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

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Jun 27, 2023 210 likes β€’536 views

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

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Wonyeol Lee Jinha Kim Hwanjo Yu Department of Computer Science & Engineering , Korea ICDM 2012

CT-IC: Continuously activated and Time-restricted Independent Cascade Model for Viral Marketing

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Introduction & Motivation

Viral Marketing Influence Maximization Problem Influence Diffusion Models Limitations of Existing Models

CT-IC model for Viral Marketing

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Viral Marketing

  • Word of mouth effect > TV advertising

CT-IC model for Viral Marketing

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Influence Maximization Problem [KDD’03]

𝜏(𝑇)

the expected number of people influenced by a seed set 𝑇

arg max

π‘‡βŠ†π‘Š,|𝑇|=𝑙 𝜏(𝑇)

Given a network 𝐻 = (π‘Š, 𝐹), and a budget 𝑙, find the 𝑙 most influential people in a social network

CT-IC model for Viral Marketing

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𝜏(𝑇) Depends On …

How influence is propagated through a graph = Influence Diffusion Model

  • We need a β€œrealistic” diffusion model to apply

influence maximization problem to a β€œreal-world” marketing.

  • Existing diffusion models

– IC (Independent Cascade) model [KDD’03] – LT (Linear Threshold) model [KDD’03]

CT-IC model for Viral Marketing

𝑣 𝑀 π‘žπ‘ž(𝑣, 𝑀)

(newly activated) activation try

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Existing Models Ignore … (1)

  • An individual can affect others multiple times.

– NOT contained in β€œIC model.”

CT-IC model for Viral Marketing

No

Yesterday

GalaxyS3 is awesome

No

Today

GalaxyS3 is awesome

Yes!

Tomorrow

GalaxyS3 is awesome

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Existing Models Ignore … (2)

  • Marketing usually has a deadline.

– NOT contained in β€œall previous models.”

CT-IC model for Viral Marketing

Yes

Yesterday

GalaxyS3 is awesome

Yes

Today

GalaxyS3 is awesome What? Don’t you know GalaxyNote2?

Tomorrow

GalaxyS3 is awesome

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Our Contributions

CT-IC model Properties of CT-IC model CT-IPA algorithm

CT-IC model for Viral Marketing

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  • We propose a new influence diffusion model β€œCT-IC” for viral

marketing, which generalizes previous models such that

– An individual can affect others multiple times. – Marketing has a deadline.

  • An efficient algorithm for influence maximization problem

under CT-IC model?

1) CT-IC model

arg max

π‘‡βŠ†π‘Š,|𝑇|=𝑙 𝜏(𝑇, π‘ˆ)

π‘žπ‘žπ‘’ 𝑣, 𝑀 = π‘žπ‘ž0 𝑣, 𝑀 𝑔

𝑣𝑀 𝑒

CT-IC model for Viral Marketing

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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 β‡’ Greedy guarantees approximation ratio (1 βˆ’ 1/𝑓).

  • CT-IC model satisfies these properties?

CT-IC model for Viral Marketing

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2) Properties of CT-IC model

  • We prove the Theorem: In CT-IC model, 𝜏 βˆ™, 𝑒 satisfies

non-negativity, monotonicity, and submodularity.

– Non-negativity: 𝜏 𝑇, 𝑒 β‰₯ 0 – Monotonicity: 𝜏 𝑇, 𝑒 ≀ 𝜏 𝑇′, 𝑒 for any 𝑇 βŠ† 𝑇′ – Submodularity: 𝜏 𝑇 βˆͺ 𝑀 , 𝑒 βˆ’ 𝜏 𝑇, 𝑒 β‰₯ 𝜏 𝑇′ βˆͺ 𝑀 , 𝑒 βˆ’ 𝜏 𝑇′, 𝑒 for any 𝑇 βŠ† 𝑇′

  • Thus, Greedy guarantees approximation ratio (1 βˆ’ 1/𝑓)

even under CT-IC model.

  • An efficient method for computing 𝜏 𝑇, π‘ˆ

under CT-IC model?

CT-IC model for Viral Marketing

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3) CT-IPA algorithm

  • Difficulties for computing 𝜏 𝑇, π‘ˆ 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 β€œCT-IPA” algorithm (an extension of IPA [ICDE’13])

for calculating 𝜏 𝑇, π‘ˆ under CT-IC model.

CT-IC model for Viral Marketing

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Experiments

Dataset Characteristic of CT-IC model Algorithm Comparison

CT-IC model for Viral Marketing

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Dataset

  • We use four real networks:

CT-IC model for Viral Marketing

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Characteristic of CT-IC model (1)

  • Model comparison between IC & CT-IC models:

CT-IC model for Viral Marketing

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Characteristic of CT-IC model (2)

  • Effect of marketing time constraint π‘ˆ:

CT-IC model for Viral Marketing

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Algorithm Comparison (1)

  • Comparison of influence spread:

CT-IC model for Viral Marketing

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Algorithm Comparison (2)

  • Comparison of processing time:

– CT-IPA is four orders of magnitude faster than Greedy while providing similar influence spread to Greedy.

CT-IC model for Viral Marketing 1.0s 7.0s 14.5s 14.3s 5.0h 10.0h

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Conclusion

CT-IC model for Viral Marketing

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Conclusion

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

CT-IC model for Viral Marketing

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Thank You!

CT-IC model for Viral Marketing

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Supplements

CT-IC model for Viral Marketing

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CT-IC model & Other Diffusion models

  • Relationship between influence diffusion models:

CT-IC model for Viral Marketing

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Properties of CT-IC model (1)

  • Difference between IC & CT-IC models:

– Here, given 𝐻 = (π‘Š, 𝐹), 𝑙, π‘ˆ, difference ratio 𝑒𝑠(𝐻, 𝑙, π‘ˆ) is defined by where

  • The Lemma tells us that

β€œFor some graphs, CT-IC model is largely different from IC model.”

CT-IC model for Viral Marketing

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Properties of CT-IC model (2)

  • Maximum probability path:

– Here, π‘žβˆ— is called a maximum probability path from 𝑣 to 𝑀 if

  • The Lemma tells us that

β€œIt is difficult to generalize PMIA algorithm into CT-IC model.”

CT-IC model for Viral Marketing

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Characteristic of CT-IC model

  • Model comparison between IC & CT-IC models:

CT-IC model for Viral Marketing

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Exact Computation of Influence Spread (1)

  • Case of an arborescence:

where π‘π‘žπ‘‡(𝑀, 𝑒) is the probability that 𝑀 is activated exactly at time 𝑒 by 𝑇.

CT-IC model for Viral Marketing

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  • Case of a simple path:

where π‘—π‘œπ‘”

π‘ž(𝑣, 𝑀) is the probability that 𝑣 activates 𝑀 in time π‘ˆ along a path π‘ž,

Exact Computation of Influence Spread (2)

CT-IC model for Viral Marketing

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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

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IPA Algorithm (1)

  • Influence spread of a single node 𝑣:

where 𝑄

𝑣→𝑀 = {π‘ž = 𝑣, … , 𝑀 |π‘—π‘œπ‘” π‘ž 𝑣, 𝑀 β‰₯ πœ„}, 𝑃𝑣 = {π‘₯|𝑄 𝑣→π‘₯ β‰  𝜚}.

Here, πœ„ is a threshold for IPA algorithm.

CT-IC model for Viral Marketing

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IPA Algorithm (2)

  • Influence spread of a seed set 𝑇:

where 𝑄

𝑇→𝑀 = {π‘ž = 𝑣, … , 𝑀 |𝑣 ∈ 𝑇, π‘—π‘œπ‘” π‘ž 𝑣, 𝑀 β‰₯ πœ„}, 𝑃𝑇 = {π‘₯|𝑄 𝑇→π‘₯ β‰  𝜚}.

Here, πœ„ is a threshold for IPA algorithm.

CT-IC model for Viral Marketing