Reconstruction of Missing Data in Social Networks Based on Temporal Patterns of Interactions

Transcription

1 Reconsrucon of Mssng Daa n Socal Neworks Based on Temporal Paerns of Ineracons Alexey Somakhn, Marn B. Shor, and Andrea L. Berozz Mahemacs Deparmen, Unversy of Calforna, Los Angeles E-mal: alexey@mah.ucla.edu, mbshor@mah.ucla.edu, berozz@mah.ucla.edu Absrac. We dscuss a mahemacal framework based on a self-excng pon process amed a analyzng emporal paerns n he seres of neracon evens beween agens n a socal nework. We hen develop a reconsrucon model ha allows one o predc he unknown parcpans n a poron of hose evens. Fnally, we apply our resuls o he Los Angeles gang nework. Keywords: Socal neworks, emporal dependence of evens, mssng daa recovery 1. Inroducon Predcon of mssng nformaon s an mporan par of daa analyss n socal scences [1, 2, 3]. The examples suded n leraure, mosly by sascans, nclude reconsrucon of he unknown connecons n a socal nework [4, 5], analyzng nongnorable non-responses n a survey samplng [6, 7], and many ohers. The mos common way o deal wh mssng values s o replace hem by some plausble esmaes usng known or model-based cross-dependences over he nework n queson. However, hese mehods do no ypcally consder neworks ha change wh me, when anoher source of nformaon s gven by he emporal paerns arsng from he nework evoluon. Such neworks are he prmary objec of sudy n he curren paper. As our man example, we consder he gang rvalry nework n he , days Fgure 1. Temporal cluserng of he neracon evens beween Clover and Eas Lake gangs n Los Angeles, durng he perod Los Angeles polcng dsrc Hollenbeck [8]. Polce daa on gang crmes from 1999 o 2002 reveal emporal cluserng of gang neracon evens, whch s demonsraed n Fgure 1. These emporal paerns can be used o solve he followng nverse problem: predc he parcpans of he gang-relaed crmes f some of hem are no known. For a gven par of agens, he neracon evens can eher be ndependen, followng a Posson process, or emporally dependen, n whch case he occurrence of one even can change he lkelhood of subsequen evens n he fuure. Such even

2 Reconsrucon of Mssng Daa Based on Temporal Paerns 2 dependency for he Los Angeles gang nework has been esablshed n [9], where a Hawkes process [10, 11], commonly used n sesmology o model earhquakes [12, 13] and defned by s nensy funcon λ() = µ + θ < g( ), (1) was compared o ner-gang volen crmes. Ths paper s organzed as follows. In Secon 2 and Secon 3 we formalze he problem and descrbe a model of neracon beween nework agens based on a Hawkes process, as n (1). In Secon 4 we propose a way of predcng he unknown parcpans of neracon evens, whch we formulae as a consraned opmzaon problem. In Secon 5 and Secon 6 we analyze our mehod and he soluon gves. Fnally, n Secon 7 we presen and dscuss he predcon resuls. 2. Problem Formulaon We model a socal nework as a graph wh nodes represenng he agens and edges, or bnary lnks [5], ndcang wheher or no he correspondng par of agens nerac. We furher look a he seres of parwse neracon evens beween he agens, characerzed by her occurrence mes and he pars nvolved. We assume ha he nework srucure represened by he graph does no change wh me, alhough each par of neracng agens can have s own prescrbed model of behavor ha mgh nvolve some me dependence. Suppose all he mes of he evens are known, bu for some of hem, daa on one or boh of he parcpans are mssng. The problem s o reconsruc he mssng daa abou he parcpans based on he behavoral model. βγ γα Fgure 2. Graphcal represenaon of he problem. Before we proceed, le us dscuss a convenen graphcal represenaon of he problem shown n Fgure 2. Here we deal wh a nework conssng of hree agens α, β, and γ, wh all pars beng acve. The black pons correspond o he evens whou any mssng nformaon. All evens are ordered n me and here s a separae melne for each par of agens. The ncomplee evens, whch are hose wh mssng daa abou he parcpans, canno be assgned o any parcular melne and are herefore represened va vercal seres of whe crcles. Our goal s o replace each vercal se of whe crcles wh a black crcle on one of he melnes n a way ha wll gve he mos plausble pcure n accordance wh he model. Reurnng o he nework of gangs n Los Angeles: here are weny-nne agens and he bnary lnks ndcae he exsng rvalres beween hem, shown n Fgure 3. In case of a rvalry, we have a seres of crmes correspondng o he neracon evens. These are ypcally murders, shos fred, ec. The daa capures he nformaon abou whch wo gangs were nvolved n a crme; however, for a large fracon of hem only vcm afflaon s provded. The problem n hs case s o esmae he afflaon of he unknown offenders.

3 Reconsrucon of Mssng Daa Based on Temporal Paerns 3 Fgure 3. Graph of he gangs nework n he Los Angeles polcng dsrc Hollenbeck [8]. Each of he weny-nne gangs s represened by a node, and he edges ndcae he presence of rvalres beween hem. 3. Agen Ineracon Model A Hawkes process s a self-excng pon process commonly used n sesmology o model earhquakes [12, 13] and defned by s nensy funcon (1). The nensy funcon λ() s paroned no he sum of a Posson background rae µ and a selfexcng componen, hrough whch evens rgger an ncrease n he nensy of he process. The elevaed rae spreads n me accordng o he kernel g, wh θ beng he scalng facor of he effec. In oher words, each even generaes a sequence of offsprng or repea evens, whch leads o emporal cluserng. Ths agrees wh he evdence ha realaons are commonplace among rval gangs. A smlar approach was used o model repea and near-repea burglary effecs n [14, 15] and emporal dynamcs of volence n Iraq n [16], where self-excaon s one of he key qualave feaures of he process. We assume ha he neracon evens for each par of agens occur ndependenly accordng o a Hawkes process. We make no exclusons for nacve pars snce for hose we smply have µ = 0, and s also useful o se θ = 0 o avod confuson n he followng analyss. For he funcon g, as n [9], we use an exponenal dsrbuon, whch gves λ() = µ + θ < ωe ω( ). (2) Here ω 1 ses he me scale over whch he overall rae λ() reurns o s baselne level µ afer an even occurs [17]. From he behavoral pon of vew, θ represens he average number of drec offsprng for each even and ω 1 s he expeced wang me unl an offsprng. To ndcae ha each par of agens has s own neracon parameers, we use ndex noaon and wre λ () = µ + θ < ω e ω ( ), (3) wh µ, θ, ω beng consans, unque for each par, and summaon over all prevous evens beween he agens α and β. If no confuson s possble, we wll om he ndces o smplfy he noaons n he fuure. In Fgure 4, we presen an example of daa generaed accordng o he descrbed model (3) for a nework conssng of hree agens α, β, and γ. Here, he same

4 Reconsrucon of Mssng Daa Based on Temporal Paerns 4 λ, 10 2 days λ βγ, 10 2 days λ γα, 10 2 days , days, days, days Fgure 4. Daa generaed accordng o a Hawkes process wh he same parameers for each par of agens: µ = 10 2 days 1, ω = 10 1 days 1, θ = 0.5. parameers are used for each par: µ = 10 2 days 1, ω = 10 1 days 1, θ = 0.5. These have approxmaely he order of magnude esmaed n [9] for he Los Angeles gang nework. 4. Reconsrucon Mehod We wll use he followng noaons: N oal number of evens n number of ncomplee evens k number of agens K oal number of pars = k(k 1)/2 To solve he predcon problem n queson one could consder he lkelhood funcon, defned on he space of all possble even lss, correspondng o dfferen ways of fllng n he mssng daa, whch s o be maxmzed n order o ge he mos lkely one. Gven any complee even ls, wh no mssng daa, s lkelhood s gven by (see, for example, [9]) ( λ ). (4) L = The frs produc s over all possble unordered pars of agens, and he second one s over all evens for a fxed par. Noe ha maxmzng (4) s a combnaoral ype problem snce he se of all agen pars s dscree. Unforunaely, here seems o be no sgnfcanly more opmal way han full search for solvng n he general case, whch s very neffcen snce s complexy depends exponenally on n. To overcome hs ssue, one could consder some smooh exenson of he lkelhood funcon and hen look for s maxmum, so ha some sandard connuous opmzaon mehod lke graden ascen could be used. Ths could be acheved by allowng each ncomplee even o move connuously beween he melnes. However, such approach an s no naurally applcable o (4) due o s mulplcave srucure.

5 Reconsrucon of Mssng Daa Based on Temporal Paerns 5 We herefore propose he followng. We desgn some reasonable approxmaon o he real lkelhood funcon (4), such ha s connuous exenson s physcally meanngful. Le us sar wh he followng smple example. Consder a nework conssng of hree agens α, β, and γ wh all pars havng he same neracon parameers. Suppose only one even s ncomplee and here s no nformaon abou s parcpans. Inuvely, because of he self-excng naure of he process, he even s less lkely o belong o he pars wh no nearby neracon, and more lkely o belong o hose for whch can be consdered as a par of a cluser. For nsance, n he suaon shown n Fgure 5, agens β and γ are he mos lkely parcpans of he ncomplee even, as hs would place whn a cluser. βγ γα Fgure 5. Example of reconsrucon based on emporal cluserng: agens β and γ are he mos lkely parcpans of he ncomplee even, as hs would place whn a cluser. To gve hs dea a quanave formulaon, we noe ha clusers correspond o he perods of me wh hgher values of he nensy funcons, whch can be seen n Fgure 4. Hence, for a mssng even would be reasonable o predc he par wh he hghes nensy a he momen of he even. I also makes sense from he probablsc pon of vew, because gven he fac ha an even happened a me he probably of par beng nvolved s proporonal o λ (). Now we consruc our energy funconal: an approxmaon o he lkelhood funcon (4) on he space of all possble even lss, correspondng o dfferen ways of fllng n he mssng daa, whch s o be maxmzed n order o ge he mos lkely one. Gven any complee even ls, wh no mssng daa, we defne s energy as Λ = λ ( ). (5) The frs summaon s over all possble unordered pars of agens, and he second one s over all evens for a fxed par. Here we bascally say ha he chances of a par o be nvolved n an neracon even are equal o s nensy funcon value a ha me. Then we ake he sum over all evens. Roughly speakng, he merc defned by (5) assgns hgher values o he even lss wh denser clusers, whch s precsely wha we need o ge a reasonable predcon. If no confuson s possble we wll replace wh n he summaon ndex, keepng n mnd ha each par of agens has her own melne and sysem of ndces for he evens on. Subsung (3) no (5) gves Λ = δ j µ (1 δ j)θ ω e ω j. (6),j Thus, Λ s decoupled no he sum of he energes of he evens hemselves, deermned by he background raes, and he sum of he parwse neracon energes beween he evens on he same melne due o self-excaon. Cluserng leads o sronger neracon, ncreasng he value of Λ. Clearly, funconal (6) s nvaran wh respec

6 Reconsrucon of Mssng Daa Based on Temporal Paerns 6 o me nverson, whch means ha each even affecs s successors and predecessors n he same way. As alernave o (5), one could normalze he nensy funcons over all pars of agens o make hem add up o 1, and defne he energy funconal as ( λ ) Λ = α β λ α β ( ), (7) an approach ha mgh seem o be more naural from he probablsc pon of vew. However makes he fnal opmzaon problem o be solved much more nonlnear and has a drawback dscussed n Secon 5. Agan, maxmzng he energy funconal (5) or (7) s a combnaoral ype problem snce he se of all agen pars s dscree, and here seems o be no sgnfcanly more opmal way han full search for solvng n he general case, whch s very neffcen. However, unlke he lkelhood funcon (4), adms a physcally meanngful smooh exenson. Ths can be obaned by dsrbung each of he ncomplee evens over he melnes wh weghs ha add up, n some sense, o 1. Thus, n Fgure 2, we would replace he whe crcles wh black ones and add weghs o each of hose; he complee evens naurally recevng wegh 1. We can nerpre hs o mean ha each ncomplee even occurred parally on every melne wh effec (he jump n he nensy funcon) proporonal o he correspondng wegh. Ths new connuous maxmzaon problem no only gves he mos lkely parcpans of an even, bu also assgns a wegh o each par showng how lkely ha par s o be nvolved. To avod msundersandng, le us specfy how we enumerae he evens on a melne, whch does maer now due o he normalzaon couplng of he pars. The reader can use Fgure 6 as a reference. We sar wh ncomplee evens and assgn Fgure 6. Evens enumeraon example. 3 hem numbers from 1 o n. The order here s no mporan, as long as s he same for all melnes. Then, for each melne we assgn numbers o he complee evens sarng from (n + 1). Thus, here s a separae even ndexng sysem for each melne, wh ndces concdng for he ncomplee evens. Usng l 2 -normalzaon for he weghs, we ge he followng formulaon of he problem { [ max,j δ j µ m + ] } (1 δ j)θ ω e ω j m m j m ( m ) 2 = 1, = 1,..., n 0, = 1,..., n,, (8)

7 Reconsrucon of Mssng Daa Based on Temporal Paerns 7 where m denoes he wegh of he even number on melne. As we menoned before, he complee evens have wegh 1, so m 1 for > n. The objecve funcon s maxmzed wh respec o m for n, gven he normalzaon and non-negavy consrans. One could alernavely choose o use l 1 -normalzaon for he weghs, whch agan mgh seem o be more naural from he probablsc pon of vew. The problem n ha case s max { m m [,j δ j µ m (1 δ j)θ ω e ω = 1, = 1,..., n 0, = 1,..., n, j m ] } m j. (9) However, hs mehod s unsable wh respec o he npu daa, as we wll see n Secon 5. Noe here ha he dscree, combnaoral verson of hs mehod can be obaned from (8) or (9) by forcng all weghs o be negers 5. Examples max { m m,j [ δ j µ m (1 δ j)θ ω e ω = 1, = 1,..., n {0, 1}, = 1,..., n, j m ] } m j. (10) The purpose of hs secon s o dscuss a few examples ha wll reveal some useful properes of he problem (8) Example 1: Tmescale Deecon Suppose N = n = 2, so we have wo ncomplee evens, and suppose we do no have any nformaon a all abou he parcpans. For smplcy we also assume µ 0 and θ 1. Then he problem o be solved accordng o (8) s max ω e ω m 1 m 2 (m ) 2 = 1, = 1, 2, (11) m 0, = 1, 2, wh beng he me nerval beween he evens. Noe ha (11) can be wren convenenly n vecor form as max m T 1 Dm 2 m 1 2 = m 2 2 = 1, (12) m 0, = 1, 2,

8 Reconsrucon of Mssng Daa Based on Temporal Paerns 8 where we have used he noaons { D = dag{ω e ω } R K K m = {m } R K, = 1, 2. (13) From lnear algebra, s well-known ha he objecve funcon n (12) s maxmzed when m 1 = m 2 = e α β, he un vecor, such ha α β = arg max {ω e ω }. (14) The maxmum of ωe ω s acheved when ω = 1. Hence he soluon of he problem (11) corresponds o he par wh self-excaon mescale closes o. Recall ha he self-excaon mescale represens he average me unl a repea even occurs. Thus, snce all background raes are equal o zero, and herefore he second even mus be an offsprng of he frs one, our mehod ndeed gves he mos lkely parcpans. Of course, for predcon purposes he dsrbuon of he weghs s no very realsc, because rules ou he possbly for all oher pars o be nvolved. Bu, as we wll see furher, here are oher mechansms ha make he soluon more regularzed, whch we do no see here due o a specfc and, n fac, unrealsc srucure of he example. Indeed, hs example s n some sense pahologcal, as here s no way o explan he occurrence of he frs even. However, we can hnk of as a lmng case when µ mn {ω e ω }. (15) Then he frs even s a background one, whch happened afer wang for suffcenly long me, and he second one s due o self-excaon, because he probably of beng a background even from some melne s much less han he probably of beng an offsprng of he prevous even, as follows from (15). Consder now he alernave energy funconal (7) wh normalzaon a each me pon, nroduced n Secon 4. Clearly, he maxmum value can acheve, for he example n queson, s 1. I happens whenever boh evens compleely belong o he same par of agens. Thus, hs model does no see he dependence of cluserng densy on self-excaon mescale, and leads o a degenerae soluon Example 2: Regularzaon Suppose N = n = 1, so we have only one even whch s ncomplee, and suppose we do no have any nformaon a all abou he parcpans. Then he problem o be solved accordng o (8) s max µ m ( ) m 2 = 1. (16) m 0, Problem (16) can be wren convenenly n vecor form as max µ T m m 2 = 1, (17) m 0,

9 Reconsrucon of Mssng Daa Based on Temporal Paerns 9 where we have used he noaons { µ = {µ } R K m = {m } R K. (18) The maxmzer of (17) s well-known from lnear algebra o be m = µ µ 2. (19) Thus, he opmal weghs, accordng o our mehod, are proporonal o he correspondng background nensy raes. Ths s exacly wha follows from he probablsc approach. Indeed, we are dealng wh he case where no self-excaon akes place, snce here s only one even. Therefore he probably of a par o be nvolved n he even s proporonal o s background nensy rae. Consder now he alernave model (9) wh l 1 -normalzaon, menoned n Secon 4. For hs example gves he followng opmzaon problem { max µ T m m 1 = 1. (20) Clearly, he objecve funcon n (20) s maxmzed when m = e α β, he un vecor, such ha α β = arg max {µ }. (21) We see ha he model pcks he par wh he hghes background rae, assgns wegh 1 o and 0 o he ohers. However, hs s no he mos desrable soluon. Suppose, for nsance, ha all background raes are approxmaely he same. Then, s no reasonable o choose one par over he ohers, snce all of hem are almos equally lkely o be nvolved. Unforunaely, hs s a general propery of model (9). I wll always eher assgn all he wegh o one par for each ncomplee even, never creang any dsrbuons, or wll gve a degenerae soluon. Indeed, he normalzaon consrans and he objecve funcon, n each of s argumens, are all lnear. Model (8) does no have such a drawback for hs example. I does no jus pck he mos lkely parcpans of he even, bu assgns weghs o all pars ndcang how lkely each of hem s o be nvolved. Ths can be hough of as some sor of regularzaon propery Dscusson As we menoned n Secon 4, he objecve funcon n (8) can be hough of as a sum of he energes of he evens. Formally, f we gnore consan erms, consss of wo pars: quadrac erms, correspondng o he neracon of he ncomplee evens, and lnear erms, correspondng o he energy of he ncomplee evens n he presence of he complee evens and background rae values. The examples above were argeed o examne hese pars separaely o reveal her roles n he reconsrucon process. In he frs example we consdered he quadrac par of he energy. We have seen ha he ncomplee evens end o gaher on hose melnes where her neracon energy s he hghes, whch leads o aggressve cluser formaon up o assgnng all he weghs o he same par of agens. On he oher hand, he lnear erms express he nfluence of he complee evens and background raes, and do no allow he ncomplee evens o devae oo much

10 Reconsrucon of Mssng Daa Based on Temporal Paerns 10 from already exsng cluserng srucure. Moreover hey regularze he soluon, whch represens he degree of uncerany n he predcon, as demonsraed n he second example. The mehods arsng from l 1 -normalzaon (9) and from he alernave energy funconal (7) have each shown some undesrable properes n hese examples, and we wll no consder hem furher. 6. Analyss Noe from Fgure 2 ha he whe crcles naurally form a K n marx and our goal s o deermne s enres. We denoe he marx as X = {x j }. For fuure reference wll be useful o express X n erms of s rows and columns X = r T 1. r T K = ( c 1 c n ). (22) Usng hese noaons, problem (8) can be wren as K =1 rt A r + r T b max c T j c j = 1, j = 1,..., n. (23) x j 0, = 1,..., K, j = 1,..., n Here A = {a jl } s he symmerc n n marx of he neracon coeffcens beween he ncomplee evens on he h melne, and b = {b j } s he column of sze n of he energy coeffcens for he ncomplee evens n he presence of he complee evens and background rae on he h melne. Clearly, he enres of A and b are nonnegave, for all = 1,..., K. Theorem. For he problem (23): () There exss a global maxmzer. () Every local maxmzer (or even a saonary pon) s a global maxmzer. () If all b j are srcly posve hen he maxmzer s unque. Proof. The objecve funcon s connuous and he admssble se, gven by he consrans, s compac. Ths proves (). Defne y j = x 2 j. Then he problem (23) becomes [ K n =1 j,l=1 ajl yj y l + ] n j=1 bj yj max K =1 y j = 1, j = 1,..., n. (24) y j 0, = 1,..., K, j = 1,..., n The admssble se n (24), gven by he consrans, s convex. We wll show ha he objecve funcon s concave on, and srcly concave f all b j are srcly posve, whch mples () and (). Noe ha a jl yj y l s concave for all = 1,..., K and j, l = 1,..., n. Ths follows from he fac ha for all a, b, c, d 0 and 0 < λ < 1 we have (λa )( ) + (1 λ)c λb + (1 λ)d λ ab + (1 λ) cd. (25)

11 Reconsrucon of Mssng Daa Based on Temporal Paerns 11 Indeed, squarng boh sdes of (25) gves a Cauchy-ype nequaly cb + ad 2 abcd, (26) afer smplfcaon. Now suffces o show ha he funcon K f j (y 1j,..., y Kj ) = b j yj (27) =1 s concave for all j = 1,..., n. Tha s K K λŷ j + (1 λ)ˇy j =1 b j =1 b j [λ ŷ j + (1 λ) ˇy j ] for all admssble dsnc {ŷ j } K =1, {ˇy j} K =1 and 0 < λ < 1. We furher wsh o show ha (27) s srcly concave, ha s he nequaly (28) mus be src, f all b j are srcly posve. Bu boh are rue snce he funcon x s srcly concave on {x : x 0}. Ths complees he proof. If all pars are acve, hen all background raes are nonzero, and we auomacally have all b j srcly posve, whch mples he unqueness of predcon n accordance wh he Theorem. When some pars are nacve par () of he Theorem s no applcable drecly. Indeed, f for example melne s nacve, hen here are no complee evens on and he correspondng background rae s 0, hence b = 0. Noe however ha n hs case addng he consran r = 0, or smply excludng he melne from consderaon, gves a problem wh a smaller unknown marx equvalen o (23). Thus, f we elmnae all nacve pars n hs way, we ge a problem wh all pars n queson beng acve, whch guaranees he unqueness of predcon. So far we mplcly assumed ha we had no nformaon a all abou he parcpans of he ncomplee evens and each par was consdered as a possble canddae for predcon. Of course, f one of he parcpans of an even s known, hen he pars whou hs agen can no be n nvolved, and he correspondng enres of X mus be equal o 0, whch means we have addonal consrans of he form x j = 0 for he problem (23). These consrans however do no affec he convexy of he admssble se n he coordnaes y j = x 2 j. Therefore all resuls of he Theorem reman vald. 7. Resuls In hs secon we presen and dscuss he resuls of varous ess of he proposed reconsrucon mehod. Snce he daa from he Los Angeles gang nework s ncomplee, and he ground ruh and neracon parameers for are unavalable, s no que suable for hs purpose. Insead, we generae synhec daa usng a Hawkes process (3), hrow ou some of he daa a random, and hen apply our algorhm o reconsruc. To evaluae he performance of our algorhm, we only focus on he orderng of he varous m for each ncomplee even. Specfcally, we deermne for each ncomplee even he weghs m for ha even on he varous melnes, order hem from lowes o hghes, and fnd he correspondng rank of he ground ruh melne for ha even. Ths s done for wo major reasons. Frs, our mehod (8) does no assgn proper probables o he varous melnes, only weghs ha should be nerpreed as beng relaed o probably n a monoonc way. Second, from an operaonal pon (28)

12 Reconsrucon of Mssng Daa Based on Temporal Paerns 12 of vew, he auhores are no very concerned wh he acual probables wh whch each gang commed a gven crme, bu raher wh a smple rankng of gangs from mos lkely o leas lkely, o prorze her nvesgaon. As a frs sep, we compare our connuous mehod (8) o wo ohers: one derved from he lkelhood funcon (4) and one usng he dscree model (10). However, noe ha he mehods (4) and (10) provde lkelhoods (or energes) only for full allocaons A of ncomplee evens, raher han one lkelhood for each melne per even. To bypass hs ssue, we smply defne he lkelhood ˆm (f) ha ncomplee even belongs o melne under merc f o be ˆm (f) = A f(a), (29) where A s mean o represen only hose allocaons n whch ncomplee even s arbued o melne, and f = L for (4) and f = Λ for (10). As menoned prevously, he mehods (4) and (10) are of combnaorc complexy, so we lm our esng here o a relavely small sysem wh N = 40, n = 4, k = 4, K = 6. Here, we assume no knowledge of he parcpans n ncomplee evens, so ha each may be assgned o any of he K = 6 melnes. Smulaons were run 10,000 mes usng parameers µ = 10 2 days 1, ω = 10 1 days 1, and θ = 0.5 for each par of agens, whch have approxmaely he order of magnude esmaed n [9]. Each smulaon generaed a rankng of he melnes for each ncomplee even, and he percenages of ncomplee evens whose ground ruh melnes were gven ceran ranks are shown n Table 1. Noe ha he hree mehods perform almos Table 1. Connuous mehod (8) compared o mehods (4) and (10), for N = 40, n = 4, k = 4, K = 6, µ = 10 2 days 1, ω = 10 1 days 1, and θ = 0.5. Mehod Top 1 Top 2 Top 3 Top 4 Top 5 (4) 47.3% 68.1% 79.8% 87.7% 94.0% (8) 47.1% 68.1% 79.7% 87.6% 94.1% (10) 47.0% 68.1% 79.7% 87.6% 94.0% dencally, each placng he correc melne a op lkelhood approxmaely 47% of he me, n he op wo lkelhoods approxmaely 68% of he me, and n he op hree lkelhoods approxmaely 80% of he me. Snce mehod (8) yelds nearly ndsngushable soluons o hose of (4) and (10), bu s vasly more compuaonally effecve, we focus only on hs connuous mehod for he remander of hs secon. We nex es our connuous mehod usng daases ha more closely mmc he gang rvalry daa. In all he expermens below, we have exacly one parcpan unknown for every ncomplee even, whch s he case for mos of he gang daa. Also, unless specfed oherwse, we assume full connecedness of he nework graph and use he same neracon parameers for each par of agens as used above: µ = 10 2 days 1, ω = 10 1 days 1, θ = 0.5. Table 2 demonsraes he performance of he connuous mehod (8). I s organzed as follows. The frs hree columns descrbe he dmensons of he nework and he daa he mehod was appled o, and he las hree ndcae how ofen, on average, a ground-ruh unknown par was n he op one, op wo, and op hree weghs of he predced dsrbuon. The value of k corresponds o he real Los

13 Reconsrucon of Mssng Daa Based on Temporal Paerns 13 Table 2. Connuous model (8) performance resuls. The frs hree columns descrbe he dmensons of he nework and he daa he mehod was appled o, and he las hree ndcae how ofen, on average, a ground-ruh unknown par was n he op one, op wo, and op hree weghs of he predced dsrbuon. The value of k corresponds o he real Los Angeles gang nework, see Fgure 3, whch s no a fully conneced graph. The Guessng rows show he resuls ha would be obaned by random guessng. k N n Top 1 Top 2 Top % 80% 92% % 79% 91% % 76% 90% 5 Guessng 25% 50% 75% % 69% 82% % 68% 80% % 65% 77% 7 Guessng 17% 33% 50% % 62% 73% % 60% 72% % 57% 69% 9 Guessng 13% 25% 38% % 72% 83% % 71% 82% % 68% 80% Angeles gang nework (see Fgure 3), whch s no a fully conneced graph. The Guessng rows show he resuls ha would be obaned by random guessng. Frs we noe ha, n erms of predcon qualy, he Los Angeles gang nework roughly corresponds o a fully conneced 6-nodes graph. Ths acually makes sense, snce each gang has abou 5 rvalres on average. Second, he predcon resuls depend raher mldly on he fracon of ncomplee evens, whch mplcly confrms he fac ha reconsrucon model (8) capures he qualave feaures of neracon process (3) raher well. As for he resuls hemselves, we can see ha hey are sgnfcanly beer han hose obaned by jus random guessng. A he same me hey are no perfec. To see why hs s so we need o have a closer look a how hey depend on he parameers of he sysem: µ, ω, and θ. If self-excaon s oo weak, ha s ω/µ 1 and θ 1, hen he rae (3) wll always say near µ and he clusers wll be vague and wdespread. Hence he mehod wll gve almos unform dsrbuons of weghs, and choosng he par wh he bgges wegh wll be equvalen o random guessng. On he oher hand, f self-excaon s very srong, ha s ω/µ 1 and θ 1, hen he clusers wll be sharp, he dsrbuon vecors wll be sparse, and choosng he par wh he bgges wegh wll gve a relable predcon. Fgure 7 confrms he above reasonng. Here we appled our mehod o a fullyconneced 6-agens nework, wh N = 400, n = 100, varyng he values of θ and τ = log 10 (ω/µ). For each dsrbuon vecor of weghs, we smply pcked he melne wh he hghes wegh and ploed average percenage of correc predcons obaned n hs way.

14 Reconsrucon of Mssng Daa Based on Temporal Paerns 14 θ 95% % % % τ 35% 20% Fgure 7. Dependence of he average percenage of correc predcons, obaned by choosng he par wh he hghes wegh for each dsrbuon vecor, on θ and τ = log 10 (ω/µ), for a fully-conneced 6-agens nework, wh N = 400, n = Concluson Realaory gang volence s a large problem n many meropolan areas around he globe, and o cural such volence, law enforcemen agences need o know who he parcpans were n a gven alercaon. We have shown ha, under he assumpons ha realaory volence on a gang nework follows a Hawkes process of he form (3), ncomplee daa on he parcpans of he offenses can be reconsruced usng a compuaonally effecve algorhm ha maxmzes an energy funconal under a se of consrans - mehod (8). Moreover, when focusng on he lkelhood rankngs of gangs for ncomplee evens, mehod (8) seems o perform on par wh a more probablybased algorhm (4) ha s oo complex o use on realscally szed daases. Fnally, we have shown ha he performance of our mehod s deeply conneced o he parameers of he Hawkes process n queson, and n ceran regmes may predc he correc parcpans wh very hgh lkelhood. Of course, here are ssues o overcome f our mehod s o be used on acual gang volence daa, raher han on smulaed evens. Frsly, for real daases, he parameers of he process mus be esmaed from he evens, raher han beng known a pror. One could magne accomplshng hs n an erave way: use he complee evens o esmae parameers, use hese parameers o esmae parcpans n unknown evens, hen use hese esmaes o re-esmae he parameers, connung he cycle unl convergence (f convergence s ndeed obaned). To mplemen hs, however, one mus choose how o use he esmaed parcpans of evens when reesmang he neracon parameers, somehng ha s no enrely clear gven ha our esmaes of he parcpans are no probables. Secondly, n real daases one mus be concerned wh sysemac devaon beween he daa and acual occurrences. Ceran ypes of gang volence may be chroncally under-repored n ways ha wll skew he deecon of self-excaon or cause evens o be allocaed n an mproper way. A horough undersandng of how

15 Reconsrucon of Mssng Daa Based on Temporal Paerns 15 hs mgh affec our esmaes should be had before rusng he resuls compleely. Acknowledgmens Ths work was suppored by NSF gran DMS , ARO gran MA, ONR gran N , and AFOSR MURI gran FA References [1] Schafer J L and Graham J W 2002 Mssng daa: our vew of he sae of he ar Psychologcal Mehods [2] Kossnes G 2006 Effecs of mssng daa n socal neworks Socal Neworks [3] Husman M 2009 Impuaon of Mssng Nework Daa: Some Smple Procedures Journal of Socal Srucure 10 [4] Hoff P D, Rafery A E and Handcock M S 2002 Laen Space Approaches o Socal Nework Analyss Journal of he Amercan Sascal Assocaon [5] Hoff P D 2009 Mulplcave laen facor models for descrpon and predcon of socal neworks Compuaonal and Mahemacal Organzaon Theory [6] Husman M and Seglch C E G 2008 Treamen of non-response n longudnal nework sudes Socal Neworks [7] Bur R S 1987 A noe on mssng nework daa n he general socal survey Socal Neworks [8] Ta George 2003 Reducng gun volence: resuls from an nervenon n Eas Los Angeles RAND Corporaon, Sana Monca, CA [9] Egesdal M, Fahauer C, Loue K and Neuman J 2010 Sascal Modelng of Gang Volence n Los Angeles SUIRO [10] Hawkes A G 1971 Specra of some self-excng and muually excng pon processes Bomerka [11] Hawkes A G and Oakes D 1974 A cluser process represenaon of a self-excng process Journal of Appled Probably [12] Ogaa Y 1988 Space-me pon process models for earhquake occurrences Ann. Ins. Sas. Mah [13] Zhuang J, Ogaa Y and Vere-Jones D 2002 Sochasc Decluserng of Space-Tme Earhquake Occurences JASA [14] Shor M B, D Orsogna M R, Branngham P J and Ta G E 2009 Measurng and modelng repea and near-repea burglary effecs Journal of Quanave Crmnology [15] Mohler G O, Shor M B, Branngham P J, Schoenberg F P and Ta G E 2010 Self-excng pon process modelng of crme JASA [16] Lews E, Mohler G, Branngham P J, Berozz A 2010 Self-Excng Pon Process Models of Insurgency n Iraq [17] Shor M, D Orsogna M, Pasour V, Ta G, Branngham P, Berozz A and Chayes L 2008 A sascal model of crmnal behavor Mahemacal Models and Mehods n Appled Scences