Submodular Optimization in Social Computing DingZhu Du UT

Submodular Optimization in Social Computing Ding-Zhu Du UT Dallas

What is a submodular function? Consider a function f on all subsets of a set E. f is submodular if Consider as Discrete Convex Function.

Discrete Convex Analysis Kazuo Murota

Similar to Convex • Unconstrained submodular minimization can be done in polynomial-time. • Unconstrained submodular maximization is NP -hard.

Similar to Concave，too! 1 2

Table of Contents 1. Submodular Maximization 2. Submodular Minimization 3. Submodular Partition.

Table of Contents 1. Submodular Maximization Influence Maximization Multi-products: Knapsack Constraint Profit Maximization: Nonmonotone Continuous Greedy 2. Submodular Minimization 3. Submodular Partition.

Influence Maximization

Kate Middleton effect “Kate Middleton effect The trend effect that Kate, Duchess of Cambridge has on others, from cosmetic surgery for brides, to sales of coral-colored jeans. ” 9

Hike in Sales of Special Products n According to Newsweek, "The Kate Effect may be worth £ 1 billion to the UK fashion industry. " n Tony Di. Masso, L. K. Bennett’s US president, stated in 2012, ". . . when she does wear something, it always seems to go on a waiting list. " 10

Influence Maximization • Given a digraph and k>0, • Find k seeds (Kates) to maximize the number of influenced persons. 11

How to Find Kate? • Influential persons often have many friends. • Kate is one of the persons that have many influenced persons in this social network. For more Kates, it’s not as easy as you might think! 12

Theorem Proof 13

Modularity of Influence 14

Submadular Function Max 15

Greedy Algorithm 16

Performance Ratio Theorem (Nemhauser et al. 1978) Corollary 17

Proof Monotone nondecreasing Submodular! Why? 18

Lower Bound 19

Multi-product: Knapsack Constraint

Knapsack Constraint

Knapsack Constraint

Submadular Function Max 23

Submadular Function Max 24

Profit Maximization

Influence and Profit: Two Sides of the Coin 13 th IEEE International Conference on Data Mining (ICDM 2013) Yuqing Zhu 5/22/2021 26

Influence vs. Profit · Classical models do not fully capture monetary aspects of product adoptions ·Being influenced Being willing to purchase · Classical models do not consider the willingness the active nodes on spreading the influence ·Being influenced Being willing to spread the influence 5/22/2021 27

Influence vs. Profit · Influence: · Profit: · In market, a famous company does not always make generous profit. E. g. Twitter, SONY, Weibo 5/22/2021 28

Product Adoption · Product adoption is a two-stage process (Kalish 85) · 1 st stage: Awareness ·Get exposed to the product ·Become familiar with its features · 2 nd stage: Actual adoption ·Only if valuation outweighs price ·Only in this case the company gains real profit · The 2 nd stage is not captured in existing work 5/22/2021 29

Price Related (PR) Frame Neutral Rules in IC or LT Influenced Active · Three node states: Neutral, Influenced, and Active · Neutral Influenced: same as in LT or IC · Influenced Active: Only if the valuation is at least the quoted price · Only active nodes will propagate influence to inactive neighbors 5/22/2021 30

Pricing Strategies ·Binar. Y pri. Cing (BYC) ·PAnoramic Pricing (PAP) 5/22/2021 31

BIPMax Problem Definition Input Problem Select a set of seeds & determine a vector of quoted price, such that the is maximized under the PR Frame Output A directed graph representing a social network, with influence weights on edges 5/22/2021 32

A Restricted Special Case · : The uniform price for every non-seed · : production cost · max: Not monotone nondecreasing submodular! 5/22/2021 33

Nonmonotone Submodular Max 1 2 3

Feige et all, FOCS 2007 For nonnegative submodular maximization, • 1/3 -approximation (deterministic local search) • 2/5 -approximation (randomized) • ½-approximation (NP-hard) for certain submodular function

Lee et al. STOC 2009 For nonnegative submodular maximization, • With k matroid constraints, • With k knapsack constraints, • With k>2 knapsack constraints,

|S|<k • a knapsack constraint • Also a matroid constraint because {S | |S|<k } form a matroid.

Feldman et al. FOCS 2011 For nonnegative submodular maximization, • (1/e-o(1))-approximation for O(1) knapsack constraints.

Can “nonnegative” be removed? • No! • With possible negative, no good approximation exists in general. • This means that the profit maximization needs some new work.

Table of Contents 1. Submodular Maximization 2. Submodular Minimization Seed Selection Positive Influence Weighted Positive Influence Active Friending 3. Submodular Partition.

Active Friending

Linked. In • Do you receive invitations from Linked. In everyday? • Does Linked. In have the following function: Linked. In may suggest a list of invitations when you want to include a target person into friend list. 42

Problem Formulation s S R t Theorem 43

Size-constrained submodular Min

Approximation Z. Svitkina and L. Fleischer, Submodular approximation: Sampling-based algorithms and lower bounds, FOCS 2008.

Problem Formulation s S R t Theorem 46

Table of Contents 1. Submodular Maximization 2. Submodular Minimization 3. Submodular Partition. Community Partition

Influence-based Community Detection

Model-Based Detection Community Detection Accurate or not? Formulation (Model) Solve formulated problem 49

Idea • People in a same community share common interests in - clothes, music, beliefs, movies, food, etc. • Influence each other strongly. 50

Definition 51

5 2 3 4 1 52

5 2 3 4 1 53

Model 54

Lemma 1 Proof 55

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Thanks end