Influence Maximization in a Social Network in the

Presence of Multiple Influences and Acceptances @2014 International Conference on Data Science and Advanced Analytics (DSAA2014)

Jun-Li Lu Mi-Yen Yeh Ling-Yin Wei

Outline

• Influence maximization (IM)

•MIMA-based IM• Multiple influences and multiple acceptances (MIMA)

•Method

• Experiment results

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Applications of influence maximization

•Product advertising

• Effective promoter (users) identification

•Popular trends/topics detection

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Multiple Influences and Multiple Acceptances (MIMA)

•Given a message m on a social network, an user accepts m multiple times, also influences other users to accept m multiple times

•MIMA is novel, compared to, • IC-model, LT-model [ICDM’12, WWW’11]• Considering time period [ICDM’12]• Pricing factor, competing [WWW’08, WINE’07]• Negative opinions [CIKM’08]

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“Zimmerman Trial” on Twitter (1/2)

•A single user retweeted multiple times

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day (in July)

USER with 80 FRIENDS

USER with 158 FRIENDS

USER with 55 FRIENDS

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“Zimmerman Trial” on Twitter (2/2)

• The increment of retweets was decreasing

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day (in July)more than 16,000 tweets in Boston, USA, July 1-22, 2013

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How to model real-world MIMA (future work)

•Which social-network messages fit for MIMA? • type of messages

• non-fit cases, e.g., one-time voting, daily highlight• user preference

• fans

•MIMA modelled by real-data?• refer to influence analysis works [KDD, SDM, ICDM, WWW,…]• which factors affect influences and acceptances of an user?

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To model MIMA by diminishing marginal utility (DMU)

• DMU, law of describing human-behaviors in economics• f: staff performance, x: number of staffs

• f: food satisffaction, x: number of foods

• Influence 𝑓𝑤𝑢(𝑥) is DMU,• 𝑓𝑤𝑢: user w’s influence on user u,

• x: number of items user w has adopted

• Acceptance threshold 𝑔𝑢(𝑥) is monotone increasing• A user is hard to get one more if the user already got many ones

• 𝑤∈𝑀(𝑢) 𝑓𝑤𝑢(𝑥𝑤𝑡 ) ≥ 𝜃𝑢,𝑧

• 𝜃𝑢,𝑧 ∈ 0, 𝑔𝑢 𝑧 , 𝑧 ∈ {𝑥𝑤𝑡 + 1, 𝑥𝑤

𝑡 + 2,… }

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f

x

DMU

MIMA-based influence maximization (1/2)

• Given an item m on a social network G, to identify a small set of users that are effective to promote m on G and each user can accept m multiple times

• Output: the promoter set that contribute the max IS• Influence spread (IS): the total # of acceptance times of all users of G

Who to promote?

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MIMA-based influence maximization (2/2)

• IM Problem• NP-hard

• Influence spread f(A) (given the set of users A)• monotone• submodular• Computing exact influence spread f(A) is #P-hard

•Greedy method

•Heuristic method

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Influence maximization (IM) problem

•Given an item m and the MIMA model, to identify a small set of users that are effective to promote m.

•NP-hard. proof, • to show each 𝐼𝑀𝑜(traditional NP-hard IM problem) can be a SP of

our problem

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• 𝐼𝑀𝑜, 𝑤 𝑏𝑤𝑢 ≤ 1, 𝜏_𝑢 ≤ 1,∀𝑢

• Influence 𝑓(𝑥) = (𝑦 1 −𝑐𝑏′𝑤𝑢

𝑐)/𝑦(1) 𝑦(𝑥 − 1) + 𝑏′𝑤𝑢

• Acceptance threshold 𝜃𝑢,𝑧 =𝑦 2 −𝜏′𝑢

𝑦 1𝑦 𝑧 − 1 + 𝜏′𝑢

𝑐 = |𝑀(𝑢)|, 𝑦(𝑥) = 1 − 𝑒−𝑥, 𝑏′𝑤𝑢 = 𝑦 1 𝑏𝑤𝑢, 𝜏′𝑢 = 𝑦(1)𝜏𝑢

• done, because 𝑤 𝑓𝑤𝑢(1) ≤ 𝜃𝑢,2, 𝜃𝑢,1 = 𝜏′𝑢

Influence spread (IS)

•Monotone: 𝜎(A ∪ 𝑢 ) ≥ 𝜎(A)

• Sub-modular:𝜎 A1 ∪ 𝑢 ≥ 𝜎 A2 ≥ 𝜎 A2 ∪ 𝑢 − 𝜎 A2 , ∀A1 ⊆ A2

•Computing exact influence spread is #P-hardidea of proof:• suppose the propagation result by user u is viewed as,• each reachable path by user u in a graph• …if user u has accepted x times, means there a x-path

reaching to u of the graph

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Greedy method

• To add the user getting best incremental IS for the selected promoters

• To simulate the IS by sampling random acceptance threshold 𝜃𝑢,𝑧, for each user u, z

• Accurate, but time-consuming

• Effect: (1 − 𝑒−1 − 𝜖) of optimal IS• 𝜖 is related to # of simulation times

• skip proof

Simulating IS by random 𝜃𝑢,𝑧

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Heuristic method (1/2)

• To avoid simulating IS

• To estimate IS by the MIMA model

• Top-rank result, and time-efficient

IS by MIMA model

Acceptance threshold

Influence

…

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Heuristic method (2/2)

• Probability of user u accepting x times:

• 𝑝𝑢 𝑥 =𝑝𝑢 𝑥−1 𝐹𝑢

𝑔𝑢 𝑥

,𝐹𝑢 = 𝑤∈𝑀(𝑢) 𝑥=1𝜆𝑤 𝑝𝑤 𝑥 [𝑓𝑤𝑢 𝑥 − 𝑓𝑤𝑢 𝑥 − 1 ]

• During propagation, incremental influence on user u that is made by user w is

• 𝐹𝑢+ = 𝑥=1

𝜆𝑣 𝑎𝑤 𝑥 [𝑓𝑤𝑢 𝑥 − 𝑓𝑤𝑢 𝑥 − 1 ]

• To speed up by dynamic programming, • by merging using the pre-computed IS of each single user

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Experiment setting

•Network data

•Acceptance threshold 𝑔𝑢(𝑥) = 𝑐𝑢(1 − exp −𝑑𝑢𝑥 )

• Influence 𝑓𝑤𝑢 𝑥 = 𝑎𝑤𝑢 1 − exp −𝑏𝑤𝑢𝑥• random influence (RI): 𝑎𝑤𝑢, 𝑏𝑤𝑢 ∈ [0,1]• uniform influence (UI): each friend give the equal influence

to you• 𝑓𝑤𝑢 = 𝑐𝑢/|𝑀 𝑢 |

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# of nodes # of edges Average degree per node

Brightkite 58K 214K 7.35

Twitter 81K 1,768K 43.49

Experiment result: influence spread

• GREEDY and HEUR got top-rank results (GREEDY best)

• In random influence (RI), GREEDY and HEUR was effective

• In uniform influence (UI), each method performed closely

• GREEDY and HEUR was effective in simple networks (e.g., Brightkite)

Brightkite, RI Twitter, RI Twitter, UI

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Experiment result: time complexity

•HEUR was much time-efficient than GREEDY

Twitter, RI

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Conclusions

•MIMA, modelled by DMU

• Investigated MIMA-based IM problem• complexity of IM problem • properties of influence spread• complexity of getting exact influence spread

• Issue: how to model MIMA by real data?

• Thanks. Q and A

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