Stability of Influence Maximization
Xinran He and David Kempe
University of Southern California
{xinranhe, dkempe}@usc.edu
08/26/2014
Diffusion In Social Networks
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• The adoption of new products can propagate in the social
network Diffusion in the social network
He & Kempe (USC)
Influence Stability
KDD 2014
IC Model & Influence Maximization
• Independent Cascade (IC) Model:
• Each newly activated node 𝑢 has a single chance to activate each inactive
neighbor 𝑣 with probability 𝑝𝑢,𝑣 .
• Influence Maximization:
• Find 𝑘 people that generate the largest influence spread (i.e. expected
number of activated nodes) [KKT 2003]
Where do parameters 𝒑𝒖,𝒗 come from?
He & Kempe (USC)
Influence Stability
KDD 2014
Uncertainty in Influence Strength
Diffusion History
Network Inference
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Questionnaire
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Does
such
instability
really
exist?
Influence Maximization
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Ground truth network
He & Kempe(USC)
Influence Stability
KDD 2014
An Extreme Example
Select one seed
𝑝= 0.0625
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𝑝= 0.0625
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He & Kempe (USC)
Influence Stability
KDD 2014
An Extreme Example (Cont.)
𝑝= 0.3
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𝑝= 0.3
0.25
He & Kempe (USC)
Influence Stability
KDD 2014
Diagnosing Instability
Given an instance of Influence Maximization, can we
diagnose
How about
thisefficiently
network?whether it is stable or unstable?
Complete answer
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⇒ Computing percolation0.8
threshold of any graph.
Partial solution 0.6
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Unstable instances ⇒ Fire an alarm correctly.
Stable instances
He & Kempe (USC)
⇒ Possible false alarms.
Influence Stability
KDD 2014
Definition of Stability
• Model of misestimation:
𝑢
𝑝𝑢,𝑣
′
𝑝𝑢,𝑣
𝑝𝑢,𝑣
𝑣
0
𝐼𝑢,𝑣
1
Definition (Stability of Influence Maximization):
An instance (𝑝𝑢,𝑣 , 𝐼𝑢,𝑣 ) is stable if the difference in influence is
′
small for all legal 𝑝𝑢,𝑣
∈ 𝐼𝑢,𝑣 and all seed sets of size 𝑘.
He & Kempe (USC)
Influence Stability
KDD 2014
Influence Difference Maximization
Optimization Problem:
max ′max |𝜎 𝑆 − 𝜎′(𝑆)|
𝑆 =𝑘 𝑝𝑢,𝑣 ∈𝐼𝑢,𝑣
Definition (Influence Difference Maximization) :
′ for all 𝑢, 𝑣,
Given two instances with probabilities 𝑝𝑢,𝑣 ≥ 𝑝𝑢,𝑣
let 𝜎 and 𝜎′ be the respective influence functions.
Find a set S of size 𝑘 maximizing 𝜎 𝑆 − 𝜎′(𝑆).
He & Kempe (USC)
Influence Stability
KDD 2014
Main Theory Result
Main Theorem: Under the IC model, 𝜎 𝑆 − 𝜎′(𝑆) is a non-negative
and submodular function of the set 𝑆 (but not monotone).
• Random Greedy Algorithm [Buchbinder et al.]
• Approximation guarantee: 0.266→ 1/𝑒 (𝑘 ≪ |𝑉|)
• Running time: 𝑂(𝑘𝑀 𝑉 2 ) (𝑀:number of Monte-Carlo Simulations)
Corollary : Assuming 𝐴 is the seed set returned by maximizing
𝜎obs 𝑆 with greedy algorithm, we have
𝜎true 𝐴 ≥ 𝑐 ⋅ 𝜎true 𝐴∗ ,
where 𝑐 is a constant depending on the given instance.
He & Kempe (USC)
Influence Stability
KDD 2014
Conclusion
• Noise is everywhere in social network data
 Influence Maximization could be unstable
 Calls into question practicality of algorithmic approaches
• Instability can be diagnosed by solving Influence Difference Maximization
• Via non-monotone submodular maximization
• Experiments on synthetic networks (2D-grid, random regular, SW, PA) and real networks
(retweet, collaboration)
• 10% relative noise ⇒ Decent approximation
• 20% relative noise ⇒ Significant Challenge
• Further extension:
• Linear Threshold Model, Triggering Model
He & Kempe (USC)
Influence Stability
KDD 2014
Future work
• Generalization to other diffusion models.
• Generalized Threshold (GT) model
• Generalization to other misestimation models.
• Current assumption: each deviation is bounded
• What if the total (squared) deviation is bounded?
• Big picture: How accurate are our diffusion models?
He & Kempe (USC)
Influence Stability
KDD 2014