Algorithms for Mining the Evolution of Conserved Relational States in Dynamic Networks

Transcription

1 Algorithms for Mining the Evolution of Conserved Reltionl Sttes in Dnmi Networks Rewn Ahmed, George Krpis Deprtment of Computer Siene & Engineering Universit of Minnesot Minnepolis, MN Emil: Astrt Dnmi networks hve reentl eing reognied s powerful strtion to model nd represent the temporl hnges nd dnmi spets of the dt underling mn omple sstems. Signifint insights regrding the stle reltionl ptterns mong the entities n e gined nling temporl evolution of the omple entit reltions. This n help identif the trnsitions from one onserved stte to the net nd m provide evidene to the eistene of eternl ftors tht re responsile for hnging the stle reltionl ptterns in these networks. This pper presents new dt mining method tht nles the time-persistent reltions or sttes etween the entities of the dnmi networks nd ptures ll miml non-redundnt evolution pths of the stle reltionl sttes. Eperimentl results sed on multiple dtsets from rel world pplitions show tht the method is effiient nd slle. Inde Terms Dnmi network; reltionl stte; evolution; I. INTRODUCTION As the pit to ehnge nd store informtion hs sored, so hs the mount nd diversit of ville dt. To represent the reltions etween vrious entities in diverse pplitions nd to pture the temporl hnges nd dnmi spets of the underling dt, dnmi networks hve een used s generi model due to its fleiilit nd vililit of theoretil nd pplied tools for effiient nlsis. Emples of some widel studied networks inludes the friendnetworks of populr soil networking sites like Feook, the Enron emil network, o-uthorship nd ittion networks, nd protein-protein intertion networks. Anlsis of temporl spets of the entit reltions in these networks n provide signifint insight out the onserved reltionl ptterns nd their evolution over time. Considerle effort hs een mde towrds the development of effiient methods to nle nd etrt useful informtion from stti networks [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. On the other hnd, the importne of dnmi networks hs een reognied onl reentl; thus, the set of methods tht re urrentl ville for their nlsis is onsiderl less developed thn for stti networks. Nevertheless, in reent ers onsiderle od of reserh hs emerged on methods for finding ptterns [17], lustering [18], hrteriing network evolution rules [19], deteting relted liques [20], [21], nd finding sugrph susequenes [22] in dnmi networks. Although the eisting tehniques n detet the frequent ptterns in dnmi network or trk relted ptterns over time, the re not designed to identif stle reltionl ptterns nd do not fous on trking the hnges of these onserved reltionl ptterns over time. Our ontriution in this pper is two folds. Firstl, we introdue new lss of ptterns referred s the evolving indued reltionl sttes tht re designed to nle the time-persistent reltions or sttes etween the entities of the dnmi networks. Seondl, we present n lgorithm to effiientl mine ll miml non-redundnt evolution pths of the stle reltionl sttes of dnmi network. We eperimentll evlute our lgorithm using three rel world dtsets. First, we evlute the performne nd slilit of the lgorithm on lrge ptent ittion network vring different input prmeters. Seond, we investigte some disovered evolving indued reltionl sttes from trde network, n emil ommunition network, nd ptent ittion network nd provide qulittive nlsis of the informtion ptured in them. II. DEFINITIONS AND NOTATION A grph G = (V, E, L[E]) is omposed of set of nodes V modeling the entities of the network, set of edges E modeling the reltions etween these entities, nd set of edge lels L[E] modeling the tpe of the reltions ( E = L[E] ). The reltions etween entities n either hve diretion or not, leding to direted or undireted edges. An indued sugrph G = (V, E, L[E ]) of G = (V, E, L[E]) is grph suh tht V V, E E nd (u, v) E suh tht v V nd u V, (u, v) E. For the rest of the disussion, n referenes to n indued sugrph will ssume it is onneted indued sugrph. A dnmi network N = G 1, G 2,..., G T is modeled s finite sequene of grphs, where eh G t is grph desriing the stte of the sstem t disrete time intervl t. The term snpshot will e used to refer to eh of the grphs in the sequene. Snpshots re ssumed to ontin the sme set of nodes, whih will lso e referred to s the nodes of N nd denoted V (N ). When nodes pper or dispper over time, the set of nodes of eh snpshot is the union of ll the nodes over ll snpshots. Also, the nodes ross the different snpshots re numered onsistentl, so tht the ith node of G k (1 k T ) will lws orrespond to the sme ith node of N.

2 G1 Fig. 1. G2 Emples of reltionl sttes. Due to its dnmi nture, the edges in N n over time pper, dispper, hnge lel nd/or diretion. To pture the sequene of snpshots for whih n edge eists in onsistent stte, we define the spn sequene of the edge s the sequene of miml-length time intervls in whih n edge is present in onsistent stte. An edge (u, v) is in onsistent stte over miml time intervl s:e if it is present in ll snpshots G s,..., G e with the sme lel nd diretion nd it is different in oth G s 1 nd G e+1 (ssuming s > 1 nd e < T ). The spn sequene of n edge will e desried sequene of time intervls of the form s 1 :e 1,..., s l :e l, where s i e i nd e i s i+1. In ddition, we define the spn sequene of verte s the sequene of miml-length time intervls in whih the verte hs t lest one edge inident on it. The spn sequene of the verte will e represented s sequene of time intervls in similr fshion tht of n edge. A persistent dnmi network N φ is derived from dnmi network N removing ll the edges from the snpshots of N tht do not our in onsistent stte in t lest φ onseutive snpshots, where 1 φ T. It n e seen tht N φ n e derived from N removing from the spn sequene of eh edge ll the intervls whose length is less thn φ. An indued sugrph tht involves the sme set of verties nd whose edges nd their lels remin onserved ross sequene of snpshots will e referred to s the indued reltionl stte (IRS). The IRS definition is illustrted in Fig. 1. The three-verte indued sugrph onsisting of the drk shded nodes tht re onneted vi the direted edges orresponds to n indued reltionl stte s it remins onserved in the three onseutive snpshots. The ke ttriute of n IRS is tht the set of verties nd edges tht ompose the indued sugrph must remin the sme in the onseutive snpshots. From this definition we see tht n IRS orresponds to time-onserved pttern of reltions mong fied set of entities (i.e., nodes) nd s suh n e thought s orresponding to stle reltionl pttern. An IRS S i will e denoted the tuple S i = (V i, s i :e i ), where V i is the set of verties of the indued sugrph tht persists from snpshot G si to snpshot G ei (s i e i ) nd it does not persist in G si 1 nd G ei+1 (ssuming s i > 1 nd E i < T ). We will refer to the time intervl s i :e i s the spn of S i, nd to the set of onseutive snpshots G si,..., G ei s its supporting set. B its definition, the spn of n IRS nd the length of its supporting set re miml. Note tht for the rest of the pper, n referenes to susequenes of snpshots will e ssumed to e onseutive. The indued sugrph orresponding to n IRS S i will e denoted s g(s i ). A snpshot G t supports n indued reltionl stte S i, if g(s i ) is n indued sugrph of G t. G3 III. EVOLVING INDUCED RELATIONAL STATE (EIRS) An emple of the tpe of evolving ptterns in dnmi networks tht the work in this pper is designed to identif is illustrted in Fig. 2. This figure shows hpothetil dnmi network onsisting of 14 onseutive snpshots, eh modeling the nnul reltions mong set of entities. The four onseutive snpshots for ers 1990 through 1993 show n indued reltionl stte S 1 tht onsists of nodes {,, e, f}. This stte ws evolved to the indued reltionl stte S 2 tht ours in ers tht ontins nodes {,, d, e, h}. Finll, in ers , the indued reltionl stte S 2 ws further evolved to the indued reltionl stte S 3 tht ontins the sme set of nodes ut hs different set of reltions. Note tht even though the sets of nodes involved in S 1 nd S 2 re different, there is high degree of overlp etween them. Moreover, the trnsition from S 1 to S 2 did not hppen in onseutive ers, ut there ws one er gp, s the snpshot for 1994 does not ontin either S 1 or S 2. Suh sequene of indued reltionl sttes S 1 S 2 S 3 represents n instne of wht we refer to s n evolving indued reltionl stte (EIRS) nd represents the tpes of ptterns tht the work in this pper is designed to identif. EIRSs identif entities whose reltions trnsition through sequene of time-persistent reltionl ptterns nd s suh n provide evidene of the eistene of eternl ftors responsile for these reltionl hnges. For emple, onsider dnmi network tht ptures the trding ptterns etween set of entities. The nodes in this trding network model the trding entities (e.g., ountries, sttes, usinesses, individuls) nd the direted edges model the trnsfer of goods nd their tpes from one entit to nother. An EIRS in this trding network n potentill identif how the trding ptterns hnge over time (e.g., ddition/deletion of edges or inlusion of new trding prtners) signlling the eistene of signifint eonomi, politil, nd environmentl ftors tht drive suh hnges (see VI-C for some emples of suh ptterns in inter-ountr trding network). Similrl, in dnmi network tht ptures the nnul ittion struture of U.S. Ptents (or other sientifi pulitions), EIRSs n identif how stle knowledge trnsfer or knowledge shring su-networks mong different siene res hve evolved nd thus filitte the identifition of trnsformtive siene developments tht hnged the stte of these su-networks. The forml definition of n EIRS is s follows. Definition 1 (Evolving Indued Reltionl Stte) Given dnmi network N ontining T snpshots, vlue φ (1 φ T ), nd vlue β (0 < β 1), n evolving indued reltionl stte of length m is sequene of indued reltionl sttes S 1, S 2,..., S m tht stisfies the following onstrints: (i) the supporting set of eh indued reltionl stte S i ontins t lest φ onseutive snpshots in the persistent dnmi network N φ of N, (ii) for eh 1 i < m, the first snpshot in S i+1 s supporting set follows the lst snpshot in S i s supporting set,

3 Fig. 2. An emple of n evolving reltionl stte. (iii) for eh 1 i < m, g(si ) is different from g(si+1 ), nd (iv) for eh 1 i < m, Vi Vi+1 / Vi Vi+1 β. The vlue φ, referred to s the support of the EIRS, is used to pture the requirement tht eh indued reltionl stte ours in suffiientl lrge numer of onseutive snpshots nd s suh it represents set of reltions mong the entities involved tht re stle. The vlue β, referred to s the interstte similrit, is used to enfore the minimum verte-level similrit etween the seleted reltionl sttes. This ensures tht the EIRS ptures the reltionl trnsitions of onsistent set of verties ut t the sme time llows for the inlusion of new verties nd/or the elimintion of eisting verties, if the re required to desrie the new reltionl stte. The third onstrint in the ove definition is used to eliminte EIRSs tht ontin onseutive IRSs with identil indued sugrphs. This is motivted our desire to find EIRSs tht pture hnges in the time-persistent reltions. However, the ove definition llows for the sme indued sugrph to our multiple times in the sme EIRS, s long s these ourrenes do not hppen one fter the other. An importnt spet of the definition of n EIRS is tht it is defined with respet to the persistent dnmi network N φ of N nd not N itself. This is euse we re interested in finding how the persistent reltions mong set of entities hve hnged over time nd s suh we first eliminte the set of reltions tht pper for short period of time. Given the ove definition, the work in this pper is designed to develop effiient lgorithms for solving the following prolem: pplition to fous on IRSs of meningful sie, wheres the minimum onstrint on the EIRS length is introdued in order to eliminte short pths. IV. F INDING E VOLVING I NDUCED R ELATIONAL S TATES The lgorithm tht we developed for finding ll miml EIRSs (Prolem 1) follows two-step pproh. In the first step, the dnmi network N is trnsformed into its persistent dnmi network N φ nd reursive enumertion lgorithm is used to identif ll the IRSs S whose supporting set is t lest φ in N φ. The N to N φ trnsformtion is done removing spns tht re less thn φ from eh edge s spn sequene nd then removing the edges with empt spn sequenes. In the seond step, the set of IRSs re mined in order to identif their miml non-redundnt sequenes tht stisf the onstrints of EIRS s definition. A. Step 1: Mining of Indued Reltionl Sttes The lgorithm tht we developed to mine ll indued reltionl sttes is sed on reursive pproh to enumerte ll (onneted) indued sugrphs of grph tht stisf minimum nd mimum sie onstrints. In the rest of this setion we first desrie the reursive lgorithm to enumerte ll indued sugrphs in simple grph nd then desrie how we modified this pproh to mine the indued reltionl sttes in dnmi network. The enumertion lgorithms ws inspired the reursive lgorithm to enumerte ll spnning Prolem 1 (Miml Evolving Indued Reltionl Stte Mining) trees [23]. Our disussion initill ssumes tht the grph Given dnmi network N ontining T snpshots, user is undireted nd the neessr modifitions tht ppl for defined support φ (1 φ T ), inter-stte similrit β direted grphs re desried fterwrds. Also, n referenes (0 < β 1), minimum sie of kmin nd mimum sie to indued sugrphs ssumes onneted indued sugrphs. of km verties per IRS, nd minimum EIRS length mmin, 1) Indued Sugrph Enumertion: Given grph G = find ll EIRSs suh tht no EIRS is susequene of nother (V, E, L[E]), let Gi = (Vi, Ei, L[Ei ]) e n indued sugrph EIRS. of G (Vi n lso e empt), Vf e suset of verties of Sine the set of miml EIRSs ontins within ll non- V stisfing Vi Vf =, nd let F (Vi, Vf ) e the set of miml EIRSs, the ove prolem will produe suint indued sugrphs of G tht ontin Vi nd ero or more set of results. Also, the minimum nd mimum onstrints verties from Vf. Given these definitions, the omplete set on the sie of the IRSs involved is introdued to llow n of indued sugrphs of G is given F (, V ) \. The set

4 F (V i, V f ) n e omputed using the reurrene reltion. F (V i, V f ) = 8 V i, >< F ({u}, V f \ u) F (, V f \ u), where u V f if sgdj(v i, V f ) = or V f = if V i = V f F (V >: i {u}, V f \ u) F (V i, V f \ u), otherwise, where u sgdj(v i, V f ) (1) where sgdj(v i, V f ) (sugrph-djent) denotes the verties in V f tht re djent to t lest one of the verties in V i. To show tht Eqution 1 orretl genertes the omplete set of indued sugrphs, is suffiient to onsider the three onditions of the reurrene reltion. The first ondition, whih orresponds to the initil ondition of the reurrene reltion, overs the situtions in whih either (i) none of the verties in V f re djent to n of the verties in V i nd s suh V i is the onl indued sugrphs tht n e generted, or (ii) V f is empt nd s suh V i nnot e etended further. The seond ondition, whih overs the sitution in whih V i is empt nd V f is not empt, deomposes F (, V f ) s the union of two sets of indued sugrphs sed on n ritrril seleted verte u V f. The first is the set of indued sugrphs tht ontin verte u (orresponding to F ({u}, V f \ u)) nd the seond is the set of indued sugrphs tht do not ontin u (orresponding to F (, V f \ u)). Sine n of the indued sugrphs in F (, V f ) will either ontin u or not ontin u, the ove deomposition overs ll possile ses nd it orretl genertes F (, V f ). Finll, the third ondition, whih orresponds to the generl se, deomposes F (V i, V f ) s the union of two sets of indued sugrphs sed on n ritrril seleted verte u V f tht is djent to t lest one verte in V i. The first is the set of indued sugrphs tht ontin V i nd u (orresponding to F (V i {u}, V f \ u)) nd the seond is the set of indued sugrphs tht ontin V i ut not u (orresponding to F (V i, V f \u)). Similrl to the seond ondition, this deomposition overs ll the ses with respet to u nd it orretl genertes F (V i, V f ). Also the requirement tht u sgdj(v i, V f ) ensures tht this ondition enumertes onl the onneted indued sugrphs. Sine eh reursive ll in Eqution 1 removes verte from V f, the reurrene reltion will terminte due to the first ondition. Finll, sine the three onditions in Eqution 1 over ll possile ses, the overll reurrene reltion is orret. In ddition to orretness, it n e seen tht the reurrene reltion of Eqution 1 does not hve n overlpping suprolems, nd s suh, eh indued sugrph of F (V i, V f ) is generted onl one, leding to n effiient pproh for generting F (V i, V f ). Constrints on the minimum nd mimum sie of the indued sugrphs n e esil inorported in Eqution 1 returning in the first ondition when V i is less thn the minimum sie nd not performing the reursive Given onneted (indued) sugrph g, it n e grown dding one verte t time while still mintining onnetivit; e.g., MST of g (whih eists due to its onnetivit) n e used to guide the order whih verties re dded. e e Si,<1:7> Fig. 3. () w,<2:4> w Sj,<2:4> <2:4> e Sm,<1:3> e k k,<1:3,5:9> <1:3,5:9> Si,<1:7> h () e h Sn,5:7 Adding verte to n indued reltionl stte. eplortion for other two ses. 2) Indued Reltionl Stte Enumertion: There re two ke hllenges in etending the indued sugrph enumertion pproh of Eqution 1 in order to enumerte the IRSs in dnmi network. First, the ddition of verte to n IRS is different from dding verte to n indued sugrph s it n n result in multiple IRSs depending on the overlpping spns etween the verte eing dded nd the originl IRS. Consider n IRS S i = (V i, s i :e i ), set of verties V f suh tht V i V f =, nd verte v V f tht is djent to t lest one of the verties in V i. If v s spn sequene ontins multiple spns tht hve overlps greter thn or equl to φ with S i s spn, then the inlusion of v leds to multiple IRSs, eh supported different disjoint spns. Fig. 3 illustrtes the verte ddition proess during n IRS epnsion. Fig. 3() shows simple se of verte ddition where n IRS S 1 = ({,, e}, 1:7) is epnded dding djent verte hving single spn of 2:4. The resultnt IRS is S 2 = ({,, e, }, 2:4) tht ontins ll the verties nd onl the overlpping spn of S 1 nd. Fig. 3() shows more omple se where the verte h tht is dded to S 1 hs the spn sequene of 1:3, 5:9. In this se, the overlpping spns 1:3 nd 5:7 form two seprte IRSs S 3 = ({,, e, h}, 1:3) nd S 4 = ({,, e, h}, 5:7), eh of whih needs to e onsidered for future epnsions in order to disover the omplete set of IRSs. Seond, the onept of removing verte from V f used in Eqution 1 to deompose the set of indued sugrphs needs to e re-visited so tht to ount for the temporl nture of dnmi networks. Filure to do so, will led to n IRS disover lgorithm tht will not disover the omplete set of IRSs nd the set of IRSs tht it disovers will e different sed on the order tht it hooses to dd verties in the IRS under onsidertion. This is illustrted in the emple of Fig. 4. The IRS S 1 = ({, }, 0:12) is epnded to S 2 = ({,, }, 1:5) dding the djent verte. In terms of Eqution 1, this orresponds to the third ondition nd leds to the reursive lls of F ({,, }, V f \) nd F ({, }, V f \) (i.e., epnd S 2 nd epnd S 1 ). It is es to see tht the set of IRSs tht will e generted from these reursions will not ontin S 4 = ({,,, d}, 6:11), sine it n onl e generted from S 2 ut its spn does not overlp with S 4 s k h

5 <1:11>,<1:5>,<6:11> v,<6:11> S1,<0:12> d <6:11> <6:11>,<1:5>,<6:11> v,<6:11> S1,<0:12> d <6:11> Fig. 4.,<6:11> v,<6:11> S1,<0:12> <6:11>,<1:5> v,<6:11> d S3,<6:11> d <6:11> Updting spn sequene of verte.,<1:5> S4,<6:11> spn. One the other hnd, if S 1 is initill epnded dding verte d, resulting in the IRS S 3 = ({,, d}, 6:11), then the reursive lls of F ({,, d}, V f \d) nd F ({, }, V f \d) will generte S 4 nd S 2, respetivel. Thus, sed on the order whih verties re seleted nd inluded in n IRS, some IRSs m e missed. Moreover, different verte inlusion orders n potentill miss different IRSs. To ddress oth of these issues the lgorithm tht we developed for enumerting the omplete set of IRSs utilies reursive deomposition pproh tht etends Eqution 1 utiliing two ke onepts. The first is the notion of the set of verte-spn tuples tht n e used to grow given IRS. Formll, given n IRS S i = (V i, s i :e i ) nd set of verties V f in N φ with V i V f =, the irsdj(s i, V f ) is the set of verte-spn tuples of the form (u, s uj :e uj ) suh tht u V f nd (V i {u}, s uj :e uj ) is n IRS whose spn is t lest φ. Note tht irsdj(s i, V f ) n ontin multiple tuples for the sme verte if tht verte n etend S i in multiple ws (eh hving different spn nd possil n indued sugrph with different sets of edges). The tuples in irsdj(s i, V f ) represent possile etensions of S i nd sine verte n our multiple times, it llows for the genertion of IRS with the sme set of verties ut different spns (ddressing the first hllenge). The seond is the notion of verte-spn deletion, whih is used to eliminte the order dependen desried erlier nd generte the omplete set of IRSs. The ke ide is when tuple (u, s uj :e uj ) is dded into S i, insted of removing u from V f, onl remove the spn s uj :e uj from u s spn sequene in V f. Verte u will onl e removed from V f iff fter the removl of s uj :e uj its spn sequene eomes empt or the remining spns hve lengths tht re smller thn φ. Formll, given V f, verte u V f, nd verte-spn tuple (u, s uj :e uj ), the spn-deletion opertion, denoted V f (u, s uj :e uj ), updtes the spn sequene of u removing the s uj :e uj spn from its spn sequene, eliminting n of its spns tht eome shorter thn φ, nd eliminting u if its updted spn sequene eomes empt. The spn-deletion opertion is the nlogous opertion to verte removl of Eqution 1. To illustrte how this opertion n ddress the seond hllenge, onsider gin the emple of Fig. 4. One is dded into S 1, the spn tht it used (i.e., 1:5) will e deleted from its spn sequene, resulting in new spn sequene ontining 6:11. Now the d v reursive ll orresponding to F ({, }, V f (, 1:5)) will e le to identif S 4. Given the ove definitions, the reursive pproh for enumerting the omplete set of IRSs n now e formll defined. Let S i = (V i, s i :e i ) e n IRS, V f set of verties in N φ with their orresponding spn-sequenes in N φ suh tht V i V f =, nd H(S i, V f ) e the set of IRSs tht (i) ontin V i nd ero or more verties from V f nd (ii) their spn is su-spn of s i :e i. Given the ove, the omplete set of IRSs in N φ is given H((, 1:T ), V (N φ )) \. The reurrene reltion for H(S i, V f ) is: H(S i, V f ) = 8 S i, if irsdj(s i, V f ) = or V f = H(({u}, s u :e u), V f (u, s u :e u)) >< H((, 1:T ), V f (u, s u :e u)), if V i = V f where u V f nd s u :e u is spn of u H((V i {u}, s u :e u), V f (u, s u :e u)) >: H(S i, V f (u, s u :e u)), otherwise. where (u, s u :e u) irsdj(s i, V f ) (2) The ove reurrene reltion shres the sme overll struture with the orresponding reurrene reltion for enumerting the indued sugrphs (Eqution 1) nd its orretness n e shown in w similr to tht used for Eqution 1. Due to spe limittions, we omit the omplete proof of Eqution 2, nd onl fous on disussing the third ondition, whih represents the generl se. This ondition deomposes H(S i, V f ) s the union of two sets of IRSs sed on n ritrril seleted verte-spn tuple (u, s u :e u ) irsdj(s i, V f ). Sine the spn s u :e u is miml overlpping spn etween u nd S i, the new IRS ontining verties (V i {u} hs the spn of s u :e u ). With respet to verte-spn tuple (u, s u :e u ), the set of IRSs in H(S i, V f ) n elong to one of the following three groups: (i) the set of IRSs tht ontin u nd hve spn tht is su-spn of s u :e u ; (ii) the set of IRSs tht ontin u nd hve spn tht is disjoint with s u :e u, nd (iii) the set of IRSs tht do not ontin u. The H((V i {u}, s u :e u ), V f (u, s u :e u )) prt of the third ondition genertes (i), wheres the H(S i, V f (u, s u :e u ) prt genertes (ii) nd (iii). Wht is missing from the ove groups is the group orresponding to the set of IRSs tht ontin u nd hve spn tht prtill overlps with s u :e u. The lim is tht this nnot hppen. Consider n IRS S j = (V j, s j :e j ) H(S i, V f ) tht ontin u nd without loss of generlit, ssume tht s u < s j < e u < e j. Sine we re deling with indued sugrphs nd stle topologies (i.e., from the definition of n IRS), the onnetivit of u to the verties in V i remins the sme during the spn of s u :e u nd lso during the spn of s j :e j, whih mens tht the onnetivit of u to the verties in V i remins the sme during the entire spn of s u :e j. This is ontrdition, sine (u, s u :e u ) irsdj(s i, V f ) nd s suh is miml length spn of stle reltions due to the ft tht (V i {u}, s u :e u ) The su-spn of spn orresponds to time intervl tht is either identil to the spn or is ontined within it.

6 Algorithm 1 mdfs(u, t, d[], p) 1: /* u is the urrent node */ 2: /* t is the urrent time */ 3: /* d[] is the disover rr */ 4: /* p is the urrent pth */ 5: d[u] = t++ 6: push u into p 7: if dj(u) = nd p > minimum EIRS-length then 8: reord p 9: else 10: for eh node v in (dj(u) sorted in inresing end-time order) do 11: if dt[v] < d[u] then 12: mdfs(v, t, d, p) 13: pop p is n IRS nd the spn of n IRS is miml. Thus, the two ses of the third ondition in Eqution 2 over ll possile ses nd it orretl generte the omplete set of IRSs. Also, sine eh reursive ll modifies t lest the set V f, none of the reursive lls led to overlpping suprolems, ensuring tht eh IRS is onl generted one. 3) Hndling Direted Edges: To hndle direted edges, we onsider eh diretion of n edge seprtel, suh tht direted edge is listed seprtel from. The diretion of n edge is stored s prt of the lel long with the spn sequene of tht edge. The diretion of n edge t ertin spn is oded s 0, 1 or 2 to represent, or. Note tht the ordering of the verties in n edge (, ) re stored in inresing verte-numer order (i.e, < ). Using the ove representtion of n edge, we determine the diretion of ll the edges of n IRS during its spn durtion. B. Step 2: Mining of Miml Evolution Pths The lgorithm tht we developed to identif the sequene of IRSs tht orrespond to the miml EIRSs is sed on modified DFS trversl of direted li grph tht is referred to s the indued reltionl stte grph nd will e denoted G RS. G RS ontins node for eh of the disovered IRSs nd root node r. For eh pir of IRSs S i = (V i, s i :e i ) nd S j = (V j, s j :e j ), G RS ontins direted edge from the node orresponding to S i to the node orresponding to S j iff V i V j, V i V j / V i V j β nd e i < s j (i.e., onstrints (ii) (iv) of Definition 1). The modified DFS lgorithm whih we will refer to s mdfs, uses onl disover rr d[] to reord the disover times of eh node, whih re initill set to 0. The DFS trversl strts from the root node nd proeeds to visit the rest of the nodes. The pseudoode is shown in Algorithm 1. The mdfs lgorithm lso keeps trk of the urrent pth from the root to the node tht it is urrentl t. If tht node hs no outgoing edges, then it outputs tht pth. The sequene of the reltionl sttes orresponding to the nodes of tht pth (the root node is eluded), represent n EIRS. The ke differene etween mdfs nd the stndrd DFS lgorithm is the method for seleting whih node to visit net (line 4). Given node u, the mdfs lgorithm selets mong its djent nodes the node v tht hs the erliest end-time nd d[v] < d[u]. The seletion of the erliest end-time is done to llow mdfs to find miml pths first, where the seletion of v tht stisfies d[v] < d[u] is done to eliminte forwrd edges ut still llow for the eplortion of ross edges. The elimintion of forwrd edges is done so tht to eliminte pths tht re supths of previousl identified longer pths (nd s suh will led to non-miml EIRSs). The inlusion of ross-edges is done so tht to llow the mdfs lgorithm to find the omplete set of miml EIRSs. Due to spe onstrints, we do not provide the forml proof tht the set of pths generted mdfs(r, 0, d, ) represent the omplete set of miml EIRSs tht we re fter. However, we hope tht the disussion ove provides evidene s to the orretness of the lim. The disussion so fr ssumes tht G RS hs een full mterilied. However, this n e epensive s it requires pirwise omprisons etween lrge numer of IRSs in order to determine if the stisf the onstrints (ii) (iv) of Definition 1. For this reson, our lgorithm strts with the mdfs trversl from the root nd mterilies the portions of G RS tht it needs during the trversl. This llows it to redue the rther epensive omputtions ssoited with some of the onstrints not hving to visit forwrd edges. Moreover it utilies the minimum EIRS length onstrint to further prune the prts of G RS tht it needs to generte. For given node u, the mdfs lgorithm needs the djent node of u tht hs the erliest end-time. Let e u e the endtime of node u. Sine node (i.e., n IRS) is required to hve t lest spn of length φ, we strt u s djent node serh mong the nodes in G RS tht hve the end-time of e k = e u + φ. Aording to onstrint (iv) of Definition 1, ertin minimum threshold of similrit is desired etween two IRSs of n EIRS. Thus, it is suffiient to ompre u with onl those nodes tht hve t lest ommon verte with u. We inde the nodes of G RS sed on the verties so tht the similr nodes of u n e essed looking up ll the nodes tht hve t lest one verte in ommon with u. If node v is similr to u nd d[v] < d[u], we dd the node to the djen list. If the serh fils to detet n djent node, we initite nother serh looking for nodes tht hve n end-time of e k + 1 nd ontinue suh inrementl serh until n djent node of u is found or ll possile end-times hve een eplored. V. RELATED WORK Over the lst ten ers, onsiderle reserh effort hs een devoted to developing lgorithms to find ptterns in stti grphs nd networks. This reserh hs resulted in the development of lgorithms for finding different tpes of ptterns suh s pths [24], [25], trees [6], [7], indued sugrphs [8], ritrril onneted sugrphs [26], [10], nd vrious tpes of liques [27], [28], [29]. While most of these methods hve een developed for mining dtses ontining reltivel smll grphs, lgorithms hve lso een developed to identif sugrphs with lrge numer of emeddings in single lrge grph (i.e., stti network) [11]. Sine dnmi networks hve onl reentl emerged s n importnt reserh re, the prolem of formll defining

7 the onserved reltionl sttes nd developing lgorithms for identifing their evolution ptterns hve not een well studied. Desikn et. l. [30] nled the importne of mining temporll evolving We grphs, ut did not provide n lgorithmi solution for deteting the stle ptterns or their evolution. Borgwrdt et. l. [17], [31] introdued the notion of the dnmi sugrph, whih etends the trditionl notion of the sugrph to inlude the sequene of sugrphs tht eist in onseutive sequene of snpshots nd developed lgorithms to identif the set of dnmi sugrphs tht our frequentl in dnmi network. Even though this work provides similr definition to the stle reltionl sttes, it uses heuristi sed pproh to detet the frequent ptterns nd it does not pture the evolution of those ptterns over time. Jin et. l. [32] foused on the prolem of finding reurrent ptterns in dnmi networks in whih time series is ssoited with eh node nd introdued the notion of the trend motif, whih is onneted sugrph in whih eh node s time series ehiits onsistent trend over time intervl. Even though the trend motif formultion is generl, their work foused onl on developing lgorithms for finding frequentl ourring trend motifs tht show either n inresing or deresing trend. Berlingerio et. l. [19] lso provided n lgorithm to detet frequent sugrphs in time-evolving grphs for deriving grph-evolution rules tht stisf minimum onfidene onstrint. Although these evolution rules re used to hrterie the hnges in the overll evolving grph, it does not determine the evolution of the onserved frequent ptterns. Rordet [21] represented the frequent ptterns of grph s pseudo-liques nd proposed n lgorithm tht first mines eh grph snpshot of dnmi grph for lol ptterns nd then omines these with ptterns from previous snpshot sed on some onstrints to form evolving ptterns. Inokuhi et. l. [33], [22] solved similr prolem of finding frequent indued sugrph susequenes from grph sequene dt nd pturing the hnges of sugrph over the susequene. However, their pproh does not fous on determining stle indued sugrphs nd does not disover the evolution ptterns of the onserved reltionl sttes. A relted od of reserh hs investigted the tsk of identifing nd trking ptterns in iologil networks [34], [35], [31] nd evolving ommunities in soil networks [36], [37]. The prolem of evolutionr lustering [18], [38] is to find lusters in dnmi network in the form of snpshots, suh tht lusters t n given time groups similr entities while lso preserving the grouping of entities in pst snpshots. Even though these methods provide vlule insights on the evolution of dnmi networks, the nture of onserved ptterns nd their evolution foused in this pper is different from the generl evolution ddressed in other ppers. A. Dtsets VI. EXPERIMENTAL DESIGN & RESULTS We evluted our lgorithm using dtsets from ptent ittion network, trde network nd n emil ommunition network. The slilit of our lgorithm ws ssessed on TABLE I DYNAMIC NETWORK DATASETS. Dtset #Verties Avg. #Edeges Spn Ptent N Ptent N Ptent N Trde Enron #Verties denotes the totl numer of verties in the dnmi network. Avg. #Edeges denotes the verge numer of edges per snpshot in the dnmi network. Spn denotes the totl numer of snpshots in the dnmi network. the ptent ittion dtset (s it ws the lrgest) wheres ll three dtsets were used in the qulittive ssessment of the identified EIRSs. ) Ptents Cittion Network: This is ittion network derived from the United Sttes Ptent nd Trdemrk offie s (USPTO) iliogrphi informtion for the ptents grnted from 1976 to The nodes orrespond to the primr rt res ssoited with the ptents sed on USPTO s lssifition nd the edges orrespond to ggregted ittions etween rt res. For emple, if ptent A of rt re α ites ptent B of rt re β in er, then direted edge is dded from α to β in the grph representing er. Cittions tht produe self referenes re removed. The lssifition of USPTO rt res is hierrhil nd forms tree struture tht n e up to 16 levels deep. The ptents re ssigned the lsses orresponding to the lef nodes of the tree. In our eperiments, in order to otin rt res tht ontin suffiientl lrge numer of ptents, we rolled up the lssifition tree to the third level, nd ll the ptents underneth eh third-level node ws ssigned the lss of tht node. The snpshots of the dnmi network tht we reted orrespond to the ittion network of eh er, leding to dnmi network onsisting of = 34 snpshots. Sine the verties t eh snpshot n potentill e onneted to ll other verties, we pre-proessed eh snpshot in order to derive set of dnmi networks tht ontin the most importnt set of outgoing edges (i.e. referenes) from eh node. This is done s follows. For eh verte of eh snpshot, we first hoose the 20 most frequent edges. The frequen of n edge (, ) is defined s the numer of ptent-to-ptent referenes (P 1, P 2 ) suh tht P 1 s lss is nd P 2 s lss is. Then, for eh edge (, ) we lulte its lift (i.e., w(, ) = p( )\p()) to use s its weight. Bsed on these weights, we onstrut three dnmi network dtsets N2, N3 nd N4 seleting the highest weighted 2, 3 nd 4 edges for eh verte in eh snpshot of the network. The sie nd densit of the networks is presented in Tl. I. ) Trde Network: This is trde network tht models the erl eport nd import reltions of 192 ountries from 1948 to 2000 sed on the Epnded Trde nd GDP Dt [40]. The nodes model the trding ountries nd the diret edges model the eport or import tivit etween two ountries for ertin er. The snpshots of the dnmi network tht we reted orresponds to the trde network of eh er, leding to dnmi network onsisting of = 53 snpshots. If the eport mount from

8 TABLE II NETWORK DENSITY. Dtset #IRS #EIRS ETime PTime TotlTime N N N The numer of verties in n IRS is 4 to 8, the mimum llowed verte differene etween two suessive IRSs is 1, nd φ is 4. Run times re in seonds. #IRS denotes the numer of disovered IRSs. #EIRS denotes the numer of disovered EIRSs. Etime denotes the mount of time spent to enumerte IRSs. PTime denotes the mount of time spent for pth enumertion. TotlTime denotes the totl mount of time spent to determine ll EIRSs in the network. ountr A to ountr B in given er is more thn 10% of the totl eport mount of A nd totl import mount of B for tht er, direted edge A B is dded to tht er s trde grph. ) Emil Communition Network: This is ommunition network tht models the emil trffi etween emploees of the ENRON ompn [41]. The nodes model the emploees who re leled their rnk nd node id to uniquel identif two nodes with sme lel (i.e. multiple emploees with the sme rnk). The direted edges model the ommunitions etween the emploees. To represent the dtset s dnmi network, the entire time spn ws divided into 30 equl-sie intervls. If n emil ws sent from node to node t ertin time intervl, direted edge is dded to the grph representing tht time intervl. This representtion ontins 130 nodes nd out 90 edges per snpshot. B. Performne Results We evluted the performne nd slilit of our lgorithm for mining the miml EIRSs using the N2, N3, nd N4 dtsets from the ptent ittion network. Our evlution is designed to ssess how the densit of the networks nd the vrious prmeters ssoited with the EIRS definition impts the performne of the lgorithm. All eperiments re onduted on Linu luster with 6-ore Intel Xeon X7542 Westmere proessors t 2.66 GH. Note tht in order to etter ssess how the interstte similrit omponent in the definition of the EIRS impts the performne of the lgorithm in ll the eperiments presented in this setion, insted of using V i V j / V i V j s mesure of inter-stte similrit (onstrint (iv) of Definition 1), we used the numer of different verties etween V i nd V j s mesure of distne. This llows us to epliitl inrese/derese the ompleit of the mining prolem hnging the numer of different verties tht is llowed etween suessive IRSs. 1) Network Densit: The performne of the lgorithm for the three dtsets is shown in Tl. II. The dtsets N2, N3 nd N4 ontin the sme numer of verties 84152, ut their densit in terms of the numer of edges present in the network inreses 1.4 times from N2 to N3 nd 1.2 times from N3 to N4. The results in Tl. II show tht s the grph densit inreses the numer of EIRSs found inreses (i.e. the numer of EIRSs inresed from 2495 in N2 to in N3 to in N4). At the sme time, the totl runtime to disover TABLE III INTER-STATE SIMILARITY. ISDiff k min k m #IRS #EIRS ETime PTime TotlTime Dtset N4 is used for this eperiment nd φ = 5. ISDiff denotes the inter-stte distne pturing the mimum llowed verte differene etween two IRSs. k min denotes the minimum numer of verties llowed in n IRS. k m denotes the mimum numer of verties llowed in n IRS. Rest of the olumn lels re desried in Tl. II TABLE IV MINIMUM SPAN (φ) STUDY. φ QEdges #IRS #EIRS ETime PTime TotlTime Dtset N3 is used for this eperiment, the numer of verties in n IRS is 5 to 7, nd the mimum llowed verte differene etween two linked IRSs is 1. QEdges denotes the numer of edges in N φ, nd rest of the olumn lels re desried in Tl. II the EIRSs inreses from 13 seonds to proess N2 to 458 seonds for N3 nd seonds for N4. Even though the IRS enumertion step is mostl the time onsuming proess, s the numer of IRS inreses the time needed to trverse the IRS grph nd disover the EIRSs strts to inrese. For N4, out 84% of the totl runtime ws spent disovering the miml pths. For sprse grphs, IRS enumertion time domintes the omputtion. For denser grphs, the diretion grph uilding requires the most mount of omputing. 2) Inter-stte Distne: Tl. III shows the performne of the lgorithm for different vlues of the mimum llowed numer of different verties. The numer of different verties is vried from 1 to 3 for two different sets of IRSs (i.e., set of IRSs with 5 to 8 verties nd other set of IRSs with 6 to 8 verties). We oserved tht the numer of disovered EIRSs inreses s the mimum llowed verte differene inreses (i.e. the similrit threshold dereses). For the 5 8 set, the inrese in the mimum llowed verte differene from 1 to 3 uses n inrese in disovered EIRSs from 315 to However, s we inrese the mimum llowed verte differene, the EIRSs will strt ontining unrelted IRSs in their pth, sine the similrit threshold etween the IRSs re lower, whih m represent less interesting EIRS. The derese in similrit threshold lso inreses the totl runtime, the pth enumertion step tkes longer to proess more edges etween IRSs. 3) Minimum Spn: The performne of the lgorithm for different vlues of minimum spn (φ) is shown in Tl. IV. The vlue of φ represents the minimum length requirement for n indued reltionl stte to e in onsistent stte nd for this eperiments is in terms of ers. From these results, we oserved tht s the vlue of φ dereses, the numer of disovered EIRSs nd the runtime inreses. The vlue of φ ontrols the numer of edges tht n qulif to e prt of the N φ nd the lower the vlue of φ is the more numer of edges will qulif. In this se, we

9 TABLE V IRS SIZE STUDY ITA ITA UKG k min k m #IRS #EIRS ETime PTime TotlTime LUX GFR FRN GFR FRN GFR FRN GFR Dtset N4 is used for this eperiment, the mimum llowed verte differene etween two linked IRSs is 1 nd φ=4. The olumn lels re desried in Tl. II see tht the numer of qulified edges inrese from 5521 to for φ=5 nd φ=3. As the numer of qulified edges in N φ inrese, the numer of IRSs inreses nd resulting in disovering higher numer of EIRSs. Similrl, the runtime inrese s the vlue of φ dereses from 34 seonds to 6028 seonds for φ=5 nd φ=3, sine the lgorithm needs to proess more numer of IRSs to find reltions nd their evolution pths. This prmeter is n importnt ftor in finding EIRSs in different dtsets, sine the onserved stte of pttern is likel to e different depending on the tpe of the dt. 4) IRS Sie: We nled the performne of the lgorithm for different sies IRSs in Tl. V. The sie of n IRS is represented s the minimum nd mimum numer of verties llowed in n IRS. For emple, the sie of 5-8 mens tht n IRS n ontin minimum of 5 verties nd mimum of 8 verties. We oserve tht s the sie inreses, the numer of disovered EIRSs inreses. Sine lrger rnge in sie llows more numer of IRSs to e deteted, the hne of finding higher numer of EIRSs is inreses. In this eperiment, the numer of IRSs nd EIRSs found for sie 5 is nd 60. When the sie ws inresed to 5-8, the numer of IRSs inresed to nd in turn resulted with EIRSs. C. Qulittive Anlsis In this setion we present some of the EIRSs tht were disovered our lgorithm in order to illustrte the tpe of informtion tht n e etrted from the dnmi networks fousing on how stle reltions hnged over time. The EIRSs tht re presented orrespond to some of the EIRSs tht show the highest hnge etween the different IRS involved. In prtiulr, given n EIRS, we omputed the rtio of the totl numer of unique edges (i.e., reltions) in ll of its onstituent IRSs over the totl numer of edges in the sme IRSs. We refer to this quntit s the totl drift. Note tht this is just one of the mn ws tht n e used to ssign quntittive interestingness mesure to n EIRS nd other mesures n e derived looking t the nodes, reltionl inversions, lilit, et. In Fig. 5 we present n EIRS generted from the trde network pturing trde reltions etween some of the Europen ountries over 30 ers period nd the hosen φ=3. The totl drift for this EIRS is 12/18 = The EIRS minl ptures trde reltions etween Belgium, Netherlnds, Germn nd Frne. The other ountries, suh s Luemourg, Itl nd United Kingdom prtiipte for prtil period of time. Bsed on the illustrtion, initill (during 1963 to 1967) Belgium nd Netherlnds were strong trde prtners s the eported BEL NTH BEL NTH Fig. 5. An EIRSs pturing trde reltion etween EU ountries. The nodes in the figure re LUX=Luemourg, BEL=Belgium, GFR=Germn Federl Repuli, NTH=Netherlnds, ITA=Itl, nd UKG=United Kingdom. JPN USA TAW CA N USA JPN SAU TAW BEL USA JPN TAW CAN BEL JPN USA Fig. 6. An EIRSs pturing trde reltion of USA. The nodes in the figure re USA=United Sttes of Ameri, CAN=Cnd, JPN=Jpn, TAW=Tiwn, SAU=Sudi Ari, nd MEX=Meio. nd imported from eh other. The period shows tht the ountries were hevil trding etween eh other. B evluting the historil events, politil nd eonomi sitution of tht period, we ould find the use of higher trde tivit. The periods nd ptures how Frne s trde reltions with Belgium nd Germn eme one sided s Frne onl imported from those ountries. The use of suh hnges ould e tht Frne ws eporting to other ountries or Belgium nd Germn deided to import from some other ountries. In Fig. 6 we present nother EIRS generted from the trde network pturing stle trde reltion of USA with other ountries over 35 ers period. The totl drift for this EIRS is 7/16 = We notie tht USA nd Cnd hve strong trde reltions over long period of time. Even though the strong tie in trding seems ovious due to the geogrphil o-lotion of the ountries, it is interesting tht the lgorithm ould disover suh reltion from the historil dt. The EIRS lso ptures sted reltion etween USA nd Jpn. In Fig. 7 we present n EIRS generted from the emil ommunition network pturing emil ehnge ptterns mong group of emploees over period of time. The totl drift for this EIRS is 7/12 = Although it is diffiult to understnd the ommunition of the emploees without the messge ontent, the diretion of the ommunition n e found in this EIRS. We see tht [id:33] hd lws initited the onverstion nd ws ver tive in emil ommunition to hve stle reltions over ll the ptured periods. It is interesting to notie tht [id:146] is not present in lst period (25-28). One n investigte nd onfirm whether he/she ws repled or terminted nd the use for suh tions. In Fig. 8 we present n EIRS generted from the ptent DIR Fig DIR EMP DIR EMP 63 An EIRSs pturing Enron emil trffi pttern. CAN CAN 33 TAW 158 MEX

10 Fig. 8. An EIRSs pturing ptent lss reltions. The nodes in figure denote ptent lsses nd the legend ptures the USPTO definition of the lsses C1 to C5. ittion network pturing the evolution of the reltions etween some ptent lsses over 15 ers period. The totl drift for this EIRS is 7/14 = 0.5. Bsed on the illustrtion, the ptent lsses C2, C3 nd C4 were iting C1 during We n interpret tht s the ptents of lss C1 re erlier inventions nd lter the ptents of lss C2, C3, nd C4 used the ides found in the ptents elonging to C1. Over time the reltions hnged s lss C5 ppers in lter ers nd oth C1 nd C5 re iting eh other. It is possile tht ptents of lss C5 represent newer tehnolog tht ited erlier ptents of lss C1 s referene nd the newer ptent of lss C1 re using C5 s referene. This ptures omple reltionl dependene etween entities in dnmi network. Moreover, we lso oserved tht lss C4 disppered in the period of This ould indite tht the tehnolog introdued in the produts of lss C4 is no longer used. VII. CONCLUSION In this pper we presented n lgorithm for finding ll miml non-redundnt evolution pths of the indued reltionl sttes in dnmi network. This n e used to disover the trnsitions of the onserved reltionl sttes over time nd to etter understnd the use of suh hnges in the stle ptterns in dnmi network. Our eperimentl evlution on multiple rel world dtsets show tht the lgorithm is le to disover interesting evolution pths from ll dtsets nd n sle well to lrge nd dense dnmi networks. ACKNOWLEDGEMENT This work ws supported in prt NSF (IIS , OCI , nd IOS ) nd the DOE grnt USDOE/DE-SC nd the Digitl Tehnolog Center t the Universit of Minnesot. Aess to reserh nd omputing filities ws provided the Digitl Tehnolog Center nd the Minnesot Superomputing Institute. REFERENCES [1] D. Bod nd N. Ellison, Soil network sites: Definition, histor, nd sholrship, Journl of Computer-Medited Communition, vol. 13, no. 11, Otoer [2] A. Chpnond, M. S. Krishnmoorth, nd B. 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