Social learning for resilient data fusion against data falsification attacks

So cial lea rning fo r resilient data fusion against data falsification attacks F ernando Rosas 1 , 2 , Kw ang-Cheng Chen 3 and Deniz G¨ und¨ uz 2 1 Centre of Co mplexity Science and Department of Ma them ati c s, Imp erial College London, UK 2 Depa rtment of Electrical and Electronic Engineering, Imp erial College London, UK 3 Depa rtment of Electrical Engineering, University of South Flo rida, USA Abstract Internet of Things (IoT) suffers f ro m vulnerabl e senso r no des, which are lik ely to endure data falsi fication att acks foll o wing physical or cyb er capture. Mo reover, centralized decision- making and data fusion schemes commo nly used b y t hese net w o rks turn these decision p oints into si ngle p oints of failure, which are li k ely to b e exploited b y smart att ack ers. In o rder to face this serious security thread, w e propose a novel s cheme that enables distributed data aggregation and decision-making b y following so cial lea rning p rinciples. Our proposed scheme mak es senso r no des to act resembling the manners of agents within a so cia l net w o rk. W e analytical ly examine how lo cal actions of ind ividual agents can pro pagate t hrough the whole net w o rk, affecting the collective b ehavi our. Fi nally , we sho w ho w socia l learning can enable net w o rk re sili ence against da ta falsificati on attacks, e ven when a significant p o rtion of the no des have b een comp romised by the adversary . Index terms— Distributed decisi on-making, data fusion, senso r net w o rks, so cial net w o rks, data fal sification a ttacks, Byzantine no des, collective b ehaviour, so cial learning, information cascades. 1 1 In tro ductio n 1.1 Motiv ation Internet of Things (IoT) is exp ected to play a central role in digit al so ciet y . How ever, b efo re adopting this technology it is crucia l to gua rantee its security , sp ecially fo r those publ ic utili- ties whose safety is crucial fo r the well-being of society [1]. Recent cyb er-attacks that created significant damage have b een widely rep o rted, e.g., the s elf-p ropagating malwa re W anna Cry that caused a fa mous wo rldwid e net w o rk hack in May 2017 [2]. Therefo re, developing tech- nologies fo r guaranteeing the safet y of large info rmation net w o rks such as IoT is a challenging but urgent need. A s info rmation net w o rks get mo re closely intert wined with our dail y li fes, ensuring t heir safety in the future wil l b ecome a n even mo re chall enging iss ue. As the level of s ecurit y is determined b y the w eak est element, a majo r dilemma of IoT securit y lies in the low-c omplexity senso r net wo rks th at a re lo cated at the edge of the system. These senso r net w o rks a re usually comp osed b y a larg e numb er of a utonomous electronic devices, which collect info rmation that is critical to the control and op eration of IoT [3, 4]. By monito ring extensive geographical areas, these net w o rks enable a wide range of services to civil so ciety , b eing a k ey element fo r the well-being of sma rt cit ies [5, 6].These net w o rks may also p erfo rm sensitive tas ks, including the surveill ance over military or secure zones, intrusi on detection to p rivate property , monitoring of drink able w ater tanks and pro tection f rom chemical attacks [7, 8]. Although the de sign of secure wireless senso r net w o rks h ave been widely studied (e.g . [9 –11] and references therein), there remain op en pro blems of b oth theo retical and engineering na- ture [12 ]. In p a rticular, as the numb er of senso rs is usually very large, a p recise management of them is therefore challenging, o r even unfeasible. A significant p o rtion of the senso rs mi ght b e deplo yed in unprote cted a reas where it is imp ossible to ensure their physical o r cyber s e- curit y (e.g., wa r zones, o r regions controlled by an adversary). F urthermo re, senso r no des are generally not t amp er-p ro of due to cost restrictions, and ha ve hence l imited computing a nd net- w o rking capabili ties. Therefo re, they ma y not b e capable of emplo ying relia ble cryptographic o r securit y functions of high complexit y . The vulnerability of s enso r no des ma k es it reasonable to exp ect that they might b e victims of cyb er/physical attacks driven b y in telligent adversa ries. A ttacks to inform ati on net w o rks a re usuall y catego rized into outside attacks and insider attacks . Outside attacks include (distributed) denial of services (DoS), which use the b roadcasting nature for wireless com- munications to disru pt the communications capabilities [10]. In contrast, in i nsider atta cks the adversa ry “recruits” senso r no des b y malwa re through cyb er/wireless means, or directly b y physical substituti on [13]. F ollo wing the classic Byzantine Generals Problem [14], these “Byzantine no des” are authenticated, a nd recognized as vali d memb ers of the netw o rk. Byzan- tine no des can hence generate false data, exhibit arbitra ry b ehaviour, and collude with others to create net w o rk malfunctions. In general, inside attacks are considered t o b e more dangerous to in fo rmation net w o rks tha n outside attacks. The effect of Byzanti ne no des and da ta falsifi cation over distribut ed senso r n et w o rks h as b een intensely studied; the impact over the net w ork p erfo rmance ha s b een cha racterized, and va rious defense mechanisms ha s b een p rop osed (c.f. [15] fo r an overview, and also [16–20] fo r some recent contributions). H o we ver, all these wo rks fo cus on netw o rks with star o r tree top ology , and rely on centralising the decision-making in sp ecial nodes, called “fusion centers” (F Cs), whi ch gather all the sensed data. Therefo re, a k ey element in these approache s is a st rong div ision of lab our: o rdina ry senso r no des merely sense and fo rwa rd data, while the p ro cessing is d one exclusively at the FC, co rresp onding to a distrib uted-sensing/centralized- p ro cessing app roach. Thi s literature impli citly a ssume that the FCs are capabl e of executing secure codi ng and protocols, a nd hence are out of the reach of attack ers. Ho w ever, l a rge info r- mation net wo rks might require an other kind of mediato r devi ces, known a s data aggregato rs (D As), which have the cap ability to access the cloud through high-bandwidth com municati on links [21]. DAs a re attra ctive ta rgets t o insider at tacks, as they might als o b e lo cated in 2 unsafe lo cations due to the limited range of senso r nodes’ radi os. Please note that a tamp ered D A can completely disable the sensing capabilit ies of all the no des whose info rmation ha s b een aggregated, generating a single p oint of fail ure that is lik ely to b e exploited b y smart adversa ries [22]. An attractive route to address this i ssue is to consider distributed-sensing/distribu ted- p ro cessing schemes, which a void centralized decisi on making while dis tributing p ro cessing tasks throughout the netw o rk [23]. Ho w ever, the design of practical distributed-sensing / distributed-processing schemes is a challenging task, as collective computati on phenomena usually exhi bit highly non-trivi al features [24 , 25]. In effect, even though the distributed sensing literature is vast (fo r clas sic references c.f. [26–28], and more mo dern su rveys see [3, 4, 29, 30]), the construction of optimal di stributed schemes is in general NP-hard [31]. More over, although in man y scena rios the optimal schemes can be characterized as a set of thresholds f o r li k eliho o d functions, the determinati on of these thresholds is usua lly an intractable p roblem [26]. F o r example, homogeneo us thresholds can b e sub optimal even fo r n et w o rks with similar sensors a rranged in star topology [32], b eing only asymptotically optimal in the netw ork size [33]. Mo reover, symme tric strategies are n ot sui table fo r more complicated net w o rk top ologies, requiring heuristic metho ds. 1.2 Distributed decision-making in so cial learning In pa rallel, si gnificant resea rch effo rts have b een dedicated to analyz ing social learning , which refers to the decision-making processes tha t t ak e place within so cia l net w o rks [34]. In t hese scena rios, agents mak e decisions b ased on tw o elements: p rivate i nfo rmation that represe nts agent’s p ersonal know ledge, and so cial inf o rmation derived from p revious decisions made by the a gent’s p eers [35]. So cial lea rning w as initially investigated b y pioneering w o rks that studied se quentia l decision- making of B a ye sian a gents over simple so cial netw ork structures [36, 37]. These mo dels sho w ed ho w, than ks to so cial interactions, ind ividuals with w eak p rivate si gnals can ha rvest i nfo rma- tion from the decisions of other agents [38]. Interestingly , i t was also found that aggregation of rati onal decisions could generate su b optimal collective resp onses, degrading th e “wisdom of the cro wds” into mere herd b ehaviour. A fter t hese in itial findi ngs, resea rchers have aimed at developing a deeper u nderstanding of information cascades extending th e o riginal mo dels b y considering mo re general cost metrics [39–41], and b y s tudying the effects of the net w o rk top ology on the a ggregated b ehaviour [42–45]. Non-bay esi an lea rning mo dels have also b een explo red, where agents use simple rul e-of-thumb metho ds to exchange in fo rmation [46 – 52]. So cial l ea rning pla ys a crucial role in many imp o rtant so cial phenomena, e.g., in the a dop- tion o r rejection of new technology , o r in the fo rmation of p olit ical opini ons [34]. So cial lea rning mo dels are pa rticularly interestin g fo r studying info rmation cascades and herd dynam- ics, whi ch arises when the s o cial info rmation pu shes a ll the su bsequent a gents to igno re their o wn p ersonal kno wledge and a dopt a homogeneo us b ehaviour [37]. More over, there have b een a renew ed int erest in understanding information cascades i n the context of e-comme rce and digital so ciet y [45]. Fo r exampl e, inf o rmation cascades might ha ve tremendous consequences in online store s where customers can see the opinions of previous customers b efo re d eciding to buy a product, o r in the emergence of viral media contents based on sequentia l actions of like o r dislik e. Therefo re, developing a deep understanding of the mechanics b ehind information cascades are t riggered, and ho w they impact so cial learning, i s fundamental for our mo d ern net w o rk ed so ciety . The main motivation b ehind this a rticle i s to explore the connections that exist b et w een so cial learning and secure senso r netw orks, bu ilding a bridge betw een the resea rch done sepa- rately by economists a nd so ciologist by one side an d electrical engineers a nd computer scien- tists b y other. A k ey insight fo r establ ishing thi s l ink is to realize that each agent’s decision co rresp ond to a comp ressed description of his/her p rivate i nfo rmation. Therefo re, the fact that agents cannot access the p rivat e i nfo rmation of others, but can only observe their de- 3 cisions, can b e understo o d as a constraint in the communication resources. In this w a y , so cial learning can b e rega rded as an inf o rmation netw ork that p erfo rms di stributed inference (see T a ble 1). Mo reover, it w ould b e natura l to use so cial lea rning principles in the design of dist ributed-sensing/distributed-processing schemes, with t he hop e that this might enables additional robustness to decision-making p ro cesses in senso r netw o rks. T able 1 : T able o f corresp ondances Distributed detection So cial learning Sensor no de So cial agen t Comm unication range So cial neighbourho o d En vironmen tal v ariables State of the world Noisy measuremen t Priv ate information Lo cal pro cessing Agen t’s decision Bandwidth constraints Decision sharing 1.3 Con tribution In contrast t o almost all the exis ting resea rch, this wo rk considers p ow erful top ology-a wa re data fals ification attacks, where the adversa ry kno ws the net wo rk top ology and leverages this kno wledge to ta k e control of the most critical no des of t he net w o rk —either regular no des, DAs o r F Cs. This rep resents a wo rst-case scena rio where the net w o rk structure has b een d isclosed, e.g. through net w o rk tomography via traffic analysi s [53]. The reason why this adversa ry mo del has not b een p opular in the literat ure might b e b ecause traditional distri buted sensing schemes do not offer any resistance against this kind of attack. In this w o rk w e explo re the use of a distrib uted-sensing/distributed-p ro cessing scheme based on s o cial lea rning principles in o rder to deal wi th a top ology-a wa re a dversa ry . The scheme is a th reshold-based data f usion s trategy , related to those considered in [26]. Ho w ever, its relationship with social decision-making allo ws an intuitive understanding of its mechanisms. F o r avoiding security threads intro duced b y fusi on centers, our scheme uses tandem or serial decision sequencing [27, 54 –57]. It i s to b e noted th at, contrast ing with some related literature, our a nalysis do es not fo cus on optimality asp ects of data fusi on, but aims to ill ustrate ho w distributed decision making can enable netw o rk resilience against p o w erful top ology-a wa re data fa lsification attacks. W e demonstrate how netw ork resil ience hold even when a significant numb er of no des have b een comp romised. Our w o rk exploits a p ositive effect of i nfo rmation cascades that has b een overlo oke d b efore : info rmation cascades mak e a la rge numb er of agents/nodes to hold equally qualified e stimat o rs, generating many lo cations where a net w o rk op erato r can collect aggregated data. Therefore , info rmation cascades are crucial in our solution f o r a voiding single p oints of failure. F o r enabling a b etter understanding of info rmation cascades, t his wo rk extends results p resented in [58] p roviding a math ematical characterization of info rmation cascades under dat a falsification attacks. In pa rticular, our results clarify the conditions up on whi ch lo cal actions of individua l agents can p ropagate across the net wo rk, comp romising the collective p erfo rmance. These results p rovides a firs t st ep in the clarification of these non-trivial so cial dynamics, enriching our understa nding of decision-making process i n bias ed so cial netw o rks. This pap er expand s the ideas p resented i n [59] b y developing a formalism that allows considering i ncomplete or imp erfect so cial information. This fo rmalism is used to overcome the st rongest limi tation of the s cheme presented in [59], na mely t he fact that each no de w as required to overhear and store all the previous transmissions i n the netw o rk. Cl ea rly thi s cannot tak e pla ce in a la rge senso r netw ork, due b oth to the sto rage constrain ts of the nodes, and to 4 the large energy consumption required to transmit and receive across all pairs of no des [60]. Therefo re, the present wo rk is an imp o rtant step tow ards making th is scheme mo re relevant fo r p ractical appli cations. The rest of this a rticle is structured as follo ws. Section 2 intro duces the system mo del, describing the net w o rk controller and the adversary b ehaviour. Our so cial lea rning data fusion scheme is then describ ed in Section 3, where some basic statistical pro p erties a re explo red and p ractical a lgo rithm fo r implementing the decision rule is derived. Section 4 ana lyses the mathematical p rop erties of the decision p ro cess, p roviding a geometrical description and a cha racterizati on of info rmation cascades. All these ideas a re then i llustrated in a concrete scena rio in Section 5. Fina lly , Section 6 summa riz es our main conclusions. 2 System mo de l and proble m statemen t This section intro duces the basic elements of our framew o rk and settles the basis for our so cial lea rning scheme , which is then intro duced in S ection 3. In the rest of the pap er upp ercase let- ters X a re used to deno te random variables and lo w ercase x realizations of them, while b oldface letters X and x rep resent vecto rs. Also, P w { X = x | Y = y } = P { X = x | Y = y , W = w } is used a s a sho rthand notation. 2.1 System mo d el W e consider a net w o rk of N no des, each co rresp onding to an info rmation-p ro cessing device that has b een deploy ed in a n area of interest. Each node is equipp ed with a sensor tha t enables the net w o rk to track variables of interest. The measurement of the senso r of the n -th node is denoted by S n , taking va lues over a set S ⊂ R that can b e discrete or continuous 1 . B ased on these signals, the netw ork needs to infer the value of an underlying binary va riabl e W . W e consider net wo rks where a ll the no des have equal sensing capabiliti es, that is, the signals S n a re a ssumed to b e identicall y distribu ted. U nfo rtunately , the general dis tributed detection p roblem fo r a rbitrarily co rrelated signals is kno wn to b e NP-ha rd [31 ]. Hence, fo r the sak e of tra ctability , it i s assumed that the variables S 1 , . . . , S N a re conditionally indep endent given the event { W = w } 2 , f ollo wing a pro babil it y dis tribution denoted b y µ w . It is al so assumed that b oth µ 0 and µ 1 a re absolut ely continuous with resp ect to each other [67], i.e., no pa rticular signal determines W unequivo cally . This property , in turn, gua rantees tha t the log-lik eliho o d ratio of these tw o distribut ions is alwa ys w ell-defined, b eing given b y the loga rithm of the co rresp onding Rad on-Nik o dym derivative 3 Λ S ( s ) = log dµ 1 dµ 0 ( s ) . In a ddition to sensing hardw are, each no de i s equipp ed wi th limited computing capab ility and a wireless radio to transit and receive data. Tw o no des in the netw o rk a re assumed to b e connected if t hey can exchange information wirelessly . Note that senso r no des usua lly have a very limited battery budget, which imp ose severe restrictions on their communication capabiliti es [68]. Therefore , it is assumed that each n o de fo rwa rd s i ts data to others only by b roadcasting a bina ry variable X n . These si mple signal s do not imp ose an add itional burden on the communication resources, a s they could b e app ended to existent wireless control pack ages and viceversa, o r could b e shared b y light, ultrasound or other al ternative media. W e fo cus on the case in whi ch the sensing capabi lities of each senso r a re limited; and hence, any inference ab out W ma de based only on the sensed data S n cannot achieve a high accuracy . Interestingly , due to t he nature of wireless broadcasting, nearb y t ransmissions can b e 1 The generalization of our framew or k a nd results to vector-v alued sensor outputs is straig ht for ward. 2 The conditional indep endence of se ns or signals is satisfied when the senso r noise is due to lo cal c a uses (e.g. thermal noise), but do not hold when there exist common noise sources (e.g. in the case of distributed ac o ustic sensors [61]). F or works that consider sensor interdependence see [62 – 66]. 3 When S n takes a finite num b er of v a lues then dµ 1 dµ 0 ( s ) = P { S n = s | W =1 } P { S n = s | W =0 } , while if S n is a contin uous r a ndom v a riable with conditio na l p.d.f. p ( S n | W = w ) then dµ 1 dµ 0 ( s ) = p ( s | W =1) p ( s | W =0) . 5 overhea rd, and therefo re the info rmation that they ca rry can b e fused with what is extracted from the lo cal s enso r. The inf o rmation that a no de can extra ct from overhea ring trans missions of other no des is called “social info rmation”, contrasting with the “senso rial inf o rmation” that is obtain ed from the sensed signal S n . Without loss of generality , no des transmit their signals sequentially acco rding to their indices (i.e., node 1 transmit s first, then no de 2, etc.). It is assumed that this sequence is randomly chosen, and can b e changed b y the net w o rk op erato r at a ny ti me (c.f. Section 2.2). In gene ral t he broadcasted signal s X 1 , . . . , X n − 1 might not b e directly observable b y th e n -th agent b ecause of various observational restrictions. Therefo re, the so cial observations obtai ned b y the n -th no de are represe nted by G n ∈ G n , which can b e a random scalar, vecto r, matrix o r other ma thematical object. S ome cases of int erest are: (i) T he k p revious decisions: G n = ( X n − k +1 , . . . , X n − 1 ) . (ii) The average value of the the previous decisions: G n = 1 n − 1 P n − 1 k =1 X k . (iii) The decisions of a gents connected by an Erdos-Renyi sto chastic netw o rk with pa rameter ξ ∈ [0 , 1] , i .e. G n = ( Z 1 , . . . , Z n − 1 ) ∈ { 0 , 1 , e } n − 1 , where Z k = ( X k with probabilit y ξ , e with probabilit y 1 − ξ . (1) Please note that in the la st example the Erdos-Renyi mo del has only b een used as an illust rative example, and it can b e easily generalized to consider the t op ology of an y s to chastic netw o rk of i nterest. In this w o rk we study t he so cial dynamics based on the p rop erties of the transi tion proba- bility from states g ′ ∈ G n − 1 to g ∈ G n , a s given by the conditional probabilities β n w ( g | x n , g ′ ) := P w  G n = g | X n − 1 = x n − 1 , G n − 1 = g ′  , (2) where x n ∈ { 0 , 1 } . W e also assu me t hat the so cial dynamics is causal, meaning tha t G n is conditionally i ndep endent of S m given W for all m ≥ n . 2.2 The net w ork op erator and th e adv ersary The netw ork is managed b y a net w o rk op erator, who is a n external agent tha t uses the netw o rk as a to ol to build an estimate of W . The netw ork operator is opp osed b y a n adversa ry , whose goal is to disrupt the inference capabiliti es of the net w o rk. F o r this aim, the adversa ry controls a group of a uthenticated Byzanti ne no des wi thout b eing noticed b y the netw o rk op erator, which have b een captured b y malw are through cyb er/wireless means , o r by physical subs titution. The overall p erformance of a net w o rk of N no des is defined b y the accuracy of the i nference of the N -th no de, which is the la st one in the decision sequence. As the d ecision sequence is generated randomly b y the net w o rk op erato r, every no de of the net w o rk is equally lik ely to b e at the end of the decision sequence. It is f urther assumed that the a dversa ry has no kno wledge of the decisi on sequence, as it can b e chosen at run-time and changed regularly . As the adversary h as no reason fo r capturing any pa rticular no de in the netw ork , it is hence reasonable to assume that the adversary captures no des randomly . T herefo re, the Byz antine no des a re considered to b e unifo rmly distributed over the net w o rk. F o r simplicity , we mo del the strength of the attack with a single pa rameter p b , which co rresp onds to the p robability of a no de of b eing comp romised 4 . M o reover, we assume that the capture pro babi lity do es not dep end on W 5 . Hence, the numb er of Byz antine no des, denoted b y N ∗ , is a Bi nomial random variable with E { N ∗ } = p b N . Due to the l a w of l a rge 4 This attack mo del as sumes implicitly that the ca pture of each no de is an independent even t. Extensio ns considering cyb e r -infection propagation prop er ties are poss ible (c.f. [69]), being left for future studies. 5 If the capture r atio would b e higher when W = 1, then detecting Byzantine nodes would provide additional evidence to detect attacks. As this w ould go ag ainst the adversary’s interest, we discard this p oss ibilit y . 6 numb ers, N ∗ ≈ p b N fo r la rge net w o rks; and hence, p b is also the ra tio of expected B yzantin e no des i n the net w o rk, which i s the tra ditional metric fo r atta ck strength used in the li terature. F o r enabling th e data processing and fo rwa rding, th e net w ork op erato r defines a strategy , i.e. a data fusi on scheme given b y a collection of (p ossi bly sto chastic) functions { π n } ∞ n =1 such that π n : S × G n → { 0 , 1 } fo r a ll n ∈ N . On the other hand, th e adversa ry can freely set the values of t he binary signals transmi tted b y Byzantine no des. This is modeled b y a random function C : { 0 , 1 } → { 0 , 1 } that co rrupts the no de’s broadcasted s ignal. Therefo re, the broadcasted signal of the n -th n o de is given by X n = ( C ( π n ( S n , G n )) with p robabili t y p b , and π n ( S n , G n ) otherwise. (3) F urthermo re, as b roadcasted signals are binary , the co rruption function C ( · ) can b e charac- terized b y the conditional p robabilit ies c 0 | 0 and c 0 | 1 , where c i | j = P { C ( π ) = i | π = j } . The rest of t his w o rk fo cuses on the cas e in whi ch the net wo rk op erato r can deduce the co rruption function and can estimate the capture risk p b . The average netw ork miss-detection and fa lse alarm rates for an att ack of intensity p b a re defined a s P { MD ; p b } := P 1 { π N ( S N , G N ) = 0 } , and (4) P { F A ; p b } := P 0 { π N ( S N , G N ) = 1 } , (5) resp ectively (note that p b implicitly affects the dis tribution of G N ). The case in which these quantities are unkno wn can b e addressed usin g the current fra mew o rk with a min-max ana lysis, which i s left fo r future studi es. 2.3 Problem statemen t Our goal is to develop a resil ient strategy , in order to p rovide a reliabl e estimation of W even under a significant numb er of un identified Byzantine no des. No te tha t in most surveil lance applications miss-detections are mo re i mp o rtant than false al a rms, b eing di fficult to estimate the cost of the w o rst-case scena rio. Therefo re, the average net w o rk p erfo rmance is evalua ted follo wing the Neyman-Pea rs on criteria, by setting an allo wable false alarm rate α and fo cusing on reducing the miss -detection rate [70]. By denoting b y P the set of all strategies, we l o ok to the follo wing optimiz ation p roblem: minimize { π n } ∞ n =1 ∈P P { MD ; p b } subject to P { FP ; p b } ≤ α. (6) Ho w ever, findi ng an optimal solution is a fo rmidable challenge, even fo r the simple case of net w o rks with sta rt top ology and no Byza ntine atta cks (see [30, 71] and references therein). Therefo re, our aim is to develop a sub-optimal strategy that enables resilience, while b eing suitable fo r implementation in senso r no des with li mited computational p o we r. 3 So cial learning as a data aggre gation scheme This section d escrib es our proposed data fusion scheme, a nd expla ins its functions against top ology-a w are data fa lsification atta cks. In the s equel, Section 3.1 describ es a nd analyses the data fusion rule, then Section 3.2 derives basi c properties of it stati stics, and finall y Section 3.3 p resents a practical al go rithm fo r its implementation. 7 3.1 Data fusion rule Let us assume tha t each senso r no de is a rational agent, who tries t o maximizes the pro fit of an inference within a so cial net wo rk. Ra tional agents follo w Ba yesian strategies 6 , which can b e elegantly describ ed by the f ollo wing decision rul e [70, Ch. 2]: P { W = 1 | S n , G n } P { W = 0 | S n , G n } π n =0 ≶ π n =1 u (0 , 0) − u (1 , 0) u (1 , 1) − u (0 , 1) . (7) Ab ove, u ( x, w ) is a cost assigned to the decision X n = x when W = w , which can b e engineered in o rder to match the relevance of miss-detections and fal se alarms [70]. Let us find a simpler expression fo r th e decision rule (7) . Due to t he causality constrain (c.f. Section 2.1 ), G n can only b e i nfluenced by S 1 , . . . , S n − 1 , and therefo re is conditionall y indep endent of S n given W . U sing t his conditional indep endence condition, one can find that P { W = 1 | S n , G n } P { W = 0 | S n , G n } = e Λ S ( S n )+Λ G n ( G n ) , (8) where Λ G n ( G n ) is the log-like lih o o d ratio of G n . Then, usi ng (8) one can re-write (7) as Λ S ( S n ) + Λ G n ( G n ) π n =0 ≶ π n =1 τ 0 , (9) where τ 0 = log P { W = 0 } P { W = 1 } + log u (0 , 0) − u (1 , 0) u (1 , 1) − u (0 , 1) . In simple wo rds , (10) states how the the n -th no de should fuse the p rivate and social kno wledge: the evidence is p rovided b y the co rresp onding log- lik eliho o d terms, which a re then s imply added and then compa red against a fixed threshold 7 . F urther understand ing of the ab ove decisi on rule can b e at tained by st udying it from the p oint of vi ew of communication theo ry [58]. We first note that the decision is mad e not over the full raw signal S n but over the “decision signal” Λ S ( S n ) , which is a processed version of it. Interestingly , this processing might serve for d imensionality reduction, as even though S n can b e a matrix o r a high-dimensional vecto r Λ S ( S n ) is alw ays a single numb er. D ue to their construction a nd the underlying assumpti ons over S n (c.f. Section 2.1), the variables Λ S ( S n ) a re identically distribut ed and conditionally indep endent given W = w . More over, b y intro ducing the shorthand notation τ n ( G n ) = τ 0 − Λ G n ( G n ) , one can re-write (10) as Λ S ( S n ) π n =0 ≶ π n =1 τ n ( G n ) . (10) Therefo re, the decision is made by compa ring the decision signal with a decision threshold τ n . Note that this rep resents a compa rison b et w een the sensed data , summa rised b y Λ S ( S n ) , and the so cial information ca rried b y τ n ( G n ) . 3.2 Decision statistics Let us find expressions fo r the p robabiliti es of the actions of the n -th a gent, first f o cusing on the case n = 1 . Note t hat P w { π 1 ( S 1 ) = 0 } = P w { Λ S ( S 1 ) < τ 0 } = F Λ w ( τ 0 ) (11) 6 Although B ay esian mo dels are elegant a nd tr actable, they a s sume ag e nts act a lwa ys r ationally [72] and mak e strong assumptions on the knowledge ag e nt s hav e abo ut poster ior probabilities [49]. How ever, Bay esian models provide a n imp ortant b enchmark, not necessa rily due to their accuracy but be cause they give an impor ta nt reference point with which other models can b e compar ed [35]. 7 As the prio r distribution of W is usually unknown, τ 0 is a free parameter of the scheme. F ollowing the discussion in Section 2.3, the net work oper ator shall sele ct the lowest v alue of τ that satisfies the require d false alarm rate given by the Neyman-Pearson criteria. 8 where F Λ w ( · ) is the c.d.f. of Λ S conditioned on W = w . Then, considering the p ossibility that the fi rst no de could b e a Byzanti ne, one can sho w that P w { X 1 = 0 } = p b P w { X 1 = 0 | Byza ntine } + (1 − p b ) P w { X 1 = 0 | not a B yzantin e } = p b ( c 0 | 0 F Λ w ( τ 0 ) + c 0 | 1 [1 − F Λ w ( τ 0 )]) + (1 − p b ) F Λ w ( τ 0 ) (12) = z 0 + z 1 F Λ w ( τ 0 ) , (13) where we a re intro ducing z 0 := p b c 0 | 1 and z 1 := 1 − p b (1 − c 0 | 0 + c 0 | 1 ) as sh o rt-hand notat ion, which a re non-negative constants that summa rize the strength of the adversa ry . In pa rticular, when the adv ersa ry is p o we rless then z 0 = 0 and z 1 = 1 and hence P w { π 1 ( S 1 ) = 0 } = P w { X 1 = 0 } . By considering the n -th no de, one can find that P w { π n ( S n , G n ) = 0 | G n = g n } = Z S P w { π n ( s n , g n ) = 0 | S n = s } µ w ( s ) d s = Z S 1 { π n ( g n , s ) = 0 } µ w ( s ) d s (14) = P w { Λ S ( s ) < τ n ( g n ) } (1 5) = F Λ w ( τ n ( g n )) . (16) The first equality is a consequence of th e fact that S n is conditi onally indep endent of G n given W = w , while the second equality is a consequence that X n can b e expre ssed as a deterministic f unction of G n and S n , and hence b ecomes conditi onally indep endent of W . Note that (16) sho ws that τ n is a sufficient statist ic fo r p redicting X n with resp ect to G n . Note that F Λ w ( x ) can b e directly computed from the st atistics of t he signal dis tribution (its p rop erties a re explo red in Ap p endix A ). Mo reover, using (16) and follo wing a s imilar derivation as in (12) , one can conclude that P w { X n = 0 | G n = g n } = z 0 + z 1 F Λ w ( τ n ( g n )) . (17) Let us now study the statistics of G n . By using the definition of the transition co efficients β n w ( g n +1 | x n , g n ) , one can find that P w  G n +1 = g n +1  = X g n ∈G n X x n ∈{ 0 , 1 } β n w ( g n +1 | x n , g n ) P w { X n = x n , G n = g n } . (18) Note that, using the ab ove derivations, th e terms P w { X n = x n , G n = g n } can b e further exp ressed as P w { X n = x n , G n = g n } = P w { X n = x n | G n = g n } P w { G n = g n } (19) = λ ( z 0 + z 1 F Λ w ( τ n ( g n )) , x n ) P w { G n = g n } , (20) where λ ( p, x ) = x (1 − p ) + (1 − x ) p . T herefo re, a closed fo rm exp ression can b e found for (18) recursively over G n . 3.3 An algorithm for computing the so c ial log-lik eliho o d The main challenge fo r implementing (10 ) as a data p ro cessing metho d in a senso r no de is to have an efficient a lgo rithm f o r computing τ n ( g n ) . Leveraging t he ab ove derivati ons, we develop Al go rithm 1 as a it erative procedure fo r computing τ n . In many cases of interest the algo rithm’s complexity scales gracefully . Fo r the particula r case of nodes with memo ry of length k (i.e. G n = ( X n − k − 1 , . . . , X n − 1 ) ), the a lgo rithmic complexit y of Algo rithm 1 is O (2 k N ) , and therefo re gro ws linearly with the siz e of the net w o rk, while b eing l imited in th e values of k that can consider. In general, the algo rithm complexit y 9 Algorithm 1 Computation of the decision threshold 1: function Compute T au ( N , F Λ 0 ( · ) , F Λ 1 ( · ) , β n w ( ·|· , · ) , τ 0 , z 0 , z 1 ) 2: τ 1 = τ 0 3: for x 1 ∈ { 0 , 1 } do 4: P 0 { X 1 = x 1 , G 1 = 0 } = λ ( z 0 + z 1 F Λ 0 ( τ 1 ) , x 1 ) 5: P 1 { X 1 = x 1 , G 1 = 0 } = λ ( z 0 + z 1 F Λ 1 ( τ 1 ) , x 1 ) 6: for n = 1 , . . . , N − 1 do 7: for ∀ g ∈ G n +1 do 8: P 0 { G n +1 = g } = P g n ∈G n P x n = { 0 , 1 } β n w ( g n +1 | x n , g n ) P 0 { X n = x n , G n = g n } 9: P 1 { G n +1 = g } = P g n ∈G n P x n = { 0 , 1 } β n w ( g n +1 | x n , g n ) P 1 { X n = x n , G n = g n } 10: Λ G n ( g ) = log P 1 { G n = g } P 0 { G n = g } 11: τ n ( g ) = ν + η − Λ G n ( g ) 12: for x n +1 ∈ { 0 , 1 } do 13: P 0 { X n +1 = x n +1 , G n +1 = g } = λ ( z 0 + z 1 F Λ 0 ( τ n ( g n )) , x n +1 ) P 0 { G n +1 = g } 14: P 1 { X n +1 = x n +1 , G n +1 = g } = λ ( z 0 + z 1 F Λ 1 ( τ n ( g n )) , x n +1 ) P 1 { G n +1 = g } 15: return τ N ( · ) scales linea rly with N a s long as the cardinalit y of G n a re b ounded, o r if a significant p ortion of the terms β n w ( g n +1 | x n , g n ) are zero. The input s that drive Algo rithm 1 can b e classi fied in t w o groups. First, the terms N , F Λ 0 ( · ) , F Λ 1 ( · ) , β n w ( ·|· , · ) are p rop erties of th e net wo rk (p osi tion of the no de wi thin the deci- sion sequence, senso r statistics and socia l observabi lity , resp ectively) tha t the net w o rk op erato r could measure. On the other hand, τ 0 , z 0 , z 1 a re properties of the ad versa ry profile tha t d e- p end on the p rio r stati stics of W , p b and the co rruption function defined b y c 0 | 0 and c 0 | 1 (c.f. Section 2.2). In most scena rios the kno wledge of the net w o rk controller ab out th ese q uantities is li mited, as attacks are ra re a nd might follow unp redictable pa tters. Limited kno wledge can still b e exploited using e.g. Bay esian estimation techniques [73]. If n o kno wledge is avail able fo r the netw ork controller, then these quantities can b e considered free pa rameters of the strategy tha t span a range of alternative bal ances b etw een miss-detections and f alse p osit ives, i.e. a receiver op erating characteristic (ROC) space. 4 Information cascade The term “so cial lea rning” refers to the fact tha t π n ( S n , G n ) b ecomes a b etter p redicto r of W as n gro ws, a nd hence larg er net w o rks tend to d evelop a mo re accurate inference. Ho w ever, as the numb er of sha red signal s grows, the co rresp onding “so cial p ressure” can make no des to igno re their individua l measurements t o blindl y f ollo w the dominant choice, triggering a cascade of homogeneous b ehaviour. It is our in terest to clarify the role of the so cial p ressure in the decision making of the agents involved in a so cial net w o rk, as info rmation cascades can intro duce severe limi tations in the a symptotic p erformance of so cial l ea rning [44]. Mo reover, an adversary can leverage the i nfo rmation cascade phenomenon. In effect, if the numb er of Byzantine nodes N ∗ is la rge enough then a misleading info rmation cascade can b e triggered almost surely , making the lea rning p ro cess to fail. Ho w ever, if N ∗ is not enough then the netw o rk ma y undo the p o ol of wrong opinions and end up triggering a co rrect cascade. In t he sequel, the effect of inf o rmation cascades is firs t studied in indivi dual no des in Section 4.1. Then, the p ropagation pro p erties of cascades is explore d i n Section 4.2. 10 4.1 Lo c al information cascades In general, the decision π n ( S n , G n ) is made bas ed in the evidence provided by b oth S n and G n . A lo cal cascade tak es pl ace in the n -th agent when t he info rmation convey ed b y S n is igno red in the decision-making p ro cess due to a dominant influence of G n . We use the t erm “lo cal” to emphasize tha t this event is related to the data fusion of an individual agent. This idea is formalized i n t he following defini tion us ing the notion of conditional mutual info rmation [74], denoted as I ( · ; ·|· ) . Definition 1. The so cial information g n ∈ G n gener ates a lo cal information cascade for the n - th agent i f I ( π n ; S n | G n = g n ) = 0 . The ab ove condition summarizes tw o p ossibilit ies: either pi n is a deterministic function of G n and hence there is no va riab ility in π n after G n has b een determined, o r there is stil l va riabi lity (i.e. π n is a sto chastic strategy) b ut it is conditionally indep endent of S n . In b oth cases, the ab ove formulation highli ghts the f act that t he decision π n contains no info rmati on coming f rom S n 8 . Lemma 1. The variables G n → τ n → π n form a Markov Chain (i.e. τ n is a sufficient statistic of G n for pr e dicting the de cision π n ). Pr o of. Using (16) one can fin d that P w { π n | τ n , G n } = λ ( F Λ w ( τ n ) , X n ) = P w { π n | τ n } , and therefore the cond itional ind ep enden cy of π n and G n giv en τ n is clear. Let us now introduce the notation U s = ess sup s ∈S Λ S ( S n = s ) and L s = ess inf s ∈S Λ S ( S n = s ) fo r the essenti al sup ermum an d infimum of Λ S ( S n ) , b eing the si gnals within S that most strongly supp ort the hyp othesis { W = 1 } over { W = 0 } and viceversa 9 . If one of these quantities di verge, this w ould imply tha t there a re si gnals s ∈ S tha t p rovide overwhelming evidence in favour of one of the competing hyp otheses. If b oth a re finite then the agents a re said to have b ounded b eli efs [44]. As senso ry si gnals of electronic devices a re ultimately p ro cessed digitall y , the numb er of different signal s that an agent can obtain are finit e and hence their suprem um is alw ays finite. Therefo re, in the sequel we asume that b oth L s and U s a re finite. Using these notions, the follo wing pro p osition provides a cha racterizati on fo r lo cal info rmation cascades. Prop osition 1. The so cial information g n ∈ G n triggers a lo c al i nf ormation c asc ade if and only if the agents have b ounde d b eliefs and τ n ( g n ) / ∈ [ L s , U s ] . Pr o of. Let us assume that the agen ts ha v e b oun ded b eliefs. F rom the definition of F Λ w , whic h is a cumulat ive densit y fu nction, it is clear that if τ n < L s then F Λ 0 ( τ n ) = F Λ 1 ( τ n ) = 0, while if τ n > U s then F Λ 0 ( τ n ) = F Λ 1 ( τ n ) = 1. Therefore, if τ n ( g n ) / ∈ [ L s , U s ] then, according to (16), it determines π n almost su rely , making π n and S n conditionally ind e- p end en t. T o prov e th e conv erse by contrapositiv e, let us assum e that L s < τ n ( g n ) < U s . Usin g again (16) and the d efi nition of U s and L s , one can conclude that th is implies that 0 < P w { π n = 0 | G n } < 1 for b oth w ∈ { 0 , 1 } . This, in tur n, implies that the sets S 0 ( τ ) = { s ∈ S | Λ S ( s ) < τ n ( G n } and S 1 ( τ ) = S − S 0 b oth h a v e p ositiv e pr ob ab ility under µ 0 and µ 1 , whic h in turn implies the existence of conditional interdep endency b et w een π n and S n in this case. 8 Recall tha t S n and G n are conditionally indep endent given W = w (c.f. Section 3.1), and hence there cannot be redundan t information ab out W that is co nvey ed by S n and also G n . F or a mor e detailed discussion ab out redundant information c.f. [75]. 9 The es sential s upremum is the smallest upp er b o und over Λ S ( S n ) that holds almost s urely , b eing the natural measure-theor etic extension of the notion of s upremum [76]. 11 Intutively , Prop osition 1 sho ws that a lo cal information cascade happ ens when the so cial info rmation goes ab ove the most i nfo rmative signal that could b e sensed. Some consequences of thi s result a re explo red in the next section. 4.2 So c ial information dynamics and global cascades It i s of great interest to pre dict when a lo cal information cascade could p ropagate across the net w o rk, dis rupting the collective b ehaviour and hence affecting t he netw ork p erfo rmance. T he follo wing definition capt ures how, during a “global inform ati on cascade”, the share d signals X n do not convey a nymo re i nfo rmation from t he co rresp onding sensor signals. Definition 2. The so cial informatio n g n ∈ G n triggers a global information cascade if I ( X m : S m | G n = g n ) = 0 holds for al l m ≥ n . A global info rmation cascade is a succession of lo cal info rmation cascades. As Prop osition 1 sho w ed t hat agents are free from lo cal cascades as long as τ n ∈ [ L s , U s ] , one can guess that global cascades are related to the dynamics of τ n . These dynamics are determined b y the transitions of G n , which follows the b ehaviour dictated b y the tra nsition co efficients β n w ( ·|· , · ) . T o further study the so cial info rmation dynamics we intro du ce the follo wing definitions. Definition 3. The c ol le ction { G n } ∞ n =1 is said to have: 1. str ongly consistent tr ansitions if, for any W = w , g ∈ G n and g ′ ∈ G n − 1 , β n w ( g | 1 , g ′ ) > 0 implies τ n ( g ) ≤ τ n − 1 ( g ′ ) , while if β n w ( g | 0 , g ′ ) > 0 implies τ n ( g ) ≥ τ n − 1 ( g ′ ) . 2. weakly consisten t transitions if, f or al l g ∈ G n and g ′ ∈ G n − 1 , τ n − 1 ( g ′ ) ≤ L s and P w { G n = g | G n − 1 = g ′ } > 0 implies τ n ( g ) ≤ L s , while τ n − 1 ( g ′ ) ≥ U s and P w { G n = g | G n − 1 = g ′ } > 0 implies τ n ( g ) ≥ U s 10 . Intuitively , strong consistency mean that the decision threshold evolves monotonically with resp ect to the bro adcast ed signals X n . Corre sp ondingly , weak consistency implies that τ n cannot return into [ L S , U S ] after going b ey ond it . Mo reover, the adjectives “strong” and “w eak” reflect the fact th at w eak consistency only takes place outside the b oundaries of the signal lik eliho o d, while the strong consistency affects all the decision s pace. Mo reover, s trong consistent transitions imply w eak consisten transitions when there a re no Byzanti ne no des, as sho wn in the next Lemma 11 . Lemma 2. Str ong c onsistent tr ansitions satisfy the we ak c onsistency c ondition if p b = 0 . Pr o of. See App end ix B. W e sho w next that if t he evolution of G n b ecomes deterministic and 1-1 after l eaving the interval [ L s , U s ] (hence forth called w eakly invertible transitions ), then it satisfies the w eak consistency condition. Lemma 3. We ekly invertible tr ansitions i mply the we akly c onsistency c ondition. Pr o of. See App end ix C. No w w e present t he main result of this section, which i s t he cha racterization of inf o rmation cascades f o r the case of so cial information that follo ws weakly consistent transitions. Theorem 1. If the so cial information ha ve we akly c onsistent tr ansitions, then a glob al information c asc ade is trigger e d by e ach lo c al information c asc ade. 10 Note that the condition P w { G n = g | G n − 1 = g ′ } > 0 is eq uiv a lent to ask either β n w ( g , | 0 , g ′ ) or β n w ( g , | 1 , g ′ ) to be strictly positive. 11 It is p ossible to build examples where weak co nsistency does not follow stro ng c o nsistency when p b > 0. 12 Pr o of. Let us consider g 0 ∈ G n suc h that it pro duce a local cascade in the n -th no de. T h en, due to Prop osition 1, th is implies that τ n ( g ) / ∈ [ L s , U s ] almost surely . This, com bined with the weakl y consistency assumption, implies that τ n +1 ( G n +1 ) / ∈ [ L s , U s ] almost s urely . A second application of Prop osition 1 mak e us to conclude that P w { π = 0 | G n +1 } is therefore equal to 0 o 1. T his, in turn , guaran tees that I ( π n +1 : S n +1 | G n = g ) = 0) almost sur ely , sho wing that the ( n + 1)-th no de exp er iences a lo cal information cascade b ecause of G n = g 0 . Finally , it is direct to see that a recur siv e application of the ab ov e argumen t allo ws one to pro ve th at I ( π n + m : S n + m | G n = g ) = 0) for all m ≥ 0, confirming the existence of a global cascade. This theo rem has a numb er of imp o rtant conseque nces. Firstly , i t pro vides an intuiti ve geometrical description ab out the nature of global cascades fo r net w o rks wi th w eak consistency . One can imagine the evolution of τ n ( G n ) as function of n as a rand om walk withi n the interval [ L s , U s ] . Because of the w eakly consist ency conditi on, if the random w alk step out of the interval, it will never come back. Mo reover, as consequence of this theo rem, the stepping out of [ L s , U s ] is a necessary and sufficient condition to trigger a global i nfo rmation cascad e over the netw o rk. Also, note that in the case where G n = X n (i.e. each no de see every p revious decision) is direct to pro ve that G n has w eakly invertible transitions. Therefo re, Theo rem 1 i s a gen- eralizati on of Theo rem 1 of [58], now b eing valid fo r the case where there are a f raction of Bizanti ne no des within the net w o rk. 5 Pro of of conc ept This section i llustrates the main results obtain ed in Sections 3 a nd 4 in a simple scena rio. In the sequel, first S ection 5.1 describ es t he scena rio, and th en Section 5.2 discusses numerical simulations. 5.1 Scenario description Let us consider a sensor netw ork that has surveillance dut ies over a sensit ive geographical are a. The sensitive a rea could co rresp ond to a facto ry , a drink abl e w ater container o r a w arzone, whose k ey variables need to b e sup ervised. The tas k of the senso r net w o rk is, through the observation of t hese va riabl es, to detect the events { W = 1 } and { W = 0 } th at co rresp ond to the p resence o r absence of an a ttack to the surveilled a rea, resp ectively . No kno wledge ab out of the p rio r dis tribution of W is a ssumed. W e consider no des that ha ve b een deplo y ed ra ndomly over the sensitive are a, a nd hence their l o cation follo w a P oisson P oint p ro cess (PPP). T he rati o of the area of interest that falls within the range of each sensor is d enoted b y r . If attacks o ccur unifo rmly over the surveilled a rea, then r is also the p robabili t y of an attack ta king place under the coverage a rea of a particula r senso r is. Note that, due to the l imited sensin g range, the miss-detection rate of indi vidual no des i s roughly equal t o 1 − r . As r is usua lly a small numb er ( 5% in our simulations), t his impl ies that without colla b o rati on each no de is extremely unreli able. Each no d e measures its environment using a digital s enso r of m levels dynamical range (i.e. S n ∈ { 0 , 1 , . . . , m − 1 } ). Under the ab sence of a n a ttack, the measured si gnal is assumed to b e no rmally distri buted with a pa rticular mean v alue and va riance. Fo r simplicity of the ana lysis, w e assume that S n conditioned i n { W = 0 } distributes follo wing a binomial distrib ution of pa rameters ( m, q ) , i .e. P 0 { S n = s n } =  m s n  q s n (1 − q ) m − s n := f ( s n ; m, q ) (21) which b ecause of the central limi t theo rem app ro ximat es a Gaussi an variable when m is relatively la rge. Moreo ver, it is assumed tha t the sensor dynamical range is ad apted to match 13 5 10 15 0 . 05 0 . 1 0 . 15 0 . 2 Digital sensor lev els Probabilit y P 0 { S n = s } P 1 { S n = s } Figure 1: Probabilit y distribution f or a digital sensor o f m = 16 lev els, conditioned on the ev en ts { W = 0 } and { W = 1 } . the mean va lue on the low er third of the senso r dynamical ra nge, i.e. E { S n | W = 0 } = m/ 3 . This na turally imp oses the requirement q = 1 / 3 . F ollo wing standa rd sta tistical app roaches, it is further assumed that the s enso rs control the environment lo oking fo r events where the measured data is significant ly h igh, i.e. when it is la rger than mean value in mo re than t w o standard deviations. This co rresp onds e.g. when a sp ecific chemical compound trespasses safe concentration values o r when to o much movement has b een detected over a given time window (see e.g. [77]). U sing the f act that V ar { S n } = mq (1 − q ) , this gives a threshold T = E { S n } + 2 p V ar { S n } = np + 2 p nq (1 − q ) . Therefo re, it i s as sumed that an attack is related to t he event of S n b eing unifo rmly dist ributed in [ T , m ] . Th erefo re, one fi nds that P 1 { S n = s n } =(1 − r ) P 1 { S n = s n | attack o ut o f range } + r P 1 { S n = s n | attack i n rang e } =(1 − r ) f ( s n ; m, q ) + r H ( s n − T ) m − T , (22) where H ( x ) is the discrete Heavisi de (step) function given b y H ( x ) = ( 1 if x ≥ 0 0 in other case. (23) In summary , S n conditioned on { W = 1 } is mo deled as a mi xture mo del b et we en a Binomial and a tru ncated unifo rm dist ribution, where the relative w eight b et w een t hem is determined b y r (c.f. Figure 1). Fi nally , using (21) and (21) , the l og-lik eliho o d f unction of the signal S n can b e determined as (see Figure 2) Λ S n ( s n ) = log P 1 { S n = s n } P 0 { S n = s n } = log  (1 − r ) + r H ( s n − T ) ( m − T ) f ( s n ; m, q )  . (24) W e fo cus on scenarios where a no de can overhea r t he tranmissi ons of a ll the other no des. Ho w ever, we consider the case where the listening p erio d is restri cted 12 . W e th erefo re st udy the case where the so cial i nfo rmation gathered by the n -th no de is G n = ( X n − k − 1 , . . . , X n − 1 ) if n > k . Here k is a design parameter, whose impact in the net w ork p erfo rmance is studied in the next section. 5.2 Discussion W e analysed the p erfo rmance of netw o rks of N = 300 senso r no des whose s enso rs can monitor r = 5% of the ta rget area. Usi ng t he definition given in (4) and (5) , combined wi th (16 ) , 12 It is well-kno wn tha t the wireless radios of small sensor no des consume a similar amount of energy while transmitting or receiving data, and hence r educing ov erhear ing p erio ds is k ey for attaining energy efficiency and hence long netw ork lifetime [6 0]. 14 5 10 15 5 10 Digital sensor level Lik eliho o d Λ S n ( s ) Figure 2: L o g-lik eliho o d of a digital signal of m = 16 lev els with resp ect to the v ariable W . miss-detection and false alarm rates w ere computed as P { MD } = X g ∈G n F Λ 1 ( τ n ( g )) P 1 { G n = g } and (25) P { F A } = X g ∈G n (1 − F Λ 0 ( τ n ( g ))) P 0 { G n = g } , (26) where the t erms P w { G n = g } are computed using A lgo rithm 1 (c.f. Section 3.3. In o rder t o favour the reduction of miss-detections over fa lse ala rms, τ 0 = 0 is chosen as is the low est value tha t still allows a non-trivial inf erence process 13 . We consider an upp er b ound over the tolerable false al a rm ra te of 5% . Simulati ons demonstrate tha t the proposed scheme enables strong net w o rk resilience in this scenario, all o wing the senso r netw ork to maintain a lo w miss-detection rate even in the p resence of an imp o rtant numb er of Byzantine no des (see Figure 3). Please recall that if a traditi onal distributed detection scheme ba sed on centrali sed decision is used, a top ology- a w are attack er can cause a miss-detection rate of 100% by just compro mising the few no des that p erfo rm data aggregation (i.e. th e F C(s)). Fi gure 3 s ho ws that no des tha t indivi dually w ould have a miss-detection rate of 95% can imp rove up t o a round 10% even when 30% of the no des are under t he control of the a ttack er. Therefo re, by making a ll the n o des t o agregate data, the net w o rk can overcome the infl uence of B izanti ne no des and hence even when some no des have b een comp romised the rest of the net wo rk can and generate a co rrect inference. Please note that, fo r the case il lustrated b y F igure 3, when there are Byzanti ne n o des the miss-detection rate imp roves unt il the net w o rk size reaches N = 500 , achieving a p erformance of ≈ 10 − 12 (not sho wn in the Figure). This result has t w o imp o rtant implications. First, this confirms the p rediction of Theo rem 1 that if the signal log-lik eliho o d is b ounded then inf o rma- tion cascades are eventuall y dominant, hence stoping the learning process of the net w o rk (fo r a mo re detailed di scussion ab out t his is sue please c.f. [58]). Secondly , this result stress a k ey difference of our approach with resp ect to the exis tent literat ure ab out i nfo rmation cascad es: even if info rmation cascades become dominant an d hence p erfect so cial l ea rning cannot b e achieved by b ounded signal s, the a chieved p erformance can still b e very high and hence useful in a p ractical inf o rmation-p ro cessing setup . The netw ork resilience provided b y our scheme is influenced by the senso r dynamical ra nge, m , as a higher sensor resolution i s lik ely to p rovide mo re discriminative p o w er. Our results sho w three sharply distinct regimes (see Figure 4). First , if m is to o small ( m ≤ 4 ) the net w o rk p erfo rmance is very p o o r, irresp ective of the numb er of Byza ntine no des. Secondly , if 8 ≤ m ≤ 32 the miss-detection rate without Byz antine no des is app ro x. 10% (cf. Figure 4) and is exp onentially degraded b y the p resence of By zantine no des. Fi nally , if m ≥ 64 then the p erfo rmance under no B yzantine no des is very high, and i s degraded sup er-exp onentially 13 Sim ulations show ed that if τ < 0 then X n = 1 for all n ∈ N independently of the v alue of W , trigger ing a premature information ca scade. 15 0 20 40 60 80 100 120 140 16 0 180 200 10 − 4 10 − 3 10 − 2 10 − 1 10 0 Agen t Rate N ∗ / N = 0 N ∗ / N = 0 . 1 N ∗ / N = 0 . 3 N ∗ / N = 0 . 5 Figure 3: Performance for the inference of eac h no de for v arious attac k in tensities, given b y the a v erage ratio of Byzantine no des N ∗ / N = p b . Agents o v erhear t he previous k = 4 bro adcasted signals, and use sensors with dynamical range of n = 64. b y the p resence of Byzanti ne no des. Interestingly , the p oint at which the miss-detection rate of this regime go es ab ove 10 − 1 is at N ∗ / N = 1 / 3 , having some resemblance with the well- kno wn 1 / 3 threshold of the Byz antine generals p roblem [14]. A lso, it is intriguing the fact that differences b et w een 8 and 32 levels in the dynamical range gives practically no p erfo rmance b enefits. 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 10 − 8 10 − 7 10 − 6 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 10 0 A ttack in tensity ( N ∗ / N ) Miss-detection rate m = 2 m = 16 m = 32 m = 64 m = 256 Figure 4 : Effect of the sensor dynamical range ov er the net w ork resilience. Our results also sho w t he effects of the memo ry size, k , show ing that larg er va lues of k p rovides great b enefits for the net wo rk resilience (see Figure 5). In effect, b y p erfo rming an optimal bay esian inference over 8 b roadcasted signals the net w o rk mis s-detection rate remains b ellow 10% up to an attack intensity of 50% of Byz antine no des. Unfo rtunately , the computation a nd sto rage requirements of Algo rithm 1 gro w exp onential ly with k , and hence usi ng memo ries b eyond k = 10 is not p ractical fo r resource-limited senso r net w o rks. Overcoming thi s li mitation is an relevant future li ne of investigation. 16 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 10 − 8 10 − 7 10 − 6 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 10 0 A ttack in tensity ( N ∗ / N ) Miss-detection rate k = 1 k = 2 k = 4 k = 6 k = 8 Figure 5: A larger no de memory , whic h allows to include more so cial signals into the inference pro cess, greatly improv es the net w ork resilience. 6 Conclus ions T radi tional approaches to data aggregation over i nfo rmation netw o rks are based on a strong division of l ab our, which discrimina tes b et we en sensing no des tha t merely sense and forw ard data, and fusion centers that monopoliz e all the p ro cessing and inference capa bilities. This generates a si ngle p oint of failure, whi ch is li k ely to b e exploited b y smart adversa ries whose interest is the disruption of the netw o rk capabili ties. This serious security thread can b e overcome by distri buting the d ecision making p ro cess across the netw ork using so cial learning principles. This appro ach a voids single p oints of failure b y generating a la rge numb er of no des from where aggregated data can b e a ccessed. In this pap er a s o cial lea rning data fusion scheme ha s b een p roposed, which is s uitable of b eing implemented i n devices with l imited computational capabili ties. W e sho w ed that i f the p rivate signals are b ounded then each lo cal info rmation cascades triggers a global cascade, extending p revious resul ts to t he case where a n adversa ry controles an numb er of B yzantine no des. This result is hi ghly relevant fo r senso r net wo rks, as digital senso rs a re intrinsically b ounded a nd hence satisf y the assumptions of these results. Ho w ever, contrasting with the literature, our appro ach do es not fo cus on the conditions that gua rantee p erfect asymptotical so cial learning (i.e. miss-detec tion and false a larm rates converging to zero), but if their limit i s sma ll enough fo r p ractical appl ications. Our results show that this is the case, even when limiting th e numb er of overheare d transmiss ions. Mo reover, our result s suggest that so cial learning principles can enable signifi cant resil ience of an info rmation net w o rk agai nst t op ology-a w are data f alsificati on atta cks, which can totally disable the detection cap abiliti es of traditional senso r net wo rks. Furthermo re, our result s illustrat e how the net w o rk resilience can p ersists even when the attack er has comp romised an imp o rtant numb er of no des. It is our hop e that these result s can motivate further explo rations on the interface b etw een distributed decision making, statistical inference and signal p ro cessing over technological and so cial net w o rks. 17 A Prop ert ies of F Λ w F o r simplicity let us consider the case of real-value s ignals, i .e. S n ∈ R . In this case, the c.d.f. of the signal l ik eliho o d is given by F Λ w ( y ) = Z S y d µ w (27) where S y = { x ∈ R | Λ s ( x ) ≤ y } . If Λ s is an increasing fun ction, then S y = { x ∈ R | x ≤ Λ − 1 s ( y ) } = ( −∞ , Λ − 1 s ( y )] and hence F Λ w ( y ) = Z Λ − 1 s ( y ) −∞ d µ w = H w (Λ − 1 s ( y )) , (28) where H w ( s ) is the cumulative densit y function (c.d.f.) of S n fo r W = w . Fo r the general case where Λ s is an arbitra ry (piece-wise continuous) functi on, then S y can b e expressed a s the uni on of interval s. Then ∪ ∞ j =1 [ a j ( y ) , b j ( y )] = S y (note that Λ s ( a j ( y )) = Λ s ( b k ( y )) = y ) and hence from (27) is clea r that F Λ w ( y ) = ∞ X j =1 Z b j ( y ) a j ( y ) d µ w = ∞ X j =1 [ H w ( b j ( y )) − H w ( a j ( y ))] . (29) B Pro of of Lemma 2 Pr o of. Lets assume that the pro cess G n has str on g consisten t transitions and consider g ′ ∈ G n − 1 suc h that τ n − 1 ( g ′ ) ≤ L s . Note that, under these cond itions F Λ w ( τ n − 1 ( g ′ )) = 0, and h en ce P w  X n − 1 = 1 | G n − 1 = g ′  = 1 − z 0 − z 1 F Λ w ( τ n − 1 ( g ′ )) = 1 − p b c 0 | 1 = 0 (30) holds for an y w ∈ { 0 , 1 } . Moreo v er, this allo ws to fi nd that P w { G n = g | G n − 1 = g ′ } = X x n ∈{ 0 , 1 } β n w ( g | x n , g ′ ) P w  X n − 1 = x n | G n − 1 = g ′  = β n w ( g | 1 , g ′ ) . (31) Therefore, d ue to the str on gly consisten t transition prop ert y , if P w { G n = g | G n − 1 = g ′ } = β n w ( g | 1 , g ′ ) > 0 then L s ≥ τ n − 1 ( g ′ ) ≥ τ n ( g ) , (32) pro ving the wea k consistent transition p rop erty . The pro of for the case of τ n − 1 ( g ′ ) ≥ U s is analogous. C Pro of of Lemma 3 Pr o of. Let us consider g 0 ∈ G n suc h that τ n ( g 0 ) / ∈ [ L s , U s ]. Then, due to the w eakly in ve rtible evol ution, f or eac h x ∈ { 0 , 1 } there exists g ( x ) ∈ G n +1 suc h that β n w ( g | x, g 0 ) = ( 1 if g = g ( x ) , 0 in other case. (33) Moreo v er, note that while the deterministic assump tion implies that the ev en t { G n = g 0 } could b e follo wed by either { G n +1 = g (0) } or { G n +1 = g (1) } , the 1-1 assumption requires 18 that g (0) = g (1). With this, note that Λ G n +1 ( g (0)) = log P 1 { G n +1 = g (0) } P 0 { G n +1 = g (0) } = log P g ′ ∈G n x ∈{ 0 , 1 } β n w ( g ( x ) | x, g ′ ) P 1 { X n = x, G n = g ′ } P g ′ ∈G n x ∈{ 0 , 1 } β n w ( g ( x ) | x, g ′ ) P 0 { X n = x, G n = g ′ } (34) = log P x ∈{ 0 , 1 } P 1 { X n = x | G n = g 0 } P 1 { G n = g 0 } P x ∈{ 0 , 1 } P 0 { X n = x | G n = g 0 } P 0 { G n = g 0 } (35) =Λ G n − 1 ( g 0 ) , (36) Ab o ve, (34) is a consequence of g (0) = g (1), wh ile (35) is b ecause of the 1-1 condition o v er the dynamic. Finally , to justify (36) let us first consider P w { X n = x | G n = g 0 } = λ ( z 0 + z 1 F Λ w ( τ n ( g 0 )) , x ) . (37) Because τ n ( g 0 ) / ∈ [ L s , U s ] then F Λ w ( τ n ( g 0 )) is either 0 or 1; in any case it do es not dep end s on W . Th is, in turn means that P 1 { X n = x | G n = g 0 } = P 0 { X n = x | G n = g 0 } , whic h explains ho w (36) is obtained. 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