{"title": "On Misinformation Containment in Online Social Networks", "book": "Advances in Neural Information Processing Systems", "page_first": 341, "page_last": 351, "abstract": "The widespread online misinformation could cause public panic and serious economic damages. The misinformation containment problem aims at limiting the spread of misinformation in online social networks by launching competing campaigns. Motivated by realistic scenarios, we present the first analysis of the misinformation containment problem for the case when an arbitrary number of cascades are allowed. This paper makes four contributions. First, we provide a formal model for multi-cascade diffusion and introduce an important concept called as cascade priority. Second, we show that the misinformation containment problem cannot be approximated within a factor of $\\Omega(2^{\\log^{1-\\epsilon}n^4})$ in polynomial time unless $NP \\subseteq DTIME(n^{\\polylog{n}})$. Third, we introduce several types of cascade priority that are frequently seen in real social networks. Finally, we design novel algorithms for solving the misinformation containment problem. The effectiveness of the proposed algorithm is supported by encouraging experimental results.", "full_text": "On Misinformation Containment in\n\nOnline Social Networks\n\nDepartment of Computer and Information Sciences\n\nGuangmo (Amo) Tong\n\nUniversity of Delaware\namotong@udel.edu\n\nWeili Wu\n\nDepartment of Computer Science\n\nUniversity of Texas at Dallas\nweiliwu@utdallas.edu\n\nDing-Zhu Du\n\nDepartment of Computer Science\n\nUniversity of Texas at Dallas\n\ndzdu@utdallas.edu\n\nAbstract\n\nThe widespread online misinformation could cause public panic and serious eco-\nnomic damages. The misinformation containment problem aims at limiting the\nspread of misinformation in online social networks by launching competing cam-\npaigns. Motivated by realistic scenarios, we present an analysis of the misinfor-\nmation containment problem for the case when an arbitrary number of cascades\nare allowed. This paper makes four contributions. First, we provide a formal\nmodel for multi-cascade diffusion and introduce an important concept called as\ncascade priority. Second, we show that the misinformation containment problem\ncannot be approximated within a factor of \u2126(2log1\u2212\u0001 n4\n) in polynomial time unless\nN P \u2286 DT IM E(npolylog n). Third, we introduce several types of cascade priority\nthat are frequently seen in real social networks. Finally, we design novel algorithms\nfor solving the misinformation containment problem. The effectiveness of the\nproposed algorithm is supported by encouraging experimental results.\n\n1\n\nIntroduction\n\nThe past years have witnessed a drastic increase in the usage of online social networks. By the end\nof April 2018, there are totally 3.03 billion active social media users and each Internet user has an\naverage of 7.6 social media accounts [24]. Despite allowing ef\ufb01cient exchange of information, online\nsocial networks have provided platforms for misinformation. Misinformation may lead to serious\neconomic consequences and even cause panics. For example, it was reported by NDTV that the\nmisinformation on social media led to Pune violence in January 2018.1 Recently, the rapid spread\nof misinformation has been on the list of top global risks according to World Economic Forum 2.\nTherefore, effective strategies on misinformation control are imperative.\nInformation propagates through social networks via cascades and each cascade starts to spread\nfrom certain seed users. When misinformation is detected, a feasible strategy is to launch counter\ncampaigns competing against the misinformation [1]. Such counter campaigns are usually called\nas positive cascades. The misinformation containment (MC) problem aims at selecting seed users\nfor positive cascades such that the misinformation can be effectively restrained. The existing works\n\n1https://www.ndtv.com/mumbai-news/misinformation-on-social-media-led-to-pune-violence-minister-\n\n1795562\n\n2http://reports.weforum.org/global-risks-2018/digital-wild\ufb01res/\n\n32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montr\u00e9al, Canada.\n\n\fhave considered this problem for the case when there is one misinformation cascade and one positive\ncascade [2, 3, 4]. In this paper, we address this problem for the general case when there are multiple\nmisinformation cascades and positive cascades. The scenario considered in this paper is more realistic\nbecause there always exists multiple cascades concerning one issue or news in a real social network.\nExample 1. In the 2016 US presidential election, the fake news that Hillary Clinton sold weapons\nto ISIS has been widely shared in online social networks. More than 20 articles spreading this fake\nnews were discovered on Facebook in October 2016 [5]. While these articles all supported the fake\nnews, they were spreading on Facebook as different information cascades because they had different\nsources and exhibited different levels of reliability. On the other hand, multiple articles aiming at\ncorrecting this fake news were being shared by the users standing for Hillary Clinton. These articles\ncan be taken as the positive cascades and, again, they spread as individual cascades. The model\nproposed in this paper applies to such a scenario.\n\nWe introduce an important concept, called as cascade priority, which de\ufb01nes how the users make\nselections when more than one cascades arrive at the same time. As shown later, the cascade priority\nis a necessary and critical setting when multiple cascades exist. The model proposed in this paper\nis a natural extension of the existing models, but the MC problem becomes very challenging under\nthe new setting. For example, as shown later in Sec. 5, adding more seed nodes for the positive\ncascade may surprisingly cause a wider spread of misinformation, i.e., the objective function is not\nmonotone nondecreasing. Our goal in this paper is to offer a systematic study, including formal\nmodel formulation, hardness analysis, and algorithm design. The contributions of this paper are\nsummarized as follows.\n\nof \u2126(2log1\u2212\u0001 n4\n\n\u2022 We provide a formal model supporting multi-cascade in\ufb02uence diffusion in online social\nnetworks. To the best of our knowledge, we are the \ufb01rst to consider the issue on cascade priority.\nBased on the proposed model, we study the MC problem by formulating it as a combinatorial\noptimization problem.\n\u2022 We prove that the MC problem under the general model cannot be approximated within a factor\n\u2022 We propose and study three types of cascade priorities, homogeneous cascade priority, M-\ndominant cascade priority, and P-dominant cascade priority, as de\ufb01ned in Sec. 5. These special\ncascade priorities are commonly seen in real social networks, and the MC problem enjoys\ndesirable combinatorial properties under these settings.\n\u2022 We design a novel algorithm for the MC problem by using nontrivial upper bound and lower\nbound. As shown in the experiments, the proposed algorithm outperforms other methods and it\nadmits a near-constant data-dependent approximation ratio on all the considered datasets.\n\n) in polynomial time unless N P \u2286 DT IM E(npolylog n).3\n\n2 Related work.\n\nIn\ufb02uence maximization (IM). The in\ufb02uence maximization (IM) problem is proposed by Kempe,\nKleinberg, and Tardos in [6] where the authors also develop two basic diffusion models, independent\ncascade (IC) model and linear threshold (LT) model. It is shown in [6] that the IM problem is\nactually a submodular maximization problem and therefore the greedy scheme provides a (1 \u2212 1/e)-\napproximation. However, Chen et al. in [7] prove that it is #P-hard to compute the in\ufb02uence and the\nnaive greedy algorithm is not scalable to large datasets. One breakthrough is made by C. Borgs et\nal. [8] who invent the reverse sampling technique and design an ef\ufb01cient algorithm. This technique\nis later improved by Tang et al. [9] and Nguyen et al. [10]. Recently, Li et al. [18] study the IM\nproblem under non-submodular threshold functions and Lynn et al. [19] consider the IM problem\nunder the Ising network. For the continuous-time generative model, N. Du et al. [30] propose a\nscalable in\ufb02uence estimation method and then study the IM problem under the continuous setting.\nMisinformation containment (MC). Based on the IC and LT model or their variants, the MC\nproblem is then proposed and extensively studied. Budak et al. [2] consider the independent cascade\nmodel and show that the MC problem is again a submodular maximization problem when there are\ntwo cascades. Tong et al. [4, 31] design an ef\ufb01cient algorithm by utilizing the reverse sampling\n\n3When there is only one misinformation cascade and one positive cascade, this problem can be approximated\nwithin a factor of 1 \u2212 1/e [2, 3, 4]. Informally, the complexity class DT IM E(f (n)) consists of the decision\nproblems that can be solved in O(f (n)).\n\n2\n\n\fFigure 1: An illustrative example of diffusion process.\n\ntechnique. He et al. [3], Fan et al. [11] and Zhang et al. [12] study the MC problem under competitive\nlinear threshold model. Nguyen et al. [13] propose the IT-Node Protector problem which limits the\nspread of misinformation by blocking the high in\ufb02uential nodes. Different from the existing works,\nwe focus on the general case when more than two cascades are allowed. In other contexts, He et al.\n[20] study the MC problem in mobile social networks and Wang et al. [21] study the MC problem\nwith the consideration of user experience. Mehrdad et al. [28] consider a point process network\nactivity model and study the fake news mitigation problem by reinforcement learning. Recently, a\ncomprehensive survey [29] regarding false information is provided by Srijan et al.\n\n3 Model and problem formulation\n\nIn this section, we formally formulate the diffusion model and the MC problem.\n\n3.1 Model\n\nA social network is given by a directed graph G = (V, E). For each edge (u, v), we say v is an\nout-neighbor of u, and u is an in-neighbor of v. Information is assumed to spread via cascades and\neach cascade spreads from seed users. Let C be the set of all the cascades, and we use \u03c4 (C) \u2286 V to\ndenote the seed set of a cascade C \u2208 C. We say a user is C-active if they are activated by cascade C.\nAll users are initially de\ufb01ned as \u2205-active. Associated with each edge (u, v), there is a real number\np(u,v) \u2208 [0, 1] denoting the propagation probability from u to v. We assume that p(u,v) = 0 iff\n(u, v) /\u2208 E. When u becomes C-active for a certain cascade C \u2208 C, they attempt once to activate\nan \u2205-active out-neighbor v with the success probability of p(u,v). We assume that a user will be\nactivated by the cascade arriving \ufb01rst and will not be activated later for another time. Associated\nwith each user v, each cascade C is given a unique priority, denoted by Fv(C), which gives a linear\norder over the cascades. Fv can be represented as a bijection between C and {1, 2, ...,| C|}, and, for\neach C1, C2 \u2208 C, Fv(C1) > Fv(C2) iff C1 has a higher priority than that of C2 at v. If two or more\ncascades reach v at the same time, v will be activated by the cascade with the highest priority. The\ncascade priority at each node is affected by many factors such as the reputation of the source, the\nreliability of the message and the user\u2019s personal opinion. Several special cascade priorities will be\nintroduced later in Sec. 5.\nFor a time step t \u2208 {0, 1, 2, ...}, we use \u03c0t(v) \u2208 C\u222a{\u2205} to denote the activation state of a user v\nafter time step t, where \u03c0t(v) = C (resp. \u03c0t(v) = \u2205) if v is C-active (resp. \u2205-active). Let \u03c0\u221e(v)\nbe the activation state of v when the diffusion process terminates. The diffusion process unfolds\nstochastically in discrete, described as follows:\n\nwhere C\u2217 = arg maxC\u2208{C|C\u2208C,v\u2208\u03c4 (C)} Fv(C) .\n\n\u2022 Time step 0. If a node v is selected as a seed node by one or more cascades, v becomes C\u2217-active\n\u2022 Time step t. Each node u activated at time step t \u2212 1 attempts to activate each of u\u2019s \u2205-active\nout-neighbor v with a success probability of p(u,v). If a node v is successfully activated by one\nor more in-neighbors, v becomes \u03c0t\u22121(u\u2217)-active where u\u2217 = arg maxu\u2208A\u2286V Fv(\u03c0t\u22121(u))\nwhere A is the set of the in-neighbors who successfully activate v at time step t.4\n\nExample 2. Consider the network shown in Fig. 1 where there are three cascades C1, C2 and C3,\nof which the seed sets are {v1}, {v2} and {v3}, respectively. Suppose that pe = 1 for each edge e,\nFv4 (C1) > Fv4(C2), and, Fv6 (C2) > Fv6(C3) > Fv6(C1). At time step 1, v4 becomes C1-active\ndue to that Fv4 (C1) > Fv4 (C2). Because Fv6 (C3) > Fv6(C1), v6 is \ufb01nally C3-active. One can see\nthat v6 would be C2-active if the cascade priority at v4 was Fv4(C2) > Fv4(C1).\n\n4Note that here \u03c0t\u22121(u) cannot be \u2205 so Fv(\u03c0t\u22121(u)) is well-de\ufb01ned.\n\n3\n\nCascade C1C2Cascade C3Cascade v1v2v3v4v6v5v3v3v1v2v3v4v6v5v3v3Step 0 Step 1 Step 2 v2v1v3v6v5v4v7Cascade P1M1Cascade Fv3(P\ue21d)>Fv3(P1)Fv5(P1)>Fv5(M1)>Fv5(P\ue21d)v1v2v3v4v6v5v3v3\f3.2 Problem formulation\n\nWe assume that, regarding one issue or topic, there are two groups of cascades: misinformation\ncascades and positive cascades. Suppose there are already some cascades in the network and their\nseed sets are known to us. For the purpose of misinformation containment, we launch a new positive\ncascade with a certain seed set. We use M and P to denote the sets of the existing misinformation\ncascades and positive cascades, respectively, and use P\u2217 to denote the newly introduced positive\ncascade. Therefore, C = M\u222a P\u222a{P\u2217}. We say a user is M-active if they are M-active for some\nM \u2208 M, otherwise they are called as M-active.5 For a seed set \u03c4 (P\u2217) of cascade P\u2217, we use\nfM(\u03c4 (P\u2217)) (resp. fM(\u03c4 (P\u2217))) to denote the expected number of the M-active (resp. M-active) nodes\nwhen the diffusion process terminates. The problems considered in this paper are shown as follows.\nProblem 1 (Min-M problem). Given a budget k \u2208 Z+ and a candidate set V \u2217 \u2286 V , select a seed set\n\u03c4 (P\u2217) \u2286 V \u2217 for P\u2217 with |\u03c4 (P\u2217)| \u2264 k such that fM(\u03c4 (P\u2217)) is minimized.\nAlternatively, we can maximize the number of the M-active users.\nProblem 2 (Max-M problem). Given a budget k \u2208 Z+ and a candidate V \u2217 \u2286 V , select a seed set\n\u03c4 (P\u2217) \u2286 V \u2217 for P\u2217 with |\u03c4 (P\u2217)| \u2264 k such that fM(\u03c4 (P\u2217)) is maximized.\nAn instance of the above problems is given by (1) G = (V, E): a network structure; (2) {pe|pe \u2208\n[0, 1], e \u2208 E}:\nthe set of the existing\ncascades together with P\u2217; (4) {Fv(C)|v \u2208 V, C \u2208 C}: the cascade priority at each node; (5)\n{\u03c4 (C) \u2286 V |C \u2208 M\u222a P}: the seed sets of the existing cascades; (6) V \u2217 \u2286 V : a candidate set of the\nseed nodes of P\u2217. The propagation probability and the cascade priority can be inferred by mining\nhistorical data [25, 26, 27].\nRemark 1. When | M| = 1, it becomes the model considered in [14]. When | M| = 1, | P| = 0\nand the cascade priority is homogeneous6, the problem considered in [2, 4] reduces to the Max-M\nproblem.\n\nthe probabilities on the edges; (3) C = M\u222a P\u222a{P\u2217}:\n\n4 Hardness result\nIn this section, we provide a hardness result for the Min-M problem. The result is obtained by a\nreduction from the positive-negative partial set cover (\u00b1PSC) problem.\nProblem 3 (\u00b1PSC problem). An instance of \u00b1PSC is a triplet (X, Y, \u03a6) where X and Y are two\nsets of elements with X \u2229 Y = \u2205, and \u03a6 = {\u03c61, ..., \u03c6m} \u2286 2X\u222aY is collection of subsets over\nX \u222a Y . For each \u03a6\u2217 \u2286 \u03a6, its cost is de\ufb01ned as |X \\ (\u222a\u03c6\u2208\u03a6\u2217 \u03c6)| + |Y \u2229 (\u222a\u03c6\u2208\u03a6\u2217 \u03c6)|. The \u00b1PSC\nproblem seeks for a \u03a6\u2217 \u2286 \u03a6 with the minimum cost.\nThe following result is presented by Miettinen [15].\nLemma 1 ([15]). There exists no polynomial-time approximation algorithm for \u00b1PSC with an\napproximation factor of \u2126(2log1\u2212\u0001 |m|4\n\n) for any \u0001 > 0, unless N P \u2286 DT IM E(npolylog n).\n\nA core result is given in the next lemma.\nLemma 2. For any \u03b1(|V \u2217|) > 1, \u00b1PSC is approximable to within a factor of 4\u00b7 \u03b1(m)\u2212 3, if Min-M\nis approximable to within a factor of \u03b1(|V \u2217|).\n\nProof. For an arbitrary instance (X, Y, \u03a6) of the \u00b1PSC problem, we construct an instance of the\nMin-M problem accordingly, as shown in Fig. 2.\nThe graph. Let us \ufb01rst construct the graph G. For each xi \u2208 X, we add a node xi to the graph, and\nfor each yi \u2208 Y we add two nodes yi and zi to the graph. For each \u03c6i \u2208 \u03a6, we add a node \u03c6i to the\ngraph. We further add four nodes a, b1, b2 and c, as shown in Fig. 2. For each \u03c6i and xj (resp. yj),\nwe add an edge (\u03c6i, xj) (resp. (\u03c6i, yj)) iff xj \u2208 \u03c6i (resp. yj \u2208 \u03c6i). For each zi, we add an edge\n(yi, zi) and an edge (c, zi). We add an edge (a, yi) for each yi \u2208 Y and an edge (b2, xi) for each\nxi \u2208 X. Finally, we add an edge (b1, c). The probability of each edge is set as 1.\n\n5Note that an \u2205-active node is M-active.\n6The de\ufb01nition of homogeneous cascade priority is given later in Sec. 5.\n\n4\n\n\fFigure 2: Reduction.\n\nCascade setting. We assume there is one misinformation cascade M1 with the seed set {b1, b2} and\none positive cascade P1 with the seed set {a}. We aim at introducing one positive cascade P\u2217 by\nselecting at most k = m seed nodes from V \u2217 = \u03a6 = {\u03c61, ...., \u03c6m}. For each yi \u2208 {y1, ..., ym}, the\ncascade priority is set as Fyi(P\u2217) > Fyi(P1). For each node zi in {z1, ..., zm}, the cascade priority\nis set as Fzi(P1) > Fzi(M1) > Fzi(P\u2217). The cascade priority at other nodes can be set arbitrarily.\nAnalysis. Each set \u03a6\u2217 \u2286 \u03a6 corresponds to a solution to the Min-M problem. We use g(\u03a6\u2217) to denote\nthe objective function of the \u00b1PSC problem, i.e.,\n\ng(\u03a6\u2217) = |X \\ (\u222a\u03c6\u2208\u03a6\u2217 \u03c6)| + |Y \u2229 (\u222a\u03c6\u2208\u03a6\u2217 \u03c6)|.\n\nNow let us \ufb01x \u03a6\u2217 and analyze the activation state of the nodes. Note that each node yi will be either\nP1-active or P\u2217-active. In particular, yi is P\u2217 active iff yi is in some \u03c6 \u2208 \u03a6\u2217. Furthermore, according\nthe cascade priority at zi, zi is M-active iff yi is P1-active. Therefore, zi is M-active iff yi is not\nin \u222a\u03c6\u2208\u03a6\u2217 \u03c6. For each node xi \u2208 X, it is M-active iff it is in some \u03c6 \u2208 \u03a6\u2217. Finally, it can be easily\nchecked that the nodes in \u03a6 \u222a Y \u222a {a} will be M-active and the nodes in {c, b1, b2} will be M-active,\nregardless of \u03a6\u2217. As a result,\n\nfM(\u03a6\u2217) = 3 + |X \\ \u222a\u03c6\u2208\u03a6\u2217 \u03c6| + |Y \u2229 \u222a\u03c6\u2208\u03a6\u2217 \u03c6| = 3 + g(\u03a6\u2217).\n\nThus, OP T \u2286 \u03a6 is an optimal solution to the MC instance iff OP T is an optimal solution to the\ninstance of the \u00b1PSC problem. Suppose that \u03a6\u2217 is an \u03b1(|V \u2217|)-approximation to the Min-M problem\nfor some \u03b1(|V \u2217|) > 1. We have\n\nfM(\u03a6\u2217) \u2264 \u03b1(|V \u2217|) \u00b7 fM(OP T ) \u21d0\u21d2 3 + g(\u03a6\u2217) \u2264 \u03b1(|V \u2217|) \u00b7 (3 + g(OP T ))\n\n\u2264 4\u03b1(|V \u2217|) \u2212 3.\n\n\u21d0\u21d2 g(\u03a6\u2217)\n\ng(OP T )\n\n\u2264 \u03b1(|V \u2217|) +\n\n3(\u03b1(|V \u2217|) \u2212 1)\n\ng(OP T )\n\n=\u21d2 g(\u03a6\u2217)\n\ng(OP T )\n\nSince |V \u2217| = |\u03a6| = m, \u03a6\u2217 is a (4 \u00b7 \u03b1(m) \u2212 3)-approximation to the instance of the \u00b1PSC\nproblem.\n\nThe following result follows immediately from Lemmas 1 and 2.\nTheorem 1. For any \u0001 > 0, there is no polynomial-time approximation algorithm for the Min-M\nproblem with an approximation factor of \u2126(2log1\u2212\u0001 |V \u2217|4\n\n) unless N P \u2286 DT IM E(npolylog n).\n\n5 Algorithms\n\nIn this section, we present algorithms for the Max-M problem. Throughout this section, we denote\nthe objective function fM as f. The technique of submodular maximization has been extensively\nused in the existing works. For a set function h() over a ground set U, the properties of monotone\nnondecreasing and submodular are de\ufb01ned as follows:\nDe\ufb01nition 1 (Monotone nondecreasing). h(A) \u2264 h(B), for each A \u2286 B \u2286 U.\nDe\ufb01nition 2 (Submodular). h(A) + h(B) \u2265 h(A \u222a B) + h(A \u2229 B), for each A, B \u2286 U.\nAs mentioned in Remark 1, the Max-M problem is a natural extension of the problem considered in\n[2, 4], but it is not submodular and even not monotone nondecreasing.\n\n5\n\nY\u03a6Xb1b2a\u03d51\u03d52\u03d5my1y2y3y4x1x2x3x4z1z2z3z4...c.........\fFigure 3: An illustrative example of non-submodularity.\n\nExample 3. Consider the network shown in Fig. 3, where there exists one positive cascade P1 and\none misinformation cascade M1. Now we deploy a new positive cascade P\u2217 and assume the candidate\nseed set V \u2217 is equal to V . Suppose that the probability on each edge is equal to 1, \u03c4 (P1) = {v1}\nand \u03c4 (M1) = {v7}, and the cascade priority at v3 and v5 is given as shown in the \ufb01gure. We can\nobserve that f ({\u2205}) = 5, f ({v2}) = f ({v4}) = f ({v2, v4}) = 4. Therefore, f ({v2}) < f (\u2205),\nand f ({v2}) + f ({v4}) < f ({v2} \u2229 {v4}) + f ({v2} \u222a {v4}). This illustrates that inappropriately\nselecting positive seed nodes may lead to a wider spread of misinformation.\n\nIn the rest of this section, we \ufb01rst study three special cascade priorities and then design an algorithm\nfor the general setting.\n\n5.1 Special cases: homogeneous, M-dominant and P-dominant cascade priority\n\nWe introduce the following types of cascade priority that frequently appear in real social networks.\nDe\ufb01nition 3 (Homogeneous cascade priority). The cascade priority is said to be homogeneous if\nFv(C) = Fu(C) for each u, v \u2208 V and C \u2208 C. That is, each cascade has the same priority at each\nnode.\nDe\ufb01nition 4 (M-dominant cascade priority). The cascade priority is said to be M-dominant if\nFv(M ) > Fv(P ) for each M \u2208 M, P \u2208 P\u222a{P\u2217} and v \u2208 V . Informally speaking, at each node,\nthe priority of each misinformation cascade is higher than that of any positive cascade.\n\nSimilarly, we have the P-dominant cascade priority.\nDe\ufb01nition 5 (P-dominant cascade priority). The cascade priority is said to be P-dominant if Fv(P ) >\nFv(M ) for each M \u2208 M, P \u2208 P\u222a{P\u2217} and v \u2208 V .\nRemark 2. The homogeneous cascade priority is capable of representing the case when the priority\nof cascade is determined by the source or the initiator of the cascade. For example, when there are\ntwo opposite cascades C1 and C2 regarding NBA on Twitter, where C1 is posted by ESPN while C2\ncomes from an unknown source, the users will all tend to believe C1 and therefore Fv(C1) > Fv(C2)\nfor each v \u2208 V . The M-dominant or P-dominant cascade priority describes the scenario when one\ngroup of the cascades are well polished and very convincing. For example, the fake news in Example\n1 was believed to be true by many online users because it was claimed to be released by WikiLeaks.\nAs a result, the fake news always had a higher cascade priority and Fv(M ) > Fv(P ) for each\nM \u2208 M, P \u2208 P\u222a{P\u2217} and v \u2208 V .\nWhile the Max-M problem does not exhibit any good property in general, it is indeed monotone\nnondecreasing and submodular under special cascade priority settings. For the above types of cascade\npriority, we have the following results.\nTheorem 2. f is monotone nondecreasing and submodular if the cascade priority is M-dominant or\nP-dominant.\nTheorem 3. f is monotone nondecreasing and submodular if the cascade priority is homogeneous.\n\nPlease see the supplementary material for the proofs of Theorems 2 and 3. Note that the greedy\nalgorithm yields a (1 \u2212 1/e)-approximation when the objective function is monotone nondecreasing\nand submodular [16]. Theorems 2 and 3 evince that special cascade priorities may admit desirable\ncombinatorial properties. In the next subsection, we will utilize these results to design an effective\nalgorithm for the Max-M problem for the general case.\n\n6\n\nCascade C1C2Cascade C3Cascade v1v2v3v4v6v5v3v3v1v2v3v4v6v5v3v3Step 0 Step 1 Step 2 v2v1v3v6v5v4v7Cascade P1M1Cascade Fv3(P\ue21d)>Fv3(P1)Fv5(P1)>Fv5(M1)>Fv5(P\ue21d)v1v2v3v4v6v5v3v3\fAlgorithm 1 Greedy scheme\n1: Input: a function h over a ground set U and a budget k;\n2: U0 \u2190 \u2205;\n3: for i = 1 : k do\n4:\n5:\n6: return U \u2190 arg maxUi h(Ui);\n\nu \u2190 arg maxu\u2208U h(Ui\u22121 \u222a {u}) \u2212 h(Ui\u22121);\nUi \u2190 Ui\u22121 \u222a {u};\n\nAlgorithm 2 Sandwich approximation strategy\n1: Input: f, f , f , V \u2217, k;\n2: S\u2217 \u2190 ALG. 1(f , V \u2217, k); S\u2217 \u2190 ALG. 1(f , V \u2217, k); S\u2217 \u2190 ALG. 1(f, V \u2217, k);\n3: return S\n\n= arg maxS\u2208{S\u2217,S\u2217,S\u2217} f (S);\n\n(cid:48)\n\n5.2 General case\n\nFor the general cascade priority, we present a data-dependent approximation algorithm based on the\nupper-lower-bound technique [22]. Each cascade priority Fv induces another two cascade priorities,\nde\ufb01ned as follows:\nDe\ufb01nition 6 (Fv). Fv is a cascade priority at node v induced by Fv, satisfying,\n(a) for each P1, P2 \u2208 P\u222a{P\u2217}, Fv(P1) < Fv(P2) \u21d0\u21d2 Fv(P1) < Fv(P2),\n(b) for each M1, M2 \u2208 M, Fv(M1) < Fv(M2) \u21d0\u21d2 Fv(M1) < Fv(M2), and,\n(c) for each P \u2208 P\u222a{P\u2217} and M \u2208 M, Fv(M ) < Fv(P ).\nDe\ufb01nition 7 (Fv). Fv is a cascade priority at node v induced by Fv(), satisfying (a) and (b) in Def.\n6, and, for each P \u2208 P\u222a{P\u2217} and M \u2208 M, Fv(P ) < Fv(M ).\nFv and Fv keep the relative priority of the cascades within the same group and adjust the relative\npriority of the cascades between groups. We can easily check that Fv and Fv are uniquely determined\nby Fv.\nExample 4. Suppose there are three positive cascades, P1, P2 and P3, and two misinformation cas-\ncades, M1 and M2. If Fv(P3) < Fv(P1) < Fv(M2) < Fv(P2) < Fv(M1), then we have Fv(M2) <\nFv(M1) < Fv(P3) < Fv(P1) < Fv(P2) and Fv(P3) < Fv(P1) < Fv(P2) < Fv(M2) < Fv(M1).\nFor a seed set \u03c4 (P\u2217) \u2286 V \u2217 of cascade P\u2217, we use f (\u03c4 (P\u2217)) (resp. f (\u03c4 (P\u2217))) to denote the expected\nnumber of the M-active nodes when each node v replaces its cascade priority Fv by Fv (resp. Fv).\nBecause Fv is P-dominant and Fv is M-dominant, the following result immediately follows from\nTheorem 2.\nCorollary 1. f and f are both monotone nondecreasing and submodular.\n\nFurthermore, f is an upper bound of f and f is a lower bound of f.\nTheorem 4. For each \u03c4 (P\u2217) \u2286 V \u2217, f (\u03c4 (P\u2217)) \u2265 f (\u03c4 (P\u2217)) \u2265 f (\u03c4 (P\u2217)).\nPlease see the supplementary material for the proof of Theorem 4. We now present an algorithm to\nsolve the Max-M problem by approximating f and f. First, we run the greedy algorithm, ALG. 1,\non all three functions, f, f and f, to obtain three solutions S\u2217, S\u2217 and S\u2217, respectively. The \ufb01nal\n= arg maxS\u2208{S\u2217,S\u2217,S\u2217} f (S). The process is formally shown in ALG. 2.\nsolution is selected as S\nAccording to [22], it has the following performance bound.\nf (OP T )} \u00b7 (1 \u2212 1/e) \u00b7 f (OP T ), where OP T is the optimal\nTheorem 5. f (S\nsolution.\nRemark 3. The performance bound of ALG. 2 depends on the closeness of the upper bound and the\nlower bound. We will experimentally examine this gap in Sec. 6.\n\n) \u2265 max{ f (S\u2217)\n\nf (OP T )\n\n(cid:48)\n\n(cid:48)\n\n,\n\nf (S\u2217)\n\n7\n\n\f6 Experiments\n\nIn this section, we evaluate the proposed algorithm by experiments. Our goal is to examine the\nperformance of ALG. 2 by (a) comparing it to baseline methods and (b) measuring the data-dependent\napproximation ratio given in Theorem 5. Our experiments are performed on a server with a 2.2 GHz\neight-core processor.\n\n6.1 Setup\n\nai\n\namax\n\nDataset. The \ufb01rst dataset, collected from Twitter, is built after monitoring the spreading process of\nthe messages posted between 1st and 7th July 2012 regarding the discovery of a new particle with\nthe features of the elusive Higgs boson [17]. It consists of a collection of activities between users,\nincluding re-tweeting action, replying action, and mentioning action. We extract two subgraphs\nfrom this dataset, where the \ufb01rst one has 10,000 nodes and the second one has 100,000 nodes. We\ndenote these two graphs by Higgs-10K and Higgs-100K, respectively. The second dataset, denoted\nby HepPh, is a citation graph from the e-print arXiv with 34,546 papers [23]. HepPh has been widely\nused in the study on in\ufb02uence diffusion in social networks. The statistics of the datasets can be found\nin the supplementary material.\nPropagation Probability. On Higss-10K, the probability of edge (u, v) is set to be proportional to\n\u00b7 pmax + pbase,\nthe frequency of the activities between u and v. In particular, we set p(u,v) as\nwhere ai is the number of activities from u to v, amax is the maximum number of the activities\namong all the edges, and, pmax = 0.2 and pbase = 0.4 are two constants. On Higgs-100K, we adopt\nthe uniform setting where the propagation probability on each edge is set as 0.1. On HepPh, we adopt\nthe wighted cascade setting and set p(u,v) as 1/deg(v) where deg(v) is the number of in-neighbors of\nv. The uniform setting and the weighted cascade are two classic settings and they have been widely\nused in the existing works [2, 4, 6, 7, 9, 10, 18].\nCascade setting. We consider three cases where there are three cascades, \ufb01ve cascades and ten\ncascades, respectively. For the case of three cascades, we deploy one existing misinformation cascade\nand one existing positive cascade, and we launch a new positive cascade P\u2217. For each existing\ncascade, the size of the seed set is set as 20 and the seed nodes are selected from the node with the\nhighest single-node in\ufb02uence. The seed sets of different cascades do not overlap with each other. The\nbudget of P\u2217 is enumerated from {1, 2, ..., 20} and the candidate set V \u2217 is equal to V . The cascade\npriority at each node is assigned randomly by generating a random permutation over {1, 2, 3}. We\nprocess the cases with \ufb01ve and ten cascades in the same way as the three cascades case. The details\ncan be found in the supplementary material.\nBaseline methods. Since there is no algorithm explicitly addressing the model considered in this\npaper, we consider three baseline methods, HighWeight, Proximity and Random. The weight of\nv p(u,v)). HighWeight\noutputs the seed set according to the decreasing order of the node weight. Proximity selects the seed\nnodes of P\u2217 from the out-neighbors of the seed nodes of the misinformation cascades, where the\npreference is given to the node with a large weight. Random is a baseline method which selects the\nseed nodes randomly. The performance of Random is evaluated by the mean over 1,000 executions.\nEstimating in\ufb02uence. The feasibility of ALG. 2 relies on the assumption that there is an ef\ufb01cient\noracle of fM. Unfortunately, it has been shown in [7] that computing the in\ufb02uence is a #P-hard\nproblem, and in fact, it is also hard to compute fM. In our experiments, the function value is estimated\nby 5,000 Monte Carlo simulations whenever fM is called, and the \ufb01nal solution of each algorithm is\nevaluated by 10,000 simulations. We note that the techniques proposed in [4, 8, 9, 10] are potentially\napplicable to the MC problem, but improving the ef\ufb01ciency of the algorithm is beyond the scope of\nthis paper.\n\na node v is de\ufb01ned as the sum of the probabilities of its out-edges (i.e.,(cid:80)\n\n6.2 Result and discussion\n\nThe experimental results are shown in Figs. 4, 5 and 6. In each \ufb01gure, the \ufb01rst three sub\ufb01gures show\nthe performance under the settings of three, \ufb01ve and ten cascades, respectively. Each sub\ufb01gure gives\nfour curves plotting the number of M-active nodes under Sandwich (ALG. 2), HighWeight, Proximity\nand Random, respectively. The last sub\ufb01gure shows the value of f (S\u2217)/f (S\u2217) in each experiment.\n\n8\n\n\f(a) Three cascades\n\n(b) Five cascades\n\n(c) Ten cascades\n\n(d) Performance bound\n\nFigure 4: Results on Higgs-10K.\n\n(a) Three cascades\n\n(b) Five cascades\n\n(c) Ten cascades\n\n(d) Performance bound\n\nFigure 5: Results on Higgs-100K.\n\n(a) Three cascades\n\n(b) Five cascades\n\n(c) Ten cascades\n\n(d) Performance bound\n\nFigure 6: Results on HepPh.\n\nMajor observations. First, as shown in the \ufb01gures, ALG. 2 consistently provides the best perfor-\nmance. Comparing it to other baseline methods, the superiority of ALG. 2 can be very signi\ufb01cant\nwhen the budget becomes large. As shown in Fig. 4a, on Higgs-10K, when there are three cascades\nand the budget is equal to 20, ALG. 2 is able to reduce the number of M-active nodes from 180 to 100,\nwhile other methods can hardly make it below 160. Another important observation is that the ratio\nf (S\u2217)/f (S\u2217) is very close to 1 in practice. For example, on HepPh, this ratio is always larger than\n0.9985. This means the performance ratio of ALG. 2 is guaranteed to be very close to 1\u2212 1/e on such\ndatasets. From Example 3 and the proofs of Theorems 2 and 3 we can see that the non-submodularity\nonly occurs in the case when two or more cascades arrive at one node at the same time. Thus, if such\na scenario does not happen frequently, the Max-M and Min-M problems will be close to submodular\noptimization problems, and consequently, the greedy algorithm is effective. While f (S\u2217)/f (S\u2217) is\ndata-dependent, we have observed that it is very close to 1 under all the considered datasets, which\nindicates that the approximation ratio is near-constant.\nMinor observations. We can also observe that Random offers no help in misinformation containment\nand HighWeight is also futile in many cases (e.g., Figs. 4a, 5a and 6b where it has the same\nperformance as that of Random). In addition, Proximity performs slightly better than HighWeight\ndoes but it can still fail to reduce the number of M-active users when budget increases, i.e., the curve\nis not monotone decreasing. We have also observed that ALG. 2 strictly outperforms that solely\nrunning ALG. 1 on fM, which means approximating the upper bound and lower bound can provide\nbetter solutions. The results of this part can be found in our supplementary material.\n\n7 Conclusion\n\nIn this paper, we study the MC problem under the general case where there is an arbitrary number of\ncascades. The considered scenario is more realistic and it applies to complicated real applications\nin online social networks. We provide a formal model and address the MC problem from the view\nof combinatorial optimization. We show the MC problem is not only NP-hard but also admits\nstrong inapproximability property. We propose three types of cascade priority and show that the MC\nproblem can be close to submodular optimization problems. An effective algorithm for solving the\nMC problem is designed and evaluated by experiments.\n\n9\n\n510152080100120140160180200BudgetNumber of M\u2212active nodes SandwichHighWeightProximityRandom5101520130140150160170180190BudgetNumber of M\u2212active nodes SandwichHighWeightProximityRandom5101520255260265270275280285290BudgetNumber of M\u2212active nodes SandwichHighWeightProximityRandom51015200.9940.9950.9960.9970.9980.9991BudgetRatio three cascadesfive cascadesten cascades5101520125130135140145150BudgetNumber of M\u2212active nodes SandwichHighWeightProximityRandom5101520170175180185190195200BudgetNumber of M\u2212active nodes SandwichHighWeightProximityRandom5101520300305310315320325330BudgetNumber of M\u2212active nodes SandwichHighWeightProximityRandom51015200.99950.99960.99970.99980.99991BudgetRatio three cascadesfive cascadesten cascades5101520100110120130140150160BudgetNumber of M\u2212active nodes SandwichHighWeightProximityRandom5101520200250300350400450BudgetNumber of M\u2212active nodes SandwichHighWeightProximityRandom5101520500550600650700750BudgetNumber of M\u2212active nodes SandwichHighWeightProximityRandom51015200.9970.99750.9980.99850.9990.999511.00051.001BudgetRatio three cascadesfive cascadesten cascades\fAcknowledgments\n\nThis work is supported in part by a start-up grant from the University of Delaware and the NSF under\ngrant #1747818.\n\nReferences\n\n[1] Kumar, KP Krishna, and G. 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