Fully Dynamic Algorithm for Top-k Densest Subgraphs

Transcription

1 Fully Dynamic Algorihm for Top-k Denses Subgraphs Muhammad Anis Uddin Nasir 1, Arisides Gionis 2, Gianmarco De Francisci Morales 3 Sarunas Girdzijauskas 4 Royal Insiue of Technology, Sweden Aalo Universiy, Finland Qaar Compuing Research Insiue, Qaar 1 anisu@kh.se, 2 arisides.gionis@aalo.fi, 3 gdfm@acm.org, 4 sarunasg@kh.se ABSTRACT Given a large graph, he denses-subgraph problem asks o find a subgraph wih maximum average degree. When considering he op-k version of his problem, a naïve soluion is o ieraively find he denses subgraph and remove i in each ieraion. However, such a soluion is impracical due o high processing cos. The problem is furher complicaed when dealing wih dynamic graphs, since adding or removing an edge requires re-running he algorihm. In his paper, we sudy he op-k denses-subgraph problem in he sliding-window model and propose an efficien fully-dynamic algorihm. The inpu of our algorihm consiss of an edge sream, and he goal is o find he node-disjoin subgraphs ha maximize he sum of heir densiies. In conras o exising sae-of-he-ar soluions ha require ieraing over he enire graph upon any updae, our algorihm profis from he observaion ha updaes only affec a limied region of he graph. Therefore, he op-k denses subgraphs are mainained by only applying local updaes. We provide a heoreical analysis of he proposed algorihm and show empirically ha he algorihm ofen generaes denser subgraphs han sae-of-he-ar compeiors. Experimens show an improvemen in efficiency of up o five orders of magniude compared o sae-of-he-ar soluions. 1 INTRODUCTION Finding a subgraph wih maximal densiy in a given graph is a fundamenal graph-mining problem, known as he denses-subgraph problem. Densiy is commonly defined as he raio beween number of edges and verices, while many oher definiions of densiy have been used in he lieraure [7, 28, 30, 31]. The denses-subgraph problem has many applicaions, for example, in communiy deeion [11, 14], even deecion [2], link-spam deecion [18], and disance query indexing [1]. In applicaions, we are ofen ineresed no only in one denses subgraph, bu in he op-k. The op-k denses subgraphs can be verex-disjoin, edge-disjoin, or overlapping [6, 17]. Differen objecive funcions and consrains give rise o differen problem formulaions [6, 17, 32]. In his work, we choose o maximize he sum of he densiies of he k subgraphs in he soluion. In addiion, Permission o make digial or hard copies of all or par of his work for personal or classroom use is graned wihou fee provided ha copies are no made or disribued for profi or commercial advanage and ha copies bear his noice and he full ciaion on he firs page. Copyrighs for componens of his work owned by ohers han ACM mus be honored. Absracing wih credi is permied. To copy oherwise, or republish, o pos on servers or o redisribue o liss, requires prior specific permission and/or a fee. Reques permissions from permissions@acm.org. CIKM 17, November 6 10, 17, Singapore. 17 ACM. ISBN /17/11...$15.00 DOI: hps://doi.org/ / (a) Figure 1: For he graph in Figure 1a, we are ineresed in exracing he op-3 denses subgraphs. Consider he arrival of an edge shown in red. Figure 1b shows he op-3 denses subgraphs afer he arrival. The objecive is o design an algorihm ha can efficienly mainain he denses subgraphs while keeping he number of updaes very low, in his case updaing only he verices in red. we seek a soluion wih disjoin subgraphs. This version of he problem is known o be NP-hard [6]. To complicae he maer, mos real-world graphs are dynamic and rapidly changing. For insance, Facebook users are coninuously creaing new connecions and removing old ones, hus changing he nework srucure. Twier users produce poss a a high rae, which makes old poss less relevan. Given he dynamic naure of many graphs, here we focus on a sliding-window model which gives more imporance o recen evens [4, 12, 13]. Finding he opk denses subgraphs in a sliding window is of ineres o several real-ime applicaions, e.g., communiy racking [33], even deecion [26], sory idenificaion [2], fraud deecion [8], and more. We assume he inpu o he sysem arrives as an edge sream, and seek o exrac he k verex-disjoin subgraphs ha maximize he sum of densiies [6]. A naïve soluion involves execuing a saic algorihm for he denses-subgraph problem k imes, while removing he denses subgraph in each ieraion. However, such a soluion is impracical as i requires o execue he algorihm k imes for each updae. An alernaive soluion o our problem is o use a dynamic denses subgraph algorihm in a pipeline manner, where he oupu of an algorihm insance serves as inpu o he following one. In his case, he graph and he insances of he algorihm are replicaed independenly across k insances of he algorihm, resuling in a high memory and processing cos. In his paper, we propose a fully-dynamic algorihm ha finds an approximae soluion. The proposed algorihm follows a greedy approach and updaes he densiies of he subgraphs conneced o verices affeced by edge operaions (addiion and removal). The (b)

2 algorihm is efficienly designed based on key properies of denses subgraphs, and i is compeiive agains oher recen dynamic algorihms [9, 15, 24]. Firs, our algorihm relies on he observaion ha only highdegree verices are relevan for he soluion. As many naural graphs have a heavy-ailed degree disribuion, he number of highdegree verices in a graph is relaively smaller han he number of low-degree ones. This simple observaion enables pruning a major porion of he inpu sream on-he-fly. Second, he verices ha are par of a denses subgraph are conneced srongly o each oher and weakly o oher pars of he graph. This enables independenly mainaining and locally updaing muliple subgraphs. Figure 1 provides an example which demonsraes his inuiion. The algorihm racks muliple subgraphs on-he-fly wih he help of a newly defined daa srucure called snowball. These subgraphs are sored in a bag, from which he k subgraphs wih maximum densiies are exraced. The algorihm runs in-place, and does no require muliple copies of he graph, hus making i memoryefficien. The one-pass naure of he algorihm allows exracing op-k denses subgraphs for larger values of k. We provide a heoreical analysis of he proposed algorihm, and show ha he algorihm guaranees 2-approximaion for he firs denses subgraph (k = 1) while providing a high-qualiy heurisic for k > 1 compared o oher soluions. Experimenal evaluaion shows ha our algorihm ofen generaes denser subgraphs compared o he sae-of-he-ar algorihms, due o he fac ha i mainains disconneced subgraphs separaely. In addiion, he algorihm provides improvemen in runime up o hree o five orders of magniude compared o he sae-of-he-ar. In summary, we make he following conribuions: We sudy he op-k denses verex-disjoin subgraphs problem in he sliding-window model. We provide a brief survey on adaping several algorihms for denses subgraph problem for he op-k case. We propose a scalable fully-dynamic algorihm for he problem, and provide a deailed analysis of i. The algorihm is open source and available online, ogeher wih he implemenaions of all he baselines. 1 We repor a comprehensive empirical evaluaion of he algorihm in which i significanly ouperforms previous sae-ofhe-ar soluions by several orders of magniude, while producing comparable or beer qualiy soluions. 2 PRELIMINARIES In his secion, we presen our noaion, revisi basic definiions, and formulae he op-k denses subgraphs problem. Consider an undireced graph G = (V, E) wih n = V verices and m = E edges. The neighborhood of v V is defined as N (v) = {u (v,u) E}, and is degree as d(v) = N (v). For a subse S V we define E(S) o be he se of edges whose boh endpoins are in S, and G(S) = (S, E(S)) he subgraph induced by S. The inernal degree of a verex v wih respec o S V is defined by d S (v) = N (v) S. 1 hps://gihub.com/anisnasir/topkdensessubgraph Finally, for a subse of verices S V we define is densiy ρ S by ρ S = E(S). (1) S Noe ha he densiy of any subgraph is equal o half of is average inernal degree. Definiion 2.1 (Denses subgraph). Given an undireced graph G = (V, E), he denses subgraph S is a se of verices ha maximizes he densiy funcion, i.e., S = arg max S V ρ S. (2) We say ha an algorihm A compues an α-approximaion of he denses subgraph if A compues a subse S V such ha ρ S 1 α ρ S, where S V is he denses subgraph of G. Nex we inroduce oher conceps relaed o denses subgraph: graph core, core decomposiion, and induced core subgraph of a verex. Definiion 2.2 (j-core). Given an undireced graph G = (V, E) and an ineger j, a j-core of G is a subse of verices C V so ha each verex v C has inernal degree d C (v) j, and C is maximal wih respec o his propery. Definiion 2.3 (Core decomposiion). A core decomposiion of a graph G = (V, E) is a nesed sequence { C i } of cores V = C 0 C 1... C l, (3) where each C i is a j-core for some j. Definiion 2.4 (Core number). Given a core decomposiion V = C 0 C 1... C l of a graph G = (V, E), he core number κ(v) of a verex v is he larges j such ha v C and C is a j-core. By overwriing noaion, he core number κ(c) of a core C is he larges j for which C is a j-core. Addiionally, we use κ S (v) o denoe he core number of a verex v in he subgraph induced by S. The larges core (or main core) of a subgraph of G(S) = (S, E(S)) is denoed by C l (S), while he main core of G is simply denoed by C l. Noe ha he densiy of a j-core is a leas j /2, as each verex in he core has degree a leas j and each edge is couned wice. This observaion implies ha he main core of a graph is a 2-approximaion of is denses subgraph. Lemma 2.5. Consider he core decomposiion of a graph G, i.e., C 1 C 2... C l. The maximum core C l is 2-approximaion of he denses subgraph of G [21]. Proof. Le S be he denses subgraph of G having densiy ρ S. Every verex in G(S ) has degree a leas ρ S ; oherwise a verex wih degree smaller han ρ S can be removed o obain an even denser subgraph. Thus, S is a ρ S -core. Given he core decomposiion of G, we know ha ρ Cl 1 2 κ(c l). We wan o show ha ρ Cl 1 2 ρ S. Assume oherwise, i.e., ρ C l < 1 2 ρ S. Then κ(c l ) < ρ S. I follows ha S is a higher-order core han C l, a conradicion. The concep of a core subgraph I(v) induced by a verex v [23, 27] is also perinen o our analysis.

3 Definiion 2.6 (Induced core subgraph). Given a graph G = (V, E) and a verex v V, he induced core subgraph of verex v, denoed by I(v), is a maximal conneced subgraph conaining he verex v s.. he core number of all he verices in I(v) is equal o κ(v). In oher words, he induced subgraph conains all verices ha are reachable from v and have he same core number κ(v). All previous definiions apply o saic graphs. Le us now focus on dynamic graphs. In paricular, we consider processing a graph in he sliding window edge-sream model [13]. According o his model, he inpu o our problem is a sream of edges. The edge e i is he i-h elemen of he sream. Equivalenly, we say ha edge e i has imesamp i. A sliding window W (x), defined a ime and of size x, is he se of all edges ha arrive beween e x+1 and e, W (x) = {e i, i [ x + 1, ]}. (4) For each edge e i = (u,v), we consider ha u and v appear a ime i, and we use V (x) o denoe he se of verices ha appear in a lengh-x sliding window a ime. The graph in a lengh-x sliding window a ime is hen defined o be G (x) = (V (x),w (x)). We are now ready o formally define he problem ha we consider in his paper, i.e., finding he op-k denses subgraphs in sliding window. We firs define he problem in a saic seing. Definiion 2.7. Given an undireced graph G = (V, E) and an ineger k > 0, he op-k denses subgraphs of G is a se of k disjoin maximal se of verices S = {S 1,..., S k } ha maximize he sum of is densiies: k ρ k (S) = max ρ Si, for all S i S subjec o i=1 here is no S j S i ρ Sj ρ Si, for all S i, S j S (5) S i S j =, for all i, j {1... k}, i j. (6) As already shown by Balalau e al. [6], he problem defined aboce is NP-hard, for any k > 1. The problem we consider in his paper is he following. Problem 2.8. Given a graph sream {e i } and a sliding window lengh x, mainain he op-k denses subgraphs ρ k (S) of he graphg (x), a any given ime. 3 BACKGROUND In his secion we presen a brief review over several algorihms for finding dense subgraphs. Addiionally, we discuss how hese mehods can be used for solving Problem 2.8. Denses subgraph in saic graphs. Finding he denses subgraph according o he densiy definiion (1) can be solved in polynomial ime. An elegan soluion involving reducion o he minimumcu problem was given by Goldberg []. As he fases algorihm o solve he minimum-cu problem runs in O(nm) ime [25], Goldberg s algorihm is no scalable o large graphs. Asahiro e al. [3] and Charikar [10] propose a linear-ime algorihm ha provides a facor-2 approximaion. This algorihm ieraively removes he verex wih he lowes degree in each ieraion, unil lef wih an empy graph. Among all subgraphs considered during his verex-removal process, he algorihm reurns he denses. The ime complexiy of his greedy algorihm is O(m + n). Bahmani e al. [5] propose a MapReduce version of he greedy algorihm, wih approximaion raio 2(1 +ϵ), while making O(log 1+ϵ n) passes over he inpu graph. Core decomposiion in saic graphs. The core decomposiion of a graph G is he process of idenifying all cores of G, as defined in 2.3. Baagelj and Zaversnik [7] propose a linear-ime algorihm o obain he core decomposiion. The algorihm firs considers he whole graph and hen repeaedly removes he verex wih he smalles degree. The core number κ(v) of a verex v is se equal o he degree of v a he momen ha v is removed from he graph. Denses subgraph in evolving graphs. There is a growing body of lieraure on finding dense subgraphs in evolving graphs [9, 15, 19, 24]. We focus mainly on he deerminisic algorihm for denses subgraph in evolving graphs. For insance, Epaso e al. [15] propose an efficien algorihm for compuing he denses subgraph in he dynamic graph model [16]. Their work assumes ha edges are insered ino he graph adversarially bu deleed randomly. Even hough he algorihm can, in pracice, handle arbirary edge deleions, is approximaion guaranees hold only under he random edge-deleion assumpion. The algorihm is similar o he one by Bahmani e al. [5], and i provides a 2(1 + ϵ) 6 -approximaion of he denses subgraph, while requiring polylogarihmic amorized cos per updae wih high probabiliy. Core mainenance in evolving graphs. Saríyüce e al. [27] propose he raversal algorihm, for efficien core mainenance. This algorihm idenifies a small se of verices ha are affeced by edge updaes and processes hese verices in linear ime in order o mainain a valid core decomposiion. Li e al. [23] propose an efficien hree-sage algorihm for core mainenance in large dynamic graphs. The algorihm mainains a core decomposiion of an evolving graph by applying updaes o very few verices in he graph. Once hese few verices have been idenified, he algorihm compues he correc core numbers via a quadraic operaion. Finding op-k denses subgraphs. The problem of finding op-k denses subgraphs has been mainly sudied for finding overlapping subgraphs in saic graphs [6, 17]. Nex, we discuss how he algorihms presened above (Charikar [10], Baagelj and Zaversnik [7], Bahmani e al. [5], Saríyüce e al. [27], Li e al. [23], and Epaso e al. [15]) can be used o produce op-k denses subgraphs. Our firs observaion is ha a se of k dense subgraphs can be obained from any algorihm ha finds he denses subgraph by k repeaed invocaions. The ime complexiy of compuing a se of k dense subgraphs in his manner is simply he runningime complexiy of he denses-subgraph algorihm muliplied by k. From a pracical poin of view, all he saic algorihms menioned are no in-place algorihms, and hus require copying he whole graph for processing. Furhermore, when a verex or edge is added or deleed from he graph, he whole k dense subgraph compuaion has o be repeaed. The second observaion is ha, by using he algorihm of Epaso e al. [15], we can obain a se of k dense subgraphs by running k insances of he fully-dynamic algorihm in a pipeline manner. The idea is o run k insances of he algorihm in which he oupu of each insance i {0... k 1} is fed ino he nex (i + 1) insance as a removal operaion. The pipeline version of he algorihm requires keeping k copies of he inpu graph and an addiional

4 O(kn) size space for bookkeeping. Noe ha he oupu of each insance of he pipeline migh cascade, which requires updaing he verices in all he insances. In paricular, verices ha cease o be par of soluion in upsream insances need o be added in downsream insances. Likewise he verices ha become par of denses subgraphs in upsream insances need o be removed from a downsream insances. The modificaion of he algorihm, as discussed above, is expensive in erms of memory, as i requires replicaing he graph and he algorihm s srucures k imes. In addiion, running and mainaining k parallel insances makes he algorihm compue-inensive. Finally, o mainain op-k denses subgraphs in evolving graphs, we can leverage algorihms for core decomposiion mainenance [23, 27]. by leveraging Lemma 2.5, Thus, he idea is o find and mainain op-k disjoin maximum cores. In order o mainain such cores we run a single insance of he algorihm by Saríyüce e al. [27] or Li e al. [23] ha mainains he core number of all he verices in he graph. We hen exrac he op-k denses subgraphs by: (i) exracing he main core, (ii) removing he verices in he main core and updaing he core number for res of he verices, and (iii) repeaing he seps unil k subgraphs are exraced. 4 ALGORITHM The main idea of our algorihm is o mainain and updae muliple dense subgraphs online. These subgraphs are candidaes for he op-k denses subgraphs. However, mainaining muliple subgraphs for fully-dynamic sreams requires answering wo ineresing quesions: (i) how o reduce he search space of he soluion, and (ii) how o spli he whole graph ino subgraphs. To answer he aforemenioned quesions, we make wo observaions. Firs, since dense subgraphs are formed by relaively highdegree verices, one can find dense subgraphs by keeping rack of hese high-degree verices only. Second, hese subgraphs can be updaed locally upon edge updaes, wihou affecing he oher pars of he graph. Based on hese observaions, we develop an algorihm ha reduces he soluion space by considering only high-degree verices, and divides he whole graph ino smaller subgraphs, each represening a dense subgraph. The op-k denses subgraphs among he candidae subgraphs provide a soluion for Problem 2.8. We begin by designing an algorihm o find he denses subgraph (op-1) and hen we exend i o find he op-k denses subgraphs. Our algorihm migh no be he mos efficien soluion for he (op-1) denses-subgraph problem per se, bu i provides efficien oucomes when exended o solve he op-k denses-subgraph problem. We sar by defining some properies of he denses subgraph ha we leverage in our algorihm. Lemma 4.1. Given an undireced graph G = (V, E), he denses subgraph S V wih densiy ρ S, all he verices v S have degree d S (v) ρ S. Proof. This lemma holds according o he definiion of opimal densiy. In an opimal soluion, each verex has degree larger han or equal o ρ S. Oherwise, removing he verex from he subgraph will increase he average degree, and hus he densiy, of he subgraph. Given Lemma 4.1, a any ime, he denses subgraph S of graph G conains only verices v ha have degree d(v) d S (v) ρ S. Then, given G and ρ S, we wan o compue he denses subgraph afer he addiion of a new verex u V a ime + 1. Le d(u) be he degree of verex u V +1 and S+1 be he denses subgraph a ime + 1. For simpliciy, assume ha he graph G +1 is conneced. According o Lemma 4.1, for any verex u o be he par of he denses subgraph, is inernal degree saisfies d S +1 (u) ρ S +1. As verex u is added o he graph he new densiy is always greaer, i.e., ρ S +1 ρ S. Therefore, for verex u o be he par of denses subgraph, he degree of verex u should saisfy d(u) ρ S. Therefore, if he degree of verex u is lower han he ρ S, i canno be par of he denses subgraph ρ S +1 and can be ignored. Now, considering he case when d(u) ρ S. Adding he verex u o denses subgraph will updae he densiy: ρ S +1 E(S ) + d S (u) S +1. (7) We also know ha ρ S +1 ρ S, which means ha E(S ) + d S (u) S +1 E(S ) S. (8) From his inequaliy i follows d S (u) ρ S. Using hese properies, we ignore he verices of he new edge ha have degree lower han he curren esimae of he densiy. Furher, we are ineresed in finding he main core in he remaining subgraph of high-degree verices, as i represens a 2- approximaion of he denses subgraph according o Lemma 2.5. To his end, we propose a new daa srucure ha relies on Lemma 4.1, he snowball. 4.1 Snowball A snowball D is an incremenal daa srucure ha sores a srongly conneced subgraph, which mainains he following invarians: The core number κ D (v) of each verex v D inside a snowball is equal o he main core (C l (D)) of he snowball. All he verices in he snowball are conneced. These invarians ensure ha all he verices in he snowball have he same core number, which is he main core of he snowball by definiion. A snowball mainains hese invarians while handling he following graph updae operaions: a) adding/removing a verex, and b) adding/removing an edge. 4.2 Bag of snowballs The high-degree verices in he graph are assigned o a snowball. As hese verices are no srongly conneced, hey migh end up in differen snowballs. We sore each of hese disconneced snowballs in a daa srucure called he bag, denoed by B. The bag ensures ha each snowball is verex disjoin. Furher, he bag provides an addiional operaion: exracing he denses snowball among he se of snowballs. The densiy of he exraced snowball is he maximal densiy, which is he hreshold separaing he high-degree verices from he low-degree ones. We denoe his esimae of he maximal densiy ρ S. The bag is a supergraph which conains a se of snowballs and all he edges beween he snowballs. We mainain all he core numbers of he nodes in he bag by leveraging a core decomposiion

5 Bag Bag D 1 D 2 D 1 D 2 Figure 2: Example showing ha he bag requires mainaining he core number of he verices. Iniially, he bag conains wo snowballs wih core number 2, i.e., D 1 and D 2. Consider he arrival of he edges shown in red. The greedy assignmen of he edges migh skip creaing a new snowball wih core number 3, using he four nodes in he middle. algorihm (see Secion 3). The core number of each node in he bag is used o ensure ha each node has he maximum possible core number. Figure 2 provides an example explaining one of he issues ha may arise. In he example, he bag conains wo snowballs, however, i is possible o produce a new snowball wih a larger core number. Nex, we define he algorihms o updae his daa srucure upon graph updaes. 4.3 Addiion operaions Verex addiion: As discussed in Secion 2, he updaes appear in he form of an edge sream. Here, we define he verex addiion algorihm ha acs as a helper for edge addiion. The algorihm is riggered when a leas one of he endpoins of a new edge is a high-degree verex. In paricular, here are wo cases o consider: 1) he bag already conains he high-degree verex, and 2) he bag does no conain he high-degree verex. In boh cases, he goal is o add he new verex o one of he snowballs (if needed). Algorihm 1 defines he algorihm for verex addiion. For he firs case, he algorihm scans he bag o find he snowball ha conains he verex and reurns i. For he second case, he algorihm firs idenifies he candidae snowballs, hen i assigns he verex o one of he candidae snowballs. The candidae snowballs are he ones having he main core number smaller han or equal o he inernal degree of he new verex (κ u (D) C l (D)). Among he candidae snowballs, he new node is assigned o he snowball wih maximum inernal degree d u (D), breaking ies randomly. Once a verex is added o a snowball, he core number of he snowball may increase. This change requires removing he verices wih core number lower han he main core of he snowball. This procedure can be implemened efficienly in linear ime by soring he verices based on heir degree similar o bin sor. 2 Verificaion. Due o he greedy assignmen of verex o he snowball, i is possible ha he verex ends up no having he highes possible core number. For example, Figure 3 shows an example where he greedy assignmen does no resul in opimal soluion. Therefore, afer addiion, he algorihm ensures ha he core number of he snowball, where he new verex is added, equals he core number of he new verex in he graph. The algorihm verifies ha he core number of he added verex by comparing i wih he core number of he verex in he bag. Noe ha he bag represens he supergraph conaining all he snowballs and edges beween he snowballs. If he core number wihin he bag is larger han he one in he snowball, he algorihm merges all he 2 The MainainInvarian mehod a line 12 of Algorihm 1. Figure 3: Example showing ha arrival of a new edge allows he verex ha is par of snowball D 2 o become par of snowball D 1, which has greaer core number (3). Algorihm 1 Verex Addiion in he Bag of Snowballs 1: procedure addobag(u) 2: S 3: for D i B do 4: if u D i hen 5: reurn D i Firs Case 6: if (d Di (u) C l (D i ) and ρ S < ρ Di ) hen 7: S D i 8: if S = hen 9: S {u } 10: else 11: S S {u } 12: MainainInvarian(S ) 13: reurn S Second Case Algorihm 2 Mainain Invarian 1: procedure MainainInvarian(D i ) 2: do 3: repea f alse 4: for u D i do 5: if ((κ Di (u) < C l (D i )) hen 6: D i D i \{u } 7: if (d(u) ρ S ) hen 8: addobag(u) 9: repea rue 10: while repea snowballs in he induced subgraph of newly added verex. As all he verices in he induced subgraph have he same core number, merging hem ensures creaing a larger snowball. 3 We leverage he core decomposiion algorihm by Saríyüce e al. [27] for he implemenaion. Algorihm 3 Fix Main Core 1: procedure fixmaincore 2: for D i B do 3: if (D i I(u) > 0) hen 4: D u D u D i 5: MainainInvarian(D u ) Theorem 4.2. Given he bag B, he algorihm ensures ha B conains he main core of he graph wihin one of he snowballs afer he verex addiion. Proof. Le us assume ha a ime he bag conains he main core of he graph. Now, we need o show ha a ime + 1, afer he node addiion, he bag sill conains he main core of he graph. In general, verex addiion mehod is called whenever here is an edge addiion. The only way for he new verex o affec he main core of he graph is ha he new verex is he par of he main core. 3 The FixMainCore mehod a line 15 of Algorihm 4.

6 Algorihm 4 Edge Addiion 1: procedure addedge((u,v)) 2: if ((d(u) < ρ S )) and (d(v) < ρ S )) hen 3: reurn 4: else if ((d(u) ρ S ) and (d(v) < ρ S )) hen 5: D u addobag(u) 6: else if ((d(u) < ρ S ) and (d(v) ρ S ) hen 7: D v addobag(v) 8: else 9: D u addobag(u) 10: D v addobag(v) 11: if (D u = D v ) hen 12: D u D u (u, v) 13: MainainInvarian(D u ) 14: else if (v I(u)) hen 15: fixmaincore(u) Afer he addiion of he verex in he bag, he algorihm verifies he core number by comparing he core number of he verex in he bag and he snowball. If he core number of he verex in he bag is greaer, he algorihm merges he snowballs conaining he verices in he induced graph of he new node in he bag. This creaes a new snowball wih a greaer core number. Edge addiion: In his case, he sae of he bag is only affeced if a leas one of he verices in he new edge is a high-degree verex. In paricular, here are wo cases o consider: a) only one of he verices is a high-degree verex and b) boh he verices are highdegree verices. For he firs case, he algorihm leverages he verex addiion mehod o add he verex o he bag of snowballs. For he second case, when boh verices are added o he bag of snowballs, he algorihm verifies ha he main core exiss in he bag. When boh verices are added o he same snowball, he algorihm adds he new edge o he same snowball and ensures ha he invarian holds. Conversely, when he wo verices are added o wo differen snowballs, he algorihm verifies if he verices exis in each ohers induced subgraphs and fixes he main core for boh he verices. Algorihm 4 describes he algorihm for edge addiion. Theorem 4.3. Algorihm 4 mainains he main core of he graph in one of he snowballs inside he bag. Proof. The proof for all he cases, oher han he case when boh he verices of he new edge are assigned o wo differen snowballs, is similar o he verex addiion algorihm. Therefore, we consider he case when boh end verices of he added edge are added o wo differen snowballs. For his case, we rely on he graph in he bag. We check if boh verices are in he same core graph in he bag, and fix he core number of he verices if hey belong o he same induced subgraph. This ensures creaing he graph wih he highes core number. 4.4 Removal operaions Verex removal: Similarly o he addiion operaions, we firs define he procedure for removing a verex from he bag of snowballs. The verex removal mehod is used as a subrouine for he edge removal process. A verex is only removed from he bag when is degree becomes lower han he maximal densiy. Therefore, according o Lemma 2.5, he removed verex canno be par of he main core. The algorihm removes he verex from he snowball wihin a bag wihou doing any oher operaion. Algorihm 5 Edge Deleion 1: procedure removeedge((u,v)) 2: if ((d(u) < ρ S ) and (d(v) < ρ S )) hen 3: reurn 4: if ((d(u) < ρ S ) and (d(u) + 1 ρ S )) hen 5: removeverex(u) 6: reurn 7: if ((d(v) < ρ S ) and (d(v) + 1 ρ S )) hen 8: removeverex(v) 9: reurn 10: for D i B do 11: if (D i (u, v) ) hen 12: D i D i \(u, v) 13: for x D i do 14: if (κ B (x) > κ Di (x)) hen 15: fixmaincore(x) 16: MainainInvarian(D i ) Edge removal: Now we urn our aenion o edge deleion, which follows he same paern as edge addiion. Again, we leverage he bag o ensure ha here exis a snowball wih a core number equal o he main core of he graph. Algorihm 5 shows he algorihm for edge deleion. The bag does no require any updae when eiher one of he verices is low-degree or if boh he verices belong o wo differen snowballs. Therefore, we consider he case when one of he verices lie a he boundary of high-degree verices. Tha is, he edge deleion moves he verex from he high-degree o low-degree. In his case, he algorihm only requires removing he verex from he bag, wihou performing any oher operaions. The ineresing case is where boh verices of he deleed edge are high-degree and belong o he same snowball. In his case, he algorihm removes he edge from he snowball. Furher, i verifies and updaes (if needed) he core number of he verices affeced by he updae in he snowball. Lasly, edge deleion migh reduce he maximal densiy, and hus require adding o he bag new verices whose degree is now greaer han he new maximal densiy. Theorem 4.4. Given he bag conaining a snowball wih he same core number as he main core of he bag, afer he edge deleion, Algorihm 5 mainains he main core of he graph in one of he snowballs inside he bag. Proof. An edge removal affecs he bag of snowballs only when boh he verices corresponding o he removed edge belong o he same snowball. In his case, edge removal migh reduce he core number of all he verices in he snowball. The algorihm ensures ha he verices in he snowball have heir maximum possible core number by comparing heir core number in he snowball wih he core number in he bag. Therefore, by verifying and fixing he core numbers, he algorihm mainains he main core of he graph in one of he snowballs inside he bag. 4.5 Fully-dynamic op-k denses subgraphs Now ha we have a fully dynamic algorihm for finding he denses subgraph in sliding windows, we move our aenion o he op-k denses-subgraph problem. Le ρ S ρ 1... ρ z 1 represen he densiies of he z subgraphs in he bag, where ρ S is he densiy of he denses subgraph. The bag conains he verices ha have a degree greaer han he densiy ρ S. To ensure ha he bag conains a leas k subgraphs,

7 (a) Figure 4: Example showing op-k version of he algorihm. The densiies of op-3 subgraphs, before he arrival of he new edge in he bag are 1.8, 1.5 and The algorihm sores all he verices in he bag wih degree greaer han equal o 1.25, as shown in Figure 4b. Afer he arrival of he new edge, he updae only affecs one of he subgraphs in he bag. we modify he algorihm o keep all he verices wih degree greaer han ρ k 1. The only modificaion required is o replace ρ S by ρ k 1 in Algorihm 2, Algorihm 4 and Algorihm 5. Furher, we leverage he prioriy queue o exrac ρ k 1 from he bag of snowballs. This simple modificaion ensures ha he bag conains a leas k subgraphs and enables accessing he op-k denses subgraphs in he sliding window model. Noe ha he new definiion of high-degree verices is relaed o he densiy of he k-h op denses subgraph. The modified algorihm guaranees ha he bag conains he verices wih a degree greaer han he ρ k 1 afer any graph updae. These graph updaes include boh edge addiions and removals. I is necessary for he algorihm o consider edge addiions, as hey migh affec he value of ρ k 1 by merging muliple subgraphs. Similarly, edge deleions have o be considered, as hey migh affec he value of ρ k 1 by reducing he densiy of any of he op-k denses subgraphs, merging muliple subgraphs, or spliing a subgraph. As he algorihm ensures keeping he main core in he bag, i guaranees 2-approximaion for he firs denses subgraph (k = 1) while providing a high-qualiy heurisic for k > 1 compared o oher soluions. We provide an example in Figure 4 for he op-k denses subgraph algorihm by expanding on he Figure 1a. We conclude his secion wih he heorem ha provides a bound for our proposed algorihm. These bounds can be generalized for any algorihm ha adaps o a soluion for denses subgraph problem for he op-k case (see secion 3 for examples). Theorem 4.5. Given an ineger k, he bag conains a se of subgraphs ha provides 2k-approximaion of he op-k denses-subgraph problem. Proof. We know ha he graph does no conain any subgraph wih densiy greaer han he opimal densiy (ρ S ). Therefore, we know ha he sum of densiies for he op-k denses-subgraph problem is upper bounded by k ρ S. Furher, he bag conains ρ S, which provides a 2-approximaion of he denses subgraph. This implies ha he sum of densiies of op-k denses subgraph in he bag 1 2 ρ S. Puing above wo observaions ogeher, we can clearly see ha he bag provides a soluion ha is a 2k-approximaion for he problem. (b) 4.6 Daa srucure The proposed algorihm requires accessing he neighborhood informaion of every node. Specifically, we are ineresed in performing hree queries on a given verex: a) exrac is degree, b) exrac all is neighbors, and c) given densiy ρ S, exrac all he verices wih degrees greaer han he densiy ρ S. Verex map: To answer he firs wo queries, we need o sore he neighborhood informaion for all verices. We sore he informaion in a hashmap wih keys being he verex idenifiers and values being he neighbors of each verex. Verex map allows performing search and updae operaions in amorized consan ime. Degree able: For he hird query, we need o order he verices by heir degrees. We use bin sor o order he verices by heir degrees, which enables exracing he verices wih degrees greaer han he densiy ρ S in consan ime. 5 DISCUSSION Mos of he soluions ha leverage a saic algorihms [5, 7, 10] require ieraing over he enire graph k imes upon any updae. Comparaively, our algorihm mosly ouches a limied region in he graph for any updaes, which makes our algorihm perform significanly faser. The op-k denses subgraph algorihm ha adaps o Epaso e al. [15] requires replicaing he graph across he k insances. In addiion, updaes across k insances migh cascade and require running he algorihm k imes, similarly o he saic op-k soluions. Our algorihm ouperforms he pipeline version of he op-k algorihm boh in erms of memory and compuaion. Finally, incremenal algorihms for k-core mainenance requires removing op-k denses subgraph upon every updae in he graph. Each removal of a denses subgraph requires updaing he core number of he remaining verices. Our algorihm efficienly avoids his removal sep by mainaining he subgraphs during execuion. Performance. The proposed algorihm leverages he skew in realworld graphs and ignores a major porion of he inpu sream on he fly. In addiion, high-degree verices are sored as small subgraphs so ha each updae is ofen applied locally on hese subgraphs. This design allows our algorihm o operae on jus small porions of he graph for each updae, raher han ieraing on he whole graph. Finally, we use he k-core algorihm [23, 27], which allows each high-degree node o updae heir core number wih a complexiy independen of he graph size. These improvemens enable our algorihm o perform significanly beer han he oher sae-ofhe-ar sreaming algorihms. Memory. Our algorihm requires only O(n 2 polylogn) memory, compared o O(kn 2 polylogn) for he op-k version of [15] and O(n 2 polylogn) for he oher algorihms discussed in Secion 3. Tigh Bounds. We show via an example ha any algorihm for denses subgraph problem can only produce a k-approximaion for op-k denses subgraph problem. Consider a graph G ha conains n m verices, for any n k. The n nodes connec o form a circle. In he circle, here is one edge connecing wo non-adjacen verices. Addiionally, each of he n nodes connecs o exacly m neighbors. The inner circle of he graph G is he denses subgraph. Removing he denses subgraph leaves he res of he verices compleely disconneced. Hence, he sum of densiy will be equal o one. Now,

8 Table 1: Daases used in he experimens. Daase Symbol n m d(v) Amazon [22] AM DBLP 1 [22] DB Youube [22] YT DBLP 2 4 DB Live Journal 5 LJ Orku [22] OT Friendser [22] FR consider he case when he denses subgraph is no removed firs. In his case, each node in he circle along wih is m neighbors creae a subgraph of densiy almos equal o one, for higher values of m. Therefore, one can reurn k such subgraphs wih he sum of densiies equal o k, which is k imes beer han he previous case. 6 EVALUATION We conduc an exensive empirical evaluaion of he proposed algorihm, and provide comparisons wih he exising soluions. In paricular, we answer he following quesions: Q1: Wha is he impac on he qualiy of subgraphs? Q2: Wha are he gains in performance? Q3: How does he algorihm perform in erms of differen inpu parameers? 6.1 Experimenal seup Daases. Table 1 shows he daases used in he experimens. The daases are seleced due o heir public availabiliy. We evaluae all he algorihms in he sliding window model. The number of edges in he sliding window is an inpu parameer. Merics. We evaluae he qualiy and efficiency of he algorihms. We assess he qualiy by he objecive funcion, i.e., he sum of densiies of he subgraphs produced by an algorihm. We evaluae he efficiency of an algorihm by reporing he average updae ime and he memory usage. The average updae ime is he average ime i akes o move he sliding window. This includes adding he new edge, removing he oldes edge, and updaing he op-k denses subgraphs. We repor he memory usage as he average percenage of occupied memory. Algorihms. Table 2 shows he noaions used for differen algorihms. The algorihms by Bahmani e al. [5] and Epaso e al. [15] require an addiional epsilon parameer for execuion, which provides a rade-off beween qualiy and execuion ime. Here we use he defauls proposed by he auhors. Sream ordering. We consider wo commonly used sream ordering schemes [29]: BFS: The ordering is a resul of a breadh-firs search saring from a random verex. DFS: The ordering is a resul of a deph-firs search saring from a random verex. Experimenal environmen. We conduc our experimens on a machine wih 2 Inel Xeon Processors E and 128GiB of 4 hp://konec.uni-koblenz.de/neworks/dblp coauhor 5 hp://konec.uni-koblenz.de/neworks/livejournal-links Table 2: Noaion for he op-k algorihms. Symbol Reference Algorihm Top-1 Approx. In-place Guaranees CH Charikar [10] 2 No V B Baagelj and Zaversnik [7] 2 No BB ϵ Bahmani e al. [5] 2(1 + ϵ) No RL Li e al. [23] 2 Yes T R Saríyüce e al. [27] 2 Yes AE ϵ Epaso e al. [15] 2(1 + ϵ) 6 No GR his paper 2 Yes memory. All he algorihms are implemened in Java and execued on JRE 7 running on Linux. The source code is available online Experimenal resuls Q1: In his experimen, we compare he qualiy of he resuls produced by he differen algorihm as measured by he objecive funcion. In order o be able o run mos of he algorihms, we use he smaller daases, i.e., AM, DB 1, YT, and DB 2. As RL and TR produce same resuls in erms of qualiy, we only repor he resuls for one of hem. Due o space consrains, we show he resuls only wih he DFS ordering. However, we achieve similar resuls also wih he BFS one. We se he size of he sliding window x = 100k. For he saic algorihms, we execue hem in micro-baches in which he op-k denses subgraph is recompued afer 1k edge updaes, which gives hem a subsanial advanage. For BB ϵ and AE ϵ, we use ϵ = We se he maximum execuion ime o 7 days, due o which he AE ϵ is no able o finish on DB 2 and YT. Finally, RL and TR conain an updae phase due o he ieraive exracion of he op-k denses subgraph. This phase is expensive, as he main core consiss of high-degree verices. To alleviae his issue, we execue he exracion phase every 10k updae operaions. Figure 5 shows he qualiy resuls. Our algorihm GR achieves compeiive qualiy, ofen generaing denser subgraphs compared o all he oher algorihms. For example, for he AM daase, GR produces subgraphs ha are 1.5 imes beer han he bes algorihm among he sae-of-he-ar soluions. All he oher algorihms produce consisen resuls across all he daases. For insance, V B, RL, and TR produce op-k dense subgraphs of lowes qualiy compared o he oher algorihms. This resul is caused by he removal of he main core (backbone) of he graph in each of he k ieraions. In addiion, i shows how he problem considered in his paper, while relaed, is differen from a simple k-core decomposiion. The resuls also validae ha BB ϵ and AE ϵ provide weaker guaranees on he qualiy compared o CH. Q2: In his experimen, we urn our aenion o evaluae he efficiency of he proposed algorihm, as measured by he average updae ime and memory usage. We again selec several daases, i.e., AM, DB 1, YT, DB 2, and LJ, and execue CH, V B, BB, RL,TR, AE, and GR. We exclude he larges daases as only a few algorihms are able o handle hem. The algorihms are execued wih boh BFS and DFS ordering of he edge sream. The sliding window size is se o x = 100k edges, and k = 10 unless oherwise specified. 6 hps://gihub.com/anisnasir/topkdensessubgraph

9 Sum of densiies DB 1 CH VB GR BB 0.1 AE 0.1 TR Edge sream (DFS) AM Edge sream (DFS) DB Edge sream (DFS) YT Edge sream (DFS) Figure 5: Sum of densiies for op-10 denses subgraphs on he DB 1, AM, DB 2 and YT daases, sliding window size 100k. Table 3: Memory consumpion as a percenage of oal memory for algorihms wih DB 2 and LJ daase, sliding window size 100k. CH V B BB ϵ RL T R AE ϵ GR DB LJ Average updae ime (s) 10 3 DFS DB 1 AM DB 2 YT LJ VB CH BB 0.1 AL TR EP 0.1 GR Algorihm 10 3 BFS VB CH BB 0.1 AL TR EP 0.1 GR Algorihm Figure 6 shows he efficiency resuls (noe he logarihmic scale). The proposed algorihm, GR, ouperforms all oher ones in erms of updae ime for all he daases and boh ordering schemes. RL and T R are he slowes algorihms. In paricular, for DFS, our algorihm achieves a performance gain of five orders of magniude. This resul is noeworhy as even hough our algorihm is dependen on core decomposiion, i is sill able o bea a naïve applicaion of hose algorihms by a wide margin. This difference is due o he fac ha boh algorihms require mainaining he core number for all he verices. Addiionally, he exracion of op-k denses subgraph furher hampers heir efficiency. AlgorihmsCH, V B, and BB 0.01 perform unfavorably due o heir saic naure. In his case, our algorihm is able o achieve more han hree orders of magniude improvemen in efficiency. Algorihm AE 0.01 is bes performing among he baselines, and even ouperforms GR in one daase for one specific ordering (AM wih BFS). However, GR sill ouperforms AE 0.01 by nearly hree orders of magniude in mos oher cases. Finally, we observe he overall memory consumpion of all he algorihms (repored in Table 3). The memory requiremen of our algorihm lies beween he saic and he dynamic algorihms, i.e., AE 0.01 requires he larges amoun of memory, while he saic algorihms require he leas. These resuls are in line wih our expecaions from he discussion in Secion 3. Q3: In his experimen, we evaluae he scalabiliy of our algorihm. Firs, we execue GR wih differen values of k. Alongside, we repor he average updae ime for BB ϵ algorihm. We choose BB ϵ, raher han AE ϵ, as i can execue in mirco-baches, i.e., op-k denses subgraph every 100 edges. We se he sliding window size x = 1M, and use he YT and DB 2 daase wih DFS ordering. Figure 7 repors he average updae ime for boh algorihms. The average updae ime of GR remains consisen even for higher values of k, whereas he execuion ime of BB 0.01 increases wih he parameer. Second, we execue GR wih differen sizes of sliding window, i.e., x = 10k, 100k, 1M, and 10M. We se k = 10 for his experimen. Again, we use he YT and DB 2 daases wih DFS ordering. Figure 6: Updae ime for he algorihms on he DB 1, AM, DB 2, YT, and LJ daases, when using boh DFS and BFS ordering and k = 10. Average Updae Time (s) GR BB 0.1 YT k 10-2 Figure 7: Updae ime for GR and BB 0.01 on he YT and DB 2 daases as a funcion of k. Noe ha BB 0.01 is execued in micro-baches of size 100. Table 4: Average updae ime for he GR algorihm wih sliding windows of differen sizes x. x = 10k x = 100k x = 1M x = 10M YT 0.80ms 91.14ms 90.33ms 95.49ms DB ms 7.58ms 8.45ms 32.21ms Table 4 repors he average updae ime for differen configuraions. Increasing he size of he sliding window does no affec he average updae ime significanly. This resul validaes our claim ha mos of he updaes are local o he subgraphs, and do no require ieraing hrough he whole graph o exrac he op-k denses subgraphs. Finally, we sudy he performance of our algorihm in on he larges daases. We selec he OT and FR daases wih DFS ordering, and execue he algorihm wih a sliding window of x = 100k, wih k = 10. Figure 8 shows he resul of he experimen. The plo is generaed by aking he moving average of he updae ime and he sum of densiies. The average updae ime of he algorihm k DB 2

10 Average updae ime (s) average updae ime sum of densiies OT Edge sream (DFS) 10-4 average updae ime sum of densiies FR Edge sream (DFS) Figure 8: Qualiy and efficiency for GR on he OT and FR daases over he sream. mosly remains consan hroughou he execuion, and our algorihm provides seady efficiency over he sream. This behavior remains consisen even when he densiies are flucuaing, as in he case for he OT daase. 7 RELATED WORK Valari e al. [32] were he firs one o sudy he op-k denses subgraph problem for a sream consising of a dynamic collecion of graphs. They proposed boh an exac and an approximaion algorihm for op-k denses subgraph discovery. Similar o our algorihm, he proposed algorihm relies on he core decomposiion o provide he approximaion guaranees. The op-k denses subgraphs produced by he algorihm are edge-disjoin. Balalau e al. [6] sudied he problem of finding he op-k denses subgraph wih limied overlap. They defined he op-k denses subgraphs as a se of k subgraphs ha maximizes he sum of densiies, while saisfying an upper bound on he pairwise Jaccard coefficien beween he se of verices of he subgraphs. The problem of finding he op-k denses subgraph as sum of densiies was shown o be NP-hard [6] and efficien heurisic was proposed o solve he problem. Furher, Galbrun e al. [17] sudied he problem of finding he op-k overlapping denses subgraphs and provided consan-facor approximaion guaranees. 8 CONCLUSION We sudied he op-k denses subgraphs problem for graph sreams, and proposed an efficien one-pass fully-dynamic algorihm. In conras o he exising sae-of-he-ar soluions ha require ieraing over he enire graph upon updae, our algorihm mainains he soluion in one-pass. Addiionally, he memory requiremen of he algorihm is independen of k. The algorihm is designed by leveraging he observaion ha graph updaes only affec a limied region. Therefore, he op-k denses subgraphs are mainained by simply applying local updaes o small subgraphs, raher han he complee graph. We provided a heoreical analysis of he proposed algorihm and showed empirically ha he algorihm ofen generaes denser subgraphs han he sae-of-he-ar soluions. Furher, we observed an improvemen in performance of up o five orders of magniude when compared o he baselines. This work gives rise o furher ineresing research quesions: Is i necessary o leverage k-core decomposiion algorihm as a backbone? Is i possible o achieve sronger bounds on he hreshold for high-degree verices? Can we design an algorihm wih a space bound on he size of he bag? Is i possible o achieve sronger approximaion guaranees for he problem? We believe ha solving hese quesions will furher enhance he proposed algorihm, making i a useful ool for numerous pracical applicaions. Sum of densiies REFERENCES [1] Takuya Akiba, Yoichi Iwaa, and Yuichi Yoshida. 13. 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