{"title": "Influence Maximization with $\\varepsilon$-Almost Submodular Threshold Functions", "book": "Advances in Neural Information Processing Systems", "page_first": 3801, "page_last": 3811, "abstract": "Influence maximization is the problem of selecting $k$ nodes in a social network to maximize their influence spread. The problem has been extensively studied but most works focus on the submodular influence diffusion models. In this paper, motivated by empirical evidences, we explore influence maximization in the non-submodular regime. In particular, we study the general threshold model in which a fraction of nodes have non-submodular threshold functions, but their threshold functions are closely upper- and lower-bounded by some submodular functions (we call them $\\varepsilon$-almost submodular). We first show a strong hardness result: there is no $1/n^{\\gamma/c}$ approximation for influence maximization (unless P = NP) for all networks with up to $n^{\\gamma}$ $\\varepsilon$-almost submodular nodes, where $\\gamma$ is in (0,1) and $c$ is a parameter depending on $\\varepsilon$. This indicates that influence maximization is still hard to approximate even though threshold functions are close to submodular. We then provide $(1-\\varepsilon)^{\\ell}(1-1/e)$ approximation algorithms when the number of $\\varepsilon$-almost submodular nodes is $\\ell$. Finally, we conduct experiments on a number of real-world datasets, and the results demonstrate that our approximation algorithms outperform other baseline algorithms.", "full_text": "In\ufb02uence Maximization with \u03b5-Almost\n\nSubmodular Threshold Functions\n\nQiang Li\u2217\u2020, Wei Chen\u2021, Xiaoming Sun\u2217\u2020, Jialin Zhang\u2217\u2020\n\u2217CAS Key Lab of Network Data Science and Technology,\n\nInstitute of Computing Technology, Chinese Academy of Sciences\n\n\u2020 University of Chinese Academy of Sciences\n\n\u2021 Microsoft Research\n\n{liqiang01,sunxiaoming,zhangjialin}@ict.ac.cn\n\nweic@microsoft.com\n\nAbstract\n\nIn\ufb02uence maximization is the problem of selecting k nodes in a social network to\nmaximize their in\ufb02uence spread. The problem has been extensively studied but\nmost works focus on the submodular in\ufb02uence diffusion models. In this paper,\nmotivated by empirical evidences, we explore in\ufb02uence maximization in the non-\nsubmodular regime. In particular, we study the general threshold model in which\na fraction of nodes have non-submodular threshold functions, but their threshold\nfunctions are closely upper- and lower-bounded by some submodular functions\n(we call them \u03b5-almost submodular). We \ufb01rst show a strong hardness result: there\nis no 1/n\u03b3/c approximation for in\ufb02uence maximization (unless P = NP) for all\nnetworks with up to n\u03b3 \u03b5-almost submodular nodes, where \u03b3 is in (0,1) and c is a\nparameter depending on \u03b5. This indicates that in\ufb02uence maximization is still hard\nto approximate even though threshold functions are close to submodular. We then\nprovide (1 \u2212 \u03b5)(cid:96)(1 \u2212 1/e) approximation algorithms when the number of \u03b5-almost\nsubmodular nodes is (cid:96). Finally, we conduct experiments on a number of real-world\ndatasets, and the results demonstrate that our approximation algorithms outperform\nother baseline algorithms.\n\n1\n\nIntroduction\n\nIn\ufb02uence maximization, proposed by Kempe, Kleinberg, and Tardos [1], considers the problem\nof selecting k seed nodes in a social network that maximizes the spread of in\ufb02uence under pre-\nde\ufb01ned diffusion model. This problem has many applications including viral marketing [2, 3], media\nadvertising [4] and rumors spreading [5] etc., and many aspects of the problem has been extensively\nstudied.\nMost existing algorithms for in\ufb02uence maximization, typically under the independent cascade (IC)\nmodel and the linear threshold (LT) model [1], utilize the submodularity of the in\ufb02uence spread as a\nset function on the set of seed nodes, because it permits a (1 \u2212 1/e)-approximation solution by the\ngreedy scheme [1, 6, 7], following the foundational work on submodular function maximization [8].\nOne important result concerning submodularity in the in\ufb02uence model is by Mossel and Roch [9],\nwho prove that in the general threshold model, the global in\ufb02uence spread function is submodular\nwhen all local threshold functions at all nodes are submodular. This result implies that \u201clocal\"\nsubmodularity ensures the submodularity of \u201cglobal\" in\ufb02uence spread.\nAlthough in\ufb02uence maximization under submodular diffusion models is dominant in the research\nliterature, in real networks, non-submodularity of in\ufb02uence spread function has been observed.\nBackstrom et al. [10] study the communities of two networks LiveJournal and DBLP and draw\n\n31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.\n\n\f\u03b3\nc\n\npictures of the impulse that a person joins a community against the number of his friends already\nin this community. The curve is concave overall, except that a drop is observed in \ufb01rst two nodes.\nYang et al. [11] track emotion contagion under Flickr and \ufb01nd that the probability that an individual\nbecomes happy is superlinear to the number of his happy friends with higher PageRank scores. These\nare all instances of non-submodular in\ufb02uence spread functions.\nIn\ufb02uence maximization under many non-submodular diffusion models are proved to be hard to\napproximate. For example, in the diffusions of rumors, innovations, or riot behaviors, the individual\nin a social network is activated only when the number of her neighbors already adopting the behavior\nexceeds her threshold. It has been shown that the in\ufb02uence maximization problem based on this \ufb01xed\nthreshold model cannot be approximated within a ratio of n1\u2212\u03b5 for any \u03b5 > 0 [1]. Meanwhile Chen\n[12] proves that the seed minimization problem, to activate the whole network with minimum size of\nseed set, is also inapproximable, in particular, within a ratio of O(2log1\u2212\u03b5 n).\nIn this paper we give the \ufb01rst attempt on the in\ufb02uence maximization under the non-submodular\ndiffusion models. We study the general threshold model in which a fraction of nodes have non-\nsubmodular threshold functions, but their threshold functions are closely upper- and lower-bounded\nby some submodular functions (we call them \u03b5-almost submodular). Such a model bears conceptual\nsimilarity to the empirical \ufb01nding in [10, 11]: both studies show that the in\ufb02uence curve is only\nslightly non-concave, and Yang et al. [11] further shows that different roles have different curves\n\u2014 some are submodular while others are not, and ordinary users usually have behaviors close to\nsubmodular while opinion leaders may not. We \ufb01rst show a strong hardness result: there is no 1/n\napproximation for in\ufb02uence maximization (unless P = NP) for all networks with up to n\u03b3 \u03b5-almost\nsubmodular nodes, where \u03b3 is in (0, 1) and c is a parameter depending on \u03b5. On the other hand, we\npropose constant approximation algorithms for networks where the number of \u03b5-almost submodular\nnodes is a small constant. The positive results imply that non-submodular problem can be partly\nsolved as long as there are only a few non-submodular nodes and the threshold function is not far\naway from submodularity. Finally, we conduct experiments on real datasets to empirically verify our\nalgorithms. Empirical results on real datasets show that our approximation algorithms outperform\nother baseline algorithms.\nRelated Work. In\ufb02uence maximization has been well studied over the past years [13, 6, 7, 14, 15].\n[6] propose a lazy-forward optimization that avoids unnecessary\nIn particular, Leskovec et al.\ncomputation of expected size. Chen et al. [7, 14] propose scalable heuristic algorithms that handle\nnetwork of million edges. Based on the technique of Reverse Reachable Set, Borgs et al. [16]\nreduce the running time of greedy algorithms to near-linear under the IC model [1]. Tang et al. [17]\nimplement the near-linear algorithm and process Twitter network with million edges. Subsequently,\nTang et al. [18] and Nguyen et al. [19] further improve the ef\ufb01ciency of algorithms. These works all\nutilize the submodularity to accelerate approximation algorithms.\nSeed minimization, as the dual problem of in\ufb02uence maximization, is to \ufb01nd a small seed set such\nthat expected in\ufb02uence coverage exceeds a desired threshold. Chen [12] provide some strong negative\nresults on seed minimization problem under \ufb01xed threshold model, which is a special case of general\nthreshold model where its threshold function has breaking points. Goyal et al. [20] propose a greedy\nalgorithm with a bicriteria approximation. Recently, Zhang et al. [21] study the probabilistic variant\nof seed minimization problem.\nDue to the limitation of independent cascade and linear threshold model, general threshold model has\nbeen proposed [1, 9]. Not much is known about the general threshold model, other than it is NP-hard\nto approximate [1]. One special case which receives many attention is k-complex contagion where a\nnode becomes active if at least k of its neighbours have been activated [22, 23, 24]. Gao et al. [25]\nmake one step further of k-complex contagion model by considering the threshold comes from a\nprobability distribution.\nOptimization of non-submodular function is another interesting direction. Du et al. [26] introduce two\ntechniques \u2014 restricted submodularity and shifted submodularity \u2014 to analyze greedy approximation\nof non-submodular functions. Recently, Horel et al.[27] study the problem of maximizing a set\nfunction that is very close to submodular. They assume that function values can be obtained from an\noracle and focused on its query complexity. In our study, the local threshold functions are close to\nsubmodular and our target is to study its effect on the global in\ufb02uence spread function, which is the\nresult of complex cascading behavior derived from the local threshold functions.\n\n2\n\n\f2 Preliminaries\nFor a set function f : 2V \u2192 R, we say that it is monotone if f (S) \u2264 f (T ) for all S \u2286 T \u2286 V ;\nwe say that it is submodular if f (S \u222a {v}) \u2212 f (S) \u2265 f (T \u222a {v}) \u2212 f (T ), for all S \u2286 T \u2286 V and\nv \u2208 V \\ T . For a directed graph G = (V, E), we use N in(v) to denote the in-neighbors of v in G.\nWe now formally de\ufb01ne the general threshold model used in the paper.\nDe\ufb01nition 1 (General Threshold Model [1]). In the general threshold model, for a social graph\nG = (V, E), every node v \u2208 V has a threshold function fv : 2N in(v) \u2192 [0, 1]. The function fv(\u00b7)\nshould be monotone and fv(\u2205) = 0. Initially at time 0, each node v \u2208 V is in the inactive state and\nchooses \u03b8v uniformly at random from the interval [0, 1]. A seed set S0 is also selected, and their\nstates are set to be active. Afterwards, the in\ufb02uence propagates in discrete time steps. At time step\nt \u2265 1, node v becomes active if fv(St\u22121 \u2229 N in(v)) \u2265 \u03b8v, where St\u22121 is the set of active nodes by\ntime step t \u2212 1. The process ends when no new node becomes active in a step.\n\nGeneral threshold model is one of the most important models in the in\ufb02uence maximization problem.\nUsually we focus on two properties of threshold function \u2013 submodularity and supermodularity.\nSubmodularity can be understood as diminishing marginal returns when adding more nodes to the\nseed set. In contrast, supermodularity means increasing marginal returns. Given a seed set S, let \u03c3(S)\ndenote the expected number of activated nodes after the process of in\ufb02uence propagation terminates,\nand we call \u03c3(S) the in\ufb02uence spread of S.\nSubmodularity is the key property that guarantees the performance of greedy algorithms [9]. In this\npaper, we would like to study the in\ufb02uence maximization with nearly submodular threshold function\n\u2014 \u03b5-almost submodular function, or in short \u03b5-AS.\nDe\ufb01nition 2 (\u03b5-Almost Submodular (\u03b5-AS)). A set function f : 2V \u2192 R is \u03b5-almost submodular\nif there exists a submodular function f sub de\ufb01ned on 2V and for any subset S \u2286 V , f sub(S) \u2265\nf (S) \u2265 (1 \u2212 \u03b5)f sub(S). Here \u03b5 is a small positive number.\n\nv\n\nv\n\n. Hence by de\ufb01nition, we have f\n\nThe de\ufb01nition of \u03b5-almost submodular here is equivalent to \"Approximate submodularity\" de\ufb01ned\nin [27]. For an \u03b5-almost submodular threshold function fv, de\ufb01ne its upper and lower submodular\n= (1 \u2212 \u03b5)f v. Given the de\ufb01nition of \u03b5-almost\nbound as f v and f\nsubmodular function, we then model the almost submodular graph. In this paper, we consider the\nin\ufb02uence maximization problem based on this kind of graphs.\nDe\ufb01nition 3 ((\u03b3, \u03b5)-Almost Submodular Graph). Given \ufb01xed parameters \u03b3, \u03b5 \u2208 [0, 1], we say that a\ngraph with n (n = |V |) nodes is a (\u03b3, \u03b5)-Almost Submodular Graph (under the general threshold\nmodel), if there are at most n\u03b3 nodes in the graph with \u03b5-almost submodular threshold functions\nwhile other nodes have submodular threshold functions.\nDe\ufb01nition 4 (\u03b5-ASIM). Given a graph containing \u03b5-almost submodular nodes and an input k,\nIn\ufb02uence Maximization problem on graph with \u03b5-Almost Submodular nodes (\u03b5-ASIM) is the problem\nto \ufb01nd k seed nodes such that the in\ufb02uence spread invoked by the k nodes is maximized.\n\n3\n\nInapproximability of \u03b5-ASIM\n\nIn this section we show that it is in general hard to approximate the in\ufb02uence maximization problem\neven if there are only sublinear number of nodes with \u03b5-almost submodular threshold functions. The\nmain reason is that even a small number of nodes with \u03b5-almost submodular threshold functions fv(\u00b7)\nwould cause the global in\ufb02uence spread function far from submodularity, making the maximization\nproblem very dif\ufb01cult. The theorem below shows our hardness result.\nTheorem 1. For any small \u03b5 > 0 and any \u03b3 \u2208 (0, 1), there is no 1/n\nmaximization algorithm for all (\u03b3, \u03b5)-almost submodular graphs where c = 3 + 3/ log 2\nP=NP.\n\nc -approximation in\ufb02uence\n2\u2212\u03b5 , unless\n\n\u03b3\n\nWe \ufb01rst construct a probabilistic-AND gate gadget by amplifying the non-submodularity through a\nbinary tree. Then we prove the lower bound of approximation ratio by the reduction from set cover\nproblem. Due to page limits, we only sketch the main technique. The proof of Theorem 1 is given in\nAppendix ??.\n\n3\n\n\fHere we construct a basic gadget with input s1, s2 and output t (see Figure 1a). We assume that\nnode t has two in-neighbours s1, s2 and the threshold function g(\u00b7) of t is \u03b5-almost submodular:\ng(S) = |S|/2, when |S| = 0 or 2; 1\u2212\u03b5\n\n2 , when |S| = 1.\n\nt\n\nd\n\nt\n\n...\n\n...\n\n...\n\n...\n\n...\n\n...\n\n...\n\n...\n\ns1\n\ns2\n\n(a) Basic gadget\n\ns1\n\ns2\n(b) Tree gadget T\u03b5\n\nFigure 1: Diagrams of gadegts\n\nLet Pa(v) be the activation probability of node v in this case. This simple gadget is obviously far away\nfrom the AND gate, and our next step is to construct a more complex gadget with input node s1, s2.\nWe hope that the output node t is active only when both s1, s2 are active, and if only one of s1 and s2\nis active, the probability that node t becomes active is close to 0. We call it a probabilistic-AND gate.\nThe main idea is to amplify the gap between submodularity and non-submodularity by binary tree\n(\ufb01gure 1b). In this gadget T\u03b5 with a complete binary tree, node t is the root of a full binary tree and\neach node holds a directed edge to its parent. For each leaf node v in the tree, both s1, s2 hold the\ndirected edges towards it. The threshold function for each node in the tree is g(\u00b7) de\ufb01ned above while\n\u03b5 is the index of gadget T\u03b5. The depth of the tree is parameter d which will be determined later. We\nuse vi to denote a node of depth i (t is in depth 1). It is obviously that Pa(t) = 1 if both s1, s2 are\nactivated, and Pa(t) = 0 if neither s1 or s2 is activated. Thus, we would like to prove, in case when\nonly one of s1, s2 is activated, the activation probability becomes smaller for inner nodes in the tree.\nLemma 2. For gadget T\u03b5 with depth d, the probability of activating output node t is less than ( 2\u2212\u03b5\n2 )d\nwhen only one of s1, s2 is activated.\n\nProof. In this case, for leaf node vd, we have Pa(vd) = 1\u2212\u03b5\n2 . Apparently, the probability of becoming\nactivated for nodes with depth d are independent with each other. Given a basic gadget, if each of\nthe two children nodes is activated with an independent probability p, then the parent node will be\nactivated with probability\n\np2 \u00d7 g(2) + 2p(1 \u2212 p) \u00d7 g(1) + (1 \u2212 p)2 \u00d7 g(0) = p2 + 2p(1 \u2212 p)\n\n= p(1 \u2212 \u03b5(1 \u2212 p)).\nSo we have Pa(vi) \u2264 Pa(vi+1)(1 \u2212 \u03b5(1 \u2212 Pa(vi+1))). Since Pa(vd) = 1\u2212\u03b5\n2 < 1/2, and Pa(vi) \u2264\nPa(vi+1) from above, we have pa(vi) < 1/2 for all i, and thus we can rewrite the recurrence as\nPa(vi) \u2264 Pa(vi+1)(1\u2212 \u03b5/2). Hence for the gadget with depth d, the probability that node t becomes\nactivated is Pa(t) = Pa(v1) \u2264 1\u2212\u03b5\n\n2 )d.\n\n1 \u2212 \u03b5\n2\n\n2 ( 2\u2212\u03b5\n\n2 )d\u22121 < ( 2\u2212\u03b5\n\nLemma 2 shows that gadget T\u03b5 is indeed a probabilistic-AND gate with two input nodes, and the\nprobability that t is activated when only one of s1 and s2 is activated approaches 0 exponentially fast\nwith the depth d. We say a gadget T\u03b5 works well if output node t stay inactive when only one of the\ninput nodes is activated.\nBy the similar method we construct multi-input-AND gates based on 2-input-AND gates. Finally,\nwe show that if the in\ufb02uence maximization problem can be approximated beyond the ratio shown\nabove, we can solve the set cover problem in polynomial time. The main idea is as follows. For any\nset cover instance, we will put all elements to be the input of our multi-input-probabilistic-AND gate,\nand connect the output with a large number of additional nodes. Thus, if k sets can cover all elements,\nall of those addition nodes will be activated, on contrast, if at least one of the elements cannot be\ncovered, almost all of the additional nodes will remain inactive.\n\n4\n\n\f4 Approximation Algorithms\n\nIn the previous section, we show that in\ufb02uence maximization is hard to approximate when the number\nof \u03b5-almost submodular nodes is sublinear but still a non-constant number. In this section, we discuss\nthe situation where only small number of nodes hold \u03b5-almost submodular threshold functions. We\n\ufb01rstly provide a greedy algorithm for small number of non-submodular nodes which may not be\n\u03b5-almost submodular, then, we restrict to the case of \u03b5-almost submodular nodes.\n\n4.1 Approximation Algorithm with Small Number of Non-submodular Nodes\n\nIn the case of (cid:96) ((cid:96) < k) non-submodular nodes, we provide an approximate algorithm as follows. We\n\ufb01rst add these non-submodular nodes into the seed set, and then generate the rest of the seed set by\nthe classical greedy algorithm. The proof of Theorem 3 can be found in Appendix ??.\nTheorem 3. Given a graph of n nodes where all nodes have submodular threshold functions except\n(cid:96) < k nodes, for in\ufb02uence maximization of k seeds with greedy scheme we can obtain a (1 \u2212 e\u2212 k\u2212(cid:96)\nk )-\napproximation ratio.\n\n4.2 Approximation Algorithm of \u03b5-ASIM\n\nIn this section, we consider the case when all non-submodular nodes have \u03b5-almost submodular thresh-\nold functions, and provide an approximation algorithm that allows more than k \u03b5-almost submodular\nnodes, with the approximation ratio close to 1 \u2212 1/e when \u03b5 is small. The main idea is based on the\nmapping between probability spaces.\nGiven a graph containing nodes with \u03b5-almost submodular threshold functions, we simply set each\nnode\u2019s threshold function to its submodular lower bound and then run classical greedy algorithm A\non this graph (Algorithm 1). Algorithm 1 takes the lower bounds of \u03b5-almost submodular threshold\nfunctions as input parameters. The following theorem analyzes the performance of Algorithm 1.\n\nSeed set S\n\nAlgorithm 1 Galg-L algorithm for In\ufb02uence Maximization\nInput: G = (V, E), A, {fv},{f\nOutput:\n1: set S = \u2205\n2: replace each nodes v\u2019s threshold function fv with f\n3: run algorithm A on G with {f\n4: return S\n\n} and obtain S\n\n}, k\n\nv\n\nv\n\nv\n\nTheorem 4. Given a graph G = (V, E), under the general threshold model, assuming that (cid:96) nodes\nhave \u03b5-almost submodular threshold functions and the other nodes have submodular threshold\nfunctions. Then the greedy algorithm Galg-L has approximation ratio of (1 \u2212 1\n\ne )(1 \u2212 \u03b5)(cid:96).\n\n} respectively.\n\nProof. Let Ve be the set of nodes with \u03b5-almost submodular threshold functions. Without loss of\ngenerality, we assume Ve = {v1, v2, . . . , v(cid:96)}. Now consider two general threshold models M , M\nwith different threshold functions. Both models hold threshold functions {fv} for v \u2208 V \u2212 Ve. For\nnode v in Ve, M , M hold {f v} and {f\nIn any threshold model, after we sample each node\u2019s threshold \u03b8v, the diffusion process be-\ncomes deterministic. A graph with threshold functions {fv} and sampled thresholds {\u03b8v} is\ncalled a possible world of the threshold model, which is similar to the live-edge graph in the\nindependent cascade model. An instance of threshold model\u2019s possible world can be written as\n{\u03b8v1 , \u03b8v2 , . . . , \u03b8vn ; fv1 , fv2, . . . , fvn}. Here we build a one-to-one mapping from all M\u2019s possible\nworlds with \u03b8v \u2264 1 \u2212 \u03b5 (v \u2208 Ve) to all M\u2019s possible worlds:\n{\u03b8v1 , . . . , \u03b8vn ; fv1 , . . . , fvn} \u2194 { \u03b8v1\n1 \u2212 \u03b5\n\n, fv(cid:96)+1 , . . . , fvn}.\n\n, \u03b8v(cid:96)+1 . . . , \u03b8vn ;\n\n\u03b8v(cid:96)\n1 \u2212 \u03b5\n\nfv(cid:96)\n1 \u2212 \u03b5\n\nfv1\n1 \u2212 \u03b5\n\nv\n\n, . . . ,\n\n, . . . ,\n\nThe above corresponding relation shows this one-to-one mapping between M and M. For any\ninstance of M\u2019s possible world with \u03b8v \u2264 1 \u2212 \u03b5 (v \u2208 Ve), we amplify the threshold of node v in\nVe to \u03b8v\n1\u2212\u03b5.\n\n1\u2212\u03b5. At the same time, we amplify the corresponding threshold function by a factor of\n\n1\n\n5\n\n\fObviously, this ampli\ufb01cation process will not effect the in\ufb02uence process under this possible world,\nbecause for each v \u2208 Ve, both its threshold value and the its threshold function are ampli\ufb01ed by the\nsame factor 1/(1 \u2212 \u03b5). Furthermore, the ampli\ufb01ed possible world is an instance of M.\n\nExpected in\ufb02uence can be computed by \u03c3(S) = (cid:82)\n\n(cid:126)\u03b8\u2208[0,1]n D((cid:126)\u03b8; (cid:126)f , S)d(cid:126)\u03b8, where D((cid:126)\u03b8; (cid:126)f , S) is the\ndeterministic in\ufb02uence size of seed set S under possible world {(cid:126)\u03b8; (cid:126)f}. We refer M , M\u2019s expected\nin\ufb02uence size functions as \u03c3, \u03c3. We de\ufb01ne (cid:126)\u03b8 \u2208 [0, 1]n as the vector of n nodes threshold, and\n(cid:126)\u03b8e \u2208 [0, 1](cid:96), (cid:126)\u03b8(cid:48) \u2208 [0, 1]n\u2212(cid:96) are the threshold vectors of Ve and V \u2212 Ve. Besides, the threshold\nfunctions of Ve and V \u2212 Ve will be represented as (cid:126)fe, (cid:126)f(cid:48). A possible world is symbolized as\n{(cid:126)\u03b8e, (cid:126)\u03b8(cid:48); (cid:126)fe, (cid:126)f(cid:48)}. For any seed set S, we have\n(cid:126)\u03b8\u2208[0,1]n D((cid:126)\u03b8; (cid:126)f , S)d(cid:126)\u03b8\n(cid:82)\n(cid:126)\u03b8e\u2208[0,1\u2212\u03b5](cid:96)\n(cid:82)\n1\u2212\u03b5 \u2208[0,1](cid:96)\n(cid:126)\u03b8e\n1\u2212\u03b5 \u2208[0,1](cid:96)\n(cid:126)\u03b8e\n(cid:126)\u03b8\u2208[0,1]n D((cid:126)\u03b8; (\n\n\u03c3(S) =(cid:82)\n\u2265(cid:82)\n= (1 \u2212 \u03b5)(cid:96)(cid:82)\n= (1 \u2212 \u03b5)(cid:96)(cid:82)\n= (1 \u2212 \u03b5)(cid:96)(cid:82)\n\n(cid:126)\u03b8(cid:48)\u2208[0,1]n\u2212(cid:96) D(((cid:126)\u03b8e, (cid:126)\u03b8(cid:48)); (cid:126)f , S)d (cid:126)\u03b8e\n(cid:126)\u03b8(cid:48)\u2208[0,1]n\u2212(cid:96) D(( (cid:126)\u03b8e\n1\u2212\u03b5 , (cid:126)f(cid:48)), S)d(cid:126)\u03b8\n\n(cid:126)\u03b8(cid:48)\u2208[0,1]n\u2212(cid:96) D(((cid:126)\u03b8e, (cid:126)\u03b8(cid:48)); (cid:126)f , S)d(cid:126)\u03b8e\n\n1\u2212\u03b5 , (cid:126)f(cid:48)), S)d (cid:126)\u03b8e\n1\u2212\u03b5\n\n1\u2212\u03b5 , (cid:126)\u03b8(cid:48)); (\n\n(cid:82)\n\nd(cid:126)\u03b8(cid:48)\n\nd(cid:126)\u03b8(cid:48)\n\nd(cid:126)\u03b8(cid:48)\n\n1\u2212\u03b5\n\n(cid:126)fe\n\n= (1 \u2212 \u03b5)(cid:96)\u03c3(S).\n\n(cid:126)fe\n\n(cid:126)\u03b8e\n\n(cid:126)fe\n\n1\u2212\u03b5 , (cid:126)\u03b8(cid:48)); (\n\n1\u2212\u03b5 , (cid:126)f(cid:48)), S) for (cid:126)\u03b8e\n\nin particular D(((cid:126)\u03b8e, (cid:126)\u03b8(cid:48)); (cid:126)f , S) =\nThe third equality utilizes our one-to-one mapping,\n1\u2212\u03b5 \u2208 [0, 1](cid:96), because they follow the same deterministic propa-\nD((\ngation process. Hence given a seed set S, the respective in\ufb02uence sizes in model M , M satisfy the\nrelation \u03c3(S) \u2265 (1 \u2212 \u03b5)(cid:96)\u03c3(S).\nLet \u03c3 be the expected in\ufb02uence size function of the original model, and assume that the optimal\n) \u2265 \u03c3(S\u2217) since for every node v,\nsolution for \u03c3, \u03c3, \u03c3 are S\nf v \u2265 fv. According to the previous analysis, we have \u03c3(S\u2217) \u2265 \u03c3(S\n). Hence for\noutput SA of the greedy algorithm for optimizing \u03c3, we have approximation ratio\n\n, S\u2217, S\u2217 respectly. Apparently, \u03c3(S\n\n) \u2265 (1 \u2212 \u03b5)(cid:96)\u03c3(S\n\n\u2217\n\n\u2217\n\n\u2217\n\n\u2217\n\n\u03c3(SA) \u2265 \u03c3(SA) \u2265 (1 \u2212 1\ne\n\n)\u03c3(S\u2217) \u2265 (1 \u2212 1\ne\n\n)(1 \u2212 \u03b5)(cid:96)\u03c3(S\n\n\u2217\n\n) \u2265 (1 \u2212 1\ne\n\n)(1 \u2212 \u03b5)(cid:96)\u03c3(S\u2217).\n\nThe theorem holds.\n\nIf we replace threshold functions by their upper bound and run the greedy algorithm, we obtain\nGalg-U. With similar analysis, Galg-U also holds approximation ratio of (1 \u2212 1\ne )(1 \u2212 \u03b5)(cid:96) on graphs\nwith (cid:96) \u03b5-almost submodular nodes. The novel technique used to prove approximation ratio is similar\nto the sandwich approximation in [28]. But their approximation ratio relies on instance-dependent\nin\ufb02uence sizes, while we utilize mapping of probabilistic space to provide instance-independent\napproximation ratio.\n\n5 Experiments\n\nIn addition to the theoretical analysis, we are curious about the performance of greedy algorithms\nGalg-U, Galg-L on real networks with non-submodular nodes. Our experiments run on a machine\nwith two 2.4GHz Intel(R) Xeon(R) E5-2620 CPUs, 4 processors (24 cores), 128GB memory and\n64bit Ubuntu 14.04.1. All algorithms tested in this paper are written in C++ and compiled with g++\n4.8.4. Some algorithms are implemented with multi-thread to decrease the running time.\n\n5.1 Experiment setup\n\nDatasets. We conduct experiments on three real networks. The \ufb01rst network is NetHEPT, an\nacademic collaboration network extracted from \"High Energy Physics - Theory\" section of arXiv\n(http://www.arXiv.org) used by many works [7, 14, 15, 19, 20]. NetHEPT is an undirected network\nwith 15233 nodes and 31376 edges, each node represents an author and each edge represents that\ntwo authors collaborated on a paper. The second one is Flixster, an American movie rating social site.\nEach node represents a user, and directed edge (u, v) means v rated the same movie shortly after u\n\n6\n\n\f\u03b2\n\nd(v)\n\n= 1\n\n\u03b2 with \u03b2 satisfying 1\n\nd(v) (1 \u2212 \u03b5); (2) fv(S) =\n\ndid. We select topic 3 with 29357 nodes and 174939 directed edges here. The last one is the DBLP\ndataset, which is a larger collaboration network mined from the computer science bibliography site\nDBLP with 654628 nodes and 1990259 undirected edges [14]. We process its edges in the same way\nas the NetHEPT dataset.\nPropagation Models. We adapt general threshold model in this paper. Our Galg-U,Galg-L are\ndesigned on submodular upper and lower bounds, respectively. Since directly applying greedy scheme\non graphs with submodular threshold function is time-consuming, we assign the submodular threshold\nfunction and submodular upper bound of \u03b5-AS function as linear function here: fv(S) = |S|/d(v),\nwhere d(v) is the in-degree of v. This makes the corresponding model an instance of the linear-\nthreshold model, and thus the greedy algorithm can be accelerated with Reverse Reachable Set\n(RRset) technique [17].\nWe construct two different \u03b5-almost submodular threshold functions in this paper: (1) a power function\n|S|\n|S|\nd(v) (1 \u2212 \u03b5) for |S| \u2264 2 and |S|/d(v)\nd(v)\notherwise. The former \u03b5-almost submodular function is a supermodular function. The supermodular\nphenomenon has been observed in Flickr [11]. The second \u03b5-almost submodular function is just\ndropping down the original threshold function for the \ufb01rst several nodes, which is consistent with the\nphenomenon observed in LiveJournal [10]. We call them \u03b5-AS-1 and \u03b5-AS-2 functions respectively.\nAlgorithms. We test our approximation Algorithm 1 and other baseline algorithms on the graphs\nwith \u03b5-almost submodular nodes.\n\u2022 TIM-U, TIM-L: Tang et al. [17] proposed a greedy algorithm TIM+ accelerated with Reverse\nReachable Set (RRset). The running time of TIM+ is O(k(m + n) log n) on graphs with n nodes\nand m edges. RRset can be sampled in live-edge graph of IC model, and with some extension\nwe can sample RRset under Triggering model [1]. LT model also belongs to Triggering model,\nbut General Threshold model with non-submodular threshold functions does not fall into the\ncategory of Triggering model. Thus TIM+ can not be directly applied on original graphs with\nnon-submodular nodes. In our experiments, we choose \u03b5-AS-1 and \u03b5-AS-2 thresholds to ensure\nthat TIM+ can run with their upper or lower bound. We then run Algorithm 1 with TIM+ as input.\nAlgorithm Galg-L based on TIM+ can be written in short as TIM-L. By using the upper bound we\nobtain TIM-U.\n\n\u2022 Greedy: We can still apply the naive greedy scheme on graph with \u03b5-almost submodular nodes\nand generate results without theoretical guarantee. The naive greedy algorithm is time consuming,\nwith running time is O(k(m + n)n).\n\n\u2022 High-degree: High-degree outputs seed set according to the decreasing order of the out-degree.\n\u2022 PageRank: PageRank is widely used to discover nodes with high in\ufb02uence. The insight of\nPageRank is that important nodes point to important nodes. In this paper, The transition probability\non edge e = (u, v) is 1/d(u). We set restart probability as 0.15 and use the power method to\ncompute the PageRank values. Finally PageRank outputs nodes with top PageRank values.\n\n\u2022 Random: Random simply selects seeds randomly from node set.\nExperiment methods. The datasets provide the structure of network, and we \ufb01rst assume each node\nholds linear threshold function as described above. In each experiment, we randomly sample some\nnodes with in-degree greater than 2, and assign those nodes with our \u03b5-almost submodular functions,\n\u03b5-AS-1 or \u03b5-AS-2. Since the naive greedy algorithm is quite time-consuming, we just run it on\nNetHEPT.\n\n5.2 Experiment results\n\nResults on NetHEPT. Our \ufb01rst set of experiments focuses on the NetHEPT dataset with the aim\nof comparing TIM-U, TIM-L and Greedy. TIM-U, TIM-L have theoretical guarantee, but the\napproximation ratio is low when the graph contains a considerable number of \u03b5-AS nodes. Figure 2\nshows the in\ufb02uence size of each method, varying from 1 to 100 seeds. Figure 2a and 2b are results\nconducted on constructed graph with \u03b5-AS-1 nodes. Observe that TIM-U, TIM-L slightly outperform\nGreedy in all cases. Compared with results of 3000 \u03b5-AS nodes, in\ufb02uence of output seeds drops\nobviously in graph with 10000 \u03b5-AS nodes. But the ratio that TIM-U, TIM-L exceed PageRank\n\n7\n\n\f(a) 3000 \u03b5-AS-1 nodes\n\n(b) 10000 \u03b5-AS-1 nodes\n\n(c) 3000 \u03b5-AS-2 nodes\n\n(d) 10000 \u03b5-AS-2 nodes\n\nFigure 2: Results of IM on NetHEPT with \u03b5 = 0.2\n\nincreases with rising fraction of \u03b5-AS nodes. In particular, \u03b5-AS-1 is indeed supermodular, TIM-U,\nTIM-L beats Greedy even when many nodes have supermodular threshold functions.\nWe remark that TIM-U, TIM-L and Greedy outperform other baseline algorithms signi\ufb01cantly. When\nk = 100, TIM-U is 6.1% better compared with PageRank and 27.2% better compared with High-\ndegree. When conducted with \u03b5-AS-2 function, Figure 2c and 2d report that TIM-U, TIM-L and\nGreedy still perform extremely well. In\ufb02uence size conducted on graphs with \u03b5-AS-2 function is\nbetter than those with \u03b5-AS-1 function. This is what we expect: supermodular function is harder to\nhandle among the class of \u03b5-almost submodular functions.\nAnother thing to notice is that TIM-U, TIM-L can output seeds on NetHEPT within seconds, while it\ntakes weeks to run the naive greedy algorithm. With RRsets technique, TIM+ dramatically reduces\nthe running time. The \u03b5-almost submodular functions selected here ensure that TIM+ can be invoked.\nSince TIM-U, TIM-L match the performance of Greedy while TIM-U, TIM-L are scalable, we do not\nrun Greedy in the following larger datasets.\nResults on Flixster. Figure 3 shows the results of experiments conducted on Flixster with\n\n(a) 3000 \u03b5-AS-1 nodes\n\n(b) 10000 \u03b5-AS-1 nodes\n(c) 3000 \u03b5-AS-2 nodes\nFigure 3: Results of IM on Flixster with \u03b5 = 0.2\n\n(d) 10000 \u03b5-AS-2 nodes\n\n\u03b5 = 0.2. We further evaluate algorithms by Flixster with \u03b5 = 0.4 (see Figure 4). Observe that\nTIM-U, TIM-L outperform other heuristic algorithms in all cases. Compared with PageRank,\n30%, 46.3%, 26%, 29.7% improvement are observed in the four experiments in Figure 3. TIM-U\nperforms closely to TIM-L consistently. The improvement is larger than that in NetHEPT. The extra\nimprovement might due to more complex network structure. The average degree is 5.95 in Flixster,\ncompared to 2.05 in NetHEPT. In dense network, nodes may be activated by multiple in\ufb02uence\nchains, which makes in\ufb02uence propagates further from seeds. Baseline algorithms only pay attention\nto the structure of the network, hence they are defeated by TIM-U, TIM-L that focus on in\ufb02uence\nspread. The more \u03b5-AS nodes in network, the more improvement is obtained.\nWhen we set \u03b5 as 0.4, Figure 4 shows that TIM-U is 37.6%, 74.2%, 28%, 35.6% better than PageR-\nank respectively. Notice that the gap between the performances of TIM-U and PageRank increases\nas \u03b5 increases. In Flixster dataset, we observe that TIM-U,TIM-L hold greater advantage in case of\nlarger number of \u03b5-AS nodes and larger \u03b5.\nResults on DBLP. For DBLP dataset, the results are shown in Figure 5. TIM-U and TIM-L are still\nthe best algorithms according to performance. But PageRank and High-degree also performs\nwell, just about 2.6% behind TIM-U and TIM-L. DBLP network has many nodes with large degree,\nwhich correspond to those active scientists. Once such active authors are activated, the in\ufb02uence will\nincrease signi\ufb01cantly. This may partly explain why TIM-U,TIM-L perform similarly to PageRank.\n\n8\n\n\f(a) 3000 \u03b5-AS-1 nodes\n\n(b) 10000 \u03b5-AS-1 nodes\n(c) 3000 \u03b5-AS-2 nodes\nFigure 4: Results of IM on Flixster with \u03b5 = 0.4\n\n(d) 10000 \u03b5-AS-2 nodes\n\n(a) 10000 \u03b5-AS-1 nodes\n\n(b) 100000 \u03b5-AS-1 nodes\n(c) 10000 \u03b5-AS-2 nodes\nFigure 5: Results of IM on DBLP with \u03b5 = 0.2\n\n(d) 100000 \u03b5-AS-2 nodes\n\n6 Conclusion and Future Work\n\nIn this paper, we study the in\ufb02uence maximization problem on propagation models with non-\nsubmodular threshold functions, which are different from most of existing studies where the threshold\nfunctions and the in\ufb02uence spread function are both submodular. We investigate the problem\nby studying a special case \u2014 the \u03b5-almost submodular threshold function. We \ufb01rst show that\nin\ufb02uence maximization problem is still hard to approximate even when the number of \u03b5-almost\nsubmodular nodes is sub-linear. Next we provide a greedy algorithm based on submodular lower\nbounds of threshold function to handle the graph with small number of \u03b5-almost submodular nodes\nand show its theoretical guarantee. We further conduct experiments on real networks and compare\nour algorithms with other baselines to evaluate our algorithms in practice. Experimental results\nshow that our algorithms not only have good theoretical guarantees on graph with small number of\n\u03b5-almost submodular nodes, they also perform well on graph with a fairly large fraction of \u03b5-almost\nsubmodular nodes.\nOur study mainly focuses on handling \u03b5-almost submodular threshold functions. One future direction\nis to investigate models with arbitrary non-submodular threshold functions. Another issue is that the\ngreedy algorithms we propose are slow when the submodular upper bound or lower bound of threshold\nfunction do not correspond to the Triggering model. It remains open whether we could utilize RRset\nor other techniques to accelerate our algorithms under this circumstance. How to accelerate the naive\ngreedy process with arbitrary submodular threshold functions is another interesting direction.\n\nAcknowledgments\n\nThis work was supported in part by the National Natural Science Foundation of China Grant 61433014,\n61502449, 61602440, the 973 Program of China Grants No. 2016YFB1000201.\n\n9\n\n\fReferences\n[1] David Kempe, Jon Kleinberg, and \u00c9va Tardos. Maximizing the spread of in\ufb02uence through a\n\nsocial network. In Proceedings of the ninth ACM SIGKDD, pages 137\u2013146. ACM, 2003.\n\n[2] Mani Subramani and Balaji Rajagopalan. Knowledge-sharing and in\ufb02uence in online social\n\nnetworks via viral marketing. 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In\n\nNIPS\u201916, pages 3045\u20133053, 2016.\n\n[28] Wei Lu, Wei Chen, and Laks VS Lakshmanan. From competition to complementarity: compar-\native in\ufb02uence diffusion and maximization. Proceedings of the VLDB Endowment, 9(2):60\u201371,\n2015.\n\n11\n\n\f", "award": [], "sourceid": 2094, "authors": [{"given_name": "Qiang", "family_name": "Li", "institution": "Institute of Computing Technol"}, {"given_name": "Wei", "family_name": "Chen", "institution": "Microsoft Research"}, {"given_name": "Institute of Computing", "family_name": "Xiaoming Sun", "institution": "Institute of Computing Technology, Chinese Academy of Sciences"}, {"given_name": "Institute of Computing", "family_name": "Jialin Zhang", "institution": "Institute of Computing Technology, Chinese Academy of Sciences"}]}