Online Model-Free Influence Maximization withPersistence

Paul Lagree, Olivier Cappe, Bogdan Cautis, Silviu Maniu

LRI, Univ. Paris-Sud, CNRS, LIMSI & Univ. Paris Saclay

May 9, 2017

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Background & Motivations

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Classic Influence Maximization [Kempe et al., 2003]

Important problem in social networks, with applications in marketing,computational advertising.

Objective: Given a promotion budget, maximize the influence spreadin the social network (word-of-mouth effect).

Select k seeds (influencers) in the social graph, given an graphG = (V ,E ) and a propagation model

Edges correspond to follow relations, friendships, etc. in the socialnetwork

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Problem

IM Optimization Problem

Denoting S(I ) the influence cascade starting from a set of seeds I , theobjective of the IM is to solve the following problem

arg maxI⊆V ,|I |=L

E[S(I )].

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Independent Cascade Model

To each edge (u, v) ∈ E , a probability p(u, v) is associated

1 at time 0 – activate seed s

2 node u activated at time t – influence is propagated at t + 1 toneighbours v independently with probability p(u, v)

3 once a node is activated, it cannot be deactivated or activated again.

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Independent Cascade Model – Example

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Independent Cascade Model – Example

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Independent Cascade Model – Example

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Independent Cascade Model – Example

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Independent Cascade Model – Example

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Approximated IM algorithms

1 Computing expected spread: Monte Carlo simulations

2 for solving the IM: greedy approximation algorithm

Multiple algorithms and estimators: TIM / TIM+ [Tang et al., 2014],IMM [Tang et al., 2015], SSA [Nguyen et al., 2016], PMC[Ohsaka et al., 2014], . . .

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Online Influence Maximization

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Online Influence Maximization

We only know the social graph, but not edge probabilities. Problemintroduced by [Lei et al., 2015] for the IC model.

1 at trial n — select a set of k seeds,

2 the diffusion happens, observe activated nodes and edge activationattempts

3 repeat to step 1 until the budget is consumed.

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Online Influence Maximization with Persistence

OIMP Problem [Lei et al., 2015]

Given a budget N, the objective of the online influence maximization withpersistence is to solve the following optimization problem

arg maxIn⊆V ,|In|=L,∀1≤n≤N

E∣∣∣⋃1≤n≤N S(In)

∣∣∣ .A node can be activated several times at different trials, but it will yieldreward only once

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Motivations

A campaign with several steps: different posts with a single semantics.

people may transfer the information several times, but “adopting” theconcept rewards only once (e.g. politics)

brand fanatics, e.g., Star Wars, Apple, etc.

social advertisement in users’ feed (e.g. Twitter / Facebook), peoplemay transfer/ like the content several times across the campaign.

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Online Model-Free Influence Maximization withPersistence

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Setting

In the following, we work in

the persistent setting

no assumption regarding the diffusion model

simple feedback: set of activated nodes

Simple, realistic, target short horizons

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Model

To simplify the graph problem, we consider the corresponding graph(depth-1 trees):

Experts

Basic Nodes

Hypothesis: empty intersection between experts

New problem: estimating the missing mass of each expert, that is,the expected number of nodes that can still be reached from a givenseed.

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Missing mass

Following the work of [Bubeck et al., 2013]

Missing mass Rn :=∑

u∈A 1 {u /∈⋃n

i=1 Si} p(u)

Corresponds to the potential of the expert

Missing mass estimator (known as the Good-Turing estimator)

Rn :=∑u∈A

Un(u)

n,

where Un(u) is the indicator equal to 1 if x has been sampled exactlyonce.

The estimator is the fraction of hapaxes!

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Confidence Bounds

Estimator Bias

E[Rn]− E[Rn] ∈[−∑

u∈A p(u)

n, 0

]

Theorem

With probability at least 1− δ, denoting λ :=∑

u∈A p(u) and

βn := (1 +√

2)√

λ log(4/δ)n + 1

3n log 4δ , the following holds:

−βn −λ

n≤ Rn − Rn ≤ βn.

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Algorithm

UCB-like algorithm

at round t, we play the expert k with largest index

bk(t) := Rk(t) + (1 +√

2)

√λk(t) log(4t)

Nk(t)+

log(4t)

3Nk(t),

where Nk(t) denotes the number of times expert k has been playedup to round t

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Optimism in Face of Uncertainty

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Experiments

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Execution time (DBLP)

100 200 300 400 500Trial

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OracleEG-CBGT-UCBRandomMaxDegree

DBLP (WC – L = 1)

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Growth of spreads (DBLP)

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DBLP (TV – L = 5)

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Thank you.

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References

David Kempe, Jon Kleinberg and Eva Tardos (2003)

Maximizing the Spread of Influence Through a Social Network.

Proceedings of the Ninth ACM SIGKDD International Conference on KnowledgeDiscovery and Data Mining, 137–146.

Siyu Lei, Silviu Maniu, Luyi Mo, Reynold Cheng and Pierre Senellart (2015)

Online Influence Maximization

SIGKDD.

Sebastien Bubeck, Damien Ernst and Aurelien Garivier (2013)

Optimal Discovery with Probabilistic Expert Advice: Finite Time Analysis andMacroscopic Optimality

Journal of Machine Learning Research, 601 – 623.

Wei Chen, Yajun Wang, Yang Yuan and Qinshi Wang (2016)

Combinatorial Multi-armed Bandit and Its Extension to Probabilistically TriggeredArms

Journal of Machine Learning Research, 1746 – 1778.

Sharan Vaswani, V.S. Lakshmanan and Mark Schmidt (2015)

Influence Maximization with Bandits

Workshop NIPS.

Zheng Wen and Branislav Kveton and Michal Valko (2016)

Influence Maximization with Semi-Bandit Feedback

Technical Report.

Naoto Ohsaka, Takuya Akiba, Yuichi Yoshida and Ken-Ichi Kawarabayashi (2014)

Fast and Accurate Influence Maximization on Large Networks with PrunedMonte-Carlo Simulations

AAAI.

Youze Tang, Xiaokui Xiao and Yanchen Shi (2014)

Influence Maximization: Near-Optimal Time Complexity Meets Practical Efficiency

SIGMOD, 75 – 86.

Youze Tang, Yanchen Shi and Xiaokui Xiao (2015)

Influence Maximization in Near-Linear Time: A Martingale Approach

SIGMOD, 1539 – 1554.

Hung T. Nguyen, My T. Thai and Thang N. Dinh (2016)

Stop-and-Stare: Optimal Sampling Algorithms for Viral Marketing in Billion-scaleNetworks

SIGMOD.

P. Lagree, O. Cappe, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 27 / 28