Distributed Detection in Tree Topologies with Byzantines

1 Distrib uted Detecti on in T r ee T opologies with Byzantines Bhavya Kailkhura, Student Member , IEEE , S w astik Brahma, Member , IEEE , Y unghsiang S. Han, F ellow , IEEE , Pr amod K. V a rshne y , F ellow , I EEE Abstract In this pap er , we consider the pr oblem of distrib u ted detection in tree topolog ies in the pr esence of By zantines. T he expression for minimu m attack ing power re quired by the Byzantines to blind the fusion center ( FC) is obtained . More specifica lly , we show that when mo re than a certain f raction of individual node de cisions are falsified, the decision fusion scheme becom es completely incap able. W e obtain closed form expressions for the optimal a ttacking strategies that minim ize the detection err or exponent at the FC. W e also lo ok at the possible coun ter - measures f rom the FC’ s perspectiv e to protect the network from these Byzantines. W e formulate the robust topology design pr oblem as a bi-le vel progr am and provide an efficient algor ithm to solve it. W e a lso p rovide some numerical results to gain insights into the solution . Index T erms Distributed Detection, Byzan tine Attacks, Kullback-Leibler Div ergence, Bounded Knap sack Prob- lem, Bi-level Pr ogramming I . I N T R O D U C T I O N Distributed detection has been a well studied topic in the detection theory li terature [1] [2] [3] and h as traditionall y focused on the parallel network topology . In di strib uted detection with Some related preliminary work was presented at the International Conference on C omp uting, Networking and Communications W orksho ps (ICNC-2013), San Diego, CA, January 2013. B. Kailkhura, S . Brahma and P . K. V arshney are with Department of EECS, Syracuse Univ ersity , Syracuse, NY 13244. (email: bkailkhu@syr .edu; sk brahma@syr .edu; v arshney@syr .edu) Y . S. Han is with EE Department, National T aiw an Unive rsity of S cience and T echno logy , T aiwan , R. O. C. (email: yshan@mail.ntust.edu.tw) DRAFT 2 parallel topology , nodes make their l ocal decisions regarding the underlying pheno menon and send them to the fusi on center (FC), where a g lobal decision is m ade. Even though the parallel topology has recei ved sign ificant attention, there are many practical situations where parallel topology cannot be implemented due to se veral f actors, such as, the FC being outside the communication range of the nodes and limited energy budget of the nodes [4]. In s uch cases, a m ulti-hop network is emp loyed, where nodes are or ganized hierarchically into multipl e levels (tree networks). W ith in telligent use of resources across levels, tree networks hav e the potential to provide a suitable b alance b etween cost, cove rage, function ality , and reli ability [5]. Some examples of tree networks include wireless s ensor and mili tary communication networks. For instance, th e IEEE 802.15.4 (Zi gbee) specifications [6] and IEEE 802.22b [7] can su pport tree- based topologies. Theses nodes are often deployed in open and unattended en vi ronments and are vulnerable to phy sical tampering . In recent years, security issues of distributed inference networks are i ncreasingly being studied. One typical attack on s uch n etw orks is a Byzanti ne attack. While Byzantine attacks (originall y proposed by [8]) may , i n general, refer to many types of malicious beha vior; our focus in this paper is on d ata-f als ification attacks [9]–[17]. In this type of attack, the comp romised node may send false (erroneous) local decisions to the FC to degrade the detection performance. This attack becom es more se vere i n tree topologies where malicious no des can alter local decisions of a lar g e part of the network and cause degradation of sys tem performance and m ay e ven make the d ecision fusion schemes to become com pletely incapable. In this paper , we refer to such a data falsification attacker as a Byzantin e. A. Related W ork Although dis trib u ted det ection has been a very active field of research in the past [1]–[3], security prob lems in dist rib uted detection networks gained att ention only very recently . In [12], the authors considered the problem of dist rib ut ed detection in the presence of Byzantines for a parallel t opology and determined the optim al attacking strategy which minimizes the detection error exponent. They assumed that the Byzantines know the true hypothesis, which o b viously is not satis fied in practice but does provide a bou nd. In [13], the authors analyzed the same problem in t he context of collaborative spectrum sensing. They relax ed t he assumpt ion of perfec t knowledge of the hypotheses by assuming that the Byzantines obtain knowledge about the true DRAFT 3 hypotheses from their own s ensing obs erv ations. The above work [12], [13] addresses t he issue of Byzantin es from t he attacker’ s perspective. Schemes t o mit igate the ef fect of Byzantines have also b een propos ed in the literature. In [13], the authors proposed a simple scheme to identify the Byzantines. The idea was to m aintain a reputation metric for every node by comparing each n ode’ s local decision to the gl obal decisio n made at the FC using the majority rule. In [16], the authors proposed another scheme to mitigate the ef fect of Byza ntines in a parallel topology . The idea behind the proposed identification scheme is to com pare e very node’ s ob serv ed beha v ior over tim e with the expected beha vior o f an honest node. The nodes wh ose obs erv ed behavior is sufficiently far from th e expected behavior are tagged as Byzantines and this inform ation is empl o yed while making a decision at the FC. In [17], the authors in vestigated the problem of dist rib ut ed detection in the presence of dif ferent types of Byzantine nodes. Each Byzantine type corresponds to a dif ferent operating point and, therefore, t he problem of identifying different Byzantine nodes along wi th their operating points was considered. O nce the Byzantine operating points are estimated, this inform ation was utilized by the FC to improve global detection performance. The problem of designing the optimal fusion rule and the l ocal sensor thresho lds with Byzantines for a parallel topology was considered in [15]. B. Main Contributions All the approaches discussed so far consi der distributed detectio n with Byzantines for parallel topologies. In contrast to previous work, we study th e problem of distributed detection with Byzantines for tree topologies. More s pecifica lly , we address the problem o f distributed detection in perfect a -ary tree net w orks 1 in the presence of Byza ntine attacks (data falsification attcks). W e assume that the cost of attacking nodes at di f ferent le vels is diff erent and analyze the problem under this assumption. In our prelimin ary work on this problem [14 ], we analyzed the problem only from an attacker’ s perspecti ve assumin g that the honest and Byza ntine nodes are identical in terms of their detection performance. In our current work, we significantly extend our previous work and inv est igate the p roblem from both the attacker’ s and the FC’ s perspective. For the analysis of the opt imal attack, we allow Byzantines to h a ve differ ent detection performance 1 For prev ious works on perfect a -ary tree networks, please see [18], [19], [20]. DRAFT 4 Fusion Center (FC) Level 3 Level 2 Level 1 Fig. 1. A distributed detection system organized as a perfect binary tree T (3 , 2) is sho wn as an example. than the honest nodes and, therefore, provide a more general and comprehensive analysis of the problem compared to our pre vi ous work [14]. The main contributions of this paper are as follows. • W e obtain a closed form expression for t he mini mum attacking power requi red by the Byzantines to blin d the FC i n a t ree network and show that when more than a certain fraction of individual node decisi ons are falsified, t he decision fusion scheme becomes completely incapable. • When the fraction of Byzantines i s not suf ficient to bli nd the FC, we pro vide closed form expressions for the optimal attacking strategies for the Byzantines that m ost degrade the detection performance. • W e also look at the problem from the network desig ner’ s (FC) perspectiv e. More specifically , we formulate the rob ust t ree topology design probl em as a bi-l e vel program and provide an ef ficient algorit hm to solve it, which is guaranteed to find an opt imal solution, if on e exists. The rest of the paper is or g anized as follows. Section II i ntroduces our sy stem model. In Section III, we study the prob lem from Byzantin e’ s perspectiv e and provide closed form expressions for opt imal attacking strategies. In Section IV, we form ulate the robust to pology design problem as a bi-level program and provide an ef ficient algorithm to s olve it in po lynomial time. Finally , Section V concludes the paper . DRAFT 5 I I . S Y S T E M M O D E L W e consider a dist rib ut ed detection system with the topology of a perfect a -tree T ( K , a ) rooted at the FC (See Fig. 1). A perfect a -tree is an a -ar y tree in which all the leaf nodes are at the same depth and all the internal nodes hav e de gree ‘ a ’. T ( K , a ) has a set N = { N k } K k =1 of trans cei ver n odes, where | N k | = N k = a k is the t otal number of nodes at level (or depth) k . W e assume that the depth of the tree is K > 1 and the numb er of chil dren is a ≥ 2 . The total number of nodes in the network is denoted as P K k =1 N k = N . B = { B k } K k =1 denotes the set of Byzantine nodes with | B k | = B k , where B k is the set of Byza ntines at le vel k . W e assu me t hat the FC is not a ware of the exact set of Byzantine n odes and considers each node at level k to be Byzantine wit h a certain probability α k . In practice, nodes operate with very limited energy and, therefore, it i s reasonable to assum e that the packet IDs (or source IDs) are no t forwarded in the tree to sav e energy . Moreover , ev en in cases where the pack et IDs (or source IDs) are forwarded, not ice that the packe t IDs (or source IDs) can be tempered too, thereby prev enting the FC to be deterministically aware of the source of a mess age. Therefore, we consider that the FC looks at messages coming from no des in a p robabilistic manner and considers each receiv ed bit to orig inate from nodes at level k with certain probabi lity β k ∈ [0 , 1] . Thi s also implies that, from the FC’ s perspective, recei ved bit s are identi cally dist rib uted. For a T ( K , a ) , β k = a k N . A. Distributed detection in a tr ee topology W e consider a binary hypothesis t esting problem wi th the two hypotheses H 0 (signal i s absent) and H 1 (signal is present). Each node i at leve l k acts as a source in that it makes a one-bit local decision v k ,i ∈ { 0 , 1 } and sends u k ,i to its parent node at lev el k − 1 , where u k ,i = v k ,i if i is an uncompromised (honest) node, b u t for a compromised (Byzantine) node i , u k ,i need not be equal to v k ,i . It also recei ves the decisions u k ′ ,j of all successors j at lev els k ′ ∈ [ k + 1 , K ] , which are forwarded to i by its imm ediate child ren. It forwards 2 these recei ved d ecisions along with u k ,i to it s parent node at lev el k − 1 . If node i is a Byzantine, then it might alter these recei ved decisions before forwarding. W e assume error -free commu nication channels between children 2 For example, IEEE 802.16j mandates tree forw arding and IE EE 802.11s standardizes a tree-based routing proto col. DRAFT 6 and the parent nodes. W e denote the probabili ties of detection and false alarm of a honest node i at level k by P H d = P ( v k ,i = 1 | H 1 , i / ∈ B k ) and P H f a = P ( v k ,i = 1 | H 0 , i / ∈ B k ) , respectiv ely . Similarly , the p robabilities of detection and false alarm of a Byzantine nod e i at level k are denoted by P B d = P ( v k ,i = 1 | H 1 , i ∈ B k ) and P B f a = P ( v k ,i = 1 | H 0 , i ∈ B k ) , respectiv ely . B. Byzantine at tac k model Now a math ematical model for t he Byzantine attack is p resented. If a node is hones t, then it forwards it s own decision and recei ved decisions wit hout altering them. Howe ver , a Byzantine node, in order to undermin e the network performance, may alter its decisi on as well as recei ved decisions from its children prior to transmission . W e define the following strategies P H j, 1 , P H j, 0 and P B j, 1 , P B j, 0 ( j ∈ { 0 , 1 } ) for the honest and Byzantine nodes, respectively: Honest nodes: P H 1 , 1 = 1 − P H 0 , 1 = P H ( x = 1 | y = 1) = 1 (1) P H 1 , 0 = 1 − P H 0 , 0 = P H ( x = 1 | y = 0) = 0 (2) Byzantine nodes: P B 1 , 1 = 1 − P B 0 , 1 = P B ( x = 1 | y = 1) (3) P B 1 , 0 = 1 − P B 0 , 0 = P B ( x = 1 | y = 0) (4) where P ( x = a | y = b ) is t he probability that a node sends a to its parent when it receiv es b from its child or its actual decision is b . Furthermore, we assume that if a node (at any level) is a Byzantine then none of its ancestors are Byzantines; otherwise, the effect of a Byzantine due to other Byzantines on th e same path may be n ullified (e.g., Byzantine ancestor re-flipping the already flipped decisions of its successor). Thi s means t hat any path from a leaf node to the FC will have at most one Byzantine. Thus, we ha ve, P K k =1 α k ≤ 1 since the aver age number of Byzantines along any path from a leaf to the root cannot be greater t han 1 . C. P erformance metric The Byzantine att ack er always wants to degrade the detection performance at the FC as much as poss ible; in con trast, the FC wants to m aximize t he detection performance. In thi s work, we empl o y the Kullback-Leibler div er gence (KLD) [21] to be th e network performance metric DRAFT 7 P ( z i = j | H 0 ) = " K X k =1 β k k X i =1 α i !# [ P B j, 0 (1 − P B f a ) + P B j, 1 P B f a ] + " K X k =1 β k 1 − k X i =1 α i !# [ P H j, 0 (1 − P H f a ) + P H j, 1 P H f a ] (7) P ( z i = j | H 1 ) = " K X k =1 β k k X i =1 α i !# [ P B j, 0 (1 − P B d ) + P B j, 1 P B d ] + " K X k =1 β k 1 − k X i =1 α i !# [ P H j, 0 (1 − P H d ) + P H j, 1 P H d ] (8) that characterizes detection performance. Th e KLD is a frequently used information-theoretic distance measure to characterize detection performance. By Stein’ s l emma, we know that i n t he Neyman-Pearson s etup for a fix ed missed detection probabili ty , the false alarm probability obeys the asympt otics lim N →∞ ln P F N = − D, for a fixed P M , (5) where P M , P F are mi ssed detection and false alarm probabi lities, respecti vely . The KLD between the distri b uti ons π j, 0 = P ( z = j | H 0 ) and π j, 1 = P ( z = j | H 1 ) can be e x pressed as D ( π j, 1 || π j, 0 ) = X j ∈{ 0 , 1 } P ( z = j | H 1 ) log P ( z = j | H 1 ) P ( z = j | H 0 ) . (6) For a K -le vel network, distributions of recei ved decisi ons at the FC z i , i = 1 , .., N , under H 0 and H 1 are given by (7) and (8), respectiv el y . In order to make the analysis tractable, we assume that the network d esigner attem pts to maximize the KLD of each node as seen by t he FC. On the other hand, the attacker attempts to m inimize t he KLD of each node as seen by the FC. Next, we explore the optimal attacking strategies for the Byzantines that most degrade the detection performance by m inimizing KLD. I I I . O P T I M A L B Y Z A N T I N E A T TAC K As dis cussed earlier , the Byzantine nod es attempt t o make their KL diver gence as small as possible. Since t he KLD is always non-negati ve, Byzantines attempt to choose P ( z = j | H 0 ) and DRAFT 8 P ( z = j | H 1 ) such that KLD is zero. In this case, an adv ersary can make t he data that the FC recei ves from the nodes such that no information is con veyed. This is pos sible when P ( z = j | H 0 ) = P ( z = j | H 1 ) ∀ j ∈ { 0 , 1 } . (9) Substitutin g (7) and (8) in (9) and after simplification, the condition t o make the K LD = 0 for a K -level network can be expressed as P B j, 1 − P B j, 0 = P K k =1 [ β k (1 − P k i =1 α i )] P K k =1 [ β k ( P k i =1 α i )]] P H d − P H f a P B d − P B f a ( P H j, 0 − P H j, 1 ) . (10) From (1) to (4), we have P B 0 , 1 − P B 0 , 0 = P K k =1 [ β k (1 − P k i =1 α i )] P K k =1 [ β k ( P k i =1 α i )]] P H d − P H f a P B d − P B f a = − ( P B 1 , 1 − P B 1 , 0 ) . (11) Hence, the attacker can degrade detection performance by i ntelligently choosing ( P B 0 , 1 , P B 1 , 0 ) , which are dependent on α k , for k = 1 , · · · , K . Observe that, 0 ≤ P B 0 , 1 − P B 0 , 0 since P k i =1 α i ≤ 1 for k ≤ K . T o make K LD = 0 , we must have P B 0 , 1 − P B 0 , 0 ≤ 1 such that ( P B j, 1 , P B j, 0 ) becomes a valid prob ability mass functi on. Notice that, when P B 0 , 1 − P B 0 , 0 > 1 there does not exist any attacking probability dis trib ut ion ( P B j, 1 , P B j, 0 ) t hat can make K LD = 0 . In the case of P B 0 , 1 − P B 0 , 0 = 1 , there exists a u nique sol ution ( P B 1 , 1 , P B 1 , 0 ) = (0 , 1) that can make K LD = 0 . F or the P B 0 , 1 − P B 0 , 0 < 1 case, t here exist an infinite num ber of attacking probability distributions ( P B j, 1 , P B j, 0 ) which can make K LD = 0 . By further assuming that the honest and Byzantine n odes are identical in terms of their detection performance, i.e., P H d = P B d and P H f a = P B f a , the above conditio n to blind the FC reduces to P K k =1 [ β k (1 − P k i =1 α i )] P K k =1 [ β k ( P k i =1 α i )]] ≤ 1 which is equivalent to K X k =1 [ β k (1 − 2( k X i =1 α i ))] ≤ 0 . (12) Recall that α k = B k N k and β k = N k P K i =1 N i . Subs tituting α k and β k into (12) and si mplifying the result, we have the following t heorem. DRAFT 9 Theor em 1. In a tr ee network with K levels, ther e e xis ts an att ac king pr obabil ity distribution ( P B 0 , 1 , P B 1 , 0 ) that can make K LD = 0 , and ther eby blind t he FC, if and only i f { B k } K k =1 satisfy K X k =1 B k N k K X i = k N i ! ≥ N 2 . (13) Dividing both sides of (13) by N , the above conditio n can be written as P K k =1 β k P k i =1 α i ≥ 0 . 5 . This i mplies t hat t o m ak e the FC blind, 50% or more nodes i n th e network need to be cove red 3 by the Byzantines. Next, to explore th e optimal attacking probability distribution ( P B 0 , 1 , P B 1 , 0 ) that min imizes K LD when (12) d oes not hold , we explore the properties of KLD. First, we sho w that attacking with symm etric flippi ng probabili ties is the optim al strategy in the region where the att ack er cannot b lind the FC. In other words, attacking with P 1 , 0 = P 0 , 1 is the optimal st rate g y for the Byzantines. For analytical t ractability , we assume P H d = P B d = P d and P H f a = P B f a = P f a in further analy sis. Lemma 1. In the r e gion wher e the attacke r cannot bli nd the FC, the optimal att ac king strate gy comprises of symmetric flipping pr obabilities. Mor e specifically , any non zer o deviation ǫ i ∈ (0 , p ] in flipping probabilities ( P B 0 , 1 , P B 1 , 0 ) = ( p − ǫ 1 , p − ǫ 2 ) , wher e ǫ 1 6 = ǫ 2 , will r esult in incr ease in the KLD. Pr oof: Let us denote, P ( z = 1 | H 1 ) = π 1 , 1 , P ( z = 1 | H 0 ) = π 1 , 0 and t = P K k =1 β k P k i =1 α i . Notice that, in the region where the attacker cannot blind the FC, th e parameter t < 0 . 5 . T o prove the lemma, we first s ho w that any positiv e de viation ǫ ∈ (0 , p ] in flipping probabiliti es ( P B 1 , 0 , P B 0 , 1 ) = ( p, p − ǫ ) will result in an increase in the KLD. After plugging in ( P B 1 , 0 , P B 0 , 1 ) = ( p, p − ǫ ) in (7) and (8), we g et π 1 , 1 = t ( p − P d (2 p − ǫ )) + P d (14) π 1 , 0 = t ( p − P f a (2 p − ǫ )) + P f a . (15) Now we show that the KLD, D , as giv e in (6) is a monotonically increasing function of the 3 Node i at lev el k ′ cov ers all it s children at le vels k ′ + 1 ≤ k ≤ K and the no de i itself an d, therefore, the total number of cov ered nodes by B k ′ , Byzantine at lev el k ′ , is B k ′ N k ′ . P K i = k ′ N i . DRAFT 10 parameter ǫ or in other words, dD dǫ > 0 . dD dǫ = π 1 , 1  π ′ 1 , 1 π 1 , 1 − π ′ 1 , 0 π 1 , 0  + π ′ 1 , 1 log π 1 , 1 π 1 , 0 + (1 − π 1 , 1 )  π ′ 1 , 0 1 − π 1 , 0 − π ′ 1 , 1 1 − π 1 , 1  − π ′ 1 , 1 log 1 − π 1 , 1 1 − π 1 , 0 (16) where dπ 1 , 1 dǫ = π ′ 1 , 1 = tP d and dπ 1 , 0 dǫ = π ′ 1 , 0 = tP f a and t is the fraction of covered nodes by t he Byzantines. After rearranging the terms in the above equati on, the condition dD dǫ > 0 b ecomes 1 − π 1 , 1 1 − π 1 , 0 + P d P f a log π 1 , 1 π 1 , 0 > π 1 , 1 π 1 , 0 + P d P f a log 1 − π 1 , 1 1 − π 1 , 0 . (17) Since P d > P f a and t < 0 . 5 , π 1 , 1 > π 1 , 0 . It can also be prov ed that P f a P d π 1 , 1 π 1 , 0 < 1 . Hence, we hav e 1 + ( π 1 , 1 − π 1 , 0 ) > P f a P d π 1 , 1 π 1 , 0 which is equivalent to 1 − π 1 , 1 1 − π 1 , 0 + P d P f a  1 − π 1 , 0 π 1 , 1  > π 1 , 1 π 1 , 0 + P d P f a  1 − π 1 , 1 1 − π 1 , 0 − 1  . (18) Applying the logarithm inequality ( x − 1) ≥ log x ≥ x − 1 x , for x > 0 to (18), one can prove that condition (17) is true. Similarly , we can show t hat any non zero de vi ation ǫ ∈ (0 , p ] in flipping prob abilities ( P B 1 , 0 , P B 0 , 1 ) = ( p − ǫ, p ) will resul t in an i ncrea se in the KLD, i.e., dD dǫ > 0 , or π 1 , 1 π 1 , 0 + 1 − P d 1 − P f a log 1 − π 1 , 1 1 − π 1 , 0 > 1 − π 1 , 1 1 − π 1 , 0 + 1 − P d 1 − P f a log π 1 , 1 π 1 , 0 . (19) Since P d > P f a and t < 0 . 5 , π 1 , 1 > π 1 , 0 . It can also be proved that 1 − π 1 , 1 1 − π 1 , 0 > 1 − P d 1 − P f a . Hence, we hav e 1 − π 1 , 1 1 − π 1 , 0 > 1 − P d 1 − P f a [1 − ( π 1 , 1 − π 1 , 0 )] (20) ⇔ 1 π 1 , 1 − π 1 , 0  π 1 , 1 π 1 , 0 − 1 − π 1 , 1 1 − π 1 , 0  > 1 − P d 1 − P f a  1 π 1 , 0 + 1 1 − π 1 , 1  (21) ⇔ π 1 , 1 π 1 , 0 − 1 − π 1 , 1 1 − π 1 , 0 > 1 − P d 1 − P f a  π 1 , 1 − π 1 , 0 π 1 , 0 + π 1 , 1 − π 1 , 0 1 − π 1 , 1  (22) ⇔ π 1 , 1 π 1 , 0 + 1 − P d 1 − P f a  1 − 1 − π 1 , 0 1 − π 1 , 1  > 1 − π 1 , 1 1 − π 1 , 0 + 1 − P d 1 − P f a  π 1 , 1 π 1 , 0 − 1  . (23) DRAFT 11 Applying the logarithm i nequality ( x − 1) ≥ log x ≥ x − 1 x , for x > 0 to (23), one can prove that condition (19) is true. Condition (17) and (19) imply that any no n zero deviation ǫ i ∈ (0 , p ] in flipping probabili ties ( P B 0 , 1 , P B 1 , 0 ) = ( p − ǫ 1 , p − ǫ 2 ) will result i n an in crea se i n the KL D. In the next theorem, we present a clos ed form expression for the optimal attacking probabil ity distribution ( P B j, 1 , P B j, 0 ) that minimizes K LD in the region where the attacker cannot b lind the FC. Theor em 2. In the r e gio n wher e the attack er cannot bli nd the FC, the optimal attacking st r ate gy is given by ( P B 0 , 1 , P B 1 , 0 ) = ( 1 , 1) . Pr oof: Observe that, i n the region where the attacker cannot blind the FC, the opti mal strategy comprises of symmetric flipping probabilities ( P B 0 , 1 = P B 1 , 0 = p ) . The proof is compl ete if we show that KLD, D , is a monotonically decreasing function of the flipping prob ability p . Let us denote, P ( z = 1 | H 1 ) = π 1 , 1 and P ( z = 1 | H 0 ) = π 1 , 0 . After plu gging in ( P B 0 , 1 , P B 1 , 0 ) = ( p, p ) in (7) and (8 ), we get π 1 , 1 = t ( p − P d (2 p )) + P d (24) π 1 , 0 = t ( p − P f a (2 p )) + P f a . (25) Now we show that the KLD, D , as gi ven in (6) is a m onotonically decreasing function of t he parameter p or in other words, dD dp < 0 . Af ter plugging in π ′ 1 , 1 = t (1 − 2 P d ) and π ′ 1 , 0 = t (1 − 2 P f a ) in the expression of dD dp and rearranging the terms , the cond ition dD dp < 0 b ecomes (1 − 2 P f a )  1 − π 1 , 1 1 − π 1 , 0 − π 1 , 1 π 1 , 0  + (1 − 2 P d ) log  1 − π 1 , 0 1 − π 1 , 1 π 1 , 1 π 1 , 0  < 0 (26) Since P d > P f a and t < 0 . 5 , we ha ve π 1 , 1 > π 1 , 0 . Now , using the fact that 1 − P d 1 − P f a > 1 − 2 P d 1 − 2 P f a and (21), we ha ve 1 π 1 , 1 − π 1 , 0  π 1 , 1 π 1 , 0 − 1 − π 1 , 1 1 − π 1 , 0  > 1 − 2 P d 1 − 2 P f a  1 π 1 , 0 + 1 1 − π 1 , 1  (27) ⇔ π 1 , 1 π 1 , 0 + 1 − 2 P d 1 − 2 P f a  1 − 1 − π 1 , 0 1 − π 1 , 1  > 1 − π 1 , 1 1 − π 1 , 0 + 1 − 2 P d 1 − 2 P f a  π 1 , 1 π 1 , 0 − 1  . (28) Applying the logarithm i nequality ( x − 1) ≥ log x ≥ x − 1 x , for x > 0 to (28), one can prove that (26) is true. DRAFT 12 0 0.5 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 P 1,0 B P 0,1 B KLD (D) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Fig. 2. KL distance vs Fl ipping Probabilities when P d = 0 . 8 , P f a = 0 . 2 , and the fraction of co vered nodes by the Byzantines is t = 0 . 4 Next, to gain insights into the soluti on, we present so me numerical result s in Figure 2 that corroborate our t heoretical resul ts. W e pl ot KLD as a function of the flipping probabilities ( P B 1 , 0 , P B 0 , 1 ) . W e assu me that the probability of detection is P d = 0 . 8 , t he probabilit y of false alarm is P f a = 0 . 2 and th e fraction of covered nodes by the Byzantines is t = 0 . 4 . It can be seen that the optimal attacking strategy comprises of sym metric flipp ing probabilities and is giv en b y ( P B 0 , 1 , P B 1 , 0 ) = (1 , 1) , which corroborate our theoretical result presented in L emma 1 and Theorem 2. Next, we explore some properties of the KLD with respect to the fraction of co vered nod es t in the region where the attack er cannot bl ind the FC, i.e., t < 0 . 5 . Lemma 2. D ∗ = min ( P B j, 1 ,P B j, 0 ) D ( π j, 1 || π j, 0 ) is a continuous, decr easing and conv e x function of fraction of cover ed nod es by the Byzant ines t = P K k =1 [ β k ( P k i =1 α i )] i n t he r e gion wher e the attacker cannot blind the FC ( t < 0 . 5 ). Pr oof: Th e continuit y of D ( π j, 1 || π j, 0 ) with respect to the in volved distributions im plies the contin uity of D ∗ . T o show that D ∗ is a decreasing function of t , we use the fact that ar gmin ( P B 0 , 1 ,P B 1 , 0 ) D ( π j, 1 || π j, 0 ) is equal to (1 , 1) for t < 0 . 5 (as s ho wn in Theorem 2). After plugging ( P B 0 , 1 , P B 1 , 0 ) = ( 1 , 1) in the KLD expression, it can be shown t hat the expression for the deri vativ e of D with respect to t , dD dt , is the same as (26). Usin g th e results of Theorem 2 , it follows DRAFT 13 0 0.1 0.2 0.3 0.4 0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Fraction of nodes covered K L distance Fig. 3. min ( P B j, 1 ,P B j, 0 ) KL distance v s Fraction of nodes co vered when P d = 0 . 8 and P f a = 0 . 2 that dD dt < 0 and, therefore, D ∗ is a monoto nically decreasing function of t in th e region where t < 0 . 5 . The con ve xity of D ∗ follows from the fact that D ∗ ( π j, 1 || π j, 0 ) is con vex in π j, 1 and π j, 0 , which are affine transformations o f t (Note th at, con ve xity holds under af fine transformation). It is worth noting that Lemma 2 suggests that by mini mizing/maximizing the f r action of co ver ed nodes t , the FC can maximize/minimize the KLD. Using t his fact, fr om now onwar ds we will consider fraction of cover ed nodes t in lieu of the KLD in further anal ysis in th e paper . Next, to gain insights into the soluti on, we present so me numerical result s in Figure 3 that corroborate our theoretical results. W e plot min ( P B j, 1 ,P B j, 0 ) KLD as a funct ion of th e fraction of covere d nodes. W e assume that the probabil ities of detection and false alarm are P d = 0 . 8 and P f a = 0 . 2 , respectiv ely . Notice th at, when 50% of t he n odes in the network are covered, KLD between the two probability distributions becomes zero and FC becomes blin d. It can be seen th at D ∗ is a continuo us, decreasing and con vex function of the fraction of cov ered nodes t in t he region t < 0 . 5 , wh ich corroborate our theoretical result presented in Lemma 2 . Until now , we ha ve e xplored the problem from the attacker’ s perspectiv e. In the rest of the paper we look into the problem from a network desi gner’ s perspecti ve and propose a t echnique to mitig ate the ef fect of the Byzantines. M ore s pecifica lly , we explore the problem of designi ng a rob us t tree topology considering the Byzantine to incur a cost for attacking t he network and the FC t o incur a cost for deploying (including t he cost of prot ection, etc.) the network. The FC DRAFT 14 (network desig ner) tries to design a perfect a -ary tree to pology under its cost budget cons traint such that the system performance metric, i.e., KLD is maxi mized. Byzantines, on the other hand, are interested in attacking or capturing nodes to cause maximal possi ble degradation in system p erformance, with the cost of attacking o r capturing nodes not to excee d the attacker’ s budget. This problem can be formu lated as a bi-le vel programming problem where the upper and the lower level problems with conflicting o bjecti ves belong to the leader (FC) and the follo wer (Byzantines), respective ly . I V . R O B U S T T O P O L O G Y D E S I G N In this problem setting, it is assumed that there is a cost associated wit h attacking each node in the tree (which may represent resources required for capturing a node or cloning a node in some cases). W e also assu me t hat the costs for att acking nodes at different levels are d if ferent. Specifically , let c k be the cost of attacking any one node at level k . Also, w e assume c k > c k +1 for k = 1 , · · · , K − 1 , i.e., it is more costly to attack nod es th at are closer to t he FC. Observe that, a no de i at lev el k covers (in other words, can alter the decisions of) all i ts successors and node i it self. It is assumed t hat the net w ork designer or the FC has a cos t budget C netw ork budg et and the attacker has a cos t budget C attacker budg et . Let P k denote the number of n odes covered b y a node at leve l k . W e refer to P k as the “profit” of a node at l e vel k . No tice t hat, P k = P K i = k +1 N i N k + 1 . Notice that, in a tree top ology , P k can be written as P k = a k × P k +1 + 1 f or k = 1 , ..., K − 1 , (29) where P k is the profit of attacking a node at level k , P k +1 is the profit of attacking a node at lev el k + 1 and a k is the number of i mmediate children of a node at level k . For a perfect a -ary tree a k = a, ∀ k and P k = a K − k +1 − 1 a − 1 . The FC designs the network, such that, giv en the attacker’ s budget, t he fraction of cov ered nodes is minim ized, and consequently a more robust perfect a -ary tree in terms of KLD (See Lemm a 2) is generated. Next, we formulate our robust topology design problem. A. Robust P erfect a -ary T r ee T opology Design Since t he attacker aims to maxim ize the fraction of covered nod es by attacking /capturing { B k } K k =1 nodes within the cost budget C attacker budg et , the FC ’ s objective is to minimi ze the fraction DRAFT 15 of cove red nodes by choosing the parameters ( K , a ) optimally in a perfect a -ary tree top ol- ogy T ( K , a ) under its cost budget C netw ork budg et . This sit uation can be interpreted as a Bi-le vel optimizatio n problem, wh ere the first decision maker (the s o-called leader) h as th e first choice, and the second o ne (the so-called follower) reac ts opt imally to the leader’ s selection. It i s the leader’ s aim to find such a decision wh ich, t ogether with the optim al response of the follower , optimizes the objective functio n of the leader . For our problem, t he upper lev el probl em (ULP) corresponds to th e FC who is the l eader of the game, wh ile t he lower leve l problem (LLP) belongs to the at tack er who is the fol lo wer . W e assum e t hat the FC has complete i nformation about th e attacker’ s problem, i.e., the objecti ve function and th e constraints of the LLP . Simi larly , the at tack er is assumed to be aw are about the FC’ s resources, i.e., cost of deploying the nodes { c k } K k =1 . Next, we formalize our robust perfect a -ary tree t opology problem as follows: minimize ( K, a ) ∈ Z + P K k =1 ( a K − k +1 − 1) B k a ( a K − 1) subject to a min ≤ a ≤ a max K ≥ K min K X k =1 a k ≥ N min K X k =1 c k a k ≤ C netw ork budg et maximize B k ∈ Z + P K k =1 ( a K − k +1 − 1) B k a ( a K − 1) subject to K X k =1 c k B k ≤ C attacker budg et B k ≤ a k , ∀ k = 1 , 2 , . . . , K (30) where Z + is the set of non-negati ve i nte gers, a min ≥ 2 and K min ≥ 2 . The objective function in ULP i s the fraction of covered nodes by the Byzantines P K k =1 P k B k P K k =1 N k , where P k = a K − k +1 − 1 a − 1 and P K k =1 N k = a ( a K − 1) a − 1 . In the con straint a min ≤ a ≤ a max , a max represents the h ardw are constraint imposed by the Medium Access Control (MA C) s cheme used and a min represents the design constraint enforced by the FC. The constraint on the number of nodes in the network P K k =1 a k ≥ N min ensures that the network satisfies pre-specified d etection performance guar- DRAFT 16 antees. In other words, N min is the m inimum number of nodes needed to guarantee a certain detection performance. The constraint on the cost expenditure P K k =1 c k a k ≤ C netw ork budg et ensures that the tot al expenditure of the network designer does not exce ed the av ailabl e budget. In the LLP , t he objective function is the same as that of the FC, but t he sense of optimization is opposite, i.e., maximization of the fraction of cove red nodes. The const raint P K k =1 c k B k ≤ C attacker budg et ensures that the total expenditure of the att ack er does no t exceed the av ail able b udget. The constraints B k ≤ a k , ∀ k are logical conditions, which prevent the attacker from attackin g non-existing resou rces. Notice that, the bi-lev el op timization problem , in general, is an NP-hard problem [22]. In fact, the optimi zation problem corresponding t o LLP is the packing formul ation of the Bounded Knapsack Problem (BKP) [23 ], which itself, in general, is NP-hard. Next, we discuss some properties of our objective function that enable our robust topology design problem to ha ve a polynomial time sol ution. Lemma 3. In a perfect a -ary tr ee to pology , the fraction of cover ed nodes P K k =1 P k B k P K k =1 N k by the attack er with the cos t budg et C attacker budg et for an opt imal attack is a non-decr easing function of t he number of levels K in th e tre e. Pr oof: Let us denote the opti mal attacking set for a K level perfect a -ary tree topol ogy T ( K, a ) by { B 1 k } K k =1 and the o ptimal attacking set for a perf ect a -ary tree topology with K + 1 lev el s b y { B 2 k } K +1 k =1 giv en the cost budget C attacker budg et . T o prove the lemma, it i s suffi cient to show that P K +1 k =1 P 2 k B 2 k P K +1 k =1 N k ≥ P K k =1 P 2 k B 1 k P K +1 k =1 N k ≥ P K k =1 P 1 k B 1 k P K k =1 N k , (31) where P 1 k is the profit of attacking a no de at le vel k in a K level perfect a -ary tree topol ogy and P 2 k is the profit of attacking a node at lev el k in a K + 1 lev el perfect a -ary tree topology . First inequality in (31) follows due to the fact t hat { B 1 k } K k =1 may not be the optimal att acking set for topol ogy T ( K + 1 , a ) . T o prov e the second inequal ity observe that, an increase in the value o f parameter K results in an in crea se in both the deno minator (number of nodes in the DRAFT 17 network) and the numerator (fraction of covered nodes). Using this fact, let us denot e P K k =1 P 2 k B 1 k P K +1 k =1 N k = x + x 1 y + y 1 (32) with x = P K k =1 P 1 k B 1 k with P 1 k = a K − k +1 − 1 a − 1 , y = P K k =1 N k = a ( a K − 1) a − 1 , x 1 = P K k =1 ( B 1 k a K − k +1 ) is the increase in t he profit by adding one more level to the topo logy and y 1 = a K +1 is the increase in th e number of nod es in t he network by adding on e more le vel t o t he topolo gy . Note that x + x 1 y + y 1 > x y if and on ly if x y < x 1 y 1 , (33) where x, y , x 1 , and y 1 are positive va lues. Hence, it is suffic ient to prove that a K +1 P K k =1  B 1 k a k  − P K k =1 B 1 k a ( a K − 1) ≤ P K k =1 ( B 1 k a K − k +1 ) a K +1 . The above equation can be further simplified to K X k =1  B 1 k a k  ≤ K X k =1  B 1 k a  which is true for al l K ≥ 1 . Next, to gain insights into the soluti on, we present so me numerical result s in Figure 4 that corroborate our theoretical results. W e plot the fraction of cov ered nodes by t he Byzantines as a function of the to tal n umber of le vels i n the tree. W e assume that a = 2 and v ary K from 2 to 9 . W e als o assume that the cost to attack nodes at differ ent le vels are giv en by [ c 1 , · · · , c 9 ] = [52 , 48 , 24 , 16 , 12 , 8 , 10 , 6 , 4 ] and the cost budget of the attacker is C attacker budg et = 50 . For each T ( K, 2) , we find the optim al attacking set { B k } K k =1 by an exhausti ve search. It can be seen that the fraction of cov ered nodes is a n on-decrea sing function of the n umber of levels K , which corroborate our theoreti cal result presented i n Lemma 3 . Next, we explore some properties of the fraction of cov ered nodes with parameter a for a perfect a -ary tree topology . Bef ore discuss ing our result, we define th e parameter a min as follows. For a fixed K and attacker’ s cost b u dget C attacker budg et , a min is defined as the minimum value of a DRAFT 18 2 3 4 5 6 7 8 9 0 0.05 0.1 0.15 0.2 0.25 Parameter ’K’ Fraction of covered nodes ’t’ Fig. 4. Fraction of nodes covered vs Parameter K when a = 2 , K is varied from 2 to 9, [ c 1 , · · · , c 9 ] = [52 , 48 , 24 , 16 , 12 , 8 , 10 , 6 , 4] , and C attack er budget = 50 for which the attacker cannot blind the net w ork or cover 50% or m ore nodes. So we can restrict our analysis to a min ≤ a ≤ a max . Notice that, the attacker cannot blind all the trees T ( K , a ) for which a ≥ a min and can bli nd all the trees T ( K , a ) for which a < a min . Lemma 4. In a perfect a -ary tr ee topology , t he fraction of cover ed nodes P K k =1 P k B k P K k =1 N k by an attack er with cost budg et C attacker budg et in an optimal attack is a decr easing fun ction of parameter a for a perfect a -ary tr ee topology for a ≥ a min ≥ 2 . Pr oof: As before, let us denote the opt imal attacking set for a K lev el perfect a -ary tree topology T ( K , a ) by { B 1 k } K k =1 and the opt imal attacking set for a perfect (a+1)-ary tree to pology T ( K, a + 1) by { B 2 k } K k =1 giv en the cost budget C attacker budg et . T o prove the lemm a, it is s uf ficient to show that P K k =1 P 2 k B 2 k P K k =1 N 2 k < P K k =1 P 1 k B 2 k P K k =1 N 1 k ≤ P K k =1 P 1 k B 1 k P K k =1 N 1 k , (34) where N 1 k is the n umber of nodes at le vel k in T ( K, a ) , N 2 k is the number o f nodes at leve l k in T ( K, a + 1) , P 1 k is the profit of att acking a node at le vel k in T ( K, a ) and P 2 k is the profit of att acking a node at lev el k in T ( K, a + 1) . Observe that, an i nterpretation of (34) is that DRAFT 19 the attacker is using th e attacking s et { B 2 k } K k =1 to attack T ( K , a ) . Howe ver , o ne m ight su spect that the set { B 2 k } k = K k =1 is not a va lid so lution. More specifically , the set { B 2 k } k = K k =1 is not a va lid solution in the fol lo wi ng two cases: 1. min ( B 2 k , N 1 k ) = N 1 k for any k : For example, if N 1 1 = 4 for T ( K , 4 ) and B 2 1 = 5 for T ( K , 5 ) then it will n ot be possibl e for the att ack er to attack 5 nodes at level 1 i n T ( K , 4) b eca use the total number of nodes at lev el 1 is 4 . In this case, { B 2 k } K k =1 might not b e a valid attacking set for the tree T ( K , a ) . 2. { B 2 k } k = K k =1 is an overlapping set 4 for T ( K , a ) : For example, for T (2 , 3) i f B 2 1 = 2 and B 2 2 = 4 , then, B 2 1 and B 2 2 are overlapping. In this case, { B 2 k } K k =1 might not be a v alid attacking s et for the tree T ( K , a ) . Howe ver , both of the above conditi ons impl y t hat the attacker can blind the network with C attacker budg et (See App endix A), which cannot be true for a ≥ a min , and, therefore, { B 2 k } K k =1 will in deed be a valid sol ution. Therefore, (34) i s sufficient to prov e the lem ma. Notice th at, the second inequality in (34) follo ws due to the fac t that { B 2 k } K k =1 may no t be the optimal attacking set for topolog y T ( K, a ) . T o prov e the first inequality in (34), we first consider the case where attacking set { B 2 k } k = K k =1 contains o nly one node, i.e., B 2 k = 1 for some k , and sho w that P 2 k P K k =1 N 2 k < P 1 k P K k =1 N 1 k . Substituti ng P 1 k = a K − k +1 − 1 a − 1 for some k and P K k =1 N 1 k = a ( a K − 1) a − 1 in the left side inequali ty of (34), we ha ve ( a ) K − k +1 − 1 ( a )(( a ) K − 1) > ( a + 1) K − k +1 − 1 ( a + 1)( ( a + 1) K − 1) . After some simp lification, the above conditio n becomes ( a + 1) K +1 [( a ) K − k +1 − 1] − ( a ) K +1 [( a + 1) K − k +1 − 1] +( a )[( a + 1 ) K − k +1 − 1] − ( a + 1)[( a ) K − k +1 − 1] > 0 . (35) In Appendix B, w e show that ( a )[( a + 1) K − k +1 − 1] − ( a + 1)[( a ) K − k +1 − 1] > 0 (36) and ( a + 1) K +1 [( a ) K − k +1 − 1] − ( a ) K +1 [( a + 1) K − k +1 − 1] ≥ 0 . (37) 4 W e call B k and B k + x are ov erl app ing, if the summation of B k + x k and B k + x is greater than N k + x , where B k + x k is the number of nodes co vered by the attacking set B k at lev el k + x . In a non-o verlapping case, the attacker can alw ays arrange nodes { B k } K k =1 such that each path in the network has at most one Byzantine. DRAFT 20 3 4 5 6 7 8 9 10 11 0 0.05 0.1 0.15 0.2 0.25 Parameter ’a’ Fraction of covered nodes ’t’ Fig. 5. Fraction of nodes cov ered vs Parameter a w hen K = 6 , parameter a is v aried from 3 to 11, [ c 1 , · · · , c 9 ] = [52 , 48 , 24 , 16 , 12 , 8 , 10 , 6 , 4] , and C attack er budget = 50 From (37) and (36), condition (35) holds. Since we have proved that P 2 k P K k =1 N 2 k < P 1 k P K k =1 N 1 k for all 1 ≤ k ≤ K , to generalize t he proo f for any arbitrary attacking set { B 2 k } K k =1 we multiply b oth sid es of the above inequality with B 2 k and sum it over all 1 ≤ k ≤ K inequalities. Now , we have P K k =1 P 2 k B 2 k P K k =1 N 2 k < P K k =1 P 1 k B 2 k P K k =1 N 1 k . Next, to gain insights into the soluti on, we present so me numerical result s in Figure 5 that corroborate our theoretical resul ts. W e plot the fraction of covered nodes by the Byzantines as a function of the parameter a in the tree. W e assume that the parameter K = 6 and v ary a from 3 to 11 . W e also assum e t hat the cost to attack nodes at di f ferent leve ls are given by [ c 1 , · · · , c 9 ] = [52 , 48 , 24 , 1 6 , 12 , 8 , 1 0 , 6 , 4] and the cost budget o f the attacker is C attacker budg et = 50 . For each T (6 , a ) we find the op timal attacking set { B k } K k =1 by an exhaustive search. It can be seen that th e fraction of covered no des is a decrea sing function of t he parameter a , which corroborate our theoretical result presented in Lemma 4 . Next, based on the above Lemmas we present an algorithm which can solve our robust perfect a -ary tree top ology design probl em (bi-level programming problem) effi ciently . DRAFT 21 B. Algorithm for solving Robust P erfect a -ary T r ee T opology Design Pr o blem Algorithm 1 Robust Perfect a -ary Tree T opology Desi gn Require: c k > c k +1 f or k = 1 , ..., K − 1 1: K ← K min ; a ← a max 2: if  P K k =1 c k a k > C networ k budget  then 3: Find the largest integer a − ℓ , ℓ ≥ 0 , such th at P K k =1 c k ( a − ℓ ) k ≤ C networ k budget 4: if ( a − ℓ < a min ) then 5: return ( φ, φ ) 6: else 7: a ← a − ℓ 8: end if 9: end if 10: if  P K k =1 a k ≥ N min  then 11: return ( K , a ) 12: else 13: K ← K + 1 14: return to Step 2 15: end if Based on Lemma 3 and Lemma 4, we present a polynomial time algorithm for solving the robust perfect a -ary t ree t opology design problem. O bserv e t hat, the rob ust network design problem is equiv alent to designing perfect a -ary t ree topology with minim um K and m aximum a that satisfy network designer’ s constraints. In Algorithm 1, we start from th e solution candidate ( a max , K min ) . If it d oes n ot satisfy the cost expenditure constraint we reduce a max by one, i.e., a max ← a max − 1 . Ne xt, the algorithm checks for the total number of nodes constraint and if it is not satisfied, we increase K min by one, i .e., K min ← K min + 1 . After these steps, the algorith m checks w hether this new solut ion candidate satisfies both the constraint s. If i t does, th is will be the solution for the problem, ot herwise, th e alg orithm solves the problem recursively unti l the hardware const raint is violated, i.e., a < a min . In t his case ( a < a min ), we wil l not ha ve any feasible solut ion which satisfies the net w ork designer’ s constraints. This procedure greatly reduces the com plexity because we do not n eed to sol v e the l o wer DRAFT 22 lev el prob lem in thi s case. Next, we prove th at Algorithm 1 indeed yields an optimal sol ution. Lemma 5. Robust P erfect a -ary T r ee T opology Design algorith m (Algorithm 1) yields an o ptimal solution ( K ∗ , a ∗ ) , if one e xists. Pr oof: Assume th at the optim al solution exists. Let us d enote by ( K ∗ , a ∗ ) , the opt imal solution given by Alg orithm 1. The main idea behin d o ur p roof i s that any solution ( K , a ) wit h K ≥ K ∗ and a ≤ a ∗ cannot perform better than ( K ∗ , a ∗ ) as suggested by Lemma 3 and Lemma 4. By transit i ve property , it can be proved that any solution ( K, a ) with K ≥ K ∗ and a ≤ a ∗ cannot perform better than ( K ∗ , a ∗ ) . Also, observe that, the only feasible solutio n in the region ( K min ≤ K ≤ K ∗ , a ∗ ≤ a ≤ a max ) is ( K ∗ , a ∗ ) . This imp lies that ( K ∗ , a ∗ ) is an optimal solution. Notice that, our algorithm searches for t he feasible solution with t he smallest K and the largest a . Any feasible soluti on ( K , a ) satisfies the follo wing two conditi ons: 1) P K k =1 c k a k ≤ C netw ork budg et ; 2) P K k =1 a k ≥ N min . By Lemma 4, if ( K, a ) is a feasible solution, then ( K, a ′ ) with a ′ < a will not be a better solution than ( K , a ) . Hence, for a given K , Step 3 only locates the solut ion with largest a for a giv en K . Furthermore, if both ( K , a ) and ( K ′ , a ′ ) sati sfy Condition 1 and K < K ′ , then a ≥ a ′ . Hence, for a giv en K , th e largest a in the current iteration s atisfying Condition 1 cannot be lar ger than the a found in the pre vi ous iteration. This verifies that ℓ ≥ 0 is a suffic ient condition to find the l ar gest a in Step 3. Next, we p ro ve that Algorithm 1 can st op when t he first feasible solutio n has been found. Let ( K 1 , a 1 ) be the first feasible solutio n found by Algo rithm 1. It is clear that the next feasible solution ( K , a ) must ha ve K > K 1 and a ≤ a 1 , since, t he alg orithm increases K and it satisfies Condition 1. Algorit hm 1 sto ps when both Condi tion 1 and Condition 2 satisfy . By the previous argument given in the beginning of the proof, we conclude that ( K, a ) does not perform better than ( K 1 , a 1 ) . Hence, ( K 1 , a 1 ) is the optimal solu tion ( K ∗ , a ∗ ) . It can be seen that if there is n o solut ion, then the algorithm will return ( ∅ , ∅ ) . This is due to the fact that if a − ℓ < a min , then no a can satisfy Condition 1 for current and furt her iterations. Hence, the algorithm terminates and returns ( ∅ , ∅ ) . Next, to gain i nsights into the solutio n, we present so me numerical resul ts in Figure 6 DRAFT 23 3 4 5 6 7 8 9 10 11 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 Parameter ’a’ Parameter ’K’ KLD Fig. 6. KLD vs Param eters ’K’ and ’a’ when ( P d , P f a ) = (0 . 8 , 0 . 2) , C networ k budget = 400000 , C attack er budget = 50 and N min = 1400 that corroborate our theoretical results. W e plot the min P 1 , 0 ,P 0 , 1 KLD for all t he combination s o f parameter K and a in the tree. W e v ary the parameter K from 2 to 10 and a from 3 t o 11 . W e also assume that the costs to attack n odes at d if ferent lev els are give n by [ c 1 , · · · , c 10 ] = [52 , 50 , 25 , 24 , 16 , 10 , 8 , 6 , 5 , 4] , and cost budgets of the network and the attacker are given by C netw ork budg et = 400000 , C attacker budg et = 50 , respective ly . The node budget constraint is ass umed to be N min = 140 0 . For each T ( K , a ) , we find the optimal attacking set { B k } K k =1 by an e x hausti ve search. All the feasible solutions are plotted in red and u nfeasible solut ions are plotted in blue. Notice that, T ( K min , a max ) which i s T (2 , 1 1) is not a feasible solutio n and, t heref ore, if we use Al gorithm 1 it will try to find th e feasible solution which has mini mum possi ble deviation from T ( K min , a max ) . It can be seen that the optimal soluti on T (3 , 11) has minimum possible deviation from the T ( K min , a max ) , which corroborate ou r algori thm. V . C O N C L U S I O N In this paper , w e hav e con sidered dist rib uted detection in perfect a -ary tree topologi es in the presence of Byzantines, and characterized the power of attack analytically . W e provided closed-form e xpressions for mini mum att acking power required by t he Byzantines to bl ind the FC. W e obtained closed form expressions for the opt imal attacking st rate g ies that m inimize the detection error exponent at th e FC . W e also looked at the pos sible counter -measures from the FC’ s perspectiv e to protect the network from these Byzantines. W e formulated the robust topology design problem as a bi-le vel program and provided an ef ficient alg orithm to solve it . DRAFT 24 There are s till m an y int ere sting questions that remain to b e explored i n th e future work su ch as an analysis of the problem for arbitrary topologi es. Note that, some analytical methodologies used in this paper are certainly e xploitable for studying the att acks in di f ferent topol ogies. Other questions such as the case where Byzantines col lude in sev eral groups (collaborate) to degra de the detection performance can also be in vestigated. A C K N O W L E D G E M E N T This work was support ed in part by AR O under Grant W911NF-09-1-0244 and AFOSR under Grants F A955 0-10-1-0458, F A9550-10-1-0263. A P P E N D I X A W e want to s ho w that the set { B k } K k =1 can blind the FC if any of following two cases is true. 1. min ( B k , N k ) = N k for any k , 2. { B k } k = K k =1 is an overlapping set In other words, set { B k } K k =1 cove rs 50% or more nodes. Let us denote by ˜ k , the k for w hich min ( B k , N k ) = N k (there can be m ultiple such k ). Then { B k } K k =1 satisfies P K k =1 P k B k P K k =1 N k ≥ P ˜ k B ˜ k P K k =1 N k ≥ P ˜ k N ˜ k P K k =1 N k ≥ P K N K P K k =1 N k . (38) Similarly , l et us assume B k ′ and B ˜ k are overlapping with ˜ k = k ′ + x (there can be m ultiple overlapping k ). Then { B k } K k =1 satisfies P K k =1 P k B k P K k =1 N k ≥ P ˜ k B ˜ k + P k ′ B k ′ P K k =1 N k ≥ P ˜ k N ˜ k P K k =1 N k ≥ P K N K P K k =1 N k . (39) Observe that, to prove o ur claim i t is sufficient t o sh o w that P K N K P K k =1 N k ≥ 0 . 5 ⇔ P K N K ≥ N 2 . (40) Using the fact that for a Perfect a -ary t ree P K = 1 , N K = a K and N = a ( a K − 1) a − 1 the condition (40) becomes 2 × a K ≥ a ( a K − 1) a − 1 . (41) DRAFT 25 When a ≥ 2 , we ha ve a × a K ≥ 2 × a K ⇔ a + a K +1 ≥ 2 × a K ⇔ 2 × a K +1 − 2 × a K ≥ a K +1 − a ⇔ 2 × a K ≥ a ( a K − 1) a − 1 . Hence, (40) holds and this comp letes our p roof. A P P E N D I X B W e skip t he proof of (36) and o nly focus on the proof of (37). T o s ho w ( a + 1) K +1 [( a ) K − k +1 − 1] − ( a ) K +1 [( a + 1) K − k +1 − 1] ≥ 0 for a ≥ 2 is equiv alent to s ho w a K +1 [( a − 1 ) K − k +1 − 1] − ( a − 1) K +1 [ a K − k +1 − 1] ≥ 0 for a ≥ 3 which can be sim plified to ( a ( a − 1 )) K − k +1 [ a k − ( a − 1) k ] ≥ [ a K +1 − ( a − 1) K +1 ] . (42) Using binom ial expansion, (42) b ecomes ( a ( a − 1)) K − k +1 [ a k − 1 + ( a − 1) a k − 2 + · · · + ( a − 1) k − 1 ] ≥ [ a K + ( a − 1) a K − 1 + · · · + ( a − 1) K − 1 a + ( a − 1) K ] ⇔ ( a − 1) K − k +1 [ a K + ( a − 1) a K − 1 + · · · + ( a − 1) k − 1 a K − k +1 ] | {z } k terms ≥ [ a K + ( a − 1) a K − 1 + · · · + ( a − 1) k − 1 a K − k +1 ] | {z } k terms + [( a − 1) k a K − k + · · · + ( a − 1) K − 1 a + ( a − 1) K ] | {z } K-k+1 terms ⇔ (( a − 1) K − k +1 − 1)[ a K + · · · + ( a − 1) k − 1 a K − k +1 ] ≥ [( a − 1) k a K − k + · · · + ( a − 1) K − 1 a + ( a − 1) K ] . (43) Since a ≥ 3 , w e have (( a − 1) K − k +1 − 1) ≥ ( K − k + 1) ≥ 1 . 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