Online Model-Free Influence Maximization with Persistence · Online Model-Free In uence...
Transcript of Online Model-Free Influence Maximization with Persistence · Online Model-Free In uence...
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
10−4
10−2
100
102
104
Runn
ing
time
(s)
OracleEG-CBGT-UCBRandomMaxDegree
DBLP (WC – L = 1)
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Growth of spreads (DBLP)
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OracleEG-CBGT-UCBRandomMaxDegree
DBLP (WC – 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.
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