Real Time Contextual Summarization of Highly Dynamic Data Streams

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1 Real Time Conexual Summarizaion of Highly Dynamic Daa Sreams Manoj K Agarwal Microsof Bing Search Technology Cener - India Hyderabad , India agarwalm@microsof.com Krihi Ramamriham Dep. of Compuer Science and Engineering IIT Bombay, India Mumbai , India krihi@cse.iib.ac.in ABSTRACT Microblogging sreams ypically conain informaion peraining o emerging real world evens. Due o he rapid pace of messages in hese daa sreams, shor message size and many concurren evens, i is ofen difficul for users o undersand he full conex behind an arriving message. Hence, users resor o he cumbersome ask of sifing hrough many messages o obain he full conex of he underlying even. To address his problem, we propose a novel noion Conexual Even Summary Threads and presen a echnique o exrac highly meaningful ye compac even summary hreads, capuring he complee conex of evens appearing in daa sream, in real ime. Our echnique is unsupervised and auomaically idenifies differen faces of live evens in an unfilered daa sream in a scalable way and presens hem o he users as evolving even hreads. Exensive experimens over real daa demonsrae ha our echnique -- while avoiding per message processing -- can summarize live daa sreams wih high accuracy and produce compac even summary hreads. The summary size of each even is dependen only on he underlying informaion and no on he number of messages peraining o ha even. Our echnique is generic and is applicable on any chronologically ordered daa sream which can be modeled in a <user: message> framework. CCS Conceps Informaion sysems Informaion rerieval Rerieval asks and goals Summarizaion Keywords Real Time Search; Daa Sreams; Even Summarizaion; Algorihm; Experimens. 1. INTRODUCTION 1.1 Moivaion Unsrucured daa sreams -- sequences of chronologically ordered messages posed by muliple users -- occur in various social media and enerprise domains. For example, on Twier, wih a large user base, messages are posed a a high rae. Twier is ofen he firs medium o repor emerging evens [14][18]. An even in a daa sream is defined by messages, posed by muliple users, in he same conex, wihin a bounded ime window, for example, messages posed by he fans during he course of a fooball mach. 2017, Copyrigh is wih he auhors. Published in Proc. 20h Inernaional Conference on Exending Daabase Technology (EDBT), March 21-24, Venice, Ialy: ISBN , on OpenProceedings.org. Disribuion of his paper is permied under he erms of he Creaive Commons license CC-by-nc-nd 4.0 An even can be a real world or an absrac aciviy, relevan for a group of people. I is only naural ha in a fas-moving world, a huge number of evens occur concurrenly. There have been many recen aemps [14][15][16] a idenifying emerging evens in real ime over live social media sreams. Exising unsupervised approaches idenify emerging evens as emporally and spaially correlaed clusers of keywords over dynamic message sreams. The emporally and spaially correlaed cluser of keywords forms an even-opic. In order o capure he even-opic, a dynamic graph is consruced using he mos recen messages, wih a sliding window model. Therefore, nodes in he dynamic graph represen emporally correlaed keywords. An edge beween wo nodes -- represening wo keywords -- indicaes ha messages wihin he recen sliding window conain boh he keywords, represening spaial correlaion. Thus, nodes of dense sub-graphs embedded in he dynamic graph represen he keyword clusers wih srong spaial and emporal correlaions [15][16]. When he wees posed during he Nairobi erroris aack [30] are fed o he sysem described in [15], one of he keyword clusers, i.e., even-opic, discovered conained he keywords: - A: UK, #kenya, #wesgae, #nairobi Clearly, he keywords are insufficien o describe he underlying even. The same is rue of anoher even-opic: - B: was, 69, kofi, among, #ghana, aacks, ghanaian, awoonor, killed, poe, prof., #kenya Sysems such as [14][15] discover even-opics like A and B. To beer undersand an even-opic, users have o search for he relevan messages in he daa sream by hemselves. This burdens he users wih he ask of undersanding he emerging evens manually or o deermine if here is a connecion beween A and B. The shorcomings of he message search-based approach are: (1) Keyword search over live daa sreams is primiive, e.g., Twier jus reurns he mos recen wees for a given search query [19]. I is no necessary ha he mos recen wees are also he mos relevan and informaive wees for he even. (2) Keyword search can produce an informaion overload for a fasmoving daa sream. Ofen a large number of wees are reurned by Twier for a search query [5]. Moreover, search resuls are coninuously updaed wih recen messages. This poses difficulies for he users o keep pace wih he evolving evens. Typically, he rae a which messages are generaed is high. Hence, even if a subse of he messages in he query response is informaive, i is challenging o idenify hese messages in realime. The high rae of message arrival, and he fac ha he Series ISSN: /002/edb

2 messages are shor ofen makes i difficul for he users o undersand he conex of a sandalone message. Thus, he firs goal of his paper is o discover a minimal se of mos informaive messages from he message sream, relaed o an emerging even. These messages represen he complee summary of he even. Furhermore, live real-world evens are no jus poin evens hey evolve. Our second goal is, for each discovered even, is summary mus be updaed every ime here is any significan change in he even. Noe ha he evens evolving in real ime may comprise several differen aspecs or faces. When hese changes in he even summary are arranged emporally, an even hread resuls, capuring he complee even conex wih he passage of ime. Given hese needs, we presen a novel mehod o auomaically exrac he Conexual Even Summary Threads in real-ime for evens unraveling in an unfilered and fas moving daa sream. In [31], we presened our mehodology o creae and updae an index over discovered Conexual Even Summary Threads in a highly dynamic daa sream in real ime and enabled keyword search over hese evens using i. In his paper, we presen our echnique o summarize he evens and o discover he Conexual Even Summary Threads. 1.2 Conribuions Mos real-world evens comprise several faces. Therefore, he summary for an even is represened as a Direced Acyclic Graph (DAG) ha reveals he way he even has evolved. Each node in an even hread, called sub-even, has an associaed even-opic similar o A and B. An even-opic is summarized using a minimal se of mos relevan messages discovered from he daa sream. The even hread in Figure 1 was produced auomaically by our sysem [31] from he wees sen during Nairobi Terroris Aack [30]. There are 13 sub-evens in he hread in Figure 1. The summary sared wih a wee abou a mall being aacked, followed by wees abou acion agains aackers, rumors, claims and couner claims by auhoriies and ciizens, ec., which were discovered in real ime. The even hread describes a meaningful chronological sequence of he even. Figure 1 depics a par of he even hread discovered over 164K wees. For each sub-even in he even hread, our mehod idenified an appropriae summary. We idenify mos appropriae message(s) in he daa sream, describing he even-opic. In Figure 1, sub-even 9 corresponds o A and sub-even 5 corresponds o B. Noe, all he keywords for boh he even-opics are presen in heir summaries. In summary, our sysem: (i) Clusers he relaed messages ogeher in he daa sream: Even-opics are idenified by discovering dense sub-graphs, called even-graphs, in he dynamic graph consruced over he message sream [15]. Our sysem explois he even-graphs o pool he relevan messages ogeher, relaed o even-opics. (ii) Idenifies imporan messages in he message pool o creae an even summary: We idenify a minimal se of he mos relevan messages from he message pool of an even-opic such ha he summary is complee and meaningful. In fac, we exploi he srucural properies of he underlying even-graph o idenify a subse of messages as summary candidaes. (iii) Discovers he even hreads for evolving evens: As he evens evolve, he underlying even-graphs go hrough srucural changes. These changes are racked in real ime. Even summary is updaed whenever is even-graph goes hrough a significan change (cf. Secion 6). The updaes in he even summaries are capured in conexual summary hreads like one shown in Figure 1. Our echnique auomaically discovers differen faces of live evens in real ime. In general, an even hread can have muliple roos. We say ha each unique pah in he even hread, from each of is roo(s) o each of is leaves, represens a differen face of he even. Even hread in Figure 1 has 6 faces. In a nushell, our research conribuions involve 1. Summarizing all he evens, discovered in a fas-moving unfilered daa sream, in real ime, in a scalable and unsupervised manner. For each discovered even-opic, a minimal se of messages is idenified o produce a meaningful, informaive, sable and complee even summary (Secion 5). 2. Tracking he evoluion, emergence and dissoluion of dense graphs (even-graphs) in a large and highly dynamic graph in he presence of node and edge inserion and deleion. 3. Exposing differen faces of live evens which are presened as a conexual even summary hread, represening one or more aspecs of he even in real ime (cf. Secion 6). 2. Day 2: Al-Shabaab Jihadiss Holding Innocen Civilians a Wesgae in Nairobi, Deah Toll a KENYA UPDATE: Deah oll in #Wesgae siege rises o 68 as 9 more bodies recovered during rescue operaion 7. Israeli forces ener Nairobi mall: securiy source hp://.co/e0nom7lxpa \u2026 #wesgae 8. Two helicopers landed on he roof of #wesgae mall where #nairobi hosage crisis coninues. 10. Kenyan forces kill wo erroriss, claim conrol of Wesgae mall: Kenyan forces assauled erroriss in Nairo... hp://.co/zojahgauun 13. Day 3: Kenyan Governmen Takes Wesgae Mall From al- Shabaab Jihadiss p://.co/e66ddy5l6y #BigTwee 1. #AlShabaab says i aacked #Wesgae mall in #Nairobi o realiae for Kenya's role in #Somalia. 3. MAJOR assaul by securiy forces ongoing o end wo-day siege a Wesgae mall. Fears abound deah oll could be higher when dus seles. 5. Ghanaian poe & auhor- Prof. Kofi #Awoonor was among he 69 killed in he aack on #Nairobi's #Wesgae mall. #Ghana #Kenya 6. Spread o all Kenyans - he wesgae siuaion may be rying o disrac Nairobi, a bigger aack may happen, STAY INDOORS- RT n SHARE 9. Speculaion ha converie 'whie widow' Samanha Lewhwaie from UK is he masermind of he aack on #Wesgae Mall in #Nairobi, #Kenya 11. Somehing I never saw in 30 yrs as journalis: civilians bringing food, coffee o journaliss covering #Nairobi's #Wesgae siege. Amazing! 12. Milians a he Wesgae mall in Nairobi, Kenya, are sill holding heir ground, Somalia's Al-Shabab group claims Figure 1: Conexual Even Summary Thread Discovered by our sysem for Nairobi Aack 169

3 To he bes of our knowledge, ours is he firs echnique ha capures he muli-faceed summary of live evens and o arrange hem in even hreads. The even opics, discovered over an unfilered and fas moving daa sream, are summarized in real ime in an unsupervised manner, in absence of any user query. The conexual even summary hread is a novel concep in conras wih he sory-line generaing echniques [4]. Our approach is generic and applicable on any daa sream ha is a sequence of <user: message>s. Experimens over real daa show, our echnique leads o compac and meaningful even summary hreads for live evens. The organizaion of he paper is: In Secion 2, we presen he relaed work. The even-graph model o capure he sae of dynamic daa sream and he challenges involved in discovering even summaries are presened in Secion 3 and Secion 4 respecively. In Secion 5 and Secion 6, we presen he mehodology o discover even summary and even summary hreads. We presen our experimenal sudy in Secion 7 followed by conclusion in Secion RELATED WORK Topic Deecion and Tracking [1][2] is relaed o our work. In [12], Yang e al., presened a mehod o summarize web documens. For hese echniques, he documen characerisics are compleely differen from microblog sreams ( large size/less in number/offline vs. small size/oo many of hem/real ime ). These mehods work on saic daa only. In [5], Shou e al., presened a echnique o summarize a Twier daa sream, filered in he conex of a given user query. In [23], auhors reduce he summarizaion problem o ha of opimizing probabilisic coverage on saic daa. In [8], Yang e al., presen a mehod o compress he Twier daa sream. Lee e al., in [24] propose a snapsho based mehod o rack he incremenal evoluion of even-opics. There is significan work [10][11][13] o idenify a fixed size summary of microblog poss on a pre-specified opic and on saic daa, i.e., daa which is no updaed wih new messages. Gao e al., [3] proposed an unsupervised approach o summarize evens, by preparing a join senence-wee level model, across news media and Twier. Too similar or oo differen senence/wee pair were discarded. In [4], Lin e al. proposed a mechanism o generae a soryline from a microblog daa sream. However, he echnique is applicable only for saic daa ses. Vosecky e al., [7] presened a mehod o idenify muliple faces of an even, i.e., imporan keywords, eniies, locaion, ec., on a saic Twier sream. The basic difference beween exising work and he problem addressed in his paper is: These echniques [1][4][5][7] work in he conex of a given query, opic and/or on saic daa. In conras, our echnique organizes he even summary ino even hreads, capuring he complee conex, for each underlying even discovered in he unfilered daa sream in real ime. The dynamic graph model is exploied o idenify he even summary. Our echnique explois he presence of Shor Cycle Propery (SCP) in he dense sub-graphs of a dynamic graph. In [15], i was shown ha dense sub-graphs of a graph possess SCP. There is a large body of work for riangle couning and riangle lising in graphs. Triangle lising is considered an imporan measure for discovering dense neighborhood [28]. A riangle in a graph induces a cycle of lengh hree. SCP exends his noion o cycle of up o lengh four, i.e., each node in a graph, possessing shor cycle propery, paricipaes in a cycle of lengh a mos four wihin he graph. Triangle lising problem can be divided ino processing saic graphs [20] and sreaming graphs [27]. Sreaming graph algorihms for riangle lising are furher divided ino edge inser graphs [27] and edge inser or delee graphs [9]. In conras, our algorihm works on a graph wih inserion and deleion of boh edges and nodes. In his paper, we idenify dense even-graphs, discovered based on SCP [15], represening emerging evens in a large dynamic graph, wih edge and node inser and delee. Our echnique keeps rack of emergence, evoluion and dissoluion of he dense sub-graphs in a large dynamic graph. This evoluion is presened as conexual even summary hread for he even. 3. EVENT-GRAPH MODEL Le S = S-wS-w+1 S represens a message sream. Si = {d i 1d i 2 d i m} is he se of messages arriving in block i. A message d j=<ui, k> in he daa sream conains message k and he userid uj of he user posing he message. A message k is an ordered se of keywords. m, called he block size, is he number of messages in a block. The sream is moved forward by expiring he messages in (-w) h block and including he messages in (+1) h block. The message imesamp is implici wih he arrival order in he daa sream. For consrucing he even hread as shown in Figure 1, we exrac he <userid: message> from he incoming daa sream. No preprocessing is done on he message, excep removing sop-words from i and okenizing he message ino keywords. Temporal and Spaial correlaion: The daa sream S is modeled as dynamic graph G (V, E ) [15]. For he problem sudied in his paper, G is an inpu o our sysem. The graph G (V, E ) capures he sae of he daa sream a he arrival of messages in block ( N ). Keywords are represened as nodes in he graph G. A ime sep -1, he graph is updaed wih he keywords presen in he las block of m messages and -1 is incremened o. V conains he bursy and acive keywords in he las w blocks a ime. Nodes V capure emporal correlaion as only emporally correlaed keywords (nodes) are presen in he graph. S; S =m G =(V, E ) G c(v c, E c) Table 1: Noaion S is he block of m messages in he h ime sep. Dynamic graph a ime sep. V represen he keywords and edges E he keyword-correlaion in he graph G. G c is he even-graph embedded in G for even c a ime sep. V c represen he keywords in even-opic. N (v); v V c N(v) represen he se of adjacen nodes o node v V c. d j=<ui, 1 j m w γ λ k>; Message d j posed by user u i during ime sep. k is se of keywords in he message. Graph G conains he bursy and acive keywords from las w message blocks. Keyword Bursiness hreshold. Edge correlaion hreshold. M c Message se represening summary of even c a ime. Uv Se of userids associaed wih node v V. usercover Se of userids, whose messages include all he keywords in an even-opic. A keyword is bursy if i is used in γ (>1) messages in he curren window of m messages. A keyword k is called acive if is corresponding node vk V is presen in G (V, E ), i.e., he keyword is used in a leas one of he messages in he las w message blocks. An acive keyword remains in he graph G for as long as i is acive. A keyword is removed from he graph G if i is inacive for w ime windows. Hence, a keyword has o be bursy a leas once o move ino he graph G. The bursiness consrain helps idenify evens relevan for a communiy or group of users. 170

4 Each node vk V is associaed wih a user lis Uk, conaining userids of he users who have used he corresponding keyword k since he ime i has moved ino he graph G. This lis is used o esablish spaial correlaion among he keywords (nodes). For a pair of nodes {vi, vj} V, if he similariy score beween heir respecive user liss Ui and Uj is above a given hreshold λ, an edge e is placed beween he nodes. Jaccard coefficien (Jc) is used as he similariy measure. Jc beween wo nodes vi, vj in graph G is calculaed as U U U U ; hence 0 λ 1. Edge e E i j i j capures he spaial correlaion beween he associaed keywords. Dense sub-graphs in graph G represen an even: Dense subgraphs, embedded in he graph G (V, E ) are called even-graphs. The nodes V c V in he even- graph G c (V c, E c) are he keywords in he even-opic c and edges E c E represen he correlaion beween hese keywords. G c represens he keywords wih srong spaial and emporal correlaion. In a daa sream, modeled as dynamic graph, emerging dense sub-graphs represen he emerging evens [15][16]. Shor cycles in dense graphs: The lengh of he Shores-cycle in a graph has a srong correlaion wih graph densiy. For example, riangle lising is a well-known mehod o measure graph densiy [27]. Each edge of a riangle induces a cycle of lengh 3. Similarly, in chordal graphs [29], each node is par of a cycle of lengh 3. In [15], auhors expose a propery for he dense sub-graphs embedded in a large graph, called he shor-cycle propery; each node in a dense sub-graph paricipaes in a leas one cycle of lengh a mos 4, wihin he sub-graph. Nex, we formally define he Shores-cycle and shor-cycle propery. Def Shores-cycle: For a graph G (V, E) and a node v V, Shores-cycle is he shores pah P hrough which one can reurn o node v; e P, e E. Def Shor Cycle Propery (SCP): For a keyword cluser c, le G c (V c, E c) represen he corresponding subgraph embedded in graph G (V, E ) ( k c v k V c ; ). V c V, E c E and e( u, v) Ec,{ u, v} Vc. G c possesses he shor-cycle propery if i saisfies he following condiions: P1. For any wo adjacen nodes u and v in he G c, here exiss a leas one more pah P beween u and v, such ha P 3 and e( u, v ) P, e( u, v ) E. Therefore, he lengh of Shores-cycle c for v V c is 4. Noe, lengh of he Shores-cycle for node v is P + 1 (for edge e(u, v)). P2. For a node v V c in graph G c, all he Shores-cycles node v paricipaes in wihin he graph are of lengh a mos 4. P3. There is no ariculaion poin in G c. An ariculaion poin is a node whose removal breaks he graph ino muliple disconneced componens (e.g. in Figure 3(a), node a4). In Figure 2 (a), in he absence of P3, clusers C1 o C4 will merge ino a single cluser since he merged cluser saisfies P1 and P2. Similarly, in he absence of P2, C1 o C5 will merge ogeher ino a single cluser. Please noe, due o SCP, hough necessary, i is no sufficien for wo even-opics o jus share wo or more keywords o merge ino a single even-opic. In Figure 2(b), C6 and C5 share wo keywords, bu hey are no merged ogeher since he merged cluser does no saisfy P2 of SCP. Thus, SCP ensures discovery of dense clusers efficienly [15]. (a) (b) Figure 2: Example even-graphs Example 1: In Figure 3(a), ses {a1, a3, a4} and {a2, a4, a5} represen an independen even each. Even-graph in Figure 3(b) represens a single even. In Figure 3(b), keywords having heir Jc 0.25, have an edge beween hem. An even-graph saisfies he SCP (Def ). In Figure 3 (b), k2 and k3 paricipae in wo Shores-cycles each. Cycle k1 k2 k4 k3 k1 is an inra-graph cycle oo bu no he Shores-cycle (Def ). In [15], auhors presen an efficien algorihm o idenify dense-graphs in a dynamic graph by exploiing SCP. SCP ensures ha keywords ha show a srong emporal and spaial correlaion are idenified as even-opics. a5 C1 a2 C2 C2 C4 a4 (a) C5 a1 Ariculaion poin a3 k3: 1,2,5,6 Figure 3: Example even-graphs 4. ISSUES in EVENT SUMMARIZATION There is a user lis Uv associaed wih each node vk V c in evengraph G c (V c, E c) for even c. vk is he node corresponding o keyword k. We represen vk by jus v when he conex is clear. The messages posed by users in lis Uv, v V c, are considered he pool of message relaed o even c. Even summary is idenified from hese messages. Def. 4.1 Even Summary: Even summary is a se of messages M c from a se of users U c ( U v; v V c ) such ha v, c u U where u has used he keyword k in is message(s). k V c Thus, he even summary is defined as a se of message(s) from a se of users whose messages covers all he keywords in he evenopic. This se of users is also called he valid usercover. Le u Uv of he userid lis of node vk V c. Le N(vk) represens he se of nodes adjacen o vk in G c (V c, E c). n N( v k ), if u U n, node n is considered covered. Thus, all neighbors of node vk ha conain u in heir respecive user liss are considered covered. Noe, only he user liss of he neighbors of node vk in he evengraph are checked for he presence of user u. The messages in he curren ime window from he users in he usercover become par of he even summary M c. The summary M c for a new even c, emerging in he h c ime window is defined as M c Su u U where Su is a se of messages from user u in h message block. M c is he ordered sequence of messages based on he message imesamp. Def. 4.3 Opimal UserCover: An opimal usercover T is he smalles subse of users such ha Uv v Vc, Uv T. C6 C5 k1: 1,2,3,4 k4: 5,6,7,8 (b) k2: 2,3,6,8 171

5 Theorem 1: Discovering opimal usercover is NP-hard. Proof Skech: Hiing se is a well-known NP-complee problem [25]. I is defined as follows: Given a collecion S of ses Sis; 1 i n, find he smalles size se H such ha Si S, Si H. By mapping each elemen in a se Si o a user id in usercover, Hiing se problem is polynomial ime reducible o usercover. If T* is a soluion for usercover and H* is a soluion for Hiing se, H* = T*. Theorem 1 saes, discovering opimal usercover is NP-hard even when he user liss associaed wih keywords are saic. However, he se of nodes V c in an even graph G c as well as he user liss Uv V c associaed wih each node v are highly dynamic due o coninuous updaes in he message sream. Thus, i is non-rivial o idenify a minimal se of mos relevan messages from he daa sream ha ensures ha he even summary be informaive, compac, complee, meaningful and sable. Idenifying users (and no messages) helps us creae a more meaningful even summary (a imes, users pos muliple messages in a small ime window o explain he full conex of heir messages). Since user liss are large for popular evens, i is nonrivial o idenify opimal UserCover. To efficienly idenify hese users, each arriving message is assigned a score in a scalable manner o rank more relevan messages higher (cf. Secion 5.3). We nex highligh how he dynamic updaes in he daa sream may resul in an unsable even summary: k k k k 4 Figure 4: Evoluion in he even-graph Live updaes can make a summary unsable: For an even presen in a live daa sream, le is iniial even-graph be as shown in Figure 4(a). The usercover had been idenified as {1, 3} by any usercover discovery algorihm (k1 in Figure 4(a) is no par of he even-graph). In he nex ime sep, he even-graph ges updaed o as shown in Figure 4(b). However, he same algorihm idenifies {2, 9} as usercover, which resuls in a differen se of messages as summary, making i difficul for users o keep rack of evolving even. Thus, due o live updaes, even summary may be unsable. The evoluion of he summary for a live even is handled as described in Secion 6. We presen our algorihm o summarize he even by idenifying an approximae usercover in Secion DISCOVERING EVENT SUMMARIES In his secion, we presen our mehod o discover meaningful even summaries, wih he aid of cenral nodes, called pivo nodes, in he dense even-graphs. In Secion 5.2, we presen our algorihm o idenify he even summaries. In Secion 5.3, we presen our mehod o rank he messages in he daa sream. 5.1 Pivo Nodes and Pivo Edges Def Pivo Edge: An edge ha paricipaes in more han one Shores-cycles wihin he even-graph is defined as a pivo edge. Higher he number of Shores-cycles a pivo edge paricipaes in, he more cenral i is in he even graph k k k k k 4 Def Pivo Node: Nodes associaed wih a pivo edge are called pivo nodes. Nodes which are no pivo nodes are called peripheral nodes. In Figure 4(b), edge (k2, k4) is a pivo edge. {k2, k4} are pivo nodes. Lemma 1: A graph possessing shor-cycle propery wih more han one shores-cycles wihin he graph, has a pivo edge. Proof: Le even-graph G c (V c, E c) has muliple shor cycles, saisfying SCP. Le C (V, E) be a cycle in he even-graph G c such ha i has no common edge wih any oher cycle in he graph. Le n C(V ) be a node common wih anoher cycle in graph G c (if cycle C has no even one node common wih any oher cycle wihin he graph; G c will be disconneced); Since no edge in cycle C paricipaes in anoher cycle, for any node v in C(V) - n, and for any node u in V c V, here exis jus one pah from v o u via node n. Hence in graph G c, node n is an ariculaion poin, violaing Def of SCP even-graph. Therefore, here mus exis anoher pah from v o u. Hence, here exis a cycle v n u v. Hence any edge e C(E) in pah v n paricipaes in anoher cycle, i.e., e is a pivo edge. Therefore, for each Shores-cycle C (V, E) in an even-graph, e C(E) ha is a pivo edge. Corollary of Lemma 1: Eiher a node iself or one of is neighbors is he pivo node in he even-graphs ha possesses SCP. Due o his corollary, we can creae a pivo edge cover (PECover), as described below. Def PECover: For G c (V c, E c), a PECover is a subse of o pivo edges E p E o c such ha, v V c, e( u, v) Ec v V p. o o Hence, a PECover (corresponding PNCover) E p ( V p ) is a subse of pivo edges (corresponding pivo nodes) in he even-graph G c (V c, E o c) such ha v ( V c Vp ), node v is adjacen o a node in o V p. Le Uv be he user lis for node v V c and C be a collecion of all he user liss in G c (V c, E c). Le Cp be a collecion of user liss o associaed wih pivo nodes V p in graph G c (Cp C). Lemma 2: For collecion Cp Ui C C p; U j Cp, U. i U j Proof: For wo nodes vi and vj and heir associaed user liss Ui and Uj in an even-graph G c(v c, E c), if edge (vi, vj) E c, U U. U U ; 0, U. Therefore, for any i j i j l { i, j} edge e E c here exiss a leas one userid which occurs in he user liss of boh he nodes. Since, each node in G c is adjacen o a node in he PECover of G c, herefore, for collecion Cp, U C C ; U C U U. i p j p i Hence, we firs idenify a se of pivo edges, called PECover. A valid usercover can be idenified only from he user liss associaed wih pivo edges (cf. Lemma 2). Thus, pivo edges help us avoid processing a large number of messages associaed wih peripheral nodes and ye ensure ha he even summary is complee. Furher moivaion o use he pivo edges is explained below: Informaive A message conaining more keywords from he even graph is considered more informaive. Naurally, a keyword wih higher j 172

6 number of neighbors in he even graph is likely o conain such messages. We capure his noion of informaive-ness as follows: Def Informaive-ness: Informaive-ness of a node v is defined as N(v), he se of nodes adjacen o v in even graph G c. Le edge e (u, v) be a pivo edge. Le s be a neighbor of u such ha e (u, s) be a non-pivo edge (s is a peripheral node). N(v) represens he se of nodes adjacen o v in G (V, E ). Lemma 3: N(u) > N(s). Proof: Omied (cf. Lemma 5). Lemma 3 shows ha he messages from he users associaed wih he pivo nodes conain more keywords from he even-opic. Idenificaion of summary from hese messages leads o a more informaive and compac summary Sable Since a pivo edge paricipaes in muliple cycles, i is more sable han he oher edges in a dynamic even-graph. Even if one or more edges/nodes, among edges and nodes adjacen o a pivo edge ge deleed in he underlying even-graph, he message se idenified as he even summary coninues o be a valid even summary (cf. Secion 6.2). Hence, idenificaion of even summary based on pivo edges, leads o more sable even summary Complee An even summary conaining messages ha cover all he keywords in an even-graph is considered a complee even summary. However, we idenify he users only from he user liss associaed wih pivo nodes, i.e., from he collecion Cp. Lemma 4: A user cover idenified only from he pivo nodes of an even-graph, possessing shor cycle propery, is a valid usercover. Proof: The corollary of Lemma 1 along wih Lemma 2 complees he proof. Lemma 4 shows ha he usercover idenified from he user liss in collecion Cp resuls in a valid usercover. Wih he help of pivo edges, we idenify a subse of user liss Cp (from he enire user lis collecion C for even c) and a valid usercover can be idenified only from hese ses. Thus, he even-graph srucure helps us idenify a small number of more relevan user liss for discovering a valid usercover. However, discovering opimal se of pivo nodes in an even-graph remains an NP-hard problem (cf. Theorem 3) Opimal Summary Size We nex show ha he summary size discovered wih he aid of opimal PECover leads o he opimal size summary. As shown in Lemma 2, once a pivo edge e(u, v) is idenified, all he nodes adjacen o he pivo nodes {u, v} can be covered using only he user liss associaed wih hese nodes. We now show ha here canno be any smaller collecion of user ids han opimal size PECover ha resuls in a valid usercover. Le T* represen he opimal collecion of user ids ha provide a valid usercover for a given even-graph G and le PECover* be he opimal PECover. Theorem 2: PECover* T*. Proof: For a given even-graph G (V, E ), a dominaing se D is subse of nodes in V such ha each node in (V D) is adjacen o a node in D [25]. Le D* V be he opimal size dominaing se. Therefore, D* is he smalles se of nodes such ha each node in V D* is adjacen o a node in D*. Sep 1: Each node in V, is adjacen o a pivo edge (corollary of Lemma 1). Therefore, for each node v in D* we ge a corresponding pivo edge as follows: eiher edge e(v, u) is a pivo edge; u N(v) or edge e (u, N(u)) is a pivo edge. In his manner, we ge a pivo edge cover pecover from D* which is a valid PECover since D* is a dominaing se for graph G ; pecover = D*. Le PECover* is he opimal PECover. Hence, PECover* pecover D*. Sep 2: Any wo nodes ha share an edge, have a leas one userid common (Lemma 2). Le T* be he opimal se of user ids, providing he valid usercover. Since D* is he opimal dominaing se, herefore, T* D* because for each node in D*, a leas one userid has o be seleced in T* (since he user liss of only he neighbors of a node in he even-graph are considered while consrucing usercover) From Sep 1 and Sep 2, PECover* T* Thus, opimal pivo edge cover leads o opimal summary size. On even graph G c we induce a graph G (V, E ) such ha each edge in G is a pivo edge in G c (wih he aid of Lemma 5, Secion 5.2). We idenify a smalles subse VPN V of nodes in G such ha every node in V VPN is adjacen o a leas one node in VPN. We call he se VPN Pivo nodes cover or PNCover. Theorem 3: Discovering opimal PNCover for a given even-graph G c(v c, E c) is NP-hard. Proof Skech: Dominaing se is a NP-complee problem [25]. I can be shown ha Domin aingse P PNCover. (a) Black nodes depicing opimal dominaing se (d=3) (b) Black nodes depicing opimal pivo nodes (c=2) Figure 5: Dominaing se Vs. Pivo nodes over even-graph An alernaive o pivo edges is o idenify he dominaing se of nodes [25] iself in an even-graph. Our primary reason o choose he pivo edges over dominaing se is, nodes in dominaing se divides a graph ino sars whereas he pivo edges divide i ino cycles (i.e., quasi-cliques). The usercover discovered from nodes ha are par of same quasi-clique leads o more informaive even summary. For insance, black nodes in Figure 5 (a) and Figure 5 (b) boh represen he dominaing ses. However, Figure 5 (b) represens he dominaing se induced due o he pivo edge. Therefore, pivo edges ensure ha a message summary discovered based on pivo edges is more informaive, sable, complee, and compac. Pivo nodes are cenral nodes in an even graph. There are oher noions of cenral nodes in a graph, for example, HITS [17] and PageRank [21] idenify cenral nodes (auhoriies). However, here are basic differences in he seings of our wo problems as: a) [17][21] echniques are ieraive in naure, herefore no amenable o rapidly changing dynamic graphs; b) The noion of compleeness (Secion 5.1.3) is no applicable for hese mehods. Due o he same reason, he noion of beweenness cenraliy [22] is no applicable; 173

7 c) These echniques exploi he link srucure o idenify he auhoriies, whereas we exploi he graph srucure o idenify he pivo nodes (since he objecives of wo problems are differen). 5.2 Approximae User Cover Nex, we presen our algorihm o idenify approximae usercover. On an even-graph G c(v c, E c), we induce a graph G (V, E ), V ' V c and ' ' ' ' E Ec ( V V ) such ha v V ; v is a pivo node in graph G c. The induced graph G is called he core evengraph. The even summary discovered from G (V, E ) follows he same noion of even-summary (Def. 4.1) excep ha i is discovered on induced graph G. Insead of idenifying he usercover for graph G c, we idenify he usercover on G. The core even-graph G does no conain he peripheral nodes in G c. However, presence of peripheral nodes in he even-graph G c defines he pivo nodes. Therefore, peripheral nodes impac he even summary only indirecly. Lemma 5: A node v wih degree Δ(v) > 2 in graph G c(v c, E c) is a pivo node. Proof: v Vc, ( v) 2 (node v is par of a cycle). Le v V c Δ (v) > 2 be a node in graph G c such ha v is no a pivo node. Since Δ(v) > 2, node v is par of more han one cycles. Le C1 and C2 be wo cycles node v is par of. Le edge e (v, n ) belong o C1 and edge e (v, n ) belong o C2. Hence, here exiss one pah from node n o node n via node v. However, if his is he only pah beween he wo nodes, node v becomes an ariculaion poin violaing he shor cycle propery. Hence here exiss one more pah p from node n o n inducing a cycle C (v n n v). Lengh of cycle C 4 (SCP). Therefore, edge e(v, n ) paricipaes in wo cycles C1 and C. Hence e(v, n ) is a pivo edge and v is a pivo node. p Wih he aid of his lemma, i is easy o induce graph G on G c. Noe, G may no follow he shor cycle propery k k k k k4 Figure 6: Discovering core graph from an even-graph Each node v in G is assigned a score p(v); p(v) dv(g c); dv(g c) is he degree of node v in graph G c. This score is used o idenify userids in he usercover. In Lemma 2, i is shown ha if wo nodes are adjacen, hey have one or more userids common. Therefore, by selecing he nodes wih higher score, even summary is likely o conain more keywords in even-graph G c. We describe our algorihm o find usercover for graph G below. Algorihm approxusercover (nodelis nl) 1. Le G c(v c, E c) be an even-graph. a. le cid be is cluserid; 2. G is he core even-graph induced on graph G c. a. ' ' ' v G ( V, E ); p v d v ( G c ) b. U v lis of userids associaed wih node v 3. S Φ 4. ' for v G( V ) {if v nl { S S v }} 5. cid.nl V ; /*we record he pivo node o efficienly updae he even summary when cluser evolves laer*/ 6. uc Φ /*usercover is se o null*/ 7. while ( S ) 9 10 k k k k k4 ' a. v arg Max ' ( pv dv ( G ) /*reurn node wih v V highes p v d v(g ) score*/ b. U U v ; c. Le U n be userids se associaed wih a node n, such ha n is neighbor of node v in G. i. T arg Max U messageran k( ) ii. iii. U n uc uc {T}; U = U U U n /*We do no selec any more userids covering same keyword*/ d. For each remaining neighbor, check if userid T exiss in heir userid lis i. A v.adjlis (G ) /*adjacency lis of v in G */ ii. 8. reurn uc; u u A n, if T U u, S S-u; Each edge in graph G is a pivo edge in G c. The degree of a node v in G, dv(g ) is a measure of he number of cycles i paricipaes in graph G c. messagerank (.) reurns he userid of he message wih highes rank. Since, a user is added in he user lis of a node one a a ime, i is kep sored in he message score efficienly. In he above algorihm, each node in he even-graph is visied exacly once. Thus, he complexiy of idenifying usercover is O( V c ). Algorihm approxusercover is greedy and achieves he same complexiy as Dominaing se [25], i.e., (1+ log (Δ)) OPT; OPT is opimal usercover size and Δ is he maximal degree in graph G. Dominaing se problem is LOG-APX-COMPLETE and no beer bound is possible. 5.3 Ranking he Messages Since our objecive is o summarize a highly dynamic daa sream in real ime, he mehodology o rank each arriving message mus be i) fas and efficien; ii) rank more meaningful messages higher; and iii) does no re-rank he already ranked messages wih fresh updaes in he daa sream. There are many sudies relaed o ranking he microblogs [18][19][6]. A common conclusion across hese sudies is ha he rank of a wee depends on he auhoriy of is auhor and he auhoriy of he message. We exploi hese feaures o esablish he rank of a wee. Le di, dj S be wo messages in he daa sream S. Le R(.) be a monoonically increasing funcion such ha if di is deemed more imporan han dj, R(di) > R(dj). To efficienly rank he messages, R(.) is applied on boh hese messages independenly. A wee is considered more meaningful i) if i is reweeed more (rewee coun RT capures he auhoriy of he wee [18]); and ii) if i is weeed by a person wih many followers (follower coun f capures he auhoriy of he user [19]). Therefore, he wee score R(d) of a wee d is compued as: R(d) = αrt.log f Since he dynamics of an even vary a much finer ime scale compared o a user s follower s coun, f is a logarihmic facor. The ranking funcion scores each arriving message efficienly, as soon as i arrives such ha he more imporan messages are likely o be ranked higher. Please noe he firs crierion o choose a message is he underlying even-graph srucure. The highes ranked messages associaed wih he pivo nodes in he even-graph are idenified as summary (Secion 5.2). Therefore a message from a user wih lesser following and/or rewees would be picked in he even summary, if i is more relevan in he conex of an even. In summary, we ranslae our goal o provide an even s summary ino one of discovering a se of users, collecively using all he keywords in he even-opic. We show, his se of users can be 174

8 discovered only from pivo nodes in he even-graph. We also show ha pivo nodes enable us o discover informaive, sable and compac summary. 6. DISCOVERING EVENT THREADS In his secion, we presen our echnique o discover he conexual even summary hreads capuring he evoluion of live evens. The evoluion of an even is racked by racking changes in is underlying even-graph G c(v c, E c). A possible approach for consrucing he even hreads is o mainain he snapshos of each even-graph in each ime window [24], bu i is no pracical o consruc summary hreads in real ime from hese snapshos, as 1) insead of incremenally processing he graph, he complee even-graph needs o be processed for each even for each snapsho, hus making i impracical for processing a fas moving daa sream in real ime; 2) i is shown in [31], ha a highly efficien mechanism is needed o keep he indexes updaed for he even hreads. Thus, i is no pracical o creae he index again for each snapsho. Hence, we mainain conexual even hreads for each even by keeping a corresponding eventree. EvenTree capures he evaluaion of he even-graph. We assign a unique even-id (called cluserid) o each even-graph. When an even-graph evolves, he summary is updaed and he change is recorded in is eventree. Wih changes in even-graph, eventree is mainained as follows: Incremenal changes in he even-graph: There are incremenal changes in he even-graph due o addiion and deleion of nodes and edges. Due o hese changes, he summary may or may no change bu he cluserid of he even-graph remains he same. Disrupive changes in he even-graph: If he srucure of an evengraph changes so much ha we need o assign i a new cluserid, such a change is called disrupive change. For example, when wo independen even-graphs wih cluserid c1 and c2 merge ino a single even-graph c due o emergence of new nodes/edges. The mapping c c1, c2 is recorded in he eventree. When wo evengraphs merge, heir corresponding hreads also merge in a single hread. Similarly, due o deleion of nodes/edges, if an even-graph c breaks ino say wo sub-graphs, c1, c2, we record c1 c and c2 c in he eventree. Thus, a eventree capures he evoluion in he corresponding even-graph. Whenever a disrupive change occurs, we record he paren cluserid (cp)and child cluserid (c)relaionship as an evoluionedge (c cp is an evoluionedge ). Each evoluionedge ha emerges in he curren ime window w, is processed a he end of he window, and he even summary is updaed (cf. Secion 6.3). We exploi he graph srucure so ha only he necessary evoluionedges are recorded. Noe ha evoluionedges rack he evoluion of even-graphs and hey are no he edges in he graph G (V, E ). Insead of mainaining complee snapshos, we jus mainain evoluionedges o capure he differences beween G and G +1. We nex illusrae how he even summary changes due o addiion and deleion of nodes. Similar process is applicable on graph edges. 6.1 Effec of Node Addiion G c(v c, E c) is an even-graph and G (V, E ) is he graph induced on G c such ha E E c be he se of pivo edges for graph G c. When a new node (keyword) n joins he even-graph, i may induce a new pivo edge in graph G +1 c. A node n can join he even-graph G c in wo possible ways: Case 1: Due o he addiion of node n, an edge e E +1 c E becomes a new pivo edge. For example, as shown in Figure 7, when a new node n joins he cluser {A, B, C, D}, a new shor cycle n A B n is induced such ha edge AB becomes a new pivo edge. In his case, he induced graph G includes an exra node (node A) and he corresponding edge(s). The exising usercover is exended o cover A. Please noe, i is possible ha he even summary does no change as he exising usercover could be sufficien due o pivo edge e(b, C). This check is done in O (1). 2, 5 n E B D Figure 7: Change in even summary wih node addiion Case 2: A node n joins he cluser such ha i induces a new cycle on an exising pivo edges ep E. In Figure 7, n induces a new shor cycle on edge BC (B n C) which was already a pivo edge. Therefore, node n is a peripheral node and no change is made in he even summary. The cases underline he way even-graph is exploied o updae he summary; due o he concep of pivo edges, summary does no change rapidly because of minor changes in he even-graph. 6.2 Effec of Node Deleion A deparing node breaks a leas one shor cycle. Deparing node is eiher a pivo node or a non-pivo node. Case 1: The deparing node n is a non-pivo node (e.g., node n in Figure 7); n V. Wih he deparure of node n, a pivo edge may no longer remain he pivo edge. Since n is no a pivo node i can impac only one pivo edge as i induces only one cycle. However, he usercover coninues o remain a valid user cover as he edge e(n, u), erswhile pivo edge, remains par of he even-graph G +1 c. Case 2: The deparing node n is a pivo node for edge e (n, v). For example, consider node B in Figure 7. Deparure of a pivo node has a significan impac on he even-graph G c and he graph may break ino muliple sub-graphs or i may ge dissolved if i no longer possesses he shor-cycle propery (in ha case, he even ceases o exis as a live even). Each surviving sub-graph mus possess SCP. Each of he surviving sub-graphs is assigned a new cluserid. The updaeevoluionedge () records he relaionship beween he old and each new cluserid. For each evoluionedge ci c, we exrac he even-graph G +1 ci, updae is summary and is even hread. Algorihm: UpdaePECover (node n) n' N (n) /*N(n) reurns neighbors of n in G c*/ If n V /* V is a se of pivo nodes in induced graph G */ u N (n ) if e(n, u) E if ep is no longer a pivo edge E E e(n, u); else /*edge is a pivo edge*/ childid genewcluserid(); updaeevoluionedge (childid, cluserid); A 0.8 C n 175

9 The merging and spliing of even-graphs, resuls in an eventree which is a Direced Acyclic Graph (DAG), represening he conexual even summary hread similar o shown in Figure Temporal Evoluion of Even Thread Id1 Node/edge addiion edge 1 Id=null Id2 Merged graph Process edge 1 edge 2 Id 1 Id= Id1 if (Id=null) updaethread(id1) Figure 8: Processing evoluionedges o updae even hreads For all he evoluionedges (childid parenid), he corresponding even hreads are updaed. Figure 8 depics how he even hreads evolve wih ime. Each eventree is idenified by a unique id, called Id. For each childid parenid edge, if he paren cluser s Id is null, i is a new even. We iniialize is eventree and idenify even summary (using approxusercover). Oherwise, we merge he eventree of he paren cluser wih he Id of he child cluser and updae he even summary. The evolving even summary is recorded in he even hread. Similarly, even hreads are updaed in case of even-graph spli. For wo cluserids c1 and c2 in an eventree, if c1 is ancesor of c2, c1 has occurred before c2 in real world ime. Each pah in an even hread, from each of is roos o each of is leaves, exposes a differen face of he even. The even hread in Figure 1 has 6 faces. 7. PERFORMANCE EVALUATION The goals of our experimens are o sudy our sysem s abiliy a) o consruc informaive, complee, meaningful, sable and compac even summaries in real ime (Secion 7.2); b) o discover he conexual even summary hreads in real ime (similar o Figure 1) and o sudy he impac of he changes in he granulariy of he even-graph on he even summary and even hreads (Secion 7.3); c) o discover even hreads efficienly and in a scalable manner (Secion 7.4). The experimens use he prooype buil by us [31] and run on a quad-core 2.61 GHz, 4GB RAM machine running Windows 8 and Java as programming language. In Table 2, we describe he Twier races used in experimens. We use wo ypes of races: a) general imeline based (ALL) which conains all he wees generaed wihin US geography wihin a ime window, provided by Twier API and b) even specific (ES) races. The even specific races were creaed as follows: For each even, we specify a se of rules. Each rule conains one or more relevan keywords and/or hashags associaed wih he even. Any wee, conaining he keywords from a rule is included in he even race. We have carefully seleced he races o cover he enire specrum of even change densiy -- from very low (22) o very high (775). Evens change densiy specifies oal number of imes all he underlying even-graphs in a race evolve every 100k wees. Traces are read in heir chronological order o mimic he real-ime arrival of he wees. Table 2: Deails of Daases if (Id!=null) mergethread(id1, Id2) Even #of Twees Evens change densiy Big Daa 200k 775 changes/100k wees Nairobi complee 720k 274 changes/100k wees Syria 1.7million 182 changes/100k wees Twier Time Line (ALL) 3.2million 22 changes/100k wees Id 2 Process edge 2 Id 1 Id 2 Id= Id Discovering Base Evens To he bes of our knowledge, no previous sysem exiss ha summarizes a complee live daa sream in he absence of any user query. Therefore, we consruc he ground ruh as follows: Live evens unraveling in he daa sream are he base evens and form he ground ruh for our sysem. The objecive of our experimens is o sudy he performance of our summarizaion echnique and is abiliy o consruc meaningful conexual summary hreads for he evens given o i as ground ruh. Any algorihm discovering dense graphs as even-opics in a highly dynamic graph can be used o provide he base evens. For our experimens, we use he algorihm in [15] i efficienly discovers he evens in a live daa sream wih high precision and recall. I is shown in [15] ha he evens discovered by his sysem correlaes highly wih real world evens repored in Google news headlines. Addiionally, i discovers many oher real world evens ha do no occur in Google news headlines. The number of even-opics as well as he number of keywords in an even-opic in a daa sream depends on he dynamic even-graph G (V, E ) a ime. The se of nodes V in he graph G depends on bursiness hreshold γ and he se of edges E depends on he edge similariy hreshold λ. The defaul values are: γ=5 and λ=0.2, unless specified oherwise. The message block size m is se o 1000 and w (cf. Table 1) is se o be 75 for all he experimens. 7.2 Qualiy of Even Summarizaion Informaive-ness: We idenify he usercover only for he pivo nodes in an even-graph (Secion 5). The premise is ha he imporan keywords in he even-graph are likely o be pivo nodes. The peripheral nodes in he even-graph may or may no be covered in he even summary. We quanify he informaive-ness of he summary as follows: If here is a keyword in an even-opic ha is a proper noun and is no presen in he even summary, we coun i owards loss of precision. Precision is compued as he fracion of proper noun keywords presen in he even summary among all he proper noun keywords in he even-opic. Le here be N evens in a given race. Le Ri be he se of proper noun words in he summary of he i h even discovered by our algorihm; 1 i N. Le Bi be he se of all he proper noun keywords presen in he even-opic of even i. The precision of informaive-ness, PI, is defined as: PI = R i B i ;1 i N Figure 9: Summary Informaive-ness and Compleeness Complee-ness: We compue he fracion of oal keywords in he even-opic covered in he even summary o compue he compleeness score PC. As shown in Figure 9, PC for various races varies from 79% o 99%. PC is marginally lower han PI, as ypically noun keywords are more cenral in he even-graphs. For he ALL race, 176

10 since almos all he keywords are covered in he even summary, here is no difference in he wo scores. We see, ha our sysem achieves very high informaive-ness and complee-ness score. Sabiliy: We sudy he sabiliy of he even summaries by our algorihm for live evens. The resuls are shown in Table 3. Since differen races have differen sizes, we repor he resuls in erms of even change densiy/100k wees. Table 3: Rae of changes in even-graphs for differen races Toal evengraph changes/ 100k wees Changes (addiion in evengraph) Changes (evengraph break) No change in summary (addiions) Big daa Nairobi Syria ALL No change in summary (deleions) We see ha across he races, more han 80% of he changes in he even-graphs do no resul in any change in he summary. For example, for Big-Daa race, for every 775 changes in he evengraphs; even summary remains he same for 411 changes when nodes/edges or messages ge added o he even-graph and for 217 changes when a node/edge ges deleed from he even-graph, i.e., no summary change for 81% of even-graph changes. Thus, he even summary remains sable for a large fracion of changes. Summary-size: Nex, we compare he message pool size of an even wih he number of messages in is summary (summary-size). The average message pool size varies from 74 wees (for ALL race) o 1394 wees (for Syria race) as shown in Figure 11, denoing ha for an emerging even in he Twier daa sream, a large number of relaed messages are posed. The number of messages is per even-graph and no per even hread (an even hread capures he evoluion of associaed even-graph(s)). We divide he evens based on he number of keywords in he evenopic, as shown in Figure 10 and Figure 11. For ALL race, no even-opic conained more han 16 keywords. Figure 10: Average number of wees in even summary for differen even-opics sizes As expeced, summary-size increases wih increasing even-opic size (Figure 10). Similarly, number of messages peraining o an even increases wih even cluser size (Figure 11). The key insighs are: 1) Summary-size is independen of he message pool size (i.e., wees associaed wih an even-graph). I depends only on he underlying informaion. For example, for Syria race, average message pool size increase from 128 o 1394 for differen size even-opics bu he summary-size remains almos sable. The even summaries were highly meaningful. 2) Average summary-size varies from 2.61 o 7.71 wees for differen races for even-opics comprising up o a few hundred wees. Thus, our sysem discovers highly compac summaries. Figure 11: Average number of wees in he even message pool for differen sizes of even-opics We consruc alernaive summary for a discovered as follows: For an even-graph, we randomly selec a message from he message pool covering an even keyword. We keep on selecing messages ill each keyword in he even graph is covered by a leas one message. This naïve mehod serves as a baseline o compare he performance of our sysem. Since we selec he messages randomly in naïve mehod, qualiaively, he summary by naïve mehod was markedly inferior for almos all he evens. The oher issues wih his naïve mehod were; 1) A significan number of redundan messages occur in he even summary. For many evens, he summary size was more han wice he summary discovered by our sysem.; 2) he naïve mehod will no discover he even hreads, exposing differen even faces. To furher compare he qualiy of even summary we compared he summary discovered based on our approach wih Google News headlines (for he evens which also appear in Google headlines). In Table 4, we presen he comparisons beween a few of he Google headlines wih our summary wees. Table 4: Google News Headlines Vs. Even Summary discovered by our approach Google Headline India vs New Zealand, 3rd Tes: Ashwin 6/81 hands India huge firs innings lead NASA Mission Tess Thrusers On Journey To Aseroid Pampore aack: Milians holed up inside gov building; combing operaions inensify NASA resupply mission o space saion posponed Obama pushes NASA o send humans o Mars by 2030s 1000 aseroids heading owards Earh; conspiracy heoriss claim end of he world is near! Summary based on our approach India vs New Zealand, 3rd Tes: Ashwin 6/81 hands India huge firs innings lead hps://.co/ih5otkoliy NASA probe ess hrusers on journey o aseroid Bennu - Zee News: NASA probe ess hrusers on jou... hps://.co/2tv0xc71cj Smoke and dus engulf a governmen building where suspeced milians have fighing wih Indian forced in Pampore Alanic Sorm Sysem Delays NASA Resupply Launch o Space Saion via NASA hps://.co/wdwmqwhz7k Can he U.S. Really Ge Asronaus o Mars by 2030?: Presiden Obama renewed his call o send Americans o h... hps://.co/zesadfvdnz Aseroid mission: 1000 space rocks heading owards Earh Daily Sar hps://.co/exdvsjozly #Aseroid 177