Efficient Algorithms for Adaptive Influence Maximization

Transcription

1 Efficient Algoithms fo Adaptive Influence Maximization Kai Han Univesity of Science and Technology of China Jing Tang National Univesity of Singapoe Keke Huang Nanyang Technological Univesity Aixin Sun Nanyang Technological Univesity Xiaokui Xiao National Univesity of Singapoe Xueyan Tang Nanyang Technological Univesity ABSTRACT Given a social netwok G, the influence maximization IM poblem seeks a set S of k seed nodes in G to maximize the expected numbe of nodes activated via anfluence cascade stating fom S. Although a lot of algoithms have been poposed fo IM, most of them only wok unde the non-adaptive setting, i.e., when all k seed nodes ae selected befoe we obseve how they influence othe uses. In this pape, we study the adaptive IM poblem, whee we select the k seed nodes in batches of equal size b, such that the choice of the i-th batch can be made afte the influence esults of the fist i batches ae obseved. We popose the fist pactical algoithms fo adaptive IM with an appoximation guaantee of expξ fo b = and expξ + /e fo b >, whee ξ is any numbe in,. Ou appoach is based on a novel AdaptGeedy famewok instantiated by non-adaptive IM algoithms, and its pefomance can be substantially impoved if the non-adaptive IM algoithm has a small expected appoximation eo. Howeve, no cuent non-adaptive IM algoithms povide such a desied popety. Theefoe, we futhe popose a non-adaptive IM algoithm called EPIC, which not only has the same wost-case pefomance bounds with that of the state-of-the-at non-adaptive IM algoithms, but also has a educed expected appoximation eo. We also povide a theoetical analysis to quantify the pefomance gain bought by instantiating AdaptGeedy using EPIC, compaed with a naive appoach using the existing IM algoithms. Finally, we use eal social netwoks to evaluate the pefomance of ou appoach though extensive expeiments, and the expeimental expeiments stongly cooboate the supeioities of ou appoach. PVLDB Refeence Fomat: Kai Han, Keke Huang, Xiaokui Xiao, Jing Tang, Aixin Sun, and Xueyan Tang. Efficient Algoithms fo Adaptive Influence Maximization. PVLDB, 9: 29-4, 28. DOI: These authos contibuted equally to the pape. Pemission to make digital o had copies of all o pat of this wok fo pesonal o classoom use is ganted without fee povided that copies ae not made o distibuted fo pofit o commecial advantage and that copies bea this notice and the full citation on the fist page. To copy othewise, to epublish, to post on seves o to edistibute to lists, equies pio specific pemission and/o a fee. Aticles fom this volume wee invited to pesent thei esults at The 44th Intenational Confeence on Vey Lage Data Bases, August 28, Rio de Janeio, Bazil. Poceedings of the VLDB Endowment, Vol., No. 9 Copyight 28 VLDB Endowment /8/5... $.. DOI: INTRODUCTION The polifeations of online social netwoks such as Facebook and Twitte have motivated consideable eseach on vial maketing as an optimization poblem. Fo example, an advetise could povide a few individuals efeed to as seed nodes in a social netwok with fee poduct samples, in exchange fo them to spead the good wod about the poduct, so as to ceate a lage cascade of influence on othe social netwok uses via wod-of-mouth ecommendations. The Influence Maximization IM poblem in such a scenaio aims to select a numbe of seed nodes to maximize the influence popagation ceated. Fomally, the input to IM consists of a social netwok G = V, E, a budget k, and a influence model M. The influence model M captues the uncetainty of influence popagation G, and it defines a set W of possible wolds, each of which epesents a possible scenaio of the influence among the nodes in G. The poblem seeks to activate i.e., influence a seed set S of k nodes that can maximize the expected numbe of influenced individuals ove all the possible wolds in W. A plethoa of techniques have been poposed fo IM [7, 2,, 5 7, 2 24]. Almost all techniques, howeve, equie that the seed set S should be decided befoe the influence popagation pocess, which means that they wok in a non-adaptive manne. In othe wods, if an advetise has k poduct samples, she would have to commit all samples to k chosen social netwok uses befoe obseving how they may influence othe uses. In pactice, howeve, an advetise could employ a moe adaptive stategy to disseminate the poduct samples. Fo example, she may choose to give out half of the samples, and then wait fo a while to find out which uses ae influenced; afte that, she could examine the set U of uses that have not beenfluenced, and then disseminate the emaining samples to k/2 uses that have a lage influence on U. This stategy is likely to be moe effective than giving out all k samples all at once, since the dissemination of the second batch of poducts is optimized using the knowledge obtained fom the fist batch s esults. In fact, the above adaptive appoach has been applied in HEALER [26], a softwae agent deployed in pactice since 26, which ecommends sequential intevention plans fo homeless sheltes. HEALER aims to aise awaeness about HIV among homeless youth by maximizing the spead of awaeness in the social netwok of the taget population. It chooses people as the seed nodes, who ae activated by paticipating the intevention plans fo HIV. The choices of seed nodes ae adaptive, i.e., they ae selected in batches and the choice of a batch depends on the obseved esults of all pevious batches. 29

2 Golovin et al. [] ae the fist to study IM unde the adaptive setting, assuming that the k seed nodes ae chosen a sequential manne, such that the selection of the i + -th node is pefomed afte the influence of the fist i nodes has been obseved. Specifically, they conside that i the social netwok confoms to a possible wold w sampled fom W, but ii w is not known to the advetise befoe the selection of the fist seed node. Then, afte the i-th seed node v i is chosen, the pat of w elevant to {v, v 2,..., v i} i.e., the nodes that they canfluence in w is evealed to the advetise, based on which she can i eliminate the possible wolds in W that contadict what she obseves, and ii select the next seed node as one that has a lage expected influence ove the emaining possible wolds. Golovin et al. [] popose a simple geedy algoithm fo adaptive IM that etuns a seed set S whose influence is at least /e of the optimum. unde the case that only one seed is selected in each batch i.e., b =. Nevetheless, the algoithm equies knowing the exact expected influence of evey node, which is impactical since the computation of expected spead is #P-had in geneal [8]. Vaswani and Lakshmanan [25] extend Golovin et al. s model by allowing selecting b seed nodes in each batch, and by accommodating eos in the estimation of expected speads. Thei method etuns an exp /e2 η -appoximation unde this setting, whee η is cetain numbe bigge than. Howeve, even this elaxed appoach is still impactical, its equiements on the accuacy of expected spead estimation cannot be met by any existing algoithms without incuing pohibitive pocessing costs see Section 2. fo a discussion. Contibutions. Motivated by the deficiency of existing techniques, we study the adaptive IM poblem unde the geneal setting that each batch contains b seed nodes, and popose the fist pactical solution fo adaptive IM. Specifically, ou contibutions include the following. Fist, We popose AdaptGeedy, a famewok that enables us to constuct stong appoximation solutions fo adaptive IM using existing non-adaptive IM methods as building blocks. In paticula, we pove that AdaptGeedy achieves an appoximation guaantee of expξ fo b = and expξ + /e fo b >, whee ξ, is a use-specified paamete. The deivation of this appoximation esult equies non-tivial extension of the existing theoetical esults on adaptive algoithms e.g., [] and [25], since AdaptGeedy imposes fa fewe constaints on the building blocks used fo adaptive IM. Second, we conduct an-depth analysis on how AdaptGeedy could be instantiated with the state-of-the-at non-adaptive IM algoithms, and povide anteesting insight: the oveall appoximation guaantee of AdaptGeedy could be impoved if the expected appoximation guaantee of the non-adaptive IM algoithm used by AdaptGeedy is much bette than the wost-case appoximation guaantee of the algoithm. Existing non-adaptive IM algoithms, howeve, do not benefit AdaptGeedy in this egad, as thee is no known esult on thei expected appoximation guaantee. Motivated by this, we develop a new non-adaptive IM method, EPIC, that povides not only an attactive expected appoximation atio, but also the same wost-case guaantees as the state-the-at nonadaptive IM techniques. We establish AdaptGeedy s pefomance guaantee whet is instantiated with EPIC, based on a non-tivial theoetical analysis utilizing Azuma s inequality []. Thid, We conduct extensive expeiments to test the pefomance of AdaptGeedy and EPIC, and the expeimental esults stongly cooboate the effectiveness and efficiency of ou appoach. 2. PRELIMINARIES 2. IM and Possible Wolds Let G = V, E be a social netwok with a node set V and an edge set E, such that V = n and E = m. We assume that the popagation of influence on G follows the independent cascade IC model [6], in which each edge u, v in G is associated with a pobability pu, v, and the influence popagation pocess is defined as a discete-time stochastic pocess as follows. At timestamp, we activate a set S of seed nodes. Then, at each subsequent timestamp t, each node u that is newly activated at timestamp t has a chance to activate each v of its neighbos, such that the pobability of activation equals pu, v. Afte that, u stays active, but cannot activate any othe nodes. The popagation pocess teminates when no node is newly activated at a cetain timestamp, and the total numbe of nodes activated thes defined as the influence spead of S, denoted as I GS. The vanilla influence maximization IM poblem asks fo a seed set S of k nodes that maximizes the expected value of influence spead E[I GS]. As demonstated in [6], the IC model also has antepetation based on possible wolds. Let w be a gaph geneated by emoving each edge u, v in G with pu, v pobability, and let W be the set of all possible choices of w. Then, w can be egaded as a possible wold sampled fom a distibution ove W that is defined by G and the edge emoval pocess. Fo example, Figue shows a social netwok and thee of its possible wolds. Fo convenience, we abuse notation and use W to denote both the univese of possible wolds and the afoementioned distibution ove it. Fo any seed set S, let I ws be the numbe of nodes in w including those in S that can be eached fom S via a diected path stating fom S, and E w W[I ws] be the expectation of I ws ove W. It is shown [6] that E w W[I ws] = E[I GS]. In othe wods, if we ae to addess the vanilla IM poblem, it suffices to identify a seed set S whose expected influence ove the possible wolds in W is the lagest. Remak. We note that the algoithms pesented in this pape can be extended to othe influence models such as the linea theshold model [6] o the topic-awae models [5]. We focus on the IC model, howeve, as it simplifies the exposition of ou technical details. 2.2 Adaptive IM Suppose that the influence popagation on G confoms to a possible wold w that is sampled fom W, i.e., fo any seed set S, the nodes that it canfluence ae exactly the nodes that it can each in w. The adaptive influence maximization IM poblem [] consides that w is unknown advanced, but can be patially evealed afte we choose some nodes as seeds. Fo example, conside the social netwok in Figue a, and suppose that the possible wold sampled fom W is w = w, as shown b. Assume that we choose v as the fist seed node. In that case, we can obseve v s influence on v 2 and v 4, since v has two outgoing edges v, v 2 and v, v 4 in w. Similaly, we can obseve v 4 s influence on v 5. In addition, we can also obseve that v esp. v 4 cannot influence v esp. v 6, as w does not contain an edge fom v to v esp. v 4 to v 6. Figue 2a shows the esults of the influence popagation fom v, with each double-line dashed-line aow denoting a successful esp. failed step of influence. In geneal, afte choosing a patial set S of seed nodes, we can lean all nodes that S can each in w, as well as the out-edges

3 v 2.6 v v. v v 2 v v v 2 v v v 2 v v5 v 4 v v 4 v5 v 6 v 4 v5 v 6 v 4 v5 v 6 a A social netwok b Possible wold w c Possible wold w 2 d Possible wold w Figue : A social netwok and thee of its possible wolds v 2 v v.2. v 5 v 4 v 6 v 6 v. v 2 v v 4 5 v v v6 v 2 v5 v 4 v6 v v.2. a v as the fist seed b Fist esidue gaph c v as the second seed d Non-adaptive IM Figue 2: Adaptive vs. non-adaptive seed selection with k = 2 of those nodes in w. This enables us to optimize the choices of the emaining seed nodes since we can focus on the nodes that have not beenfluenced by S. Fo instance, conside that selecting anothe seed node based on the esult in Figue 2a. In that case, we can omit the nodes that have beenfluenced i.e., v, v 2, v 4, and v 5, and focus on the subgaph induced by the emaining nodes, as shown Figue 2b. Based on this, we can choose v as the second seed node, which yields the esults in Figue 2c, whee we have 6 nodes influenced in total. In contast, if we ae to non-adaptively choose two seed nodes fom the social netwok in Figue a, we may end up choosing v and v 4, in which case we would obtain the esult in Figue 2d when the undelying possible wold is w in Figue b. In othe wods, we can only influence 4 nodes instead of 6. Assume that we ae to choose k seed nodes in batches of equal size b = k/, and that we ae allowed to obseve the influence popagation w fo times in total, once afte the selection of each batch. The adaptive IM poblem asks fo seed sets S, S 2,..., S, such that selecting S i i [, ] as the i-th batch maximizes the expected influence spead ove the choices of w W. Obseve that when b = k i.e., =, the poblem degeneates to the vanilla IM poblem. We aim to develop algoithms fo adaptive IM that povide nontivial wost-case guaantees in tems of both accuacy i.e., the expected influence of i Si and efficiency i.e., the time equied to identify S i. We do not conside the waiting time equied to obseve the influence of a seed node batch S i befoe the selection of the next batch S i+, since it is independent of the algoithms used. That is, we taget at helping the advetise to identify S i+ as quickly as possible afte the effects of S i have been obseved. Table lists the notations that ae fequently used in the emainde of the pape. 2. Existing Solutions The fist solution to adaptive IM is by []. It assumes that b = i.e., each batch consists of only one seed node, and adopts a geedy appoach as follows. Given G, it fist identifies the node v whose expected spead E[I G{v }] on G is the lagest, and selects it as the fist seed. Then, it obseves the nodes that ae influenced by v which ae in accodance to the possible wold w, and emoves them fom G. Let G 2 denote the subgaph of G induced by the emaining nodes. Afte that, fo the i-th i > Table : Fequently used notations Notation Desciption G = V, E A social netwok with node set V and edge set E. n, m the numbes of nodes and edges in G, espectively k the total numbe of selected seed nodes b the numbe of nodes selected in each batch G i the i-th esidue gaph, m i the numbes of nodes and edges in G i, espectively S i the seed set selected fom G i Si o the optimal seed set in G i c c = when b = and c = /e othewise. OP T k,b the optimal expected influence spead of k seed nodes unde the setting of selecting b nodes in each batch. OP T b G i the optimal expected influence spead of b seed nodes in G i. I G S the numbe of nodes activated by S in G ɛ i, δ i the paametes fo the wost-case appoximation guaantee in the ith batch. ξ i the absolute eo facto in the ith batch Cov R S the numbe of RR-sets in R that ovelap S F R S the faction of RR-sets in R that ovelap S batch, it i selects the node v i with the maximum expected spead E[I Gi {v i}] on G i, ii obseves the influence of v i on G i, and then iii geneates a new gaph G i+ by emoving fom G i those nodes that ae influenced by v i. Fo convenience, we efe to G i as the i-th esidue gaph, and let G = G. Let OP T k,b denote the expected spead of the optimal solution to the adaptive IM poblem paameteized with k and b. Golovin et al. [] show that the above geedy appoach etuns a solution whose expected spead is at least /e OP T k,. This appoximation guaantee, howeve, cannot be achieved in polynomial time because i in the i-th batch, it equies identifying a node v i with the maximum lagest expected spead E[I Gi {v i}] on G i, but ii computing the exact expected spead of a node is #P-had in geneal [8]. To emedy the above deficiency, Vaswani and Lakshmanan [25] popose a elaxed appoach that allows eos in the estimation of expected speads. In paticula, they assume that fo any node set S and any esidue gaph G i, we can deive an estimation Ẽ[IG i S]

4 Algoithm : AdaptGeedy Input: social netwok G, seed set size k, batch numbe Output: adaptively selected seed sets S,, S b k/; 2 G G; if = k then 4 c ; 5 else 6 c /e; 7 fo i = to do 8 Identify a size-b seed set S i fom G i, such that the expected spead of S i on G i is at least c ξ i times the lagest expected spead of any size-b seed set on G i; 9 Obseve the influence of S i in G i; Remove all nodes in G i that ae influenced by S i, and denote the esulting gaph as G i+; etun S,, S of E[I Gi S], such that c E[I Gi S] Ẽ[IG i S] c E[I Gi S], with c /c bounded fom above by a paamete η. They show that, by feeding such estimated expected speads to the geedy appoach in [], it can achieve an appoximation guaantee of exp /η. In addition, they show that the geedy appoach can be extended to the case when b >, with one simple change: in the i-th batch, instead of selecting only one node, we select a sizeb seed set S i whose estimated expected spead on G i is at least /e faction of the lagest estimated expected spead on G i. In that case, they show that the esulting appoximation guaantee is exp /e2 η. Unfotunately, the accuacy equiement in Equation is so stingent that no existing algoithm fo evaluating expected spead can meet the equiement without incuing pohibitive computation costs. To undestand this, obseve that when E[I Gi S] is vey small, Equation allows only a tiny amount of estimation eo in Ẽ[I Gi S], in which case the deivation of Ẽ[IG i S] is extemely challenging. Due to this issue, Vaswani and Lakshmanan [25] popose to tade accuacy fo efficiency and adopt algoithms that do not enfoce Equation, but fail to establish any non-tivial appoximation guaantees accodingly.. SOLUTION FRAMEWORK Algoithm illustates the famewok of ou solution fo adaptive IM, efeed to as AdaptGeedy. At the fist glance, AdaptGeedy may seem simila to Vaswani and Lakshmanan s method [25], since both techniques i adaptively select seed nodes in batches and ii do not equie exact computation of expected speads. Howeve, thee is a cucial diffeence between the two: Vaswani and Lakshmanan s method equies that the expected spead of evey node set should be estimated with a small elative eo, wheeas AdaptGeedy allows a andom absolute eo of ξ i OP T b G i, whee OP T b G i denotes the maximum expected spead of any size-b seed set on G i. The eo equiement of AdaptGeedy is much moe lenient than that of Vaswani and Lakshmanan s method, and it can be achieved by seveal state-ofthe-at solutions [7,2,24] fo vanilla influence maximization, i.e., it admits pactical implementations. In addition, AdaptGeedy povides a stong appoximation guaantee, as shown the following theoem. THEOREM. Let G i be the set of possible choices fo G i. Let P[ξ i G,..., G i] be the pobability that S i achieves an appoximation atio of c ξ i conditioned on the event that the fist i esidue gaphs ae G,..., G i, and ξ = ξ i P[ξ i G,..., G i ] P[G,..., G i ]. G G,...,G i G i 2 Then, the appoximation guaantee of AdaptGeedy is at least { expξ, if b = exp ξ + e, othewise Intuitively, Theoem states that the appoximation guaantee of ξ depends on the aveage value of ξ i i [, ] conditioned on the possibilities of G,..., G. Now ecall that the adaptive IM method in [25] povides the following appoximation guaantee fo cetain η > : { exp /η, if b = exp /e2 η, othewise In compaison, the appoximation guaantee of AdaptGeedy is significantly bette when b >, and is compaable when b =. In addition, AdaptGeedy is flexible in that it allows each batch of seed nodes S i to be selected with diffeent even andom appoximation guaantee c ξ i, wheeas the existing solutions e.g., [] fo adaptive IM equie that all seed sets S,..., S should be pocessed with identical accuacy assuance. As we show in Section 4, the flexibility of AdaptGeedy is cucial impoving the efficiency of ou adaptive IM algoithms. The poof of Theoem is athe inticate as it equies nontivial extensions of the theoetical esults developed fo adaptive submodula optimization []. We efe inteested eades to Appendix A fo the details. 4. INSTANTIATIONS OF ADAPTGREEDY 4. Instantiation using Existing Algoithms As shown Algoithm, AdaptGeedy equies identifying a size-b seed set S i fom the i-th esidue gaph G i, such that E[I Gi S i] c ξ i OP T b G i, whee E[I Gi S i] is the expected spead of S i on G i, OP T b G i is the maximum spead of any size-b seed set on G i, and c equals if b = and /e othewise. Such an appoach achieves a povable appoximation guaantee epesented by ξ as long as ξ,, ξ i, ξ satisfy the condition shown Theoem. We obseve that such a seed set S i could be obtained by applying the state-of-the-at algoithms e.g., [7, 2, 24] fo vanilla influence maximization IM on G i. In paticula, these algoithms ae andomized, and they povide a wost-case appoximation guaantee as follows: given a seed set size b, an eo theshold ɛ i and a failue pobability δ i, they output a size-b seed set S i in G i whose expected spead is c ρ i times the maximum expected spead of any size-b seed set on G i, such that ρ i ɛ i with at least δ i pobability. Fo convenience, we efe to ρ i as the absolute eo facto. By applying such algoithms on each esidue gaph G i with any given paametes ɛ i and δ i, we obtain anstantiation of 2

5 AdaptGeedy achieving an appoximation atio of expξ c see Theoem, with P [ ξ > ] [ ɛi = P ɛi < ] G,...,G i ρ i P [ρ i, G,..., G i] [ɛ P i < ] G,...,G i ρ i P[ρ i, G,..., G i ] G,...,G i P [ɛ i < ρ i G,..., G i] P[G,..., G i] G,...,G i δ i P[G,..., G i] = δi In othe wods, the instantiation yields an appoximation guaantee of exp ɛi c with at least δi pobability. But how efficient is the above instantiation? To answe the above question, we need to investigate the time complexity of the vanilla IM algoithms in [2]. The theoetical analysis in [2] show that if we ae to achieve c ɛ i-appoximation on G i with at least δ i pobability, then the expected computation cost is Ob log + log δ i m i + /ɛ 2 i, whee and m i denotes the numbes of nodes and edges in G i. Since n and m i m, the expected time equied to pocess G i is Ob log n + log δ i m + n/ɛ 2 i. As such, all batches of seed nodes can be identified in O b log n + log δ i m + n/ɛ 2 i expected time. 4.2 Rationale fo an Impoved Appoach The instantiation of AdaptGeedy mentioned in Section 4. is simple and intuitive, but is fa fom optimized in tems of its appoximation guaantee. To explain, ecall that it equies each seed set S i to achieve c ɛ i-appoximation on G i with at least δ i, based on which it povides an oveall appoximation atio of exp ɛi c with at least δi pobability. In othe wods, it imposes a stingent wost-case appoximation guaantee on each seed set S i. This, howeve, might be ovely consevative. Fo example, suppose that one S j of the seed sets has an expected spead that is c ρ j times the optimum, with ρ j > ɛ j, i.e., it fails to achieve c ɛ j-appoximation. Even that case, the oveall appoximation atio of AdaptGeedy could still be c ɛi, as long as thee exists anothe seed set Si whose expected spead is c ρ i times the maximum, with ρ i ɛ i ɛ j ρ j. In othe wods, the deficiency of one seed set can be compensated, as long as thee exist othe seed sets whose quality is above the ba by a sufficient magin. Fomally, if we egad each seed set S i s appoximation atio c ρ i as a andom vaiable, then the oveall appoximation guaantee of AdaptGeedy, namely, exp ρi c, depends on the mean of all vaiables. Intuitively, when is sizable, ρi should be concentated to its expectation, i.e., E[ρi]. That is, instead of fomulating the appoximation atio of AdaptGeedy based on the wost-case guaantee of each S i, we might deive it based on each S i s expected appoximation atio, which could lead to much tighte esults. To make the above idea wok, howeve, thee ae seveal challenges that we need to addess. Fist, thee is no known esult fo vanilla IM with expected appoximation guaantees. This motivates us to develop a vanilla IM method that is tailoed fo AdaptGeedy, as we show in Section 4.. Second, as the selection of the i-th seed set S i is dependent on the esults of the i -th seed set S i, the andom vaiables ρ, ρ 2,..., ρ ae coelated, which makes it athe non-tivial to deive concentation esults fo ρi. We cicumvent this issue with a theoetical analysis leveaging Azuma s inequality [] in Section 4.4. Finally, evef we ae given a concentation bound on ρi, we still need to caefully tune each ρ i, so as to yield a stong theoetical guaantee while achieving supeio pactical efficiency. 4. Vanilla IM with Expected Appoximation As discussed in Section 4.2, the existing IM algoithms povide only a wost-case appoximation guaantee c ɛ i, i.e., they ensue that thei absolute eo facto ρ i is no moe than the input theshold ɛ i with high pobability. To optimize the pefomance of AdaptGeedy, howeve, we ae in need of non-adaptive IM algoithm A with two popeties:. The wost-case appoximation guaantee and time complexity of A should be at least as good as those of the state-ofthe-at non-adaptive IM algoithms. 2. The expected value of A s absolute eo facto ρ i should be much smalle than the input theshold ɛ i. In the following, we pesent a new non-adaptive IM algoithm, efeed to as EPIC, that satisfies both of the above equiements. Towads that end, we fist intoduce the concept of evese eachable sets RR-sets [7], which is the basis of ou algoithm. RR-Sets. In a nutshell, RR-sets ae subgaph samples of G that can be used to efficiently estimate the expected speads of any given seed sets. Specifically, a andom RR-set of G is geneated by fist selecting a node v V unifomly at andom, and then taking the nodes that can each v in a andom gaph geneated by by independently emoving each edge e E with pobability pe. If a seed node set S has lage expected influence spead, then the pobability that S intesects with a andom RR-set is high, as shown the following equation [7]: E{I GS} = n P{R S }, 4 whee R is a andom RR-set. This esult suggests a simple method fo estimating the expected influence spead of any node set S: we can use a set R of andom RR-sets to estimate the value of P{R A } and hence E{I GS}. In paticula, let Cov RS denote the numbe of RR-sets in R that ovelap S. Then the value of E{I GS} can be unbiasedly estimated by n F RS, whee F RS = Cov RS/ R 5 By the law of lage numbes, n F RS should convege to E{I GS} when R is lage, which povides a way to estimate E{I GS} to any desied accuacy level. Howeve, due to the cost of geneating RR-sets, thee is a tadeoff between accuacy and efficiency in any algoithm using RR-set sampling. The EPIC Algoithm Algoithm 2 shows the pseudo-code of ou EPIC algoithm. At the fist glance, EPIC is simila to the SSA algoithm in [7] as they both i stat fom a small numbe of RR-sets and ii iteatively incease the RR-set numbe until a satisfactoy solutios identified. The main diffeence between the two algoithm lies in the way that they geneate RR-sets in each iteation. In paticula, in SSA [7], the numbe of RR-sets geneated in each iteatios a andom numbe, which makes it athe difficult to deive the algoithm s time complexity o its expected appoximation guaantee. This could explain the absence of fomal time complexity analysis in [7]. In contast, in each iteation of EPIC, it uses a numbe of RR-sets that is fixed based on the numbe of peceding iteations, which enables us to deive igoous bounds on its wost-case appoximation guaantee, expected time complexity, Expected appoximation fo influence maximization.

6 Algoithm 2: The EPIC Algoithm input : G i, ɛ i, δ i, and b. output: The seed set S i selected in the ith batch. γ i, = ɛ i 6, γ i, = ɛ i 2, γ i,2 = ɛ i γ i, cγ i, +γ i, 2 Υ = 4e 8+γ i,+γ i,2 ln/δ i γ 2 i, T max = 8+2ɛ i bɛ 2 i 4 Υ 2 = + 4e 8+γ i,2 γ 2 i,2 ln 2 δ i + ln b, ω = ln ω δ i 5 Geneate a set R of Υ andom RR sets 6 epeat 7 S i, F R S i Max-CoveageR, b 8 if R F R S i Υ then 9 Geneate R andom RR sets into R 2 log 2 Tmax Υ Calculate F R2 S i of S i in R 2 if R 2 F R2 S i Υ 2 then 2 if F R S i + γ i,f R2 S i then etun S i 4 R = R R 2 5 until R T max ; 6 etun S i Algoithm : The MaxCove Algoithm input : A set R of andom RR sets, and b. output: A node set S i, and the faction of RR sets in R coveed by S i. S i = 2 fo i = to b do v = ag max v V Cov RS i {v } Cov RS i 4 Inset v into S i 5 etun S i, Cov RS i/r and expected appoximation atio. In what follows, we discuss the details of EPIC and its suboutine MaxCove in Algoithm. Based on the RR-set sampling method descibed peviously, a simple appoach fo selecting S i with a lage expected influence spead is to fist geneate a set R of RR-sets, and thenvoke the MaxCove algoithm on R. In paticula, MaxCove uses a simple geedy appoach to identify S i V such that S i ovelaps with as many RR-sets in R as possible. Since F R is a submodula function fo any set R of RR-sets [7], the set S i found by such an appoach ensues that F RS i cf RS o i, 6 whee S o i is an optimal seed set in G i. Note that n F RS i and n F RS o i ae unbiased estimations of the expected influence spead of S i and S o i, espectively. Theefoe, when R is lage, the appoximation guaantee of S i conveges to c accoding to Equation 6. To stike a balance between the quality of S i and the numbe of RR-sets used to deive S i, EPIC iteates in a caeful manne as follows. In each iteation, it maintains two sets of andom RR-sets R and R 2 with R = R 2. It invokes MaxCove on R to identify a seed set S i, and then utilizes R 2 to test whethe S i povides a good appoximation guaantee. Initially, the cadinalities of R and R 2 ae small constants detemined by the paamete Υ in Line 2 in the fist iteation of EPIC. Then, wheneve EPIC finds that the quality of the seed set S i geneated in ateatios not satisfactoy, it doubles the sizes of R and R 2. This pocess epeats until a satisfying solutios found o R and R 2 eaches an uppe bound T max Line 5. As explained befoe, one of the main designing goals fo EPIC is to achieve a wost-case appoximation atio of c ɛ i, as with the state-of-the-at IM algoithms. EPIC achieves this goal by a seies of opeations in each iteation, whose implications ae biefly explained in the following. In each iteation, EPIC fist applies MaxCove on R Line 7, which etuns a seed set S i satisfying F R S i cf R S o i 7 Afte that, EPIC uses R 2 to estimate the expected spead of S i i.e., E{I Gi S i}. Obseve that R 2 F R2 S i is a binomial andom vaiable due to Equation 5. Accodingly, EPIC uses the Chenoff bound to set a theshold Υ 2 Line 4 such that, if the condition R 2 F R2 S i Υ 2 in Line is satisfied, then F R2 S i + γ i,2e{i Gi S i} 8 should hold with high pobability. Intuitively, Equation 8 implies that R 2 is lage enough such that F R2 S i is a sufficiently accuate estimation of the expected influence spead of S i in G i. Afte that, EPIC futhe checks whethe F R S i + γ i,f R2 S i 9 holds in Line 2. Intuitively, if Equation 9 is tue, then we know that F R S i is also a sufficiently accuate estimation of the expected spead of S i in G i. Note that E{I Gi S i} OP T b G i. Theefoe, if the estimation of E{I Gi S i} using R is sufficiently accuate, then the estimation of OP T b G i using R should also be sufficiently accuate due to the Chenoff Bound. Thus, when Equation 9 and the inequality R F R S i Υ in Line 8 hold, then F R S o i γ i,op T b G i holds with high pobability. Combining Equations. 7, we can deive a quantitative elationship between E{I Gi S i} and OP T b G i when S i is etuned: + γ i,2e{i Gi S i} F R2 S i nifr S i + γ i, cnifr S o i + γ i, c γi,op T bg i + γ i,. This poves the c ɛ i wost-case appoximation atio of E{I Gi S i} as ɛ i = γ i, + γ i,2 + γ i,γ i,2 + cγ i,. 4.4 Theoetical Analysis Based on the discussions in Section 4., we pove the wost-case appoximation guaantee and time complexity of EPIC as follows. THEOREM 2. With a pobability of at least δ i, EPIC etuns a seed set S i satisfying E{I Gi S i} c ɛ iop T b G i fo any G i. In addition, the expected time complexity of EPIC is Ob log + log δ i m i + /ɛ 2 i, whee and m i ae the numbes of nodes and edges of G i, espectively. Due to the space constaint, we omit the poofs of Theoem 2 and its coollay i.e., Lemma 2, and include them in ou technical 4

7 epot []. In what follows, we analyze the expected appoximation guaantee of EPIC, and show that instantiating AdaptGeedy using EPIC can lead to impoved pefomance fo adaptive IM. R, E{I G i S o i and 4.4. Expected Appoximation Guaantee To deive the expected appoximation guaantee of EPIC, we would need to compute the expectation of the absolute eo facto ξ i of EPIC. We obseve that ξ i consists of two components: the estimation eo on I Gi S i and the estimation eo on I Gi Si o, whee Si o is an optimal seed set in G i. In what follows, we analyze these two components in detail. Recall that EPIC utilizes the set R of andom RR-sets to find S i, and then employs R 2 to estimate E{I Gi S i}. This ensues that R 2 is independent of S i and R is independent of Si o. Based on this popety and the definition of F R in Equation 5, we know that R F R Si o and R 2 F R2 S i ae both andom vaiables following binomial distibutions Bin Bin R 2, E{I G i S i }, espectively. By the well known bound on the mean absolute deviation of binomial andom vaiables see Lemma 4 in the appendix, we have { E F RSi o OP T } bg i OP T b G i 2 R { E F R2S i E{IG Si} } i OP T b G i R 2 Intuitively, Equation 2 povides an uppe bound of the estimation eo of E{I Gi S o i } using R, while Equation gives an uppe bound of the estimation eo of E{I Gi S i} using R 2. Moeove, the absolute eo facto ξ i of EPIC can be epesented by these estimation eos, due to Equations 7. Combining these esults, we obtain the following lemma: LEMMA. Suppose that R is the set of andom RR-sets used by EPIC to identify S i in the last iteation of EPIC. Let γ i, be the paamete set in Line of EPIC. Then, E{ξ i R } c + R OP T b G i + cγi, 4 Meanwhile, the following theoem shows that, when EPIC teminates, the size of R is likely to be lage. LEMMA 2. Let γ i, be the paamete set in Line of EPIC. When EPIC stops, we must have R ni4e 8 γ 2 i, OP T bg i ln/δi with the pobability of at least δ i/. Combining Lemma and Lemma 2, we immediately get the following bound on the expected appoximation atio of EPIC in each batch i: THEOREM. Fo each batch i, define λ i, β i as λ i = c + γi, 2ɛ i e 2 ln/δi + γi,c ɛ i 5 β i = λ i + c λ iɛ iδ i/ɛ i 6 whee γ i, and γ i, ae the paametes set in Line of EPIC, and ɛ i, δ i ae the input paametes to EPIC in batch i. Then, we have E{ξ i} β iɛ i, and hence, the expected appoximation guaantee of EPIC is at least c β iɛ i. PROOF. Using Lemma and Lemma 2, we can get Hence, the lemma follows. E{ξ i} = E{E{ξ i R }} λ iɛ i δ i/ + cδ i/ = β iɛ i 7 By Theoem, the expected appoximation guaantee of EPIC can be much bette thats wost-case appoximation guaantee i.e., c ɛ i, as long as β i is consideably smalle than. In Section 4.4.2, we investigate how this popety can be exploited to develop ampoved instantiation of AdaptGeedy based on EPIC Pefomance Impovement fo AdaptGeedy In this section, we conside the instantiation of AdaptGeedy with EPIC, and aim to deive ampoved appoximation guaantee fo AdaptGeedy based on the esults in Section Towads this end, we utilize Azuma s inequality: LEMMA. Azuma s inequality [] Let Y,, Y be any sequence of andom vaiables satisfying Y i α and E{Y i Y,, Y i } ϑ fo evey i []. Then we have [ ] P Yi > ϑ + z α exp{ z 2 /2} 8 Obseve that Azuma s inequality povides a concentation bound fo possibly coelated andom vaiables. Recall that we have shown Theoem that the appoximation guaantee of AdaptGeedy is detemined by the summation of the absolute eo factos of the non-adaptive IM algoithm i.e., EPIC in ou case used to select each batch of seed nodes, and these absolute eo factos could be coelated. Theefoe, if we conside them as the andom vaiables in Azuma s inequality, we can get a bound on thei summation, based on which we can deive the oveall appoximation guaantee by Theoem. Howeve, thee is one issue in this appoach: the Azuma s inequality equies that the andom vaiables consideed have a deteministic uppe bound i.e., α in Lemma, which is not the case fo ou absolute eo factos. Fotunately, due to the wost-case appoximation guaantee of EPIC, its absolute eo facto is bounded by anput paamete i.e., ɛ i with high pobability. We leveage this popety to apply Azuma s inequality, by consideing a tuncated vesion of each absolute eo facto, which is defined as the minimum of the absolute eo facto and the input paamete ɛ i. We deive a concentation esult fo such tuncated vaiables, and then extend it to pove the following theoem that establishes the appoximation guaantee of AdaptGeedy whenstantiated by EPIC. THEOREM 4. Suppose that we instantiate AdaptGeedy using EPIC with the paametes ɛ i, δ i in each batch i. Define ɛ = max{ɛ,, ɛ } and β = max{β,, β }, whee β i is defined in Theoem. Fo any given δ, and ξ,, if ɛ = max{β + 2/ ln/δ ξ, ξ} 9 then AdaptGeedy achieves the appoximation atio shown Theoem with a pobability of at least δ δi. Recall that the expected appoximation guaantee of EPIC can be bette thats wost-case appoximation guaantee, in which case the paamete β in Equation 9 can be much smalle than. In that case, Equation 9 indicates that ɛ can be lage than ξ, especially when the ound numbe 2 ln/δ = 2 ln n assuming 5

8 δ = /n. Note that ɛ = max{ɛ,, ɛ }, and we have poved in Theoem 2 that EPIC has the same time complexity with the existing IM algoithms unde the same input paametes. This indicates that we can use some ɛ,, ɛ satisfying ɛi/ > ξ as the input of EPIC in each batch i, while still achieving the appoximation atio shown Theoem, but unde a smalle time complexity compaed with that of using the existing IM algoithms. In othe wods, Theoem 4 shows that instantiating AdaptGeedy using EPIC can achieve tighte appoximation guaantee unde the same time complexity compaed with that of the existing IM algoithms. 5. RELATED WORK Non-Adaptive Influence Maximization: The IM poblem unde the non-adaptive setting has been extensively studied. The seminal wok of Kempe et al. [6] shows that thee is a /e ɛ appoximation guaantee fo the non-adaptive IM poblem, and it poposes a monte calo simulation algoithm to achieve this appoximation atio with high time complexity. Afte that, a lot of studies have appeaed to impove Kempe et al. s wok in tems of time efficiency. Among these woks, Bogs. et al. [7] popose the RR-set sampling method fo influence spead estimation, and seveal late studies [7, 2, 2, 24] use this method to find moe efficient algoithms fo the IM poblem. Howeve, all these studies concentate on the non-adaptive IM poblem, and hence thei appoximation guaantees do not hold fo the adaptive IM poblem. Adaptive Influence Maximization: Compaed with the studies on non-adaptive IM, the studies on adaptive IM ae elatively few. Golovin et al. [] deive a /e appoximation atio unde the case that only one seed node can be selected in each batch. Chen et al. [9], Vaswani and Lakshmanan [25] study adaptive seed selection unde the case that moe than one seed nodes can be selected in each batch. Nevetheless, Chen et al. [9] aim to minimize the cost of the selected seeds unde the constaint that the influence spead is lage than a given theshold, which is a diffeent goal fom ous. Vaswani and Lakshmanan [25] deive an appoximation guaantee exp /e2 η fo cetain η >. Unfotunately, none of the studies listed above povide a pactical algoithm to achieve the claimed appoximation atios. Moe specifically, Golovin et al. [] and Chen et al. [9] assume that the expected influence spead can be exactly computed in polynomial time which is not tue due to [8], while Vaswani and Lakshmanan [25] did not povide a method to bound the key paamete η appeaing in thei appoximation atio. We also notice that Seeman et al. [9], Hoel et al. [4] and Badanidiyuu et al. [4] conside anfluence maximization poblem called adaptive seeding, but with totally diffeent implication fom ous. Moe specifically, they assume that the seed nodes can be selected in two stages. In the fist stage, a set S can be selected fom a given node set X V. In the second stage, anothe seed set T can be selected fom the influenced neighboing nodes of S. The goal of thei poblem is to maximize the expected influence spead of T, unde the constaint that the total numbe of nodes in S T is no moe than k. Howeve, the poblem model and optimization goal of these studies ae both vey diffeent fom ous, and hence thei methods cannot be applied to ou poblem. 6. PERFORMANCE EVALUATION In this section, we evaluate the pefomance of ou poposed appoach using extensive expeiments. The goal of ou expeiments is to test the efficiency and effectiveness of AdaptGeedy using eal Table 2: Dataset details. K =, M = 6, G = 9 Dataset n m Type Avg. degee NetHEPT 5.2K.4K undiected 4.8 Epinions 2K 84K diected.4 DBLP 655K.99M undiected 6.8 LiveJounal 4.85M 69.M diected 28.5 Okut.7M 7M undiected 76.2 social netwoks. All of ou expeiments ae conducted on a Linux machine with an Intel Xeon 2.6GHz CPU and 256GB RAM. 6. Expeimental Setting Datasets: We use five eal datasets in ou expeiments, as shown by Table 2. All these datasets ae downloaded fom [2]. To the best of ou knowledge, only Vaswani and Lakshmanan [25] have used eal social netwoks to test adaptive IM algoithms but without an appoximation guaantee, and the lagest netwok used by them only has 75k nodes and 5k edges [25]. Note that the numbe of edges in Okut is about 24 times of that of the lagest dataset in [25]. Theefoe, as fa as we know, ou datasets ae the lagest ones fo testing adaptive IM algoithms in the liteatue. We also geneate 2 possible wolds fo each dataset to test the pefomance of ou algoithms, and the epoted data ae the aveage esults on these possible wolds. Algoithms: As we discuss in Section 2., thee ae only two existing methods [, 25] fo adaptive IM. They both equie using a non-adaptive IM algoithm that is both efficeint and extemely accuate, but none of the existing non-adaptive IM algoithms satisfy such equiements. Consequently, the methods in [, 25] do not allow pactical implementations without invlidating thei theoetical esults. Instead, we implement two adaptive IM algoithms AdaptIM- and AdaptIM-2 by instantiating AdaptGeedy using EPIC. The diffeence between AdaptIM- and AdaptIM- 2 is that AdaptIM- achieves the appoximation guaantee shown in Theoem by leveaging the wost-case appoximation guaantee of EPIC in the way explained by Sec. 4., while AdaptIM-2 leveages Theoem 4 to achieve the same appoximation guaantee. The pupose of implementing AdaptIM- and AdaptIM-2 is to test whethe the method poposed in Sec is effective, i.e., whethe the pefomance of AdaptGeedy can be impoved by leveaging the small appoximation eo of EPIC. We also test two state-of-the-at non-adaptive IM algoithms i.e., D-SSA [8] and IMM [2] in ou expeiments. IMM is obtained fom [], and D-SSA is obtained fom [8]. The pupose of using D-SSA and IMM in ou expeiments is to test whethe we can achieve lage influence spead by adaptively selecting seed nodes, compaed with the non-adaptive IM algoithms such as D-SSA and IMM. d in v Paamete settings: We use the popula independent cascade IC model [6] in ou expeiments. Following a lage body of existing wok onfluence maximization [7,6,7,2,24], we set the popagation pobability of each edge e = u, v to, whee dinv is the in-degee of the node v. Given any ξ, and δ,, the goal of both AdaptIM- and AdaptIM-2 is to find a exp{ξ c} appoximation solution with pobability of at least δ. To achieve this goal, we set δ = = δ = δ/ and ɛ = = ɛ = ξ fo AdaptIM-. As AdaptIM-2 leveages Theoem 4 to impove its pefomance, we set δ = = δ = δ/2, δ = δ/2 and ɛ = = ɛ = ɛ 6

9 unning time 2 sec unning time.2 sec IMM D-SSA AdaptIM- AdaptIM-2 unning time 6 sec unning time.2 4 sec unning time 4 sec a NetHEPT b Epinions c DBLP d LiveJounal e Okut Figue : Running time vs. unning time 7.5 sec unning time 2.5 sec unning time sec unning time.2 sec unning time sec numbe of seeds k numbe of seeds k numbe of seeds k numbe of seeds k numbe of seeds k a NetHEPT b Epinions c DBLP d LiveJounal e Okut spead spead Figue 4: Running time vs. seed size spead spead a NetHEPT b Epinions c DBLP d LiveJounal e Okut spead spead 4 Figue 5: Spead vs. spead spead spead numbe of seeds k numbe of seeds k numbe of seeds k numbe of seeds k numbe of seeds k a NetHEPT b Epinions c DBLP d LiveJounal e Okut Figue 6: Spead vs. seed size 2.2. spead 5 fo AdaptIM-2, whee ɛ, δ and ξ satisfy the elationship shown Eqn. 9. We also set δ = /n and ξ =. in all ou expeiments. Recall that we need to select k nodes in batches in adaptive IM, whee b = k/ nodes ae selected in each batch. To see how the pefomance of ou algoithms is impacted by k, b and, we set these paametes accoding to the b-setting and k-setting explained as follows. Unde the b-setting, we fix k = 5 and vay b such that b {, 2, 5,, 2, 5, 5}. Unde the k-setting, we fix = 5 and vay k such that k {5,, 2,, 5}. 6.2 Compaing the Running Time In this section, we compae the time efficiency of the implemented algoithms by vaying b, k and. We fist plot ou expeimental esults unde the b-setting in Fig., whee k is fixed to 5 and b scales fom to 5. As both IMM and D-SSA ae non-adaptive IM algoithms that select all 5 seed nodes in one batch, we can only test thei pefomance unde the b-setting fo b = 5. Fo AdaptIM- and AdaptIM-2, we test thei unning time unde the case of b < 5 in Fig.. The expeimental esults in Fig. show that IMM and D-SSA have the smallest unning time, which is not supising given that they only un fo one batch, and hence, geneate a smalle numbe of RR-sets. Besides, it can be seen that the time efficiency of AdaptIM-2 is significantly bette than AdaptIM-, especially when b is small i.e., is lage. Fo example, AdaptIM-2 uns almost 4 times faste than AdaptIM- on the Epinions dataset when b =. We also notice that, AdaptIM- even cannot finish unde the case of b < 5 fo the lagest datasets LiveJounal and Okut, due to the memoy oveflow. To explain, eall that AdaptIM-2 leveages Theoem 4 to set its input paamete ɛ, due to which ɛ can be much lage than ξ, especially when is lage. Consequently, the numbe of RR-sets geneated in AdaptIM-2 can be much smalle than that in AdaptIM-, and hence AdaptIM-2 achieves bette time efficiency and less memoy consumption. In Fig. 4, we plot ou expeimental esults unde the k-setting, whee is fixed to 5 and k vaies fom 5 to 5. The esults show simila tends as those in Fig., which can be explained by simila eason as that fo Fig.. Besides, Fig. 4 eveals that the pefomance gain of AdaptIM-2 with espect to AdaptIM- can be moe pominent when k is small unde the k-setting. This can be 7

10 explained as follows. As is fixed to 5 unde the k-setting, b must incease with k, due to which the optimal influence spead OP T b G i also tends to incease with k fo each batch i. Consequently, AdaptIM- and AdaptIM-2 ae both less sensitive to thei input paamete ɛ i when k gets lage, as they both geneate fewe RR-sets in each batch; thus, they both achieve bette time efficiency when OP T b G i gets lage. 6. Compaing the Influence Spead In this section, we study the pefomance of the implemented algoithms on the influence spead, and the expeimental esults ae shown Fig. 5 and Fig. 6. The paamete settings in Fig. 5 and Fig. 6 ae the same with those in Fig. and Fig. 4, espectively. We fist study the pefomance of the implemented algoithms unde the b-setting in Fig. 5, whee k is fixed to 5 and b scales fom to 5. It can be seen that AdaptIM- and AdaptIM-2 achieve simila influence speads, which poves the effectiveness of AdaptIM-2, as AdaptIM-2 achieves bette time efficiency than AdaptIM-. Moeove, the influence speads of AdaptIM- and AdaptIM-2 both tend to decease when b inceases. This can be explained as follows. Unde the b-setting, deceases when b inceases, which implies that both AdaptIM- and AdaptIM-2 become less adaptive when b inceases. Consequently, they both could activate fewe nodes. In fact, when b = 5, all the seed nodes must be non-adaptively selected in one batch. This explains why IMM and D-SSA achieve much wose influence spead than AdaptIM- and AdaptIM-2 do. Finally, we study the influence spead unde the k-setting in Fig. 6, whee is fixed to 5 and k scales fom 5 to 5. It can be seen that all the influence speads of the implemented algoithms incease with k, which is due to the eason that selecting moe seed nodes causes a lage influence spead. Moeove, both the influence speads of AdaptIM- and AdaptIM-2 outpefom those of IMM and D-SSA, as they can activate moe nodes by adaptively selecting seed nodes. Indeed, Fig. 6 shows that AdaptIM- and AdaptIM-2 can achieve moe than ten pecentage gain on the influence spead fo LiveJounal. We also note that the gain on the influence spead bought by adaptively selecting seed nodes can be affected by the netwok itself, as diffeent social netwoks have diffeent topologies and diffeent possible wolds. 7. CONCLUSION We have studied the adaptive Influence Maximization IM poblem, whee the seed nodes can be selected in multiple batches to maximize thei influence spead. We have poposed the fist pactical algoithms to addess the adaptive IM poblem that achieve both time efficiency and povable appoximation guaantee. Ou appoach is based on a novel AdaptGeedy famewok instantiated by a new non-adaptive IM algoithm EPIC, which has a povable expected appoximation guaantee. We also have conducted extensive expeiments using eal social netwok to test the pefomance of ou algoithms, and the expeimental esults stongly cooboate the supeioities and effectiveness of ou appoach. 8. ACKNOWLEDGEMENT This wok is patially suppoted by National Natual Science Foundation of China unde Gant No , No , by Natual Science Foundation of Jiangsu Povince unde Gant No.BK26256, by MOE, Singapoe unde gant MOE27-T- 2-24, MOE25-T2-2-69, by NUS, Singapoe unde an SUG, and by NRF, Singapoe unde gant NRF-RSS26-4. APPENDIX A. MISSING LEMMAS AND PROOFS LEMMA 4. [6] Suppose that X Binn, p is a binomial andom vaiable. Then the mean absolute deviation MAD of X i.e., E{ X E{X} } is no moe than np p. A. Poof of Theoem In this section, we fist intoduce some definitions and lemmas, and then use them to pove Theoem. Ou theoetical analysis boows some concepts fom [], but is consideably diffeent fom that in []. This is mainly due to the eason that ou AdaptGeedy famewok allows diffeent and even andom appoximation guaantee in diffeent batches, while [] equies that the appoximation guaantees in all batches ae identical and fixed constants. Fo convenience, we call any stategy that selects seed nodes in the way explained by Section. 2.2 as an adaptive seeding policy. To analyze the pefomance atio of AdaptGeedy, we fomally define the expected influence spead of any adaptive seeding policy in Definition, and thentoduce the tuncation and concatenation opeations on policies in Definition 2 and Definition, espectively. DEFINITION. Influence spead of policy Given any adaptive seeding policy Λ, let NΛ denote the set of all seed nodes that would be selected by Λ. The expected influence spead of Λ is defined as πλ = E w W{I wnλ}, whee w and W ae explained in Section 2.. DEFINITION 2. Policy tuncation Fo any adaptive seeding policy Λ, the policy tuncation Λ [i] denotes an adaptive policy that pefoms exactly the same as Λ, except that Λ [i] only selects the fist i i batches of nodes. DEFINITION. Policy concatenation Fo any two adaptive seeding policy Λ and Λ, the policy concactenation Λ Λ denotes an adaptive policy that fist executes the policy Λ, and then executes Λ on the esidue gaph output by Λ, but without any knowledge about Λ. With the above definitions, we futhe intoduce the concept of optimal maginal gain of any adaptive seeding policy in Definition 4, and thentoduce Lemma 5 and Lemma 6, which ae useful fo poving Theoem. DEFINITION 4. Optimal maginal gain Fo any adaptive policy Λ, define G[Λ] as the gaph geneated by emoving the nodes activated by Λ in G. Define Λ as the maximum expected influence spead of any b seed nodes in G[Λ], which is called the optimal maginal gain of Λ. LEMMA 5. Fo any adaptive seeding policy Λ and any i, we have πλ i πλ [i ] E w W{ Λ [i ] } 2 PROOF. Suppose that C is the set of nodes selected by Λ in the ith batch. Let K = E{I G[Λ[i ] ]C} be the expected influence spead of C in G[Λ [i ] ]. So K is always no moe than than Λ [i ]. Theefoe, we have πλ [i] πλ [i ] = E{K} E{ Λ [i ] } 2 = E w W{ Λ [i ] } whee the expectations in 2 ae taken with espect to the andomness of G[Λ [i ] ]. 8