Autonomous Robotshttps://doi.org/10.1007/s10514-018-9716-7

Information-sharing and decision-making in networks of radiationdetectors

Indrajeet Yadav1 · Chetan D. Pahlajani2 · Herbert G. Tanner1 · Ioannis Poulakakis1

Received: 15 February 2017 / Accepted: 9 February 2018© Springer Science+Business Media, LLC, part of Springer Nature 2018

AbstractAnetwork of sensors observes a time-inhomo-geneousPoisson signal andwithin afixed time interval has to decide between twohypotheses regarding the signal’s intensity. The paper reveals an interplay between network topology, essentially determiningthe quantity of information available to different sensors, and the quality of individual sensor information as captured by thesensor’s likelihood ratio. Armed with analytic expressions of bounds on the error probabilities associated with the binaryhypothesis test regarding the intensity of the observed signal, the insight into the interplay between sensor communicationand data quality helps in deciding which sensor is better positioned to make a decision on behalf of the network, and linksthe analysis to network centrality concepts. The analysis is illustrated on networked radiation detection examples, first insimulation and then on cases utilizing field measurement data available through a U.S. Domestic Nuclear Detection Office(dndo) database.

Keywords Radiation sensor networks · Error Probability bounds · Networked detection · Binary hypothesis testing

1 Introduction

Spatially distributed sensor networks are already deployedto help authorities mobilize and respond in a timely fashionto emergency situations, including natural disasters such ashurricanes (Morreale et al. 2011), earthquakes (Ogata 1999)and tsunamis (González et al. 2005). Decision-making inthis type of networks is currently not automated; typically,the infrastructure merely provides the information for humandecision-makers. The latter, however, can definitely be sup-ported by automated systems that process quickly the large

This is one of several papers published in Autonomous Robots compris-ing the “Special Issue on Distributed Robotics: From Fundamentals toApplications”.

B Indrajeet Yadavindragt@udel.edu

Chetan D. Pahlajanicdpahlajani@iitgn.ac.in

Herbert G. Tannerbtanner@udel.edu

Ioannis Poulakakispoulakas@udel.edu

1 University of Delaware, Newark, USA

2 Indian Institute of Technology, Gandhinagar, India

volume of information collected by the network to indicateoptimal choices based on the constraints of the case at hand.The results in this paper find application in the problemof detecting nuclear material in transit, using a distributednetwork of radiation sensors. The severity of the threat asso-ciated with radioactivematerials falling into the wrong handshas been recognized (Byrd et al. 2005). Possible mitigationstrategies include networks of detectors deployed along roadsand highways, tasked with detecting illicit material that hasslipped through border checks or portal alarms (Byrd et al.2005).

Existing work on decision-making over networks ofobservers that monitor a physical process and have to decideon its state, suggests that the performance of decision-makingis affected by the structure of the network. Interestingly,in an application context quite different from the one con-sidered in this paper, one encounters neurophysiologicalmodels of decision-making that represent neurons as stochas-tic evidence accumulators, each described by a drift-diffusionprocess accruing evidence in continuous time by observinga noisy signal; the information exchange among individ-ual nodes affects the certainty of each node in a way thatis governed by information centrality (Poulakakis et al.2012, 2016). Information centrality is a structural prop-erty of the underlying interconnection graph that depends

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on the totality of paths connecting that node with the restof the network (Stephenson and Zelen 1989). Althoughthese results refer to continuous-time implementations ofsequential probability ratio tests inspired by human decision-making models (Bogacz et al. 2006), they still highlightthe effect of general network topologies on decision-makingperformance.

Due to the explosive combinatorial complexity ofdecision-making in distributed sensor systems (Tsitsiklis andAthans 1985), only specific communication topologies forthe network have been explored. There is emphasis on par-ticular directed rooted trees (Tsitsiklis 1993; Viswanathanand Varshney 1997; Varshney 1997; Tenney and Sandell1981), where the root is a designated fusion center nodethat makes the final decision. For this limited range oftopologies where information from all observers eventuallyreaches the designated fusion center, results show that deci-sion performance is indeed affected by the structure of thenetwork. To curb the complexity associated with distributeddecision-making, themajority of results formore general treetopologies focus on asymptotic analyses, where the order ofthe graph grows unbounded. For particular applications, suchas nuclear detection, practical and economic considerationspreclude the deployment of large-scale networks (Sundare-san et al. 2007). In that smaller-scale regime, and particularlyfor the case where decision makers do not have direct accessto (compressed) information from all other observers, thereis not enough knowledge and insight to determine the spe-cific effect of general network topologies on decision-makingperformance.

Remote nuclear detection in particular is challenging for atleast two reasons. Firstly, and assuming that there is a radioac-tive source in the vicinity of the radiation detectors, thosedetectors pick up not only this source’s radioactivity, but alsoubiquitous, naturally occurring, background radiation; froman inexpensive counter’s (not spectrometer’s) perspective,

the two signals are of identical nature and indistinguishableonce superimposed. The second reason relates to attenuation:although a kilogram of Highly Enriched Uranium (heu) canemit as many as 4× 107 gamma rays per second (Byrd et al.2005), shielding and attenuation can limit the effective detec-tion range to a few feet, and require detection times that canrange from several minutes to hours. To put these in perspec-tive, the gamma-ray emission of nuclear missiles containingheu becomes comparable to background just 25cm awayfrom the warhead (Srikrishna et al. 2005). The problem isexacerbated by the motion of the signal source. Not onlydoes the mathematical model of the physical phenomenonchange (becoming time-inhomogeneous), but now detectorshave limited time to decide before the target disappears fromsight: the sensors are faced with a problem of detecting in amatter of seconds, a weak time-varying signal, buried insideanother signal of the same nature.

Significant improvements in decision-making perfor-mance are achievable by allowing limited sensor mobility(Pahlajani et al. 2014b). Radiation sensors mounted onmobile robots could regulate their distance to a source andthus increase the Signal-to-Noise-Ratio (snr) of their obser-vations. This paper therefore focuses on the case of a sensornetwork observing a time-inhomogeneous Poisson process,just as the one modeling the arrival of gamma-rays on aradiation counter when the distance between sensor andsource is changing over time, and has to decide within afixed time interval, between two hypotheses concerning theintensity of the observed process. In a rather typical fixed-interval networked decision-making scenario [cf. (Pahlajaniet al. 2014a)], after observing this process for the giventime period, all sensors would submit a statistic to a pre-determined (data) fusion center that makes a decision forthe whole network; see Fig. 1a. This paper explores alterna-tive cases where the topology of the network can vary andmay be a tunable parameter of the decision-making algo-

Fig. 1 A source (star) is moving through a network of sensors which share their statistics over different topologies. a All sensors send their locallikelihood ratios LT (i) to a fusion center. b Sensors share their local likelihood ratios LT (i) over a directed graph. Communication occurs at timeT

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rithm, sensors may relay information to other sensors overone or multiple network hops, and any sensor can potentiallybecome the decisionmaker. The goal here is twofold. First, toexpose the interplay between information sharing and infor-mation quality on the capacity of individual sensors in thenetwork tomake accurate decisions. Second to develop a net-working strategy that while minimizing transmission energyoverhead, permits sensors that aremore advantageously posi-tioned in the network to make the most accurate decisions onbehalf of the whole group. The networking strategy referredto here is not a networking protocol, but is rather developed atthe application layer, assuming that a physical layer providesthe necessary channel capacity and bandwidth.

The interdependencies between robot motion and infor-mation sharing in robotic groups has been exploited in thecontext of several applications. Early seminal work hasexposed the effect of the topology of information flow onthe stability properties of vehicle formations (Fax and Mur-ray 2004). Similar connections between the topology ofinformation exchange in networks of robotic vehicles havebeen drawn in studies of controllability (Mesbahi 2005) andcooperative localization (Howard et al. 2003; Roumeliotisand Bekey 2002; Stroupe et al. 2001). Somewhat similarin spirit, in terms of control of the flow of information, iswork that appears in the context of mobile sensor networkdeployment and distributed map building (Li et al. 2003);there, each sensor is considered as an information reposi-tory, and control of flow of information is important in orderto achieve a goal. Although robotic-assisted networked radi-ation detection has been investigated (Sun and Tanner 2015),the connection between the routing of sensor informationand detection accuracy has not been adequately explored.Preliminary work that formed the basis of this paper (Pahla-jani et al. 2016) touched on this question but stopped shortof examining networked decision-making accuracy from agraph-theoretic and complex network viewpoint. This paperproceeds with this extension. The problem setting is thata group of mobile robotic sensor platforms equipped withradiation counters have to collectively decide whether a tar-get they are tracking is benign or radioactive. The roboticplatforms need to have a minimal payload capacity to carryhand-held radiation detectors, be able to establish an ad-hocwireless network, and develop spatial velocities that wouldallow them to reasonably keep upwith themoving target theyare tracking, at least for a relatively short time period. Ques-tions of interest include finding which one of these roboticagents is best positioned to make the decision on behalf ofthe whole group, and what information should it utilize fromother agents when there are considerations of transmissionenergy cost. Whereas earlier work (Pahlajani et al. 2016)indicated sensor Likelihood Ratio Test (lrt) thresholds asindicators of individual decision-making accuracy, this papershows that network centrality concepts are at the core of

answering the question of how to route the sensor informa-tion in the network.

2 Background

The analysis focuses on the effect of information sharing ondecision-making by a network of sensors observing a time-inhomogeneous Poisson process, and as such, is applicablein a variety of domains. That said, it will be convenient fromthis point on to frame the discussion largely in terms of appli-cations to nuclear detection. It goes without saying that thetreatment can be carried over to other applied problems aftersuitably reinterpreting various mathematical quantities.

With the aforementioned applicationdomain inmind, con-sider a network of k radiation sensors that is deployed over aspatial region of interest for the purpose of detecting nuclearmaterials in transit; see Fig. 1. The typical detection scenarioinvolves a vehicle (target) suspected of being a carrier ofnuclear material (source) moving through the sensing fieldof this network. The objective is to decide, at the end of afixed time interval [0, T ], whether the counts recorded at thesensors can be attributed solely to ubiquitous backgroundradiation (hypothesis H0) or whether they contain, in addi-tion, radiation from a source carried by the moving target(hypothesis H1). Local decision-making can be enhancedthrough allowing, at the terminal time T , limited communi-cation among sensors according to a suitable communicationtopology, following which each sensor is enabled to act as aDecisionMaker (dm) operating on the information availableto it.

The topology of the sensor network is modeled by adirected graph; each sensor is represented by a node, anda directed edge from node i to node j signifies that thereis directed flow of information from sensor i to sensor j .Sensors do not need to communicate until time instant T , atwhich a decision needs to be made by the network. At thattime instant each sensor sends a locally computed statisticto its network neighbors along outgoing edges in the com-munication graph. In this way, over some fixed number oftransmission phases, sensor statistics can propagate throughthe network. Once the transmission phases are completed,each sensor—now referred to as a dm—implements a lrt

by appropriately fusing the information that has been madeavailable to it. The test compares the combined statistic tosome threshold and depending on the outcome of this com-parison, a (local) decision is made.

Intuitively, (local) decision-making accuracy is affectedby how much information is available to a given dm (that is,its own statistics, the number of incoming edges in the infor-mation sharing graph, and the number of network hops localstatistics can take in the transmission phase) and the expectedquality of information observed individually by dms (the lat-

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ter captured in their threshold constants, computed basedon the specific sensor-source position dynamics during theobservationwindow). The principal challenge now lies in for-malizing exactly how these factors mathematically influencethe decision error probabilities, given that these probabili-ties are, in all but perhaps the simplest cases, intractable toanalytic computation. The paper circumvents this difficultyusing Chernoff upper bounds on the error probabilities foreach dm, borrowing analytic expressions for these boundsfrom existing work (Pahlajani et al. 2014a, b).

A review of some existing notation and results (Pahlajaniet al. 2014a) helps set the stage for the decision-making rulesconsidered in this paper. The probabilistic setup is as follows.On the measurable space (Ω,F ), there is a k-dimensionalvector of counting processes N t � (Nt (1), . . . , Nt (k)),t ∈ [0, T ]. Here, Nt (i) represents the number of countsrecorded at sensor i ∈ {1, 2, . . . , k} up to (and including)time t ∈ [0, T ]. The two hypotheses, H0 and H1, regardingthe state of the environment correspond, respectively, to twodistinct probability measures P0 and P1 on (Ω,F ). Withrespect to P0, the Nt (i)’s, 1 ≤ i ≤ k, are assumed to beindependent Poisson processes with Nt (i) possessing inten-sity βi (t), while with respect to P1, the Nt (i)’s, 1 ≤ i ≤ k,are assumed to be independent Poisson processes with inten-sities βi (t) + νi (t), respectively. The functions βi (t), νi (t),1 ≤ i ≤ k, defined for t ∈ [0, T ] are assumed to be bounded,continuous and strictly positive as in Pahlajani et al. (2014a).Here, βi (t) is the (possibly time-varying) intensity at timet due to background radiation at the spatial location of sen-sor i , while νi (t) represents the intensity due to the source(if present) as perceived by sensor i at time t . The time-dependence of νi (t) arises from relative motion between thesource and the sensor; indeed, in the context of radiationmea-surement it is generally accepted that νi (t) is proportional tothe inverse square of the distance ri (t) between the sourceand sensor i (Nemzek et al. 2004).

A test for deciding between two hypotheses H0 and H1

is an event B1, whose occurrence can be ascertained on thebasis of sensor observations over [0, T ], and has the follow-ing significance: If the outcome ω ∈ B1, decide H1; if theoutcome ω ∈ B0 � Ω \ B1, decide H0. For such a test, twotypes of errors can occur. A false alarm occurs ifω ∈ B1 withH0 being the correct hypothesis; this occurs with probabilityP0(B1). A miss occurs if the outcome ω ∈ B0 while H1 isthe true hypothesis; this occurs with probability P1(Ω \ B1).

In the context of theNeyman–Pearson lemma, one is givenan acceptable upper bound α ∈ (0, 1) on the probability offalse alarm, and the problem is to find an optimal test: aset B∗

1 ∈ F which maximizes the probability of detection1

1 For a test B1, the probability of detection is given by P1(B1); ofcourse, this equals 1−P1(Ω \ B1) where P1(Ω \ B1) is the probabilityof miss.

over all tests whose probability of false alarm is less thanor equal to α. If P1 is the probability measure on (Ω,F )

corresponding to hypothesis H1, and P1 is absolutely contin-uous with respect to probability measure P0 correspondingto H0, the optimal test is an lrt (Brémaud 1981) with thelikelihood ratio defined as the Radon-Nikodym derivative ofP1 with respect to P0. For the present problem, it can beseen (Pahlajani et al. 2014a) that dP1

dP0= LT , with the latter

being defined by Eqs. (1), (2) below. Intuitively, LT captureshow much more likely a given set of observations is underone hypothesis versus the other. The likelihood ratio is thencompared to a suitably selected threshold to decide betweenthe two hypotheses, based on whether or not it is above orbelow the threshold. The challenge lies in selecting an opti-mal threshold.

The optimal lrt for deciding between H0 and H1 isobtained as follows (Brémaud 1981; Pahlajani et al. 2014a).For i ∈ {1, 2, . . . , k}, let τn(i) for n ≥ 1 denote the n-th jumptime (when the sensor generates a count) of Nt (i). Then, asshown in Brémaud (1981); Pahlajani et al. (2014a), the like-lihood ratio LT (i) of sensor i up to and including time T isgiven by

LT (i) � exp

(−∫ T

0νi (s)ds

)NT (i)∏n=1

[1 + νi (τn(i))

βi (τn(i))

], (1)

where the convention that∏0

n=1(·) = 1 is used. Assumingthat P1 is absolutely continuous with respect to P0, the like-lihood ratio of the fusion center LT is computed, and thebinary hypothesis test

{LT

H1≷H0

γ

}where γ ∈ R and LT �

k∏i=1

LT (i) (2)

becomes optimal in the Neyman–Pearson sense; that is, ifB is any test whose probability of false alarm P0(B) ≤P0(LT ≥ γ ), then the probability of miss for the test (2)is at least as low as that for B, i.e., we have P1(LT < γ ) ≤P1(Ω \ B).

In the context of the test (2), the decision is made at asingle network node that receives all sensory informationfrom the network, and processes it by computing the productLT to issue the decision. This node is called the fusion center;see Fig. 1a. Before proceeding further, note that the onlyinformation needed from sensor i ∈ {1, 2, . . . , k} includesthe functions βi (·) and νi (·) together with the single realnumber LT (i). Put anotherway, once the problemparametersβi (·), νi (·) and the local likelihood ratios LT (i) are known,there is no increase in accuracy that can be obtained throughknowledge of the sample path t �→ Nt (i).

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3 Deciding without a fusion center

One of the drawbacks of the setup of Fig. 1a described aboveis the vulnerability of the system to targeted attacks or fail-ures: if the fusion center is lost, the entire detection systemcollapses. This motivates one to study the following prob-lem. Suppose that at the terminal time T (or just after), thereis some sharing of the LT (i)’s (and the problem parametersβi (·), νi (·)) among the sensors according to a directed graph,as depicted inFig. 1b.Assume that no single sensor has accessto all local likelihood ratios, i.e., there is no obvious choiceof fusion center. If each sensor is now enabled to act as adm operating on the information available to it, one can askwhich dm is the most reliable in terms of decision-makingaccuracy.

To formulate this problem precisely, a directed graph isfirst specified. This graph encodes the allowed communica-tion among the sensors. Recall that a directed graph is anordered pair G = (V , E) where V is a set of vertices ornodes, and E is a set of ordered pairs of nodes, referred toas arcs or edges. For the problem at hand, the vertex setV = {1, 2, . . . , k} indexes the set of observers (sensors).The arcs in E correspond to inter-sensor communication inthe sense that (i, j) ∈ E if, and only if, there is directionalflow of information from sensor i to sensor j . Thus, if thereis two-way communication between sensors i and j , both(i, j) and ( j, i) are included in E . Self-loops are meaning-less in this context and therefore excluded. Finally, at thisstage, it will be convenient to assume that information trav-els exactly one edge, and no further, thus realizing single-hoptransmissions. In other words, if there are edges (i, j) and( j, ) in E , it is assumed that LT (i) is sent from sensor i tosensor j , and LT ( j) is sent from j to , but does not getLT (i) because the latter becomes available to j after the endof the single-hop communication cycle. Section 5 considersmore general scenarios where this assumption is lifted, andmulti-hop transmissions allow the dissemination of informa-tion pertinent to decision-making over appropriate networktopologies.

For each sensor i , letSi comprise the set of sensors whoseinformation is made available to sensor i just after time T(including itself). Thus, for 1 ≤ i ≤ k,

Si � { j ∈ {1, 2, . . . , k} : Sensor i knows LT ( j) and

β j (·), ν j (·) just after time T } .

Since a sensor always has access to its own information, wehave i ∈ Si for all 1 ≤ i ≤ k. Thus, Si consists of theindex i , together with the indices corresponding to incom-ing edges. Once inter-sensor communication has taken place,each sensor can be considered a dm. Thus, for 1 ≤ i ≤ k,dm(i) refers to sensor i once it has access to the quantities{LT ( j), β j (·), ν j (·) : j ∈ Si }. Letting dm(i) use the test

{LT (i)

H1≷H0

γi

}; LT (i) �

∏j∈Si

LT ( j) , γi > 0 (3)

the probabilities of false alarm and miss for dm(i) are

PF,i � P0 {LT (i) ≥ γi } , PM,i � P1 {LT (i) < γi } ,

respectively.However, an analytical computation of these error proba-

bilities is often extremely hard, except in simple Gaussiansetups. These—intractable, in general—true probabilitiescan be bounded using Chernoff bounds expressed in termsof moment generating functions of the underlying randomvariables (Trees 2001). For the application setup utilized inthis paper, these bounds have been derived as functions ofthe time varying distance between the source and the sensors(Pahlajani et al. 2014b).

Take a sensor i , let the logarithm of the threshold it usesbe denoted ηi � log γi , and define μi (t) = 1 + νi (t)

βi (t)as

an indicator of the sensor’s snr, which depends on the timevarying relative distance between the source and the sensor[cf. (Nemzek et al. 2004)]. For the function Λi : R → R;p �→ Λi (p) � logE0

[(LT (i))p

]it follows Pahlajani et al.

(2014b, Theorem 8) that the Chernoff bounds on the falsealarm and missed detection probabilities associated with thelrt performed at sensor i are (Pahlajani et al. 2014b)

PF,i ≤ exp

(infp>0

[Λi (p) − p ηi ])

PM,i ≤ exp

(infp<1

[Λi (p) + (1 − p)ηi ])

. (4)

The functions Λi (p) can be computed explicitly via

Λi (p) =∑j∈Si

∫ T

0[μ j (s)

p − pμ j (s) + p− 1]β j (s)ds .

(5)

The bounds (4)–(5) hold for any ηi = log γi ∈ R.In a fixed-interval lrt like the one performed here, the

selection of terminal time T is critical to the decision-makingprocess. In principle, T is determined by the relative speedof the target with respect to the robotic sensor platformsthat pursue it. Clearly, a fast moving target would allow lessobservation time to the sensors compared to a slow movingone. An estimate for T may be obtained with some knowl-edge of sensor network coverage area, and the velocity ofthe target. Alternatively, detection of stationary sources hasbeen attempted (Jarman et al. 2004; Rao et al. 2008) usingSequential Probability Ratio Test (sprt) approaches, whichhave no restriction for a terminal decision time T . However,in an sprt case, the decision time T—which is closely related

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to the number of observations required to achieve the desireddetection and false alarm probabilities—is random and mayturn out to be significantly high (Rao et al. 2008), particularlyfor low intensity sources.

In order to effectively use these bounds, one would needto know where, if at all, the infima in (4) are realized,and further, whether the infima are negative—to ensure thatthe bounds are non-trivial. One proceeds here by makingrepeated use of convexity of the functions p �→ Λi (p).Indeed, it follows Pahlajani et al. (2014b, Lemmas 16–19)that the function p �→ Λi (p) is C2 with derivatives given bydifferentiating under the integral sign; further, Λ′′

i (p) > 0for all p ∈ R, implying that the function p �→ Λi (p) isstrictly convex; and finally that Λ′

i (0) < 0, Λ′i (1) > 0. It

now follows Pahlajani et al. (2014b, Proposition 13) that ifηi is chosen to lie in (Λ′

i (0),Λ′i (1)), then the infima in (4) are

attained at a unique p∗i ∈ (0, 1) and the infima are negative.

More precisely, there exists a unique p∗i ∈ (0, 1) given by

Λ′i (p

∗i ) = ηi such that

infp>0

[Λi (p) − pηi ] = EF,i (p∗i ) < 0

infp<1

[Λi (p) + (1 − p)ηi ] = EM,i (p∗i ) < 0 ,

where the error exponents EF,i (p), EM,i (p) mapping (0, 1)to R are

EF,i (p) � Λi (p) − pΛ′i (p) and

EM,i (p) � Λi (p) + (1 − p)Λ′i (p) . (6)

Thus, if ηi ∈ (Λ′i (0),Λ

′i (1)), then the tightest error prob-

ability Chernoff bounds for dm(i) are given by

PF,i ≤ exp[EF,i (p∗i )]< 1, PM,i ≤ exp[EM,i (p

∗i )]< 1. (7)

To rank the dms in terms of their capacity to make accu-rate decisions, one ideally solves the following problem: Letα ∈ (0, 1) (acceptable upper boundon the probability of falsealarm) be given. Allowing each dm to choose its own thresh-old to comply with the constraints that PF,i ≤ α and PM,i

is minimized, rank the nodes in increasing order of PM,i .The node with the smallest PM,i is the best dm, at least forthe particular α. The challenge, as noted earlier, is that theerror probabilities PF,i and PM,i are not amenable to ana-lytic computation.We thereforeworkwith the correspondingChernoff bounds (7) and study the problem stated above withPF,i and PM,i replaced by the corresponding tightest upperbounds.

To solve this problem, the threshold selection algorithmPahlajani et al. (2014b, Proposition 14) is employed. Thealgorithm implies that if, for some 1 ≤ i ≤ k, we havelogα > −Λ′

i (1), then there exists a unique p†i ∈ (0, 1)which solves the equation

EF,i (p†i ) = logα . (8)

Moreover, p†i minimizes EM,i (p) over all p ∈ (0, 1) whichsatisfy EF,i (p) ≤ logα. This minimum value of EM,i (p) isgiven by

EM,i

(p†i

)= logα + Λ′

i

(p†i

).

Thus, choosing ηi = Λ′i

(p†i

), i.e. letting dm (i) select the

threshold γi = exp(

Λ′i

(p†i

) ), yields

PF,i ≤ α , PM,i ≤ α exp(

Λ′i

(p†i

) )= α γi .

To provide some insight on the foregoing expressions [see(Pahlajani et al. 2014b) for details] one can use convexity toshow that on (0, 1), the functionsEF,i ,EM,i are differentiableandnegative,withEF,i being strictly decreasingwhileEM,i isstrictly increasing. The threshold selection algorithm can besummarized as follows:Take the thresholdγi = exp(Λ′

i (p) )

and select p ∈ (0, 1) as small as possible to make EM,i assmall as possible, while ensuring that the false alarm con-straint is met, i.e., EF,i (p) ≤ logα. The latter is possibleonly if logα is strictly greater than the infimum of EF,i on(0, 1), which is easily computed to be −Λ′

i (1). One should

now find p†i by solving (8).With this information, the dms can be ranked from most

reliable to least reliable by ranking the quantities EM,i (p†i ),

1 ≤ i ≤ k (or, equivalently, exp(EM,i (p†i ) ) = αγi ) from

smallest to largest. Thesefindings are summarized as follows.

Theorem 1 For logα > max1≤i≤k(−Λ′i (1))withα ∈ (0, 1),

and for p†i ∈ (0, 1) being the solution of EF,i (p†i ) = logα,

let i range in {1, . . . , k} and define the lrt threshold forDM(i) as

Mi (α) � exp(

Λ′i

(p†i

) ). (9)

Then these thresholds are indicators of decision-makingaccuracy at their corresponding dms, in the sense that anordering (i1, i2, . . . , ik) such that

Mi1(α) ≤ Mi2(α) ≤ · · · ≤ Mik (α) ,

ranks the dm’s from most accurate to least accurate.

Remark 1 Note that the accuracy index Mi (α) is tied to thespecific α ∈ (0, 1). This naturally prompts the question: fora given network and set of problem parameters, does thesensor which has the smallest Mi (α) vary with α. It is alsonatural to ask, for a given network and set of problem param-eters, whether the ranking based on Mi (α) coincides with

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the ranking based on minimizing PM,i subject to PF,i ≤ α,i.e., do the Chernoff bounds faithfully capture the relativedecision-making accuracies of various sensors. Monte-Carlosimulation results presented in Sect. 4.2 below confirm thefaithfulness of the bounds in capturing the ranking basedon true probabilities—caution needs to be exercised whenneighboring sensors exhibit very similar performance; a the-oretical treatment of this issue and the effect of varying α aresubjects of ongoing work.

4 Accuracy, networking, and quality ofinformation

Let A denote the adjacency matrix of the graph that cap-tures the topology of the communication network. As wasmentioned in Sect. 3, we assume here that sensor informa-tion travels only for a single (network) hop; this assumptionwill be relaxed in the following section. The lrt (3) can bewritten in matrix form as

(I + A)

⎡⎢⎣log LT (1)

...

log LT (k)

⎤⎥⎦ H1

≷H0

⎡⎢⎣

Λ′1(p

†1)

...

Λ′k(p

†k )

⎤⎥⎦ (10)

where the inequalities are interpreted element-wise. In (10)each sensor i performs individually its lrt using its own

threshold Λ′i

(p†i

), provided of course that its constraint on

the probability of false alarm can be satisfied—for if not,the particular network node has to abstain. Note now that

the test thresholds γi = exp(

Λ′i (p

†i )), which have been

used in Sect. 3 to rank dms according to their perceiveddecision-making capacity, are similarly related to the net-work topology and information quality via their exponentsas follows

decision-making accuracy︷ ︸︸ ︷⎡⎢⎣

Λ′1(p

†1)

...

Λ′k(p

†k )

⎤⎥⎦

= (I + A)︸ ︷︷ ︸network topology

⎡⎢⎢⎢⎣

∫ T0 [μp†1

1 logμ1 − μ1 + 1]β1ds...∫ T

0 [μp†1k logμk − μk + 1]βkds

⎤⎥⎥⎥⎦

︸ ︷︷ ︸sensor information quality

.

(11)

Equation (11) encodes the way that information sharingand data quality contribute to decision-making accuracy onthe part of dms. Essentially, decision-making accuracy is

factored into network topology and data quality. The adja-cencymatrix of the graph indicates how sensor information isshared in one hop, and the vector of integrals in the right handof (11) expresses how the history of sensor-target distancedetermines the signal-to-noise ratio of the particular sensornode. Both network topology and data quality contribute toaccuracy, the former by increasing the amount of data, thelatter by the information content of the data.

The next two subsections present case studies that offerevidence in support of the claim that the network topol-ogy in a network of radiation sensors significantly affectstheir decision-making performance. The first case, specifi-cally, verifies the theoretical prediction that in instances ofuniform sensing quality among all observers, the one thatreceives themost information fromother nodes is the one bestsuited to make the decision on behalf of the whole network.The second example addresses the case of heterogeneity indetector measurement characteristics, and demonstrates hownetworking may compensate for lack of sensing quality, inthe sense of (11).

4.1 Uniform sensor information quality

This section demonstrates that all else being equal, infor-mation quantity determines the best decision maker. Specif-ically, if all detectors collect information with the samesensing characteristics, then the sensor receiving the mosttransmissions from other sensors is best suited to performthe hypothesis test.

Suppose that all sensors 1 ≤ i ≤ k are subject tothe same background noise βi ≡ β, and their perceivedsource intensities are identical, i.e., νi ≡ ν. Then obviouslyμi ≡ μ = 1 + ν/β. Let Si denote the set of nodes incident(communicating to) node i and |Si | be the cardinality of thatset. Define

g(p) � −β(μp − pμp logμ − 1) T

h(p) � −β [μp + (1 − p) μp logμ − μ] T ,

and observe that (6) now reduces to

EF,i (p) = −g(p) · |Si |EM,i (p) = −h(p) · |Si |.

Since EF,i (p), EM,i (p) are negative for p ∈ (0, 1) with theformer being strictly decreasing and the latter strictly increas-ing [see Pahlajani et al. (2014b, Lemma 19)], it follows thatg(p) is positive and strictly increasing for p ∈ (0, 1), whileh(p) is positive and strictly decreasing for p ∈ (0, 1).

Fix α as in Theorem 1. Let i, j ∈ {1, 2, . . . , k} with i =j . The statement made in the beginning of this section istechnically translated as follows:

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(i) If |Si | > |S j |, then Mi (α) < M j (α),(ii) If |Si | = |S j |, then Mi (α) = M j (α),(iii) If |Si | < |S j |, then Mi (α) > M j (α).

Only claim (i) is proven here; the arguments used in this proofcanbe easilymodified to prove the other two. Suppose |Si | >

|S j |. By (8), one has EF,i

(p†i

)= logα and EF, j

(p†j

)=

logα, and recalling the notation above,

g(p†i

)= − logα

|Si | , g(p†j

)= − logα

|S j | .

Noting that − logα > 0 and |Si | > |S j |, one writes

g(p†i

)< g

(p†j

). Since g is strictly increasing, it follows

that p†i < p†j . Since function h is strictly decreasing, it must

be h(p†i

)> h

(p†j

). Given that h is positive and by assump-

tion |Si | > |S j |, it follows that |Si | ·h(p†i ) > |S j | ·h(p†j ).

Taking negatives, yields EM,i (p†i ) < EM, j (p

†j ). By Theorem

1 now it follows Mi (α) < M j (α).Thus under the assumption that the measurement process

has identical statistical characteristics at each node, the nodeswith the largest in-degree are the most accurate dms. It isinteresting to note that this observation is in contrast to thediffusive models studied in Poulakakis et al. (2012, 2016), inwhich local centrality measures such as those based on nodaldegrees cannot capture the certainty of each unit in terms ofthe collected evidence.

4.2 Non-uniform sensing quality

Similarly to (Nemzek et al. 2004), five sensors are arrangedalong a straight line at fixed locations with 0.5m distancefrom each other (Fig. 2). The first sensor positioned at 0.5m.A radioactive source moves parallel to this array of sen-sors with a constant speed of 0.03m/s. As a result of thesource’s motion, the measurement characteristics among dif-ferent sensors are not the same. The source is moving along astraight line which runs parallel to the line where the sensorsare arranged at a distance of 0.5m, starting from an initialposition 0.5m behind the sensor array.

Fig. 2 Detecting a source moving over a sensor array

Fig. 3 Different communication topologies studied in Example 2. Adirected edge from node i to node j indicates that at time T , the locallycomputed LT and μ at node i is transmitted to node j

The activity of the source and background radiation aremeasured in gamma rays emitted per second, i.e., counts persecond (cps). For the source, this activity is taken at a = 3cps, while for the background it is assumed that β = 0.167cps. Independent arrival processes are numerically generated(Lewis and Shedler 1979; Pasupathy 2009) with the afore-mentioned intensities. The maximum acceptable probabilityof false alarm is set to α = 10−3, and just for numericalpurposes the sensor cross-section is assumed to be 1 m2 [cf.(Pahlajani et al. 2014b)].

Three different topologies over which locally processedinformation is shared among sensors, are shown in Fig. 3.The arrows indicate unidirectional flow of likelihood ratiosLT and the histories of the intensities β, ν, between (net-work) adjacent sensors at timeT . These three communicationtopologies (graphs) result in different sensor network perfor-mance in terms of accuracy of decision making, as measuredby the computed bounds on the probability of missed detec-tion. Various detection scenarios are compared in Figs. 4, 5, 6and 7, which provide evidence that for constant backgroundactivity, the performance of each dm naturally depends onthe perceived signal strength and on the sensors’ integrationwindow T . At the very top of Figs. 4, 5, 6 and 7 a dottedline segment ending at a dot is shown: this is the path of thesource during the time interval from 0 to T seconds. Rightbelow, the arrangement of the five sensors in the sensor arrayis depicted, with each sensor represented by a red star. Thecolor-coded curves appearing below the sensor array expressthe variation in the logarithm of the Chernoff bound over theprobability of missed detection for each individual dm.

Variables Mi (α) derived in (9) are directly proportionalto the bounds on the probability of missed detection, (PM,i ),with in the role of theα as the proportionality constant. There-fore, PM,i , or its exponent for that matter, can serve as a

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Fig. 4 The communication topologies of Fig. 3 result in differentbounds for the probability of missed detection for sensors S1–S5. Thebound on the probability of missed detection varies within two extremecases: the one where no information is shared (magenta curve), andthe one where there is a fusion center having information from all sen-sors (black curve). Missing line segments mark sensors that do notsatisfy the constraint on the probability of false alarm. a Inter-sensordistance=0.5m. b Inter-sensor distance=1m (Color figure online)

metric for dm ranking. Whenever the constraint on probabil-ity of false alarm (PF,i ) cannot be satisfied, PM,i andMi (α)

cannot be obtained, and therefore the particular sensor is notranked.

Comparing graphs 1 and 2, Fig. 4a suggests that increasinginformation sharing improves decision making accuracy (asquantified by the logarithm of the Chernoff bound on theprobability of missed detection). Observing the performanceof sensor 3 in the topology of graph 3 in Fig. 3 is particularlyrevealing. In graph 3, sensor 3 does not receive informationfrom sensor 1, nor does make good quality measurementsdue to the large distance from the source. Yet, because of theadditional information received by sensors 4 and5, sensor 3 isa better dm compared to sensors 1 and 2 which were closer tothe source. In the topology of graph 2, sensor 3 does not havemore incoming edges compared to nodes 1 and 2. However,it receives better information over its communication linkscompared to sensors 1 and 2—statistics from sensors whichwere closer to the source—and thus it again outperformssensors 1 and 2.

Fig. 5 Decision-makingperformance as a functionof sensor integrationtime in the topology of graph 1

That last point is reinfored in Fig. 4b: more information(incoming edges) to a sensor does not always result in betterperformance. In Fig. 4b the spatial arrangement is scaled upby 2 such that all the distances are doubled (e.g. sensors are1 m apart). Doubling the distances between sensors weak-ens the observed signal: the intensity of the source perceivedby each sensor decreases by a factor of four. In fact, somesensors are now too far away to make decisions with thedesired confidence (bound on the probability of false alarm)even when the decision interval is increased to T = 40 s.Because now sensors 4 and 5 are so far away, their informa-tion contribution to sensor 3 along the topologies of graphs 1and 3 is not sufficient to compensate for the sensors’ spatialdisadvantage compared to sensors 1 and 2. The only casewhere sensor 3 outperforms sensors 1 and 2 is in the topol-ogy of graph 2, where information sharing is balanced, butinformation quality gives sensor 3 again the advantage.

A single topology, that of graph 1, is examined in Fig. 5,which illustrates how sensor integration window lengthaffects dm performance. With increasing decision deadlineT , the source penetrates deeper into the sensor array area, andits proximity to different sensors over time changes accord-ingly. Previously “distant” sensors 3, 4, and 5 now have anincreasingly “closer look.” The result is that as T increases,and the source ends up further to the right, the location of thebest dm shifts to the right too.

The probability of missed detection is shown in Fig. 6to vary between different sensors when the connectivity ingraph 3 of Fig. 3 is altered while preserving the number ofnodes and edges. Intuitively, sensor 3 performs best undergraph 3 (red line), despite not having the best data from sen-sor 1, because it has the data of all other sensors. With theavailability of data from sensor 3, sensor 2 is best perform-ing under graph 3∗ (blue line), while performance of sensor3 deteriorates due to unavailability of data from 4 and 5.Sensor 4 performs best under graph 3∗∗ (green line) due toavailability of data from sensors 1, 2 and 3.

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Fig. 6 Variations on the connectivity of graph 3 of Fig. 3 result inchanges in the ranking of dms. a Variation in edge-placement of graph2, b Inter-sensor distance=0.5m (Color figure online)

Fig. 7 Monte-Carlo estimates of the probability of missed detectionfor different topologies of Fig. 3 for an inter-sensor distance of 0.5m.Dashed lines track Monte-Carlo estimates while continuous lines markthe Chernoff bounds. Sensor cross-section is adjusted to 3×10−2 m2 toreduce theMonte-Carlo runs required to capture very small probabilities

A different issue is addressed in Fig. 7: the conserva-tiveness of the Chernoff bounds themselves and its effecton the ranking of the sensor nodes. It tracks the variationover the nodes of different topologies of Fig. 3, for both a(MonteCarlo) empirical estimate of the probability ofmisseddetection, and its Chernoff boundwhich is used as a decision-

making performance metric. Although the two quantitiesare consistently separated, with the Chernoff bound on top,the monotonicity of the variation curves matches. What thisseems to imply is that the ranking of dms suggested by theChernoff bounds is in agreement with their actual decision-making capability. Caution needs to be exercised, however,because due to the bounds not being tight, in cases whereneighboring sensors exhibit comparatively similar decision-making performance metrics, these fine differences may notbe reflected in the bounds.

5 Networking topology selection

Having gained some insight on the interplay between dataquality (governed by observed signal at each sensor) andquantity (governed by the communication topology), thenatural question one may ask is if it is possible to designa communication topology which maximizes the decisionmaking accuracy of the dms.

In this light, (11) suggests a natural choice for the radia-tion data communication topology in order to boost decisionmaking performance: the complete graph. This is the casewhere each sensor shares its information directly with everyother sensor. All sensors end up having a complete copy ofthe whole network’s data in one single hop, and every sensoris equally posed to perform the function of the fusion cen-ter optimally. This configuration has obvious advantages interms of robustness and resilience to node failures. It doesintroduce, however, maximal redundancy and it thereforecomes at a cost, primarily in terms of communication over-head and energy required for data transmission. But even ifthis solution is acceptable in energetic terms, it may not befeasible, due to the limits on the power and communicationcapabilities of the sensor nodes.

If one wishes to depart from complete connectivity, butpreserve the assumption of single-hop information transmis-sion (as in Sect. 4), a fusion center cannot be realized unlessthe network possesses a node to which all other nodes sendtheir likelihood ratios in a single hop. This section, there-fore, relaxes the assumption of sensor communication over asingle network hop, and argues for designing the communi-cation topology based on transmission energy. The rationaleis similar to the one used in the area of transmission energyrouting (Shepard 1996; Ettus 1998), where multi-hop trans-fers are allowed in order to reduce transmission energy. Sincethe latter is associated with the spatial distance between sen-sors (Shepard 1996; Ettus 1998), edges inG are distinguishedbased on the distance between their adjacent nodes. Thus, thedescription of graphG introduced in Sect. 3 is augmented bya setW of edgeweights. Each edge (i, j) ∈ E ofG = (V , E)

is now assigned a weight wi j ∈ W which is equal to thespatial distance between the sensors i and j . The working

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assumption for the rest of the paper is that each sensor hasknowledge of G.

AMinimumSpanningTree (mst) topology is used tomin-imize the sum of transmission distances. A small caveat inthe particular application considered here is that the amountof information that is transferred increases with each hop, aseach sensor along a transmission path will augment the mes-sage it received with the statistic of its own measurementsbefore passing it on, so the proportionality is not necessarilypreserved.

With this noted, the first question that this section willaddress is how to determine a subgraph X of G, and thedirection of information flow on this subgraph, so thatdecision-making accuracy is maximized and the data trans-mission cost is minimized. The second question is relatedto the selection of a single dm to perform (2) on behalfof the whole network. Given that now there is no restric-tion to how far information can travel in the network, it isclear that the optimality of the centralized lrt can be recov-ered by funneling all sensor statistics to that strategicallyselected fusion center node. The central questions there-fore are (i) what is the optimal topology to convey data tothe fusion center, and (ii) which node should be the fusioncenter.

5.1 Fusion center and optimal information flow

As long as information from all dms propagates to someof them, then the latter can serve as a fusion center, andthe (optimal) performance of a centralized lrt is recovered.Given now that the decision-making accuracy is by defaultmaximized, the intuitive answer to the first question in termsof X is a mst with edge weights proportional to the distancebetween the sensors associated with the incident nodes. [Infact, the square of this distance is related to the required radiotransmission energy (Heinzelman et al. 2002).]

If A now denotes the (unweighted) adjacency matrix ofX , it is known (Godsil and Royle 2001) that the (i, j) entryof matrix An expresses the number of walks of length n fromvertex i to vertex j . By nature of X being a mst, all simplepaths of lengthn between twonodes (whenever they exist) areunique. Therefore, the entries of An will either be 0 or 1, anduniqueness of simple paths also ensures that A+A2+· · ·+Ak

will share the same property. By extension, if informationin the sensor network is allowed to be propagated over atmost k hops, the matrix I + A + A2 + · · · + Ak will be theadjacency matrix of a graph that expresses the feasible flowof information between sensors, assuming trivially that allsensorsmaintain access to the data that they have individuallyacquired. In this light, that is, by allowing sensor informationto propagate through the network over some directed mst,(10) is restated as

(I + A + · · · + Ak)

⎛⎜⎝log LT (1)

...

log LT (k)

⎞⎟⎠ H1

≷H0

⎛⎜⎝

Λ′1(p

†1)

...

Λ′k(p

†k )

⎞⎟⎠ . (12)

Remark 2 This observation has parallels in literature Pon-stein (1966, Theorem 4): in an acyclic graph, all simple pathsstarting from node i and ending at node j are counted inentry (i, j) of matrix [I − A(x)]−1, where A(x) is a pos-sibly variable (weighted) adjacency matrix of the acyclicgraph, and I is the identity matrix. Given that [I − A(x)]−1

is equal to∑∞

i=0 Ai (x), and over the mst X there are no

paths of length greater than k, in the case considered here,I + A + A2 + · · · + Ak = (I − A)−1.

For this mst X , and for some n ≤ k there will be a row ofones in I + A + A2 + · · · + An . The index of this row iden-tifies the (first) node in the network that will have receivedinformation from every other node after n ≤ k network hops.That node is the fusion center that requires the least numberof transmissions on behalf of the sensor nodes. The node isdesignated as the root of X , an orientation on X can be set sothat all other nodes are (weakly) connected to the root, andthe second question is now also resolved.

5.2 Network centrality

In network theory (Newman 2010), closeness centrality indi-cates the centrality of a node in terms of access to theinformation (Newman 2001). The closeness centrality of agiven node i is defined as the inverse ofmean geodesic lengthd(i, j) between nodes i and j , averaged over all nodes j ina graph G with N nodes (Newman 2010):

CL(i) = N − 1∑j∈V (G)\{i} d( j, i)

. (13)

Geodesic distances can be replaced by edge weights toyield weighted variations of closeness centrality. This sec-tion shows that the process of selection of the fusion centerin the mst of Sect. 5.1 actually corresponds to identifyingthe node u with the maximum closeness centrality.

Take as an example the graph of Fig. 8. There, a com-plete graph with nine randomly placed nodes and with edgeweights equal to physical distances between nodes is repre-sented; anmst generated usingKruskal’s algorithm (Kruskal1956) is highlighted. Theweighted closeness centrality of thenodes in the graph of Fig. 8a is shown in Fig. 8b, where theinduced ordering, from highest to lowest, is used to label thenodes in the figure’s planar representation.

Intuitively, the reasoning behind the fact that (13) identi-fies the same node as Sect. 5.1, is that in the sensor networks

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(a) (b) (c)

Fig. 8 Information sharing with minimum communication cost. a Graph G and itsmst. b ordering and labeling of the graph’s nodes based on theirweighted closeness centrality. c orientation induced on the mst of G so that the node with the highest centrality to become the network’s sink

where links are established based on node proximity con-ditions, the first node which will fill its row with ones inmatrix I + A + A2 + · · · + An , is essentially the one that is“closer,” on average to every other node, compared to all othernodes.

The spatial distribution of sensors can be arbitrary as longas the network graph is strongly connected. The existence,however, of multiple connected clusters in the radiation datanetwork may present implementation problems. The fusioncenter would most likely be either within the largest cluster,or outside clusters altogether, inwhich case, depending on thesize of the network, the volume of data transmitted outside ofa cluster and on route to the fusion center could overwhelmthe available bandwidth. In addition, transmitting data overa large number of small hops within the cluster may not beas energy efficient as directly transmitting it to the nodes atthe intersection of the cluster andmst. There exist dedicatedclustering-based routing protocols that alleviate such issues,but this falls outside the scope of this present paper whichfocuses on algorithm design at an application layer level. Theproposed algorithm is not a communication protocol, rather,it is a networking strategy that controls the amount of datasharing within the network so that the sensors collectivelymake the best possible decision regarding potential radioac-tivity of the suspected target.

The case study presented next utilizes actual radiationmeasurements from one of dndo’s Intelligent RadiationSensing System (irss) program (Intelligent RadiationSensing Systems https://www.dhs.gov/intelligent-radiation-sensing-system) datasets (Canonical IRSS Data-Sets https://github.com/raonsv/canonical-datasets). By implementingthe networked decision making methodology reported in thispaper under the assumption that the source intensity and tar-get trajectory is known, it is demonstrated that a particularsensor network realized in irss could have picked up thesource an order of magnitude faster than individual sensorswould for an independent decision with probabilities of missand false alarm less than 10−10.

5.3 The IRSS dataset

Under dndo’s irss program (Intelligent Radiation SensingSystems https://www.dhs.gov/intelligent-radiation-sensing-system), a series of indoor and outdoor tests were performedto detect, localize and identify radiation sources using Com-mercial Off The Shelf (cots) radiation detectors. These testswere performed using multiple source strength and types,different background profiles and various source and sensormovement, of which the first batch of the data-set includes 10indoor and 2 outdoor measurements (Canonical IRSS Data-Sets https://github.com/raonsv/canonical-datasets).

The calculations in this section are based on the experi-mental dataset titled Outdoor-C1 that includes a movingsource and 16 NaI 2"× 2" detectors arranged in a regu-lar square grid configuration. A 175 μCi Cs-137 sourcewas transported along a straight line through the sensorgrid, and the total counts on each sensor was recorded.For every one-second time interval over an integration win-dow of 147 seconds, a normalized count rate βi (t)+νi (t) iscomputed.

The movement of the source among a subset of ten,arbitrarily picked out of the total sixteen, sensors in theOutdoor-C1 dataset is shown in Fig. 9. The location ofthese ten sensors is determined by averaging their gps mea-surements from a single (the first out of six) measurement“run.” The normalized background count rate βi (t) for eachsensor is taken fromdata in test LSI_A_Background. Thevalues used in the calculations of this section are the all-timeaverages over all 18 runs in this test.

For a known source intensity a, background intensity βi ,sensor cross-section χi and relative distance between thesource and the sensor ri , the total perceived intensity at sensori can be approximated as (Sun and Tanner 2015):

νi (t) + βi = χi a

2χi + r2i+ βi (14)

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Fig. 9 Sensor placement and source movement obtained from run1of the irss test Outdoor-C1

Fig. 10 Observed background and total count rate for detector 12 and13 in the irss tests LSI_A_Background and Outdoor-C1, respec-tively. The numerical fit for the perceived and background intensitiesare shown by solid lines; background data is only available for the first60 seconds

Fig. 11 The minimum spanning tree computed based on physical sen-sor distances from the subset of sensors depicted in Fig. 9

Given average background intensities βi based on datasetLSI_A_Background, values for the cross-section coef-ficient χi and the actual source intensity a are estimatednumerically based on Outdoor-C1/run1, in order to tune(14) to match closely experimental observations (Fig. 10).

The mst calculated for the sensor network of Fig. 9 isshown in Fig. 11, while the closeness centrality for the nodesin that mst is shown in Fig. 12.

Fig. 12 Weighted closeness centrality for the nodes in the mst high-lighted in Fig. 11

Fig. 13 Integration time required by each sensor in the network ofFig. 11 to individually decide having bounds on the probability ofmissed detection and false alarm below 10−10. The straight black lineindicate absolute lower bound for an optimal centralized hypothesis test(2)

Node 8 is indicated as the best choice for a fusion cen-ter in the network over the mst of Fig. 11 in Fig. 12. On theother hand, Fig. 13 shows the integration time (time period ofdata collection) required by each sensor if it were to make anindividual decision based only on its own data, with a boundon probability of missed detection and false alarm below10−10. The straight line at the bottom of the figure indicatesthe optimum (minimum) time required when a fusion cen-ter performs the test (2) utilizing information from all tensensors.

The detection times shown in Fig. 13 are indicative of thequality of information that each individual sensor possessesbefore any communication occurs. If nodes were to decideindividually, it appears that node 12 can make the decisionfaster than anyone else—thismay have to dowith the fact thatthe source passes closer to node 12 compared to any othersensor. On the other hand, node 8 can receive everybody’sstatistics and make an equally confident decision through (2)at a third of that time, and with the least communication costcompared to every other node.

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6 Conclusions

The decision-making accuracy of individual nodes in a net-work of radiation detectors can be quantified using explicitlycomputed Chernoff upper bounds on error probabilities. Thecapability of a sensor to make the decision depends on theinterplay between the quantity (governed by the location ofthe sensor in the underlying network topology) and the qual-ity (controlled by the spatial location of sensors relative tosource) of the information about the observed process avail-able.

Conceptually, therefore, decision-making accuracy =information quality × information quantity, as reflectedin (11), with information quantity determined by the sensornetwork’s topology, assuming that sensors share their statis-tics only once with their network neighbors. If, however,information is allowed to sufficiently disseminate throughthe network, then the optimal performance of a centralizedhypothesis test is recovered, and the relevant question iswhich is the sensor best suited to perform the role of thefusion center.

If one wishes to minimize the inter-sensor transmissionenergy, the solution can be expressed in network-theoreticterms, since the node best suited to be the decision-makeris the one that features the highest closeness centrality. Theoptimal topology for information exchange is that of a min-imum spanning tree with orientation such that data flows tothat designated fusion center node.

Acknowledgements This work is supported in part by DTRA underaward #HDTRA1-16-1-0039.

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Indrajeet Yadav received his Bach-elor of Engineering degree inMechanical Engineering from ShriG S Institute of Technology andScience, India in 2003. Up until2015 he worked on mechanicaldesign and analysis of nuclear rea-ctor components and systems withvarious Indian and UK nuclearorganizations. He is currently aPh.D. student in the Departmentof Mechanical Engineering, Uni-versity of Delaware. His researchinterests include optimal controland multi-agent coordination.

Chetan D. Pahlajani receivedhis Ph.D. in Mathematics from theUniversity of Illinois at Urbana–Champaign in 2007. From 2007to 2010, he was a Visiting Assis-tant Professor in Mathematics atthe University of California SantaBarbara. He then held a VisitingAssistant Professor/ PostdoctoralResearcher position in Mathemat-ical Sciences at the University ofDelaware from 2010 to 2014. Heis currently an Assistant Profes-sor at IIT-Gandhinagar, India. Hisresearch interests are in Probabil-

ity Theory and Stochastic Processes. Dr. Pahlajani is currently anAssistant Professor of Mathematics at the Indian Institute of Technol-ogy Gandhinagar.

Herbert G. Tanner received hisEng. Diploma and his Ph.D. inmechanical engineering from theNational Technical University ofAthens, Greece, in 1996 and 2001,respectively. From 2001 to 2003he was a post doctoral fellow withthe Department of Electrical andSystems Engineering at the Uni-versity of Pennsylvania. From2003 to 2005 he was an assis-tant professor with the Depart-ment of Mechanical Engineeringat the University of New Mexico,and he held a secondary appoint-

ment with the Electrical and Computer Engineering Department atUNM. In 2008 he joined the Department of Mechanical Engineeringat the University of Delaware, where he is currently an associate pro-fessor. Since 2012 he is serving as a director of the graduate certificateprogram in cognitive science at the University of Delaware. Dr. Tannerreceived the 2005 National Science Foundation Career award. He is amember of the ASME, and a senior member of IEEE. He has served inthe editorial boards of the IEEE Robotics and Automation Magazineand the IEEE Transactions on Automation Science and Engineering,and the conference editorial boards of both IEEE Control Systems andIEEE Robotics and Automation Societies.

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Ioannis Poulakakis received hisPh.D. in Electrical Engineering(Systems) from the University ofMichigan, MI in 2009. From 2009to 2010 he was a post-doctoralresearcher with the Department ofMechanical and Aerospace Engi-neering at Princeton University,NJ. Since September 2010 he hasbeen with the Department ofMechanical Engineering at theUniversity of Delaware, where heis currently an Associate Profes-sor. His research interests lie inthe area of dynamical systems and

control with applications to robotic systems, particularly to dynam-ically dexterous legged robots. Part of his recent work deals withaspects of networked decision making. Dr. Poulakakis received theNational Science Foundation Career Award in 2014 for his researchon robotic legged locomotion.

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