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Online Model-Free Influence Maximization with Persistence Paul Lagr ee, Olivier Capp e, Bogdan Cautis, Silviu Maniu LRI, Univ. Paris-Sud, CNRS, LIMSI & Univ. Paris Saclay May 9, 2017 P. Lagr ee, O. Capp e, B. Cautis & S.

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Online Model-Free Influence Maximization with Persistence

Paul Lagr´ ee, Olivier Capp´ e, Bogdan Cautis, Silviu Maniu

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

May 9, 2017

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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 1 / 28

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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 spread in the social network (word-of-mouth effect). Select k seeds (influencers) in the social graph, given an graph G = (V , E) and a propagation model Edges correspond to follow relations, friendships, etc. in the social network

  • P. Lagr´

ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 3 / 28

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Problem

IM Optimization Problem

Denoting S(I) the influence cascade starting from a set of seeds I, the

  • bjective of the IM is to solve the following problem

arg max

I⊆V ,|I|=L

E[S(I)].

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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 4 / 28

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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 to

neighbours 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

0.3 0.5 0.1 0.8 0.2 0.05 0.5 0.1 0.3

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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 6 / 28

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Independent Cascade Model – Example

0.3 0.5 0.1 0.8 0.2 0.05 0.5 0.1 0.3

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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 7 / 28

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Independent Cascade Model – Example

0.3 0.5 0.1 0.8 0.2 0.05 0.5 0.1 0.3

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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 8 / 28

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Independent Cascade Model – Example

0.3 0.5 0.1 0.8 0.2 0.05 0.5 0.1 0.3

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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 9 / 28

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Independent Cascade Model – Example

0.3 0.5 0.1 0.8 0.2 0.05 0.5 0.1 0.3

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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 10 / 28

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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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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 11 / 28

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Online Influence Maximization

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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 12 / 28

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Online Influence Maximization

We only know the social graph, but not edge probabilities. Problem introduced 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 activation

attempts

3 repeat to step 1 until the budget is consumed.

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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 13 / 28

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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 with persistence is to solve the following optimization problem arg max

In⊆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 yield reward only once

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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 14 / 28

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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” the concept rewards only once (e.g. politics) brand fanatics, e.g., Star Wars, Apple, etc. social advertisement in users’ feed (e.g. Twitter / Facebook), people may transfer/ like the content several times across the campaign.

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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 15 / 28

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Online Model-Free Influence Maximization with Persistence

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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 16 / 28

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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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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 17 / 28

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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 given seed.

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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 18 / 28

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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) ˆ Rn :=

  • u∈A

Un(u) n , where Un(u) is the indicator equal to 1 if x has been sampled exactly

  • nce.

The estimator is the fraction of hapaxes!

  • P. Lagr´

ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 19 / 28

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Confidence Bounds

Estimator Bias

E[Rn] − E[ ˆ 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 4 δ, the following holds:

−βn − λ n ≤ Rn − ˆ Rn ≤ βn.

  • P. Lagr´

ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 20 / 28

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Algorithm

UCB-like algorithm at round t, we play the expert k with largest index bk(t) := ˆ Rk(t) + (1 + √ 2)

  • ˆ

λk(t) log(4t) Nk(t) + log(4t) 3Nk(t), where Nk(t) denotes the number of times expert k has been played up to round t

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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 21 / 28

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Optimism in Face of Uncertainty

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Experiments

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Execution time (DBLP)

100 200 300 400 500 Trial 10−4 10−2 100 102 104 Running time (s) Oracle EG-CB GT-UCB Random MaxDegree

DBLP (WC – L = 1)

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ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 24 / 28

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Growth of spreads (DBLP)

100 200 300 400 500 Trial 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Influence Spread ×105 Oracle EG-CB GT-UCB Random MaxDegree

DBLP (WC – L = 5)

100 200 300 400 500 Trial 0.0 0.2 0.4 0.6 0.8 1.0 Influence Spread ×105

DBLP (TV – L = 5)

100 200 300 400 500 Trial 0.0 1.0 2.0 3.0 4.0 5.0 Influence Spread ×104

DBLP (LT – L = 5)

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Thank you.

  • P. Lagr´

ee, O. Capp´ e, B. Cautis & S. Maniu Online Model-Free IM May 9, 2017 26 / 28

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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 Knowledge Discovery and Data Mining, 137–146. Siyu Lei, Silviu Maniu, Luyi Mo, Reynold Cheng and Pierre Senellart (2015) Online Influence Maximization SIGKDD. S´ ebastien Bubeck, Damien Ernst and Aur´ elien Garivier (2013) Optimal Discovery with Probabilistic Expert Advice: Finite Time Analysis and Macroscopic 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 Triggered Arms Journal of Machine Learning Research, 1746 – 1778. Sharan Vaswani, V.S. Lakshmanan and Mark Schmidt (2015) Influence Maximization with Bandits Workshop NIPS.

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