood estimation Wireless sensor network (WSN)Abbreviations
 BP
Belief propagation
 CMLE
Centralized maximum likelihood estimator
 DECM
Distributed expectationconditional maximization
 DML
Distributed maximum likelihood
 ECDF
Empirical cumulative distribution function
 ECM
Expectationconditional maximization
 EM
Expectationmaximization
 LoS
Lineofsight
 LS
Least squares
 MC
Monte Carlo
 ML
Maximum likelihood
 MLE
Maximum likelihood estimator
 NBP
Nonparametric belief propagation
 NLoS
Nonlineofsight
 RDML
Robust distributed maximum likelihood
 RF
Radiofrequency
 RMSE
Root mean square error
 RSS
Received signal strength
 RSSI
Received signal strength indicator
 SDP
Semidefiniteprogramming
 SPAWN
Sumproduct algorithm over wireless networks
 WLS
Weighted least squares
 WSN
Wireless sensor network
 2D
2dimensional
1 Introduction
The wide spread of telecommunication systems has led to the pervasiveness of radiofrequency (RF) signals in almost every environment of daily life. Knowledge of the location of mobile devices is required or beneficial in many applications [1], and numerous localization techniques have been proposed over the years [1, 2, 3, 4]. Techniques based on the received signal strength (RSS) are the preferred option when low cost, simplicity, and technology obliviousness are the main requirements. In some standards, e.g., IEEE 802.15.4, an RSS indicator (RSSI) is encoded directly into the protocol stack [5]. In addition, RSS is readily available from any radio interface through a simple energy detector and can be modeled by the wellknown path loss model [6] regardless of the particular communication scheme. Based on that, RSS can be exploited to implement “opportunistic” localization for different wireless technologies, e.g., WiFi [7], FM radio [8], or cellular networks [9]. In the context of wireless sensor networks (WSNs), nodes with known positions (anchors) can be used to localize the nodes with unknown positions (agents). Generally speaking, localization algorithms can be classified according to three important categories.
Centralized vs. distributed. Centralized algorithms, e.g., [10, 11, 12, 13, 14], require a data fusion center that carries out the computation after collecting information from the nodes, while distributed algorithms, such as [15, 16], rely on selflocalization and the computation is spread throughout the network. Centralized algorithms are likely to provide more accurate estimates, but they suffer from scalability problems, especially for largescale WSNs, while distributed algorithms have the advantage of being scalable and more robust to node failures [17].
Cooperative vs. noncooperative. In a noncooperative algorithm, e.g., [18], each agent receives information only from anchors. For all agents to obtain sufficient information to perform localization, noncooperative algorithms necessitate either long range (and highpower) anchor transmission or a highdensity of anchors [17]. In cooperative algorithms, such as [19, 20], interagent communication removes the need for all agents to be in range of one (or more) anchors [17].
Bayesian vs. nonBayesian. In nonBayesian algorithms, e.g., expectationmaximization (EM) [21], and its variant, expectationconditional maximization (ECM) [22], the unknown positions are treated as deterministic, while in Bayesian algorithms, e.g., nonparametric belief propagation (NBP) [23] and sumproduct algorithm over wireless networks (SPAWN) [24, 25] and its variant SigmaPoint SPAWN [26], the unknown positions are assumed to be random variables with a known prior distribution.
Many existing works on localization using RSS measures, such as [27] and [28], are based on the assumption that the classical path loss propagation model is perfectly known, mostly via a calibration process. However, this assumption is impractical for two reasons. Firstly, conducting the calibration process requires intensive human assistance, which may be not affordable, or may even be impossible in some inaccessible areas. Secondly, the channel characteristics vary due to multipath (fading), and nonnegligible modifications occur also due to mid to longterm changes in the environment, leading to nonstationary channel parameters [29]. This implies that the calibration must be performed continuously [29, 30] because otherwise the resulting mismatch between design assumptions and actual operating conditions leads to severe performance degradation. These facts highlight the need for algorithms that adaptively estimate the environment and the locations. A further difficulty is due to the existence of nonlineofsight (NLoS) propagation in practical localization environments. Among various works handling the NLoS effect, a majority of them have treated the NLoS meaures as outliers and tried to neglect or mitigate their effect, including the maximum likelihood (ML)based approach [31, 32], the weighted leastsquares (WLS) estimator [32, 33], the constrained localization techniques [34, 35], robust estimators [36, 37], and the method based on virtual stations [38]. In contrast to these works, several approaches, including [21, 22, 39], have proposed specific probabilistic models for the NLoS measures, therewith exploiting the NLoS measures for the localization purpose. In the light of these considerations, our aim is to develop an RSSbased, cooperative localization framework that works in mixed LoS/NLoS environments, requires no knowledge on parameters of the propagation model, and can be realized in a distributed manner.
Succinct characterization of the related works and the proposed work
Original contributions: We address the problem of RSSbased cooperative localization in a mixed LoS/NLoS propagation environment, requiring no calibration. To characterize such a mixed LoS/NLoS environment, we assume a modedependent propagation model with unknown parameters. We derive and analyze a robust, calibrationfree, RSSbased distributed cooperative algorithm, based on the ML framework, which is capable of coping with mixed LoS/NLoS conditions. Simulation results, carried out in comparison with a centralized ML algorithm that serves as a benchmark, show that the proposed approach has good overall performance. Moreover, it adaptively estimates the channel parameters, has acceptable communication overhead and computation costs, thus satisfying the major requirements of a practically viable localization algorithm. The convergence analysis of the proposed algorithm is conducted by restating the problem as a graph coloring problem. In particular, we formulate a graph compatibility test and show that for compatible network structures, the convergence is guaranteed. Unlike existing algorithms, the proposed method naturally takes into account the (possible) asymmetry of links between nodes.
The paper is organized as follows. Section 2 formulates the problem and details the algorithms. Section 3 discusses convergence. Section 4 presents the simulations results, while Section 5 concludes the paper. Finally, Appendices A and B contain some analytical derivations which would otherwise burden the reading of the paper.
2 Methods/experimental
2.1 Problem formulation
Consider^{1} a directed graph with N_{a} anchor nodes and N_{u} agent nodes, for a total of N=N_{a}+N_{u} nodes. In a twodimensional (2D) scenario, we denote the position of node i by \(\boldsymbol {x}_{i} = [\!x_{i} \ y_{i}]^{\top } \in \mathbb {R}^{2 \times 1}\), where ^{⊤} denotes transpose. Between two distinct nodes i and j, the binary variable o_{j→i} indicates if a measure, onto direction j→i, is observed (o_{j→i}=1) or not (o_{j→i}=0). In the case when i=j, since a node does not selfmeasure, we have o_{i→i}=0. This allows us to define the observation matrix \(\boldsymbol {\mathcal {O}} \in \mathbb {B}^{N \times N}\) with elements \(o_{i,j} \triangleq o_{i \rightarrow j}\) as above. The aforementioned directed graph has connection matrix \(\boldsymbol {\mathcal {O}}\). It is important to remark that, for a directed graph, \(\boldsymbol {\mathcal {O}}\) is not necessarily symmetric; physically, this models possible channel anisotropies, missdetections and, more generally, link failures. Let m_{j→i} be a binary variable, which denotes if the link j→i is LoS (m_{j→i}=1) or NLoS (m_{j→i}=0). Due to physical reasons, m_{j→i}=m_{i→j}. We define the LoS/NLoS matrix^{2}\(\mathbf {L} \in \mathbb {B}^{N \times N}\) of elements \(l_{i,j} \triangleq m_{i \rightarrow j}\), and we observe that, since m_{j→i}=m_{i→j}, the matrix is symmetric, i.e., L^{⊤}=L. We stress that this symmetry is preserved regardless of \(\mathbf {\mathcal {O}}\), as it derives from physical reasons only. Let Γ(i) be the (open) neighborhood of node i, i.e., the set of all nodes from which node i receives observables (RSS measures), formally: \(\Gamma (i) \triangleq \{ j \neq i : o_{j \rightarrow i}=1 \}\). We define Γ_{a}(i) as the anchorneighborhood of node i, i.e., the subset of Γ(i) which contains only anchor nodes as neighbors of node i. We also define Γ_{u}(i) as the agentneighborhood of node i, i.e., the subset of Γ(i) which contains only agent nodes as neighbors of node i. In general, Γ(i)=Γ_{a}(i)∪Γ_{u}(i).
2.2 Data model

i,u, with u∈Γ_{u}(i), are the indexes for the unknown nodes;

a∈Γ_{a}(i) is an index for anchors;

k=1,…,K is the discrete time index, with K samples for each link;

\(p_{0_{\text {LOS/NLOS}}}\) is the reference power (in dBm) for the LoS or NLoS case;

α_{LOS/NLOS} is the path loss exponent for the LoS or NLoS case;

x_{a} is the known position of anchor a;

x_{u} is the unknown position of agent u (similarly for x_{i});

Γ_{a}(i), Γ_{u}(i) are the anchor and agentneighborhoods of node i, respectively;

The noise terms w_{a→i}(k),v_{a→i}(k),w_{u→i}(k), and v_{u→i}(k) are modeled as serially independent and identically distributed (i.i.d.), zeromean, Gaussian random variables, independent from each other (see below), with variances:
\(\text {Var} [\! w_{a \rightarrow i}(k) ] = \text {Var} [\! w_{u \rightarrow i}(k) ] = \sigma ^{2}_{\text {LOS}}\),
\(\text {Var}[\!v_{a \rightarrow i}(k)] = \text {Var}[\!v_{u \rightarrow i}(k)] = \sigma ^{2}_{\text {NLOS}}\),
and \(\sigma ^{2}_{\text {NLOS}} > \sigma ^{2}_{\text {LOS}} > 0\).
for any k_{1},k_{2},i_{1},i_{2},j_{1}∈Γ(i_{1}),j_{2}∈Γ(i_{2}). The previous equations imply that two different links are always independent, regardless of the considered time instant. In this paper, we call this property link independence. If only one link is considered, i.e., j_{2}=j_{1} and i_{2}=i_{1}, then independence is preserved by choosing different time instants, implying that the sequence \(\left \{ w_{j \rightarrow i} \right \}_{k} \triangleq \left \{ w_{j \rightarrow i}(1), w_{j \rightarrow i}(2), \dots \right \}\) is white. The same reasoning applies to the (similarly defined) sequence {v_{j→i}}_{k}. As a matter of notation, we denote the unknown positions (indexing the agents before the anchors) by \(\boldsymbol {x} \triangleq \left [\boldsymbol {x}^{\top }_{1} \ \cdots \ \boldsymbol {x}^{\top }_{N_{u}}\right ]^{\top } \in \mathbb {R}^{2 N_{u} \times 1}\) and we define η as the collection of all channel parameters, i.e., \(\boldsymbol {\eta } \triangleq \left [\boldsymbol {\eta }^{\top }_{\text {LOS}} \ \boldsymbol {\eta }^{\top }_{\text {NLOS}}\right ]^{\top }\), with \(\boldsymbol {\eta }_{\text {LOS}} \triangleq \left [p_{0_{\text {LOS}}} \ \alpha _{\text {LOS}} \ \sigma ^{2}_{\text {LOS}}\right ]^{\top } \in \mathbb {R}^{3 \times 1}\), \(\boldsymbol {\eta }_{\text {NLOS}} \triangleq \left [p_{0_{\text {NLOS}}} \ \alpha _{\text {NLOS}} \ \sigma ^{2}_{\text {NLOS}}\right ]^{\top } \in \mathbb {R}^{3 \times 1}\).
It is important to stress that, in a more realistic scenario, channel parameters may vary from link to link and also across time. However, such a generalization would produce an underdetermined system of equations, thus giving up uniqueness of the solution and, more generally, analytical tractability of the problem. For the purposes of this paper, the observation model above is sufficiently general to solve the localization task while retaining analytical tractability.
2.3 Timeaveraged RSS measures
as our new observables^{4}. While it would have been preferable to work with the original data from a theoretical standpoint, several considerations lead to the preference of timeaveraged data, most notably: (1) comparison with other algorithms present in the literature, where the data model assumes only one sample per link, i.e., K=1, which is simply a special case in this paper; (2) reduced computational complexity in the subsequent algorithms; (3) if the RSS measures onto a given link needs to be communicated between two nodes, the communication cost is notably reduced, since only one scalar, instead of K samples, needs to be communicated; (4) formal simplicity of the subsequent equations.
where \(\sigma _{j \rightarrow i}^{2}\) is either \(\sigma ^{2}_{\text {LOS}}\) or \(\sigma ^{2}_{\text {NLOS}}\) and the general result follows from link independence.
which represents the information available to the whole network.
2.4 Singleagent robust maximum likelihood (ML)

λ_{i}∈(0,1) is the mixing coefficient for anchoragent links of node i;

ζ_{i}∈(0,1) is the mixing coefficient for agentagent links of node i.
Empirically, we can intuitively interpret λ_{i} as the fraction of anchoragent links in LoS (for node i), while ζ_{i} as the fraction of agentagent links in LoS (for node i). As in [21], the Markov chain induced by our model is regular and timehomogeneous. From this, it follows that the Markov chain will converge to a twocomponent Gaussian mixture, giving a theoretical justification to the proposed approach.
where the maximization is subject to several constraints: λ_{i}∈(0,1), α_{LOS} > 0, α_{NLOS}>0, \(\sigma ^{2}_{\text {LOS}} > 0\), and \(\sigma ^{2}_{\text {NLOS}} > 0\). In general, the previous maximization admits no closedform solution, so we must resort to numerical procedures.
2.5 Multiagent robust MLbased scheme
where \(\hat {\Gamma }_{u}(i)\) is the set of all agent neighbors of node i for which estimated positions exist. We can iteratively construct (and update) the set \(\hat {\Gamma }_{u}(i)\), in order to obtain a fully distributed algorithm, as summarized in Algorithm 1.
A few remarks are now in order. First, this algorithm imposes some restrictions on the arbitrariness of the network topology, since the information spreads starting from the agents which were able to selflocalize during initialization; in practice, this requires the network to be sufficiently connected. Second, convergence of the algorithm is actually a matter of compatibility: if the network is sufficiently connected (compatible), convergence is guaranteed. Given a directed graph, compatibility can be tested a priori and necessary and sufficient conditions can be found (see Section 4). Third, unlike many algorithms present in the literature, symmetrical links are not necessary, nor do we resort to symmetrization (like NBP): this algorithm naturally takes into account the (possible) asymmetrical links of directed graphs.
2.6 Distributed maximum likelihood (DML)
which, in general, does not admit a closedform solution, but can be solved numerically. After obtaining \(\hat {\boldsymbol {x}}_{i}\), node i can estimate p_{0} and α using (23) and (25).
where (again) an initialization phase is required and the set of estimated agentsneighbors \(\hat {\Gamma }_{u}(i)\) is iteratively updated. The key difference with RDML is that, due to the assumption of the links being all of the same type, the estimates of p_{0} and α are broadcasted and a common consensus is reached by averaging. This increases the communication overhead, but lowers the computational complexity, operating a tradeoff. The DML algorithm is summarized in Algorithm 2.
Similar remarks as for the RDML can be made for the DML. Again, the network’s topology cannot be completely arbitrary, as the information must spread throughout the network starting from the agents which selflocalized, implying that the graph must be sufficiently connected. Necessary and sufficient conditions to answer the compatibility question are the same as RDML. Secondly, the (strong) hypothesis behind the DML derivation (i.e., all links of the same type) allows for a more analytical derivation, up to position estimation, which is a nonlinear leastsquares problem. However, it is also its weakness since, as will be shown later, it is not a good choice for mixed LoS/NLoS scenarios.
2.7 Centralized MLE with known nuisance parameters (CMLE)
where \(\left (p_{0_{j \rightarrow i}}, \alpha _{j \rightarrow i}, \sigma ^{2}_{j \rightarrow i}\right)\) are either LoS or NLoS depending on the considered link. It is important to observe that, if all links are of the same type, the dependence from \(\sigma ^{2}_{j \rightarrow i}\) in (28) disappears. From standard ML theory [45], CMLE is asymptotically (K→+∞) optimal. The optimization problem (28) is computationally challenging, as it requires a minimization in a 2N_{u}dimensional space, but still feasible for small values of N_{u}.
3 Convergence analysis
The convergence test of our proposed algorithm (and also of DML) can be restated as a graph coloring problem: if all the graph can be colored, then it is compatible and convergence is guaranteed. As it is common in the literature on graph theory, let G=(V,E) be a directed graph, with V denoting the set of nodes and E the set of directed edges. The set of nodes is such that V=V_{a}∪V_{u}, where V_{a} is the (nonempty) set of anchor nodes and V_{u} is the (nonempty) set of agent nodes.
Definition 1
(RDMLinitializable) A directed graph G is said to be RDMLinitializable if and only if there exists at least one agent node, say x, such that Γ_{a}(x)≥3.
 Initialization (k=0)
 1
All anchors are colored black and all agents white;
 2
Every agent i with Γ_{a}(i)≥3 is colored black;
 1
 Iterative coloring: Start with k=1
 1
Every agent with \(\left  \Gamma _{a}(i) + \hat {\Gamma }^{(k1)}_{u}(i) \right  \geq 3\), where \(\hat {\Gamma }^{(k)}_{u}(i)\) is the set \(\hat {\Gamma }_{u}(i)\) at step k, is colored black;
 2
Every agent j updates is own \(\hat {\Gamma }^{(k)}_{u}(j)\) with the new colored nodes;
 3
The set \(B^{(k)}_{u}\) is updated, where \(B^{(k)}_{u}\) is the set B_{u} at step k;
 4
Set k←k+1 and repeat the previous steps until V_{u} contains only black nodes.
 1
Suppose that the previous algorithm can color the entire graph black in a finite amount of steps, say n. Then, n is called RDMLlifetime.
Definition 2
(RDMLlifetime) A directed graph G is said to have RDMLlifetime equal to n if and only if the RDMLcoloring algorithm colors black the set V_{u} in exactly n steps. If no such integer exists, by convention, n=+∞.
This allows us to formally define compatibility:
Definition 3
 1
G is RDMLinitializable;
 2
the RDMLlifetime of G is finite.
Otherwise, G is said to be RDMLincompatible.
In practice, there are only two ways for which a graph is RDMLincompatible: either G cannot be initialized, or the RDMLlifetime of G is infinite. Testing the first condition is trivial; the interesting result is that testing the second condition is made simple thanks to the following.
Theorem 1
that is, if there is a step h in which no more agents can be colored black and at least one agent is still left white.
Proof
Since \(B^{(k1)}_{u} \subseteq B^{(k)}_{u}\), C_{G} is nondecreasing. An RDMLinitializable graph is RDMLcompatible if and only if it has finite RDMLlifetime, i.e., there must exist n such that C_{G}(n)=V_{u}. But condition (29) implies that there exists h such that C_{G}(h)=C_{G}(h−1)<V_{u}. At step h+1 and all successive steps, C_{G} cannot increase since the set \(B^{(k)}_{u}\) cannot change. To show this, the key observation is that, as the graph G at step h−1 did not satisfy the conditions for \(B^{(h1)}_{u}\) to grow (by hypothesis), the set was equal to itself at step h, i.e., \(B^{(h1)}_{u} = B^{(h)}_{u}\). But since no color change happened in V_{u} at step h, the graph G still does not satisfy the conditions for \(B^{(k)}_{u}\) to grow for k≥h. Thus, C_{G}(k) becomes a constant function for k≥h and can never reach the value V_{u}.
(⇒) Since G is an RDMLincompatible graph by hypothesis, at least one agent must be white, so \(\left B_{u}^{(h)}\right  < V_{u}\) is true for any h. Since C_{G}(k) is nondecreasing, it must become constant for some h>k, but this implies that, for some h, \(\left B^{(h)}_{u}\right  = \left B^{(h1)}_{u}\right \). This implies that, since \(B^{(k1)}_{u} \subseteq B^{(k)}_{u}\) by construction, \(B^{(h)}_{u} = B^{(h1)}_{u}\) for some h. □
Definition 4
is called the RDMLdepth of G.
A complete graph has h_{G}=0, as all agents are colored black during the initialization phase of the RDMLcoloring algorithm.
Corollary 1
Let G be a directed graph. Then, h_{G}≤n, where n is the RDMLlifetime of G.
Proof
If G is not RDMLinitializable, h_{G}=0 as \(B^{(0)}_{u} = \emptyset \). If G is RDMLinitializable, there are two cases: either n is finite or not. In the latter, n=+∞ and h_{G} is finite by previous theorem. If n is finite, h_{G}=n since \(\left B^{(n)}_{u}\right  = V_{u}\) by definition of RDMLlifetime. □
The previous corollary proves that h_{G} is always finite, regardless of G. This allows us to write the graph compatibility test, shown in Algorithm 3. Thanks to the previous results, this algorithm always converges and can be used to test a priori if a graph is RDMLcompatible or not.
Remark Algorithm 3 can be intuitively explained via a physical metaphor, where, in a metal grid (representing the graph), “heat” (information) spreads out starting from some initial “hot spots” (nodes that are colored black in the first iteration). This spreading is continued, reaching more and more locations on the grid, until the event occurs that further spreading of “heat” does not change the “heat map.” If, at this point, there are cold spots (nodes that have not been colored black), the graph is RDMLincompatible. By contrast, if heat spreads throughout the grid, the graph is RDMLinitializable.
4 Results and discussion
The optimization problems (27) and (19) have been solved as follows. For DML, a 2D grid search has been used, while, for RDML and CMLE, the optimization has been performed with the MATLAB solver fmincon.
4.1 Gaussian noise
4.2 NonGaussian noise for NLoS
4.3 Cooperative gain
4.4 Variable K and NLoS fraction
4.5 Communication and computational costs
Regarding computational complexity, both RDML and DML scale linearly with N_{u} and they benefit from parallelization, as the main optimization task can be executed independently for each involved node. As already mentioned, DML operates a tradeoff between communication and computational complexity; in fact, the DML optimization problem (27) is easier to solve than the RDML optimization problem (19). Both problems are nonconvex and may have local minima/maxima, so care must be taken in the optimization procedure.
5 Conclusions
We have developed a novel, robust MLbased scheme for RSSbased distributed cooperative localization in mixed LoS/NLoS scenarios, which, while not being optimal, has good overall accuracy, is adaptive to the environment changes, is robust to NLoS propagation, including nonGaussian noise, and has communication overhead upperbounded by a quadratic function of the number of agents and computational complexity scaling linearly with the number of agents, also benefiting from parallelization. The main original contributions are that, unlike many algorithms present in the literature, the proposed approach (a) does not require calibration and (b) does not require symmetrical links (nor does it resort to symmetrization), thus naturally accounting for the topology of directed graphs with asymmetrical links as a result of missdetections and channel anisotropies. To the best of the authors’ knowledge, this is the first distributed cooperative RSSbased algorithm for directed graphs. The main disadvantage is imposing some restrictions on the arbitrariness of networks’ topology, but these restrictions disappear for sufficiently connected networks. We also derive a compatibility test based on graph coloring, which allows to determine whether the given network is compatible. If it is compatible, convergence of the algorithm is guaranteed. Future work may include the consideration of possible approximations, in order to extend this approach to more general networks, and of alternative models, overcoming the limitations of the standard path loss model.
6 Appendix A: On using the timeaveraged sample mean
Proof
which completes the proof. □
7 Appendix B: CMLE derivation
which completes the derivation.
Footnotes
 1.
Throughout the paper, vectors and matrices will be denoted in bold, ∥v∥ denotes the Euclidean norm of vector v, \( \mathcal {A} \) denotes the cardinality of set \(\mathcal {A}\). We denote by \(\mathbb {E}[X]\) and Var[X] the statistical expectation and variance, respectively, of random variable X. Finally, \(\mathbb {B} = \{0,1\}\) is the Boolean set.
 2.
The values on the main diagonal are arbitrary. Here we choose m_{i→i}=1.
 3.
δ_{i,j}=1 if and only if i=j, zero otherwise.
 4.
For better readability, the notation ^{(m)} has not been carried over, as it implicit in the formalism.
 5.
The reason for this is that localizing a node in 2D requires at least three anchors.
 6.
Hereafter, we omit the conditioning on the set {o_{n→i}} of actually observed RSS measures (received by node i) in the joint likelihood function, since it is implicit in the neighborhood formalism.
 7.
This easily follows by observing that the deterministic (noiseless) version of all the relevant equations for agent X admit infinite solutions.
 8.
This is due to the fact that RSME (Root Mean Square Error) is not a suitable metric when the Error ECDFs are very longtailed, as in our case.
Notes
Acknowledgements
The authors would like to thank prof. F. Bandiera from University of Salento for having started the collaboration which lead to this work.
Funding
The work of L. Carlino was supported by the “Erasmus+ Traineeship” programme of University of Salento. The work of M. Muma was supported by the “Athene Young Investigator Programme” of Technische Universität Darmstadt.
Availability of data and materials
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
Authors’ contributions
LC has contributed towards the development of the proposed algorithms and the performance analysis. DJ has contributed towards the introduction, the related work, and with other minor revisions throughout the paper. MM has contributed towards the example networks regarding graph connectivity, the overall organization of the paper, and with other minor revisions throughout the paper. As the supervisor, AMZ has proofread the paper several times and provided guidance throughout the whole preparation of the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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