Lecture 3-1 Independent Cascade Weili Wu Ding-Zhu Du University of Texas at Dallas

Outline Influence Max Independent Cascade 2

Given a digraph and k>0, Find k seeds (Kates) to maximize the number of influenced persons. Influence Maximization 3

4 Theorem Proof

Modularity of Influence 5

Submadular Function Max 6

Greedy Algorithm 7

Performance Ratio 8 Theorem (Nemhauser et al. 1978) Proof

Theorem 9

Outline Influence Max Independent Cascade 10

Deterministic Model both 1 and 6 are source nodes. Step 1: 1--2,3; 6--2,4.. 12/13/201511

Step 2: Example 12/13/201512

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.

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).

15 Important understanding

Example v w Inactive Node Active Node Newly active node Successful attempt Unsuccessful attempt Stop! U X Y

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.

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.

Proof of Submodularity 19

20

Decision Version of InfMax in IC 21 Theorem Corollary Is it in NP?

22 Theorem (Chen et al., 2010) Proof

23

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?

Monte-Carlo Method 25 Buffon's needle

References 26

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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Yuqing Zhu, Zaixin Lu, Yuanjun Bi, Weili Wu, Yiwei Jiang, Deying Li: Influence and Profit: Two Sides of the Coin. ICDM 2013:

Lidan Fan, Zaixin Lu, Weili Wu, Yuanjun Bi, Ailian Wang: A New Model for Product Adoption over Social Networks. COCOON 2013: COCOON

Songsong Li, Yuqing Zhu, Deying Li, Donghyun Kim, Huan Ma, Hejiao Huang: Influence maximization in social networks with user attitude modification. ICC 2014: ICC

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

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: ICDCS Workshops