![Paper, We Propose the Optimal Influential Pieces Assignment (OIPA) Problem, Which Is to Assign K Distinct Pieces Aiming to Achieve Overlapping Adoption Results [1]](http://data.docslib.org/img/e2d3123feddc77b795629c631f99800e-1.webp)
2019 IEEE 35th International Conference on Data Engineering (ICDE)
Maximizing Multifaceted Network Influence
Yuchen Li ,JuFan†, George V. Ovchinnikov#, Panagiotis Karras∗ School of Information Systems, Singapore Management University [email protected] †DEKE Lab & School of Information, Renmin University of China †[email protected] #Skolkovo Institute of Science and Technology #[email protected] ∗ Department of Computer Science, Aarhus University ∗ [email protected]
Abstract—An information dissemination campaign is often [11], [18], [29]. In another example, a Youtube channel may multifaceted, involving several facets or pieces of information try to increase the number of subscribers by spreading viral disseminating from different sources. The question then arises, videos on SM like Facebook or Twitter. Due to the short-lived how should we assign such pieces to eligible sources so as to achieve the best viral dissemination results? Past research has effect of SM contents [6], the viral video could quickly fade studied the problem of Influence Maximization (IM), which is to out from a user’s memory even if she has watched it and shared select a set of k promoters that maximizes the expected reach of it with friends. Only upon watching multiple videos from a message over a network. However, in this classical IM problem, the same channel would the user turn to a subscriber. Thus, each promoter spreads out the same unitary piece of information. a campaign strategy should spread multiple viral messages In this paper, we propose the Optimal Influential Pieces Assignment (OIPA) problem, which is to assign k distinct pieces aiming to achieve overlapping adoption results [1]. An early of an information campaign T to k promoters, so as to achieve the example of an such approach was a 1919 marketing campaign highest viral adoption in a network. We express adoption by users for a deodorant Odorono by Walter Thompson. Initial research with a logistic model, and show that approximating OIPA within indicated that consumers insisted that they did not need such any constant factor is NP-hard. Even so, we propose a branch- a product. Thereafter, the campaign set a goal of consumer and-bound framework for OIPA with an (1 − 1/e) approxima- tion ratio. We further optimize this framework with a pruning- education independently of product promotion, suggesting that intensive progressive upper-bound estimation approach, yielding a nonuser “may offend without knowing it”; that caused an a (1 − 1/e − ε) approximation ratio and significantly lower time outrage at the time, but enormously improved sales. In our complexity, as it relies on the power-law properties of real-world terms, a multifaceted promotion campaign was used, as the social networks to run efficiently. Our extensive experiments on focus shifted from the product to a wider social concern. several real-world datasets show that the proposed approaches consistently outperform intuitive baselines, adopted from state- In this paper, we propose the Optimal Influential Pieces of-the-art IM algorithms. Furthermore, the progressive approach Assignment (OIPA) problem. Given a multifaceted campaign k demonstrates superior efficiency with an up to 24-fold speedup T that consists of pieces, we need to assign i promoters over the plain branch-and-bound approach. i k k to spread the th piece, where i i = , so as to maximize the number of users who adopt the goal of T after receiving I. INTRODUCTION a subset of the pieces. We assume that each piece is char- Campaigning in Social Media (SM) is an indispensable tool acterized by a unique topic distribution and spreads in the for promoting business, advocating political standpoints, and network in a topic-aware manner, governed by topic-dependent spreading information for the social good. The problem of influence probabilities between any two users, as in [31], [3]. influence Maximization (IM) [16] asks to find a set of k initial The influence spread of a promoter with respect to a piece promoters that maximize the spread of a unitary message t ∈ T depends on the piece’s unique topic distribution; it propagating among users in a network. This problem has been follows that each promoter may be better positioned to spread extensively studied over the last decade [16], [33], [7], [32] in one piece than others. We model a user’s adoption behavior the context of viral marketing, network monitoring [19] and when receiving multiple pieces by a logistic activation model, recommendation [30]; a survey is given in [21]. as in consumer behavior studies [34], [13], [11], [18], [29]: However, real-world campaigns often need to be multi- the adoption probability is small when a user receives little faceted: composed of multiple constituent message pieces. For information, but increases drastically upon seeing more; after instance, an election campaign aims to inform voters regarding exposure to a sufficient number of pieces, the probability a candidate’s policy statements on multiple issues, including gradient declines as the effect of extra information diminishes. taxation, immigration, and healthcare. Studies in consumer As an effect of the logistic activation model, the objective behavior confirm that it is unlikely to trigger any meaningful function in OIPA is non-submodular and thus techniques actions (e.g., vote for a candidate or adopt a product) when a based on a submodularity assumption are inapplicable. Further, user only receives a single element of the campaign [34], [13], we show that it is NP-hard to approximate OIPA within
2375-026X/19/$31.00 ©2019 IEEE 446 DOI 10.1109/ICDE.2019.00047
Authorized licensed use limited to: Renmin University. Downloaded on June 01,2021 at 07:12:03 UTC from IEEE Xplore. Restrictions apply. any constant factor, using a novel reduction from the maxi- Native IM: The objective of the well-studied IM problem [21] mum clique problem. Still, we propose a branch-and-bound is to find a set S of k promoters in a network that maximizes framework and introduce a novel formulation of monotone the expected number of influenced users, denoted as σ(S). submodular optimization, which can be approximated with While IM is NP-hard under the popular independent cascade factor of (1 − 1/e) by means of a greedy heuristic, to obtain (IC) and linear threshold (LT) influence models, σ(·) is a a tight upper bound on the unexplored search space. Thereby, monotone1 and submodular2 function for both IC and LT, we iteratively solve a tractable optimization problem until we hence a constant approximation factor holds for a simple compute an upper bound smaller than or equal to the best greedy algorithm [16]. obtained solution. Thus, this branch-and-bound framework Classical influence models treat different viral messages thus guarantees an (1 − 1/e) approximation ratio. Still, the as interchangeable in terms of their influence spread under upper bound estimation procedure is expensive, as it repeatedly the model. In contrast, topic-aware models differentiate each scans O(kn) users to select a set of promoters (where n is the unique message’s influence in a topic-dependent manner [31], number of SM users). We render it more efficient by means of [3]. The topic-aware IM problem [2] aims to find promoters a novel progressive upper bound estimation approach. Instead under such a topic-aware influence model. of scanning a large number candidate promoters, we select a All aforementioned works are concerned with the problem promoter only if its marginal gain in terms of user adoption is of spreading a unitary message, as opposed to a message larger than a predefined threshold h. By progressively lowering consisting of multiple parts as in OIPA. Still, topic-aware this threshold, we select more promoters as the budget k influence models are useful in describing the propagation of allows. Due to the power-law principle of social influence, we message parts. Yet, solutions for native IM problems cannot be show that this progressive approach significantly lowers the extended to address OIPA, in which the adoption probability computational complexity, while ensuring an approximation function is non-submodular. ratio of (1 − 1/e − ε) where ε is a tunable parameter that Competitive IM: Competitive IM considers a scenario in trades effectiveness with efficiency. which each of several competitors spreads a message in We summarize our contributions as follows: the same network, and each user adopts at most one mes- • We propose a novel OIPA problem, which is to assign sage. Past research has proposed three main objectives for candidate promoters to the propagated facets of a mul- competitive IM problems: (1) maximizing one competitor’s tifaceted campaign in a way that maximizes adoption, influence spread given the opponents’ strategies in choosing while capturing user behavior with a logistic model. promoters [4], [8], [14]; (2) finding the equilibrium of a com- • We show that OIPA is NP-hard, and also NP-hard to petition using game-theoretical concepts, when the opponents’ be approximated within any constant factor. strategies are unknown beforehand [20], [23]; (3) maximizing • We propose a branch-and-bound framework for OIPA, the total influence spread of all competitors, which is desirable and introduce a novel formulation of monotone submod- from the perspective of a network host who is interested to ular optimization to tightly bound the unexplored search allocate competing campaigns fairly [24], [17]. Competitive IM considers multiple propagating units, like space for effective pruning. This framework provides an OIPA (1 − 1/e) approximation guarantee. does, yet each of these units is a message belonging • We devise an efficient progressive approach for the upper to a different competitive campaign; they are not pieces con- stituting a single multifaceted campaign; thus, the competitive bound evaluation, which achieves an approximation ratio OIPA of (1 − 1/e − ε) and drastically reduces time complexity. IM problem is fundamentally different from . Comparative IM: Some works have considered viral market- • We experimentally evaluate the effectiveness and effi- ciency of our approaches on three real world datasets. The ing with multiple non-competing messages. Datta et al. [10] proposed approaches achieve over 215% quality improve- study the case of campaigns whose influence spreads are ment against two intuitive baselines adapted from state- independent. Narayanam et al. [27] study a scenario with of-the-art IM algorithms, while the progressive approach two sets of complementary products, where a product can achieves a 24-fold speedup compared to the plain branch- be adopted only by a user who has already adopted its and-bound approach. corresponding product in the other set. Lu et al. [25] introduce a comparative influence model which subsumes both com- The rest of the paper is organized as follows. We present petitive and complementary IM: they consider two different related work in Section II. Section III introduces preliminaries kinds of relationships between two campaigns, A and B:in and the OIPA problem definition. We present the hardness a competitive relationship, a user’s adoption of A lowers the result in Section IV. We propose the branch-and-bound frame- probability to adopt B;inacomplementary relationship, a work and the progressive upper bound estimation approach in user’s adoption of A raises the probability to adopt B.TwoIM Section V. Section VI reports on our experimental study, and problems arise with this comparative influence model: SELF- Section VII concludes the paper. INFMAX, which is to maximize a campaign’s own influence; II. RELATED WORK 1A set function g is monotone if for all A ⊆ B, g(A) ≤ g(B). In this section, we review related works in native, compet- 2A set function g is submodular if for all A ⊆ B and any element x ∈ B, itive, and comparative Influence Maximization (IM). g(A ∪{x}) − g(A) ≥ g(B ∪{x}) − g(B).
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Authorized licensed use limited to: Renmin University. Downloaded on June 01,2021 at 07:12:03 UTC from IEEE Xplore. Restrictions apply. TABLE I: Frequently used notations in the paper G(V,E) random variable that equals 1 if the item t propagating from the social graph with vertex and edge sets S u n, m, Z n = |V |,m= |E| and Z is the set of topics the seed set successfully activates user , and 0 otherwise. T,l the viral campaign with l viral pieces Then the expected influence spread of t spreading from S is: V p ⊆ V the set of available promoters σ S E Iv v v im( )= v∈V [ S ]. β the preference vector of v and βj is the preference v of on the viral piece tj B. Problem Definition rv the parameter to control the adoption probability of user v We consider a viral campaign T consisting of viral pieces, k the number of possible assignments T = {t1, .., t }. We assume that each piece t propagates in the S¯,S ∈ S¯ j j j an assignment plan for T and the th seed set for network independently3 of others. We model the user adoption tj ∈ T Iv v S Sj an indicator for if a node is activated by j behavior upon receiving some of the viral pieces in a campaign σ(·) the adoption utility (AU) function with a logistic model [34], [13], [11], [18], [29]. In this model, the probability that user v adopts the campaign T is a Bernoulli random variable Xv, captured by the logistic function: and COMPINFMAX, which is to maximize the incremental ⎧ ⎨ 1 v 1+exp{ − · v } if ∃I =1 influence of one campaign on another. v v α β j=1.. IS Sj p[Xv =1|I .., I ]= j Still, works on comparative IM focus on tuning the order in S1 S ⎩ 0 otherwise which a user receives the messages of typically two campaigns (1) with interdependent propagation processes. In contrast, the where each Iv is a Bernoulli random variable that indicates OIPA problem concerns the adoption of a single campaign Sj whether the piece t propagating from seed set S reaches v, consisting of several pieces whose propagation processes are j j and α, β > 0 are parameters that control the users’ turning mutually independent, yet serve a common goal. point for adoption. The larger α is, the harder it is for a user β III. PRELIMINARIES to adopt T, while weighs the the effect each piece tj has on the adoption probability for any user. In this section, we introduce a topic-aware influence model As each promoter has a varying potential in spreading that describes how different message pieces spread in a message pieces on different topics, our goal is to assign the network (Section III-A) and the Optimal Influential Pieces pieces of T to a judiciously selected subset of the promoters Assignment (OIPA) problem (Section III-B). Before moving in S, so that the overall adoption utility over all users about on, all frequently used notations are listed in Table I. T is maximized. We first formally define the adoption utility. A. Topic-aware Influence Model We assign the propagation of each piece tj in T to a subset of promoters S ⊆ S. Let S¯ denote the collection of We model a social network (SN) as a directed graph j all seed subsets that compose our overall assignment plan, G(V,E), where V is a set of users and each edge e =(u, v) ∈ S¯ = {S , ..., S }. We express the overall effectiveness of the E captures the (bidirectional) friend or (unidirectional) fol- 1 assignment plan S¯ via an adoption utility function σ(S¯): lower relationship from u to v, |V | = n, and |E| = m.To model how an item propagates in an SN, we adopt the well- σ S¯ E X E X p X studied topic-aware propagation model [3], [2], [9], [22]. This ( )= v = [ v]= [ v =1] (2) v∈V v∈V v∈V model extracts a set of hidden topics Z = {z1,z2,...,z|Z|} from social activities (e.g., tweets and replies) propagated on The Optimal Influential Pieces Assignment (OIPA) prob- an SN, and explains social influence by means of these topics. lem is to extract the assignment plan that yields the largest Formally, given an edge e =(u, v),atopic-aware influence adoption utility for a campaign T. probability p(e|z) models how u influences v under topic Definition 1 (OIPA). Given a social graph G, a multifaceted z ∈ Z; we denote p(e) as the topic-wise influence vector campaign T containing viral pieces, a pool of promoters V p for e. These probabilities can be learned from logs of past and a budget k, the Optimal Influential Pieces Assignment propagation activities [31], [12], [3]. (OIPA) problem is to find an assignment plan S¯∗ that We describe a piece of a viral message by the vector t = maximizes the overall adoption utility of T over G. Formally, (t1,t2,...,t|Z|), where tz is the probability that this piece relates to topic z. Given a seed set of promoters S ⊆ V , the S¯∗ =argmaxσ(S¯) (3) influence process of t starting out from S under the topic- |S|≤¯ k aware influence model unfolds as follows. Initially, all users where |S|¯ = |S |. in V \ S are inactive and the seed promoters in S are active. j=1,.., i Each promoter gets one chance to activate each of its neighbors Example 1. Figure 1 presents a running example of an via the corresponding edge. As in [9], [2], we compute the OIPA instance, using two topics z1, z2 with the topic- probability that message t goes through edge e as p(t,e)= aware influential probabilities shown on every graph edge. For t·p(e). Subsequently, any newly activated user u also gets one chance to activate its neighbors. The process terminates when 3While this independence assumption is not general, we show (Section IV) Iu no more users can be activated. Let S denote a Bernoulli that the problem is intractable even in this restricted case.
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(a) Topic-aware influence graph (b) Influence graph viral piece t1 (c) Influence graph viral piece t2 Fig. 1: A running example for OIPA. In (a), each edge in contains a topic-aware influence probability vector. In (b), we show the influence graph for t1 which has a topic distribution of (1, 0). In (c), we show the influence graph for t2 which has a topic distribution of (0, 1).
a instance, the topics could be z1 =“tax” and z2 =“healthcare” Definition 4. The i-union of an assignment plan S¯ with a a in a political campaign. We run a campaign on the network seed set S is a new assignment plan S¯ = S¯ ∪i S where , S Sa j i S Sa ∪ S i with two messages, t1 t2, having topic distributions t1 = j = j , = and i = i . Then the -marginal gain a (1.0, 0.0) and t2 =(0.0, 1.0) respectively, i.e., one is only of adding S to S¯ is: about tax and the other only about healthcare. Given a budget i ¯a ¯a δ ¯a (S)=σ(S ∪ S) − σ(S ) (5) of two promoter assignments, OIPA finds the assignment S i plan that maximizes the overall adoption utility in the network. We study the monotonicity and submodularity of σ(·) under This optimal assignment plan is to assign t1 to user a and these containment and union relationships. e Ia Ib Ic Id t2 to user . Under this plan, {a} = {a} = {a} = {a} =1 e a Definition 5. We say that σ(·) is monotone iff, for any two and I =0for t1 (see Figure 1 (b)), while I =0 {a} {e} assignment plans S¯a and S¯b such that S¯a ⊆ S¯b, it holds that and Ib = Ic = Id = Ie =1(see Figure 1 (c)). {e} {e} {e} {e} σ(S¯a) ≤ σ(S¯b). We say that σ(·) is submodular iff, for any Assume the parameters α =3,β =1, the adoption utility of two such plans and any S¯, δS¯b (S¯) ≤ δS¯a (S¯). each user is given by Equation (1). For instance, p(Xb)= 1 . σ · 1+exp{3−(1+1)} =027. Eventually, the overall utility of It is trivial to show that ( ) is monotone. However, as the σ {{a}, {e}} p X σ(·) the assignments plan is ( )= i=a,b,c,d,e [ i] following counterexample shows, is not submodular. =0.12 + 0.27 · 3+0.12 = 1.05 a 1 e as user receives t , user Example 2. Building on Example 1, consider assignment 2 1 receives t , and others receive both, with probability . plans S¯x = {∅, ∅}, S¯y = {{a}, ∅}, and S¯ = {∅, {e}}.It x y holds that S¯ ⊆ S¯ . Yet, δS¯y (S¯)=1.05 − 0.48 = 0.57 > IV. PROBLEM ANALYSIS δS¯x (S¯)=0.48 − 0.00 = 0.48. Thus, σ(·) is not submodular. This section provides a theoretical analysis of OIPA. First, we study the properties of the adoption utility function σ(S¯) The absence of submodularity precludes any greedy approx- (Section IV-A). Then, we study the hardness and approxima- imation algorithm based on that property. Given this negative bility of OIPA (Section IV-B). result, we study the problem’s hardness and approximability. B. Hardness and Approximability A. Properties of Adoption Utility Function When a campaign consists of one viral piece, T = {t0}, OIPA Here, we study the properties of the objective OIPA is reduced to a variant of the classical IM problem [16] function, the adoption utility function σ(S¯), to gain insights in which a user v adopts T with probability p[Xv =1]= into the optimization problem. While σ(S¯) measures the 1 1+exp{ − · v } ,ifv is activated by the seed set S for t0. S¯ α β IS effectiveness of an assignment plan , i.e., a set of seed sets α → c · β c< p X → σ(·) Further, if for any 1, then [ v =1] 1 assigned to different pieces; thus, is not a proper set func- β →∞ v σ(·) as for a activated by t0. Thus, the classical IM tion. To analyze , we define a containment relationship OIPA between assignment plans. problem is fully reduced to this special variant of . In effect, OIPA is straightforwardly NP-hard, since IM is Definition 2. An assignment plan S¯a contains another plan NP-hard. Nevertheless, as we have seen, the AU function is S¯b S¯b ⊆ S¯a Sb ⊆ Sa ∀j ,..., , i.e., , if and only if j j =1 . not submodular in the general OIPA case, as the influence spread function is in the IM setting, leading to an efficient We further define the union of one assignment plan with greedy approximation algorithm [16]. This fact indicates that another, and the associated marginal utility gain, as follows. OIPA is harder than IM. That motivates us to study how Definition 3. The union of two assignment plans, S¯a and S¯b, hard it is to approximate OIPA within a constant factor. S¯ S¯a ∪ S¯b S Sa ∪ Sb is a new assignment plan = with j = j j , We study approximability by reduction from the Maximum b a j =1,..., . The marginal gain of adding S¯ to S¯ is: Clique (MC) problem. Given an instance of MC, Πa,on GΠ (VΠ ,EΠ ) |VΠ | = n b a b a a graph a a a where a , we construct a δS¯a (S¯ )=σ(S¯ ∪ S¯ ) − σ(S¯ ) (4) corresponding instance of OIPA, Πb, in polynomial time:
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Authorized licensed use limited to: Renmin University. Downloaded on June 01,2021 at 07:12:03 UTC from IEEE Xplore. Restrictions apply. 1) We create 3n vertices for Πb which we denote each vertex promoter; thus, it should be in the from {{u1},...,{un}} as xi, yi and ri, i =1,...,n. where ui is either xi or yi, i.e., one of the only eligible ∗ 2) We set n topics, Z = {z1,...,zn} and a campaign promoters of piece ti. Even so, only nodes ri in C(S¯ ) can ∗ consisting of = n pieces, T = {t1,...,tn}, each piece possibly receive all n pieces, and will do so if and only if S¯ th ti having a topic vector where the i entry is 1 and others selects the xi corresponding to each of the OPT(Πa) vertices 0. In other words, piece ti is about topic zi only. vi in a maximum clique in Πa, and the yi corresponding to 3) We set an edge e =(xi,rj) for all j ∈{j|j = any other vertex vi in Πa, provided there exists at least one i ∨ v ,v ∈ E } e ( i j) Πa and set p( )=ti. In other words, edge (i.e., a clique of size 2)inΠa. Then it follows that: we connect each xi to the r-vertices standing for vi and G OPT p X p X its neighbors in Πa , with a topic vector consisting of (Πb)= [ r =1]+ [ r =1] ¯∗ ¯∗ exactly topic zi. ∀r∈C(S ) ∀r∈C(S ) e =(y ,r ) j ∈{1,...,i− 1,i+ n 4) We set an edge i j for all ≤ 1OPT 2 (Πa)+ 2 1,...n} and set p(e)=ti. In other words, we connect 2 1+(2n) each yi to all r-vertices except the one standing for vi in 1 1 G z ≤ OPT(Πa)+ Πa , with a topic vector consisting of exactly topic i. 2 2n 5) We set α =ln(2n)2n and β =ln(2n)2. With these · OPT − 1 ≤ OPT settings, for a vertex v that receives all n pieces it is Therefore 2 (Πb) n (Πa). 1 p[Xv =1]= 2 , while for a vertex v that receives at n − p X ≤ 1 Lemma 1 leads to the following hardness result. most 1 pieces it is [ v =1] 1+(2n)2 . V {x |i 6) We set the set of available promoters as p = i = Theorem 1. There is no polynomial-time algorithm to approx- 1,...,n}∪{yi|i =1,...,n} with budget k = n. 1 1−ε imate the optimal solution of OIPA within a factor of 2 n In the constructed Πb OIPA instance, p(ti,e)=1if for any ε>0, unless NP = ZPP. and only if e starts from xi or yi, hence xi and yi are the only eligible promoters for piece ti. The following lemma Proof. Let k ≥ 1. By Lemma 1, if OPT(Πa) ≥ k then k k establishes the connection between instances Πa and Πb. OPT ≥ OPT ≤ OPT ≤ (Πb) 2 , and if (Πa) n1−ε then (Πb) k 1 ≤ 1 · k 2 1−ε + 2 1 1−ε 2 . Thus, we have a gap-preserving Lemma 1. Let clique size OPT(Πa) and adoption utility n n 2 n reduction from Π to Π [35]. If we can approximate OIPA OPT(Πb) denote the optimal solutions for the MC in- a b in polynomial time within a factor of 1 n1−ε, then we can stance Πa and the OIPA instance Πb, respectively. Then 2 1 approximate MC within a factor of n1−ε. This can not be 2 · OPT(Πb) − ≤ OPT(Πa) ≤ 2 · OPT(Πb). n true unless P = ZPP [15]. C Proof. Let Πa be the maximum clique vertex set in Πa.We S¯ Π can then deploy an assignment plan for b as follows: for V. A PPROXIMATION ALGORITHMS i v ∈ C x y each ,if i Πa , we choose i to promote ti, and i v ∈ C r n otherwise. By this plan, if i Πa , than i receives all Given the results of Section IV, there is no known way to σ S¯ ≥ p X 1 OPT pieces. Thus, ( ) vi∈CΠ [ ri =1]= 2 (Πa), develop a polynomial-time approximation algorithm for the a OIPA hence OPT(Πa) ≤ 2 · OPT(Πb), the second inequality. general case. Still, social influence follows a power On the other hand, any plan S¯ for Πb will have the form law principle [28]: a few people have significantly larger {{u },...,{u }} u x i1 in where each ij is either some ij or some influence than others. Based on this principle, we propose a y C S¯ N u ∩ N u ∩ ... ∩ N u ij . Let ( )= ( i1 ) ( i2 ) ( in ) where branch-and-bound framework that prioritizes promoters with N u u ( ij ) is the neighbor set of j in Πb. Given the way edges large influence and enables early termination when necessary. in Πb are set, since ri is never a neighbor of yi, yet ri is We maintain partial candidate plans in a max-heap, sorted always a neighbor of xi, only contributions by xi’s survive by their estimated AU score upper bound. If the upper bound all intersections in C(S¯). As all vertices rj corresponding to of a partial plan is smaller than the exact AU score of the best neighbors of vi in Πa are neighbors of xi in Πb, any pair of currently obtained plan, we safely prune the partial plan. Yet vertices in Πa that corresponds to a pair ri,rj ∈ C forms an this solution brings nontrivial technical challenges, as we need v ,v ∈ E C S¯ edge in Πa, i.e., ( i j) Πa . Thus, ( ) induces a clique to: (1) quickly compute the AU score of a candidate plan; (2) G OPT in Πa , hence the largest size it can get is (Πa). derive an effective upper bound for the AU score of a partial If all n pieces are propagated in the network and at least one plan, and (3) efficiently compute the upper bound function. ¯ 1 vertex ri receives them all, then adoption utility is σ(S) ≥ 2 . This section addresses these challenges. A branch-and- On the other hand, if less than n pieces are propagated in the bound technique that efficiently evaluates the AU of any network, then, even if all vertices ri, i =1,...,n, receive candidate assignment plan is the subject of Section V-A. We σ S¯ < 1 them all, adoption utility is ( ) 2n . Thus, an optimal present a tight upper bound function for effective pruning that ∗ plan S¯ for Πb should ensure all n pieces are propagated in ensures a (1 − 1/e) approximation ratio in Section V-B. Last, the network and at least one vertex receives them all (as long we propose an efficient progressive estimation scheme that has as that is possible). To include all n pieces, since the budget is a slightly worse (1 − 1/e − ε) approximation ratio with huge k = n promoters, S¯∗ should assign each piece to exactly one performance boost in Section V-C.
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Authorized licensed use limited to: Renmin University. Downloaded on June 01,2021 at 07:12:03 UTC from IEEE Xplore. Restrictions apply. TABLE II: Example of MRR samples and AU estimation A. AU Estimation 1 2 vertex Ri for t1 Ri for t2 p(Xi =1|{{a}, {e}}) R 1 =027 To employ a branch-and-bound framework, we need to 1 c ac cde 1+exp {3−2} . R 1 =012 evaluate the AU for a large number of candidate assignment 2 a a a 1+exp {3−1} . R 1 =027 plans. Since evaluating the influence spread of any user set in 3 b ab be 1+exp {3−2} . R 1 =027 the classical IM problem is #P -hard, it follows that computing 4 c ac cde 1+exp {3−2} . the AU for any candidate plan is also #P -hard. Yet we can evaluate AU with good accuracy using an extension of the reverse reachable (RR) sets method [7], [33], [32]. We briefly B. Upper Bound Function by Branch-and-Bound review the RR set method in the following. A naive OIPA solution would enumerate all possible Reverse-Reachable (RR) Sets. Given a homogeneous in- candidate plans and compute their AU scores using MRR sets. fluence graph G =(V ,E ), |V | = n , where a single · Yet that would incur n AU score estimations. Instead, we value p(e) characterizes the activation probability through k estimate an AU upper bound function by branch-and-bound. edge e =(u, v) ∈ E , the RR set method estimates the expected influence of any seed set S ∈ V .ARandom RR set rests on two random choices: (i) an initial node x is Algorithm 1 Branch-and-Bound R ← θ G randomly chosen from the graph, and (ii) the graph is sampled 1: Generate MRR sets for T on . S←∅¯ ∅ {S ∅,S ∅,...,S ∅} by keeping each edge e ∈ E with probability p(e). The RR 2: = 1 = 2 = = Vp ←{V V p,V V p, ..., V V p} set comprises all vertices from which we can reach x in the 3: 1 = 2 = = L ← U ←∞ S¯∗ ← S¯ sampled graph. Let I[R ∩ S = ∅] be a boolean variable that 4: 0, , i H ← S¯, Vp,U indicates whether RR set R intersects with S. Then, after 5: Initialize max heap ( ) i LLthen ∗ select θ users from V uniformly at random. Yet now, each 15: L ← La and S¯ ← S¯c U a >L viral piece tj induces a homogeneous influence graph where 16: if then H ← H ∪ S¯a, Vp,Ua the influence probability of edge e is computed as p(tj,e)= 17: ( ) tj ·p(e). Then, for each selected user vi, we generate a multi-set 18: Repeat Lines 13-17 for S¯b R {R1,...,R } of RR sets, one for each viral piece tj, i = i i . i Given an assignment plan S¯ = {S1,...,S }, we denote I = Sj Algorithm 1 presents this framework. We first generate θ I[Rj ∩ S = ∅]. Then an unbiased estimator of σ(S¯) is: i j MRR sets for campaign T (Line 1). Then we initialize a max n 1 heap with each entry denoted as (S¯, Vp,U) (Lines 2–5). S¯ is σ(S¯) ← (6) θ {α − β · Ii } a partial plan, Vp the set of promoters that have not yet been =1 1+exp j=1,..., S ] i ,...,θ j considered and U the upper bound of the corresponding search Lemma 2. Eqn. 6 gives an unbiased estimator for σ(S¯). space. In each iteration of the search loop, we get the top entry of the heap by upper bound value, (S¯, Vp), and update the We refer the reader to the appendix for the proof. This global upper bound U. As long as the partial plan has not θ X estimator is the sum of i.i.d random variables i =[1+ been filled up, we select a promoter v∗ from Vp, and branch i −1 exp{α − β · =1 I }]] . We can adopt the Chernoff ∗ j ..., Sj the search space to examine the cases of including v and bound used in the RR sets method [26] to derive the MRR excluding v∗ from S¯, to form two new partial plans S¯a and S¯b θ convergence speed. In practice, a large ensures the estimated (Lines 9–12). For each such partial plan, e.g., S¯a,weinvoke S¯ AU score for any is accurate with a high probability. ComputeBound(·) to get a triple (S¯c,La,Ua), where U a S¯c Example 3. Table II shows an example of using MRR samples is the upper bound for the partial plan and is a complete La S¯c to estimate AU, following the running example in Figure 1. We candidate plan with AU score (Line 13). If is better than S¯∗ sample four vertices, c, a, b, c, and then build the RR sets for the current best solution , we update the global lower bound L S¯a t1 and t2. We compute the adoption probability for each MRR and add the partial solution back to the heap if its upper set shown in the table, and estimate the AU score for the bound is better than the global lower bound (Lines 14–17). ¯ ¯ 5 We repeat the process for the other partial solution S¯b. The assignment plan S = {{a}, {e}} as σ(S)= 4 (0.27 + 0.12 + 0.27 + 0.27) = 1.16. algorithms terminates when the global lower bound is larger or equal to the global upper bound.
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Authorized licensed use limited to: Renmin University. Downloaded on June 01,2021 at 07:12:03 UTC from IEEE Xplore. Restrictions apply. The ComputeBound(S¯a, Vp, R) function estimates the po- Algorithm 2 ComputeBound(S¯a, Vp, R) S¯a a a tential of a partial plan , i.e., the maximum AU score of all 1: Refine τ(·|S¯ ) according to the updates in S¯ ¯a possible complete plans that contain S . If this bound is lower /** Run greedy selection from Vp **/ ¯a than the global lower bound, we can safely prune S from the 2: S←∅¯ ∅ search space. Formally, the ComputeBound functions solves 3: while |S|¯ ≤ k −|S¯a| do ∗ a a the following optimization problem: v ,V p τ S∪¯ {v}|S¯ −τ S|¯ S¯ 4: ( j) = argmaxv∈Vj ∈V ( j ) ( ) p 1 5: Vj ← Vj \{vj} and update V with Vj ¯c ∗ max|S |≤k i S←¯ S∪¯ {v } {α − β · I c } 6: j =1 1+exp j=1..l S (7) i ,..,θ j ¯ ¯a ¯ ¯a ¯ ¯a Sc ⊇ Sa ∀Sc ∈ S¯c 7: return (S∪S ,σ(S∪S ),τ(S|S )) s.t. j j j
While this optimization is still non-submodular, we propose a bound estimation method based on adaptively solving a function obtained by the tangent-line method, this greedy submodular optimization problem. Figure 2 illustrates the main strategy provides an approximation guarantee. In more detail, idea: Although each individual logistic function is not sub- Algorithm 2 first refines the upper bound function τ(·|S¯a),in th modular, we choose the submodular function that most tightly case S¯a has a promoter who appears in the i MRR set. Since upper bounds the function value, by means of a tangent line at most one promoter, say v, is added to S¯a in the branch- that intersects with the logistic S-curve. Since the objective and-bound framework (Lines 11–12, Algorithm 1), we only ¯a function in Equation 7 is a sum of logistic functions, the sum update those τi(·|S ) functions that correspond to MRR sets of these upper bound functions is submodular. Whenever the in which v appears. Then the algorithm greedily selects the branch-and-bound routine searches a subspace which contains remaining k −|S¯a| promoters to form a complete plan S¯ with a partial assignment plan, we update the upper-bound function k assignments (Lines 2–6), and returns S∪¯ S¯a as a candidate as in Figure 2. For example, if we find a user v activated for solution with its AU score, computed by Equation 6, as a lower a τ S|¯ S¯a viral piece tj by a partial assignment plan S¯ , we refine the bound and ( ) as an upper bound. submodular upper bound function by shifting the tangent line Theorem 2. The branch-and-bound framework using Algo- to a larger gradient. rithm 2 for upper bound estimation achieves a (1 − 1/e) approximation ratio to the MRR-based solution to OIPA. Æ ´Ì Ɣ̊ Æ ´Ì Ɣ̊ Proof. Since τ(·) is monotone and submodular, and ComputeBound(·) selects a candidate solution S¯ by the greedy approach, it is τ(S|¯ S¯a) ≥ (1 − 1/e)τ(S¯∗|S¯a) ≥ (1 − 1/e)σ(S¯∗ ∪ S¯a) for all |S¯∗| = |S|¯ . Let S¯g denote the returned assignment plan. For any τ(S|¯ S¯a) that has not been evaluated when the search terminates, we deduce that Ǝë Ǝë ìƎë ÎƔ ȕ ìʜ¥Ì Ǝë ÎƔ ȕ ìʜ¥Ì Ǝë g a g ∗ a ¯À ¯À σ S¯ ≥ τ S|¯ S¯ σ S¯ ≥ − /e σ S¯ ∪ S¯ ÀͰ̊TT ÀͰ̊TT ( ) ( ). Thus, ( ) (1 1 ) ( ) for any unexplored partial solution S¯a. S¯g is the best solution v Fig. 2: Example of upper-bound refinement when user is among all evaluated ones, hence the guarantee holds. found to be activated by piece tj. The approximation ratio in Theorem 2 holds with respect to the OIPA solution computed by estimating AU scores σ(·) τ S|¯ S¯a Definition 6. The upper bound function ( ) is defined from sampled MRR sets, which incurs a sampling error. Still, S∈¯ V p × V p × over a space of all possible assignment plans when generating millions of MRR sets, this error is negligible. ...× V p s.t. a a C. Progressive Upper Bound Estimation τ(S|¯ S¯ )= τi(S|¯ S¯ ) i=1,...,θ The branch-and-bound framework repeatedly invokes ComputeBound(·) to compute an upper bound for each τ S|¯ S¯a where i( ) is a minimal monotone submodular function sub-search space that corresponds to a partial assignment ¯ ¯a ¯ for S s.t. S ⊆ S and plan. Although the greedy selection in Algorithm 2 gives an 1 (1 − 1/e) approximation ratio, it scans all available promoters τ (S|¯ S¯a) ≥ i {α − β · Ii } in each iteration, incurring O(kn) evaluations of AU marginal 1+exp j=1..l S j gain over τ(·|S¯a) (Line 4), each performed on millions of We derive τ(S|¯ S¯a) (i.e., discuss how to efficiently obtain MRR sets (Definition 6). Nevertheless, due to the power law the tangent line) in the appendix due to space constraints. principle of social influence, most users have small social Armed with this upper bound function, Algorithm 2 influence, hence it is unnecessary to scan all promoters when presents the pseudocode for ComputeBound(S¯a, Vp, R), constructing partial plans. Motivated by this observation, we which greedily finds a promoter of maximum marginal gain propose a progressive upper bound estimation method that a a on τi(·|S¯ ) in each iteration. As τi(·|S¯ ) is a submodular maintains a (1 − 1/e − ε) approximation ratio, where ε is
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Authorized licensed use limited to: Renmin University. Downloaded on June 01,2021 at 07:12:03 UTC from IEEE Xplore. Restrictions apply. a tunable parameter that trades efficiency with accuracy. This simplify the presentation, we denote v ∈ S¯ if there exists x y method is based on an asymptotic bound for the number of a j =1, .., s.t. v ∈ Sj ∈ S¯, and denote S¯ \ S¯ = τ ·|S¯a {Sx \ Sy, ..., Sx \ Sy} S¯x, S¯y vi ( ) evaluations, based on the power law principle. 1 1 for any two plans . Last, The idea of the progressive estimation method is as fol- denotes the ith promoter selected in Algorithm 3. lows. Instead of scanning all promoters to find the one with k k −|S¯a| S¯ τ ·|S¯a Lemma 3. Let = . The solution obtained by maximum marginal gain in ( ) in each iteration, we ¯ a Algorithm 3 with |S| = d ≤ k has an approximation ratio of sort promoters v by their individual τ(·|S¯ ). Then, we set 1 − e−d/((1+ε)k ) to the optimal solution w.r.t. (S¯a, Vp, R). a threshold h and include a promoter in the candidate plan if its marginal gain is larger than h. We progressively lower Proof. Let S¯∗ be the optimal solution for the instance the threshold so as to include more promoters. This method (S¯a, Vp, R) and assume vi is selected at a given threshold accelerates upper bound estimation thanks to two features: h. Due to the submodularity of τ(·|S¯a), it follows that: First, the sorting process does not need to rerun in each ≥ h if v = vi iteration. We only need to update the position of promoters δS¯(v)= ∗ (8) ∗ ≤ h · (1 + ε) v ∈ S¯ \ (S∪¯ {vi}) v whose score is affected by the new promoter v in S¯a, if j v∗ i.e., who coexist in an MRR set with . Second, when the where S¯ is the current partial plan and δS¯(v) is as defined i threshold is small enough, the algorithm terminates and returns in Line 8 of Algorithm 3. Equation 8 implies that δS¯(v ) ≥ a plan even if the number of assignments is lower than k.As ¯∗ ¯ i δS¯(v)/(1 + ε) for v ∈ S \ S. Thus, we have δS¯(v ) ≥ 1 1 we will discuss, this second feature tightly bounds the number δ ¯ v δ ¯ v (1+ )|S¯∗\S|¯ v∈S¯∗\S¯ S ( )= (1+ ) v∈S¯∗\S¯ S ( ). of τ(·) evaluations. ε ε k Let S¯i be partial plan obtained by the algorithm after including i promoters and vi+1 be the promoter included at Algorithm 3 ComputeBoundPro(S¯a, Vp, R) the (i +1)th step. Then we have: 1: Refine τ(·|S¯a) according to the updates in S¯a p a ¯i+1 ¯a ¯i ¯a i+1 2: Reorder v ∈ V ∈ V by δ∅(v)=τ(∅∅∪ {v}|S¯ ) − τ(S |S ) − τ(S |S )=δS¯i (v ) j j τ(∅∅|S¯a) for j =1,..., 1 ≥ δS¯i (v) maxinf ← p δ v 3: maxv∈Vj ∈V ∅( ) (1 + ε)k v∈S¯∗\S¯i 4: h ← maxinf ¯ 1 ∗ 5: S←∅∅ ≥ (τ(S¯ ∪ S¯i|S¯a) − τ(S¯i|S¯a)) ε k 6: while |S|¯ ≤ k −|S¯a| do (1 + ) v ∈ V ∈ p 1 ∗ 7: for j V do ≥ (τ(S¯ |S¯a) − τ(S¯i|S¯a)) a a ε k 8: δS¯(v)=τ(S∪¯ j {v}|S¯ ) − τ(S|¯ S¯ ) (1 + ) 9: if δS¯(v) ≥ h then 10: S←¯ S∪¯ j {v} where the first inequality is obtained from Equation 8, the 11: if δ∅(v) 453 Authorized licensed use limited to: Renmin University. Downloaded on June 01,2021 at 07:12:03 UTC from IEEE Xplore. Restrictions apply. First, d = k. The approximation ratio then follows directly The proof follows from Equation (9) and Lemma 4. In prac- from Lemma 3, as we have: tice, τ(S|¯ S¯a) is large especially for large k, i.e., τ(S|¯ S¯a) ∼ n, τ ·|S¯a −1 ∗ −1 ∗ hence the number of ( ) evaluations is small. Compared to τ S|¯ S¯a ≥ − e 1+ε τ S¯ |S¯a ≥ − e − ε τ S¯ |S¯a ( ) 1 ( ) (1 ) ( ) the greedy selection that performs O(nk) evaluations, the pro- ∗ gressive approach effectively avoids unnecessary evaluations where S¯ is the optimal solution. while maintaining a (1 − 1/e − ε) approximation ratio. Second, d 454 Authorized licensed use limited to: Renmin University. Downloaded on June 01,2021 at 07:12:03 UTC from IEEE Xplore. Restrictions apply. • BAB-P: The progress upper bound estimation techniques more promoters would enable a wider spread of viral messages proposed in Section V-C to speedup BAB. and lead to higher adoption utility of the campaign. IM and Note that for all compared approaches, we need to generate TIM show inferior utilities than those of BAB and BAB-P. θ RR sets for each viral pieces. For a fair comparison, we IM has the worst utility because it completely ignores influ- fix θ =106 across all experiments. Note that when comparing ence behavior of viral pieces with different topic distribution. the efficiency of different approaches, we exclude the sampling TIM has better results compared with IM since it chooses the time for generating RR sets since the time is the same for all promoters who collectively maximize the spread of a single compared approaches. The sampling overheads can be found viral piece. However, users have a low probability to adopt in Table III. the campaign if they only receive one viral piece thus the Parameter and Query. The parameters in this experiments overall adoption probability could be low. Our proposed BAB are: (1) k is the number of promoters selected for the campaign and BAB-P achieve superior adoption utility as the branch T. (2) is the number of viral pieces in T. For each viral piece, and bound framework ensures the theoretical guarantee. On we generate the topic vector by uniformly sampling a non- top of that, BAB-P demonstrates competitive solution quality zero topic dimension. (3) α, β are parameters in the logistic with BAB with near-equivalent adoption utilities. Thus, the model. We fix β =1and vary β/α to test the performance progressive upper bound estimation technique introduced in against increasing difficulty levels for the user to adopt the BAB-P does not show significant quality degradation even idea promoted by T. (4) For Vp, we select 10% users from for a larger k empirically. V since in reality not all users are eligible for promoting ads. The efficiency results of all proposed approaches are also The parameter setup can be found in Table IV. shown in Fig. 4 (y-axis is plotted in log-scale). The run time increases for all compared approaches as more promoters TABLE IV: Parameters in the experiments are selected into a plan. IM and TIM are efficient since Parameter Values k 10,20,..., 50, ..., 100 they simply run the greedy maximum cover algorithm on the 1, 2, 3,4,5 generated samples. BAB takes much longer run time as it β/α 0.3, 0.5, 0.7 needs to repeatedly invoke the ComputeBound(·) functions, ε 0.1, 0.2, ..., 0.5, ..., 0.9 each of which is a greedy procedure to select the promoters, Experiment Settings. All the methods are implemented with until the procedure stops (Algorithm 1). BAB-P optimizes C++ and ran on a Ubuntu 14.04 server (Intel(R) Xeon(R) CPU the computation of ComputeBound(·) via the progressive E5-2650 v2 with 256G memory). upper bound estimation technique with early termination and demonstrates superior efficiency (Algorithm 2), achieving up B. Tunning Parameter ε to 24x, 22x, 8.1x speedups compared with BAB in lastfm, Recall that BAB-P progressively lowers the threshold h dblp and tweet respectively. BAB-P shows great scalability for a promoter to be included in the solution (Sec. V-C) by for a larger k and has converging performance with TIM. a ratio of (1 + ε). We examine how the parameter ε affects This is because BAB-P does not need to scan all candidate the solution quality of BAB-P. The results are illustrated in promoters to obtain an upper bound estimation and could early Figure 3. The adoption utility shows a descending trend when terminate even when there are less than k promoters selected. ε rises. Thus, the larger ε is, the easier an promoter is included, On the contrary, TIM, IM and BAB need to iteratively scan all which could potentially degrade utility. When lowering ε from candidates and thus their performance degrades more severely 0.1 to 0.9, the adoption utility drops by 0.08%, 6.6% and 1.4% than BAB-P for larger k. for lastfm, dblp and tweet respectively, a result which aligns with the theoretical result presented in Theorem 3. For D. Varying the Number of Viral Pieces the remaining part of the experiments, we fix the parameter ε We examine the effect of conducting a campaign with to be 0.5. varying the number of viral pieces (Fig. 5). The utility of all lastfm dblp tweet the compared approaches increases as more viral pieces are 15.574 92 6110 BAB-P BAB-P BAB-P 6100 15.572 91 being promoted. This is because, the adoption probability of 6090 15.57 90 6080 a user u increases with the number of viral pieces influencing 15.568 89 6070 6060 u when β =1, as the model defined by Eqn. 1. The qualities utility utility utility 15.566 88 6050 15.564 87 6040 of IM and TIM degrade with larger compared with BAB 6030 15.562 86 6020 and BAB-P since they do not optimize towards multiple 15.56 85 6010 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9 viral pieces. Take tweet for an example, BAB achieves ε ε ε quality gain of 71x, 2.9x against IM and TIM respectively Fig. 3: Tunning parameter ε for BAB-P. when l =5, meanwhile BAB-P has competitive qualities k against BAB. It is worth-noting that the quality of IM and C. Varying the Number of Promoters TIM is severely poor in tweet. This is because the average The experimental results for the compared approaches with number of non-zero topic influence probability (i.e., p(e|z))) in varying k are shown in Fig. 4. The utility of all proposed tweet is only 1.5 across all edges in the dataset and tweet methods increases with a larger k, which is expected since more topics than lastfm and dblp. Under this scenario, 455 Authorized licensed use limited to: Renmin University. Downloaded on June 01,2021 at 07:12:03 UTC from IEEE Xplore. Restrictions apply. lastfm dblp tweet lastfm dblp tweet 30 160 8000 IM IM IM IM IM IM TIM TIM TIM TIM TIM TIM 140 7000 100 25 BAB BAB BAB BAB 100 BAB 100 BAB BAB-P 120 BAB-P 6000 BAB-P BAB-P BAB-P BAB-P 20 100 5000 10 10 10 15 80 4000 time time time utility utility utility 1 60 3000 10 1 1 40 2000 0.1 5 20 1000 0 0 0 0.01 0.1 0.1 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 k k k k k k t Fig. 4: Varying number k of promoters. lastfm dblp tweet lastfm dblp tweet 17 200 9000 IM IM IM IM IM IM TIM 180 TIM 8000 TIM 1000 TIM TIM TIM 16.5 BAB BAB BAB BAB 1000 BAB 1000 BAB BAB-P 160 BAB-P 7000 BAB-P BAB-P BAB-P BAB-P 140 100 16 6000 100 100 120 5000 15.5 100 10 time time time utility utility utility 4000 80 10 10 1 15 60 3000 2000 1 1 14.5 40 0.1 20 1000 14 0 0 0.01 0.1 0.1 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 l l l l l l Fig. 5: Varying number of viral pieces . lastfm dblp tweet VII. CONCLUSION &FUTURE WORK 24 140 8000 IM IM IM 22 TIM TIM TIM 120 7000 20 BAB BAB BAB We introduced a novel formulation of a real-world setting BAB-P BAB-P 6000 BAB-P 18 100 5000 16 80 where a network influence campaign T is multifaceted, con- 14 4000 utility utility utility 60 12 3000 sisting of various pieces pertaining to its different facets. We 10 40 2000 8 described user adoption behavior in this setting by a logistic 20 6 1000 4 0 0 model and introduced the Optimal Influential Piece Assign- 0.3 0.5 0.7 0.3 0.5 0.7 0.3 0.5 0.7 OIPA k β/α β/α β/α ment ( ) that assigns campaign elements to promoters in a way to maximizes adoption utility. We showed that it is Fig. 6: Varying the ratio β/α. NP-hard to approximate OIPA within any constant factor. Nevertheless, we developed a branch-and-bound framework optimizing a single viral piece results in low influence spread that solves OIPA with an (1 − 1/e) approximation ratio, a thus produces poor overall adaption utility. novel formulation of monotone and submodular optimization The runtime of all the compared approaches increases with to compute an upper bound of the unexplored search space, larger and the performance trend follows previous observa- and a progressive pruning-intensive approach to efficiently tions: given competitive solution qualities, BAB-P is able to compute this bound. We show that this progressive approach achieve substantial efficiency improve over BAB. Although has an approximation factor of (1 − 1/e − ε) and much lower IM and TIM have better overall performance, this is at the time complexity than the plain branch-and-bound approach, expense of significant lower utility. by virtue of the power law principle. Our experiments show β/α that our solutions achieve adoption utility superior to that E. Varying the Ratio of two intuitive baselines adapted from state-of-the-art IM The experimental results for the compared approaches with approaches, while the progressive approach achieves up to 24- varying β/α are shown in Fig. 6. The utility shows an fold speedups over the plain approach. increasing trend when the ratio is set to larger values. Since we In this work, the viral pieces are spread in the network fix β =1, β/α rises when α drops. When that happens, the independently. It would be interesting to study the interde- probability of a user v adopting a campaign increases (Eqn. 1), pendence of different viral pieces while still optimizing the which leads to higher overall utility. We note that BAB and adoption utility. However, as we have shown the problem to BAB-P has significant better utilities than IM and TIM, with be intractable in general, a promising future direction would greater improvement ratio for a smaller β/α (larger α). For be to relax the adoption behavior model in a way that would example, in tweet, BAB has 190% utility improvement over render the problem tractable, i.e., monotone and submodular, TIM when β/α =0.7, whereas the improvements pump to yet add interdependence considerations as well. 280% when β/α =0.3. This is because for a smaller β/α,itis Acknowledgement. Yuchen Li was supported by the Singa- harder for a user to adopt the campaign for receiving one viral pore MOE Tier 1 grant MSS18C001. Ju Fan was supported by piece only. Thus, it requires a more sophisticated optimization National Natural Science Foundation of China (No. 61602488, technique to find promoters for different viral pieces. We omit No. 61632016), the Research Funds of Renmin University of the efficiency results since they demonstrate similar patterns China (No. 18XNLG18) and the Tencent Social Ads Rhino- with previous results. Bird Focused Research Grant. 456 Authorized licensed use limited to: Renmin University. Downloaded on June 01,2021 at 07:12:03 UTC from IEEE Xplore. Restrictions apply. VIII. APPENDIX [6] J. Bollen, H. Mao, and A. Pepe. Modeling public mood and emotion: Twitter sentiment and socio-economic phenomena. In ICWSM, 2011. Proof of Lemma⎡ 2. We show the lemma as follows: ⎤ [7] C. Borgs, M. Brautbar, J. Chayes, and B. Lucier. Maximizing social influence in nearly optimal time. In SODA, pages 946–957, 2014. n 1 [8] C. Budak, D. Agrawal, and A. El Abbadi. Limiting the spread of Eqn. 6=E ⎣ ⎦ θ {α − β · Ii } misinformation in social networks. In WWW, pages 665–674, 2011. =1 1+exp j=1..l S ] i ,..,θ j [9] S. Chen, J. Fan, G. Li, J. Feng, K. Tan, and J. Tang. 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