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Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas
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Outline Influence Max Independent Cascade 2
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Given a digraph and k>0, Find k seeds (Kates) to maximize the number of influenced persons. Influence Maximization 3
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4 Theorem Proof
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Modularity of Influence 5
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Submadular Function Max 6
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Greedy Algorithm 7
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Performance Ratio 8 Theorem (Nemhauser et al. 1978) Proof
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Theorem 9
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Outline Influence Max Independent Cascade 10
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Deterministic Model 1 3 4 5 2 6 both 1 and 6 are source nodes. Step 1: 1--2,3; 6--2,4.. 12/13/201511
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1 3 5 2 4 6 Step 2: 4--5. Example 12/13/201512
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Models of Influence Diffusion Two basic classes of probabilistic diffusion models: – threshold and cascade General operational view: – A social network is represented as a directed graph, with each person (customer) as a node. – Nodes start either active or inactive. – An active node may trigger activation of neighboring nodes – Monotonicity assumption: active nodes never deactivate.
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Independent Cascade (IC) Model When node v becomes active, it has a single chance of activating each currently inactive neighbor w. The activation attempt succeeds with probability p vw. The deterministic model is a special case of IC model. In this case, p vw =1 for all (v,w).
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15 Important understanding
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Example v w 0.5 0.3 0.2 0.5 0.1 0.4 0.3 0.2 0.6 0.2 Inactive Node Active Node Newly active node Successful attempt Unsuccessful attempt Stop! U X Y
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Influence Maximization Problem Influence spread of node set S: σ(S) – expected number of active nodes at the end of diffusion process, if set S is the initial active set. Problem Definition (by Kempe et al., 2003): (Influence Maximization). Given a directed and edge-weighted social graph G = (V,E, p), a diffusion model m, and an integer k ≤ |V |, find a set S ⊆ V, |S| = k, such that the expected influence spread σ m (S) is maximum.
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Known Results Bad news: NP-hard optimization problem for both IC and LT models. Good news: σ m (S) is monotone and submodular. We can use Greedy algorithm! Theorem: The resulting set S activates at least (1-1/e) (>63%) of the number of nodes that any size-k set could activate.
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Proof of Submodularity 19
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Decision Version of InfMax in IC 21 Theorem Corollary Is it in NP?
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22 Theorem (Chen et al., 2010) Proof
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Disadvantage Lack of efficiency. – Computing σ m (S) is # P-hard under both IC and LT models. – Selecting a new vertex u that provides the largest marginal gain σ m (S+u) - σ m (S), which can only be approximated by Monte-Carlo simulations (10,000 trials). Assume a weighted social graph as input. – How to learn influence probabilities from history?
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Monte-Carlo Method 25 Buffon's needle
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References 26
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Editor-in-Chief: Ding-Zhu Du My T. Thai Computational Social Networks 27 A New Springer Journal Welcome to Submit Papers
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THANK YOU!
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Yuqing Zhu, Zaixin Lu, Yuanjun Bi, Weili Wu, Yiwei Jiang, Deying Li: Influence and Profit: Two Sides of the Coin. ICDM 2013: 1301-1306 29
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Lidan Fan, Zaixin Lu, Weili Wu, Yuanjun Bi, Ailian Wang: A New Model for Product Adoption over Social Networks. COCOON 2013: 737-746COCOON 2013 30
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Songsong Li, Yuqing Zhu, Deying Li, Donghyun Kim, Huan Ma, Hejiao Huang: Influence maximization in social networks with user attitude modification. ICC 2014: 3913-3918ICC 2014 31
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Zaixin Lu, Lidan Fan, Weili Wu, Bhavani Thuraisingham and Kai Yang, Efficient influence spread estimation for influence maximization under the linear threshold model, Computational Social Networks, 1 (2014) 32
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Zaixin Lu, Wei Zhang, Weili Wu, Bin Fu, Ding- Zhu Du: Approximation and Inapproximation for the Influence Maximization Problem in Social Networks under Deterministic Linear Threshold Model. ICDCS Workshops 2011: 160-165ICDCS Workshops 2011 33
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