Assessing the Performance of ad hoc Sensor Networks

Thomas V. Huynh John S. OsmundsonThe Naval Postgraduate School The Naval Postgraduate School

1 University Circle 1 University CircleMonterey, CA 93943 Monterey, CA 93943thuynh@&nps.edu

A specific problem of interest is an ad hoc mobileAbstract - As an exploratory effort to understand sensor network, which is a complex network with

effects of network complexity coupled with distributed information fusion used in the detection,

interoperability on performance of ad hoc sensor tracking, and classification of targets [2]. Complexity is

networks, we treat ad hoc sensor networks as random tied to the evolving large size of the network and the real-

networks. In this work complexity is reflected by the time, unplanned, evolving topology of the network.

average degree of the network and interoperability by the Interoperability is related the nature to distributed

mean number of sensors in the network that are information fusion and attending activities that are

interoperable. The average degree of the network in this affected by, for instance, bandwidth incompatibility,

case depends on both the probability ofsensor discovery fusion rules, data association failures of sensors,

and the probability ofsensor connection. For illustration interfaces to interactively task and query the sensor

purposes, we consider the binary hypothesis distributed network.

fusion problem and a simple fusion rule. The result Due to varying operational life, sensors fail or appear

shows that the criterion for successful fusion may be in an ad hoc sensor network. The topology of the ad hoc

violated by the network conditions that depend on the sensor network is therefore not known a priori and must

network complexity and interoperability. be established in real time and updated periodically [3],

[4]. Network discovery is thus necessary, in the senseKeywords: Complexity, interoperability, ad hoc sensor that fixed and mobile sensors in a topologically dynamic

network, binary hypothesis distributed fusion. mobile network must be able to discover each other [4].

Success or failure in any two nodes discovering each1 Introduction

other affects the connectivity of the network. Also, in adOur ongoing research aims at understanding network hoc networks, connectivity is not planned but must

complexity and eventually translating it to complex emerge as needed, and since communication links are

systems engineering. An area of interest is to gain unreliable and shadow fading may eliminate links [4]insights into impacts created by these network attributes connectivity between any two nodes can be stochastic.on sensor network performance. Our view is that not Distributed information fusion in an ad hoc sensor

only does complexity induce vulnerability of a network or network involves collaborative processing of data andnetwork [1] but also lacking interoperability among the information. A sensor in the network fuses its localsubsystems or sensors that make up the network degradesinomtnwthnfraonrmaohendernds

or~~~~~~~~~~~~~imeehefrac ftentok n the network. Local fusion rules may not be identical.Nodes may have different processing power and limited

1-4244-1 160-2/07/$25.OO ©2007 IEEE.

and communication resources, which would adversely network is evolving and the maximum size of the network

affect data association. All of these hence may not yield is N. Two sensors in the network are connected if they

comparable results and are tied to interoperability. discover each other and, upon discovery, the

Interoperability is also tied to interfaces among the nodes communication channel between them is established. We

to allow them to interactively task and query the sensor assume two sensors discover each other with a constant

network. probability Pd. Every pair of sensors upon discovery is

With focus on wireless ad hoc sensor networks inassumed to be connected with a constant probability pK

this work, we address effects of complexity (in terms of K

connectivity) and interoperability on distributed Then the probability pC that a pair of sensors is

information fusion in a wireless ad hoc sensor network. connected is Pc = P,KPd . Let I ={1,2,3, ---, N}be the

Huynh and Osmundson [5] adopted and extended index set. Let L be the number of links node i has withGheorghe and Vanamu' s model [1] to incorporate

complexity and interoperability and assess the t ,iaknown as the degree of sensor i [6], follows a binomialperformance of a complex, distributed fusion sensor

network. In this paper we model the ad hoc sensor distribution with parameters pc and N - 1 [7]. That is,

network as a random network [6], [7]. We employ P(f)P(L= = e)=( )PC (l- PC As in [8],random network theory to incorporate complexity and f

interoperability and, for simplicity, we address the binary XO denotes the complexity of a network defined as the

hypothesis distributed fusion problem mean number of links supported by a sensor of theOur goals in this paper are: N-1

* Explain our exploratory work toward network; that is, xO = E P(?) Xo is thus a

understanding of impacts of complexity and

interoperability on the performance of ad connectivity measure of the network. Then a simple

hoc wireless sensor networks. computation leads to X =PC (N - 1), or

* Elucidate the incorporation of complexity O =PKPd (N -1). The more complex the network is,

and interoperability in a random network. the higher the value of%o. It is also known as the

* Illustrate the results with an ad hoc wireless average degree of the system [9].sensor network for information fusion. 3 Representation of Interoperability

The rest of the paper is organized as follows. InWe can think of each sensor as having a "spin"

Section 2 network complexity is defined. Section 3

discusses network interoperability. Section 4 describes denoted by s, where s E {-1,+1}. A (+1) - spin sensor

the binary hypothesis distributed fusion problem and theis interoperable with any other (+1) - spin sensor; it is

impact of complexity and interoperability on the

performance of the sensor fusion network.. not interoperable with a (-1)- spin sensor. Likewise,

2 Representation of Network Complexity2 Represetation ofNetwork Cmplexitytwo (-1)- spin sensors are not interoperable. Letps beA random graph [6] is a collection of points, or

vertices, with lines, or edges, connecting pairs of them at tepoaiiy(sue osat htasno a

random. Consider an ad hoc network of N sensors. The (+1) - spin. Let Xl be the number of (+1) - spin

sensors. Then XI follows a binomial distribution with the value of 3 is I if the state of the ith sensor is

parameters p5 and N, and, hence, (> i,+i), with probability

P(k) =P(X=k) KkpSk((-ps )N-k, which is the ST ___ ,O and that

probability that k sensors are interoperable. Let X% denoteof Hi is 0 if the state of the ith sensor is either

the interoperability of the network, which is defined=(>.-), do+ (0,+4), or =(0,-1), with

N

asX = kP(k). Again, a simple computation then Nn=O probability -To -1-1 1L- NI-xI thus

leads to X/ = ps (N - I) . XI is thus the mean number of

admits a binomial distribution with parameters N andinteroperable sensors.

Let each sensor be characterized by a doubletparameter ;T]. The average number of interoperable

(degree, spin); that is, =(C,s), where nodes K is then K E(X1) z1N, or

e=0,1,2, ,N-I, and s E {-1,+1}. A sensor is said to N N_ tK=(,) -In the N-c'

be in state 1, denoted by t+ (. i,+i), if f >l and

s=+l. It is in state 2, if either f.Iand s=-1or limit E(X,)=I-(I- x ) X]-N-> -e )X

= 0 and s = +1 state 2 can thus be denoted by either

A summary of the results follows.qt_ (>.1,-1) orq:0+1 (0,±i). In this case, the

probability that a sensor is in state 1 is PC PKpd;' % Pc (N -1) z , i(N 1)

(l - P which, by virtue of (| Ir||| I

XO =p (N - 1) and X1 ps (N - 1), can be written as In the N-o limit,%0 p- N m, - ,pN, ad

N L@X l{eZ'k.* If N is known, any For a network with N sensors, the complexity

increases with the probability of connection between aproperty which can be expressed in termns of pC and

pair of sensors. The probability of connection between aps can also be expressed in terms of XO and X,1.

pair of sensors increases with the probability ofLet X1 denote the number of sensors in state 1;

discovery, for a fixed probability of a communicationsX/ then takes the values of 0,1,2, ,.~N. We can channel existing between the sensor pair. The

consider X/ as Xl=~ + ~ N where 3" ,:E {0,1}; interoperability measure of the N -sensor network

increases with the probability of interoperability between FCN] N 1

a pair of sensors. As the size of the network becomes !CD l1PD j

large (i.e., N - oc ) the expected value of the number of In[Pp ( -PD)sensors in state 1 increases with the interoperability

This criterion overlooks the fact that some of themeasure, for a fixed complexity of the network, while it

sensors may not be available and, even when they aredecreases with the interoperability measure, for a given

available, some interoperability condition may not be met,interoperability measure. Given the size of the network

and the required number of sensors therefore may not beand the probability of discovery and the probability of

achieved, hence the failure of the network. When wecommunication between two sensors, the complexity and

assess the performance of the fusion network using K0p1,interoperability of the network can be determined. The

mean number of the sensors in state 1 can then be readily we must therefore impose FE(X1)1 . Kopt, i.e.,

obtained.N

4 Distributed Sensor Fusion Problem N N C (1PD>_

As in [5], we consider a network of N sensors that I[P KIPF]observe a common phenomenon (e.g., a space object, a

threat, etc.), pass their observations among themselves,or, in the N X~cc limit,

and independently process all the observations. Each

sensor is a fusion center, whose output is binary - either C (1PN 1

the phenomenon is present or absent. This is thus a binary LCD - PD)hypothesis distributed fusion problem. For simplicity, the LDe KzP)>local decision rules are assumed to be identical, and all InlPD ('PF J

sensors are assumed to have the same probability of LPF W'PD)]detection, PD, and the same probability of false-alarm,

5 ConclusionPF. Let CF denote the cost of making a false-alarm

As an exploratory effort to understand effects ofdecision and CD the cost of making a correct decision.

network complexity coupled with interoperability onThere are many fusion rules. In this work, we invoke the performance of ad hoc sensor networks, we treat ad hocK-out-of-N fusion rule. The optimal value ofK , Kopt, sensor networks as random networks. In this work

for the K-out-of-N fusion rule is given by [10] complexity is reflected by the average degree of the

K {FK *1 if K* > 0 network and interoperability by the mean number ofKopt o otherwise,

+- F-opt - l Ootherwise, ~~~sensors in the network that are interoperable. The

average degree of the network in this case depends onwhere []1 denotes the standard ceiling function, and

both the probability of sensor discovery and the

probability of sensor connection. For illustration

4

purposes, we consider the binary hypothesis distributed [9] Newman, M.E.J., "The Structure and function of

fusion problem and a simple fusion rule. The model complex networks," arXiv:cond-mat/0303516, vl, 25

shows that the criterion for successful fusion may be March 2003.

violated by the network conditions that depend on the [10] P.K. Varsney, Distributed Detection and Data

network complexity and interoperability. Fusion, Springer, 1996.

REFERENCES

[1] Gheorghe, A. V. and Vamanu, D. V., "Towards QVA -

Quantitative Vulnerabiltiy Assessment: A Generic

Practical Model," (Preprint, 2004).

[2] C.Y. Chong, S. Mori, and K. C. Chang, "Distributed

Multitarget Multisensor Tracking," in Multitarget

Multisensor Tracking. Advanced Applications, Y. Bar-

Shalom, Ed. Norwood, NIA: Artech House, 1990, pp.

247-295.

[3] B. Deb, S. Bhatnagar, and B. Nath, "A topology

discovery algorithm for sensor networks with

applications to network management," Dept. Comput.

Sci., Rutgers Univ., Tech. Rep. DCS-TR-441, 2001.

[4] Chong, C-Y and Kumar, S. P., "Sensor Networks:

Evolution, Opportunities, and Challenges,"

Proceedings ofthe IEEE, Vol. 91, No. 8, August 2003,

pp. 1247-1256.

[5] T.V. Huynh and J. S. Osmundson, "A Model for

Assessing the Performance of Interoperable, Complex

Systems," Proceedings of the 4th Annual Conference

on Systems Engineering Research, Los Angeles, CA,

April 6-9, 2006.

[6] B. Bollobas, Random Graphs, Academic Press, New

York (1985).

[7] M. E. J. Newman, S. H. Strogatz, and D. J. Watts,

"Random graphs with arbitrary degree distributions

and their applications," Preprint arXiv:cond-

mat/0007235 v2 7 May 2001.

[8] Aslaksen, E.W., "Thermodynamics: A Model

Illustrating Complexity Emerging from Simplicity,"

Systems Engineering, Vol. 7, No. 3, 2004, pp. 271-

284.

5