Independent Cascade Model for Viral Marketing Wonyeol Lee Jinha Kim - PowerPoint PPT Presentation

0/19 CT-IC: Continuously activated and Time-restricted Independent Cascade Model for Viral Marketing Wonyeol Lee Jinha Kim Hwanjo Yu Department of Computer Science & Engineering , Korea ICDM 2012

CT-IC model for Viral Marketing 1/19 Viral Marketing Influence Maximization Problem Influence Diffusion Models Limitations of Existing Models Introduction & Motivation

CT-IC model for Viral Marketing 2/19 Viral Marketing • Word of mouth effect > TV advertising

CT-IC model for Viral Marketing 3/19 Influence Maximization Problem [KDD’03] 𝜏(𝑇) the expected number of people influenced by a seed set 𝑇 arg 𝑇⊆𝑊,|𝑇|=𝑙 𝜏(𝑇) max Given a network 𝐻 = (𝑊, 𝐹) , and a budget 𝑙 , find the 𝑙 most influential people in a social network

CT-IC model for Viral Marketing 4/19 𝜏(𝑇) Depends On … How influence is propagated through a graph = Influence Diffusion Model • We need a “ realistic ” diffusion model to apply influence maximization problem to a “ real-world ” marketing. • Existing diffusion models 𝑞𝑞(𝑣, 𝑤) – IC (Independent Cascade) model [KDD’03] 𝑣 𝑤 – LT (Linear Threshold) model [KDD’03 ] (newly activated) activation try

CT-IC model for Viral Marketing 5/19 Existing Models Ignore … (1) • An individual can affect others multiple times. GalaxyS3 is GalaxyS3 is GalaxyS3 is No No Yes! awesome awesome awesome Yesterday Today Tomorrow – NOT contained in “IC model.”

CT-IC model for Viral Marketing 6/19 Existing Models Ignore … (2) • Marketing usually has a deadline . What? Don’t you know GalaxyS3 is GalaxyS3 is GalaxyS3 is GalaxyNote2? Yes Yes awesome awesome awesome Yesterday Today Tomorrow – NOT contained in “all previous models.”

CT-IC model for Viral Marketing 7/19 CT-IC model Properties of CT-IC model CT-IPA algorithm Our Contributions

CT-IC model for Viral Marketing 8/19 1) CT-IC model • We propose a new influence diffusion model “CT - IC” for viral marketing, which generalizes previous models such that – An individual can affect others multiple times. – Marketing has a deadline . 𝑞𝑞 𝑢 𝑣, 𝑤 = 𝑞𝑞 0 𝑣, 𝑤 𝑔 𝑣𝑤 𝑢 arg 𝑇⊆𝑊,|𝑇|=𝑙 𝜏(𝑇, 𝑈) max • An efficient algorithm for influence maximization problem under CT-IC model?

CT-IC model for Viral Marketing 9/19 Greedy Algorithm [KDD’03] • Influence maximization even under IC model is NP-Hard. • Greedy algorithm: – Repeatedly select the node which gives the most marginal gain of 𝜏 𝑇 • Theorem: 𝜏 𝑇 satisfies non-negativity, monotonicity, submodularity ⇒ Greedy guarantees approximation ratio (1 − 1/𝑓) . • CT-IC model satisfies these properties?

CT-IC model for Viral Marketing 10/19 2) Properties of CT-IC model • We prove the Theorem: In CT-IC model, 𝜏 ∙, 𝑢 satisfies non-negativity, monotonicity, and submodularity. – Non-negativity: 𝜏 𝑇, 𝑢 ≥ 0 – Monotonicity: 𝜏 𝑇, 𝑢 ≤ 𝜏 𝑇′, 𝑢 for any 𝑇 ⊆ 𝑇′ – Submodularity: 𝜏 𝑇 ∪ 𝑤 , 𝑢 − 𝜏 𝑇, 𝑢 ≥ 𝜏 𝑇′ ∪ 𝑤 , 𝑢 − 𝜏 𝑇′, 𝑢 for any 𝑇 ⊆ 𝑇′ • Thus, Greedy guarantees approximation ratio (1 − 1/𝑓) even under CT-IC model. • An efficient method for computing 𝜏 𝑇, 𝑈 under CT-IC model?

CT-IC model for Viral Marketing 11/19 3) CT-IPA algorithm • Difficulties for computing 𝜏 𝑇, 𝑈 under CT-IC model – Monte Carlo simulation is not scalable. [KDD’10] – Evaluating 𝜏(𝑇) is #P-Hard even under IC model. [KDD’10] – We show that it is difficult to extend PMIA (the state-of-the-art algorithm for IC model) to CT-IC model! • We propose “ CT- IPA” algorithm (an extension of IPA [ICDE’13] ) for calculating 𝜏 𝑇, 𝑈 under CT-IC model.

CT-IC model for Viral Marketing 12/19 Dataset Characteristic of CT-IC model Algorithm Comparison Experiments

CT-IC model for Viral Marketing 13/19 Dataset • We use four real networks:

CT-IC model for Viral Marketing 14/19 Characteristic of CT-IC model (1) • Model comparison between IC & CT-IC models:

CT-IC model for Viral Marketing 15/19 Characteristic of CT-IC model (2) • Effect of marketing time constraint 𝑈 :

CT-IC model for Viral Marketing 16/19 Algorithm Comparison (1) • Comparison of influence spread:

CT-IC model for Viral Marketing 17/19 Algorithm Comparison (2) • Comparison of processing time: 10.0h 5.0h 14.5s 14.3s 7.0s 1.0s – CT-IPA is four orders of magnitude faster than Greedy while providing similar influence spread to Greedy .

CT-IC model for Viral Marketing 18/19 Conclusion

CT-IC model for Viral Marketing 19/19 Conclusion Existing diffusion models ignore important aspects of real marketing. 1) Propose a realistic influence diffusion model “CT - IC” for viral marketing. 2) Prove that CT-IC model satisfies non-negativity, monotonicity, and submodularity. 3) Propose a scalable algorithm “ CT-IPA ” for CT-IC model.

CT-IC model for Viral Marketing 20/19 Thank You!

CT-IC model for Viral Marketing 21/19 Supplements

CT-IC model for Viral Marketing 22/19 CT-IC model & Other Diffusion models • Relationship between influence diffusion models:

CT-IC model for Viral Marketing 23/19 Properties of CT-IC model (1) • Difference between IC & CT-IC models: – Here, given 𝐻 = (𝑊, 𝐹), 𝑙, 𝑈 , difference ratio 𝑒𝑠(𝐻, 𝑙, 𝑈) is defined by where • The Lemma tells us that “For some graphs, CT -IC model is largely different from IC model .”

CT-IC model for Viral Marketing 24/19 Properties of CT-IC model (2) • Maximum probability path: – Here, 𝑞 ∗ is called a maximum probability path from 𝑣 to 𝑤 if • The Lemma tells us that “It is difficult to generalize PMIA algorithm into CT- IC model.”

CT-IC model for Viral Marketing 25/19 Characteristic of CT-IC model • Model comparison between IC & CT-IC models:

CT-IC model for Viral Marketing 26/19 Exact Computation of Influence Spread (1) • Case of an arborescence: where 𝑏𝑞 𝑇 (𝑤, 𝑢) is the probability that 𝑤 is activated exactly at time 𝑢 by 𝑇 .

CT-IC model for Viral Marketing 27/19 Exact Computation of Influence Spread (2) • Case of a simple path: where 𝑗𝑜𝑔 𝑞 (𝑣, 𝑤) is the probability that 𝑣 activates 𝑤 in time 𝑈 along a path 𝑞 ,

CT-IC model for Viral Marketing 28/19 Exact Computation of Influence Spread (3) • Case of a simple path: (proof) By Lemma 2, By gathering in a matrix,

CT-IC model for Viral Marketing 29/19 IPA Algorithm (1) • Influence spread of a single node 𝑣 : where 𝑄 𝑣→𝑤 = {𝑞 = 𝑣, … , 𝑤 |𝑗𝑜𝑔 𝑞 𝑣, 𝑤 ≥ 𝜄} , 𝑃 𝑣 = {𝑥|𝑄 𝑣→𝑥 ≠ 𝜚} . Here, 𝜄 is a threshold for IPA algorithm.

CT-IC model for Viral Marketing 30/19 IPA Algorithm (2) • Influence spread of a seed set 𝑇 : where 𝑄 𝑇→𝑤 = {𝑞 = 𝑣, … , 𝑤 |𝑣 ∈ 𝑇, 𝑗𝑜𝑔 𝑞 𝑣, 𝑤 ≥ 𝜄} , 𝑃 𝑇 = {𝑥|𝑄 𝑇→𝑥 ≠ 𝜚} . Here, 𝜄 is a threshold for IPA algorithm.