1. Stability of Influence Maximization
Xinran He and David Kempe
University of Southern California
{xinranhe,
08/26/2014

2. Diffusion In Social Networks
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Some friend did and someone did not
The adoption of new products can propagate in the social network
Diffusion in the social network
He & Kempe (USC)
Influence Stability
KDD 2014

3. IC Model & Influence Maximization
He & Kempe (USC)
Influence Stability
KDD 2014
Independent Cascade (IC) Model:
Each newly activated node 𝑢 has a single chance to activate each inactive neighbor 𝑣 with probability 𝑝 𝑢,𝑣 .
Influence Maximization:
Find 𝑘 people that generate the largest influence spread (i.e. expected number of activated nodes) [KKT 2003]
Where do parameters 𝒑 𝒖,𝒗 come from?

4. Uncertainty in Influence Strength
He & Kempe(USC)
Influence Stability
KDD 2014
Diffusion History
Questionnaire
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Network Inference
Influence Maximization
Does such instability really exist?
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Ground truth network

5. An Extreme Example 0.0625 0.0625 0.055 0.07 Select one seed 𝑝= 𝑝=
He & Kempe (USC)
Influence Stability
KDD 2014
Select one seed 𝑝=
0.0625
0.0625
0.055
0.07 𝑝=

6. An Extreme Example (Cont.)
He & Kempe (USC)
Influence Stability
KDD 2014
Simple slide with only one animation 𝑝=
0.3
0.25 𝑝=
0.3
0.35

7. Diagnosing Instability
He & Kempe (USC)
Influence Stability
KDD 2014
Given an instance of Influence Maximization, can we
diagnose efficiently whether it is stable or unstable?
How about this network?
Complete answer
⇒ Computing percolation threshold of any graph.
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Partial solution
Unstable instances ⇒ Fire an alarm correctly.
Instability diagnosis correctly
Stable instances ⇒ Possible false alarms.

8. Definition of Stability
He & Kempe (USC)
Influence Stability
KDD 2014
Definition of Stability
Model of misestimation: 1
𝑝 𝑢,𝑣
𝑝 𝑢,𝑣 ′ 𝑢 𝑣
𝑝 𝑢,𝑣
𝐼 𝑢,𝑣
Definition (Stability of Influence Maximization):
An instance ( 𝑝 𝑢,𝑣 , 𝐼 𝑢,𝑣 ) is stable if the difference in influence is small for all legal 𝑝 𝑢,𝑣 ′ ∈ 𝐼 𝑢,𝑣 and all seed sets of size 𝑘.

9. Influence Difference Maximization
He & Kempe (USC)
Influence Stability
KDD 2014
max 𝑆 =𝑘 max 𝑝 𝑢,𝑣 ′ ∈ 𝐼 𝑢,𝑣 |𝜎 𝑆 −𝜎′(𝑆)|
Optimization Problem:
Definition (Influence Difference Maximization) :
Given two instances with probabilities 𝑝 𝑢,𝑣 ≥ 𝑝 𝑢,𝑣 ′ for all 𝑢, 𝑣,
let 𝜎 and 𝜎′ be the respective influence functions.
Find a set S of size 𝑘 maximizing 𝜎 𝑆 −𝜎′(𝑆).
Get rid of the animation for the optimization problem

10. Main Theory Result
He & Kempe (USC)
Influence Stability
KDD 2014
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
Running time: 𝑂(𝑘𝑀 𝑉 2 )
(𝑘≪|𝑉|)
(𝑀:number of Monte-Carlo Simulations)
Corollary : Assuming 𝐴 is the seed set returned by maximizing 𝜎 obs 𝑆 with greedy algorithm, we have
𝜎 true 𝐴 ≥𝑐⋅ 𝜎 true 𝐴 ∗ ,
where 𝑐 is a constant depending on the given instance.

11. Approximation Guarantee for InfMax
He & Kempe (USC)
Influence Stability
KDD 2014 1
𝑝 𝑢,𝑣
𝑝 𝑢,𝑣 ′
𝑝 𝑢,𝑣 −
𝑝 𝑢,𝑣 +
𝐼 𝑢,𝑣
𝜽={ 𝑝 𝑢,𝑣 } , 𝜎(𝑆): the observed/inferred probabilities.
𝜽 = 𝑝 𝑢,𝑣 , 𝜎 𝑆 : the ground truth probabilities.
𝜽 + , 𝜽 − , 𝜎 + 𝑆 , 𝜎 − (𝑆) : making each probability as large/small as possible.
Theorem: Assuming 𝐴 is the seed set returned by maximizng 𝜎 𝑆 with greedy algorithm, we have 𝜎 𝐴 ≥𝑐⋅ 𝜎 𝐴 , where
𝑐= 1− 𝛼 − ⋅ 1−1/𝑒 1+ 𝛼 + ⋅ 1−1/𝑒
𝛼 − = 𝛿 𝜽, 𝜽 − 𝐴 𝜎(𝐴)
(By calculation)
𝛼 + = max 𝑆 𝛿 𝜽 + ,𝜽 𝑆 𝜎(𝐴)
(By Inf Diff Max)

12. Experiments: Setting 𝑝 𝑢,𝑣 Networks
He & Kempe (USC)
Influence Stability
KDD 2014
Networks
Synthetic:
2D-grid, random regular graphs, small world, preferential attachment
Real:
STOCFOCS: co-authorship network, |V|=1768, |E|=10024
HAITI: retweet network, |V|=274, |E|=383
Set 𝐼 𝑢,𝑣 =[ 1−Δ 𝑝 𝑢,𝑣 , (1+Δ) 𝑝 𝑢,𝑣 ]
Vary 𝑝 𝑢,𝑣 ∈{0.01,0.02,0.05,0.1}
Vary Δ∈ 1%,5%,10%,20%,50%
Metrics:
Inf Diff Max: max 𝑆 𝛿 𝑆
Approximation guarantee: 𝑐 such that 𝜎 ob𝑠 𝐴 ≥𝑐⋅ 𝜎 true 𝐴 ∗ 1
𝑝 𝑢,𝑣
𝑝 𝑢,𝑣 Δ

13. Experiments: PA network
He & Kempe (USC)
Influence Stability
KDD 2014
𝒎𝒂𝒙 𝑺 𝜹 𝑺
Approximation guarantee 𝒄

14. Experiments: STOCFOCS
He & Kempe (USC)
Influence Stability
KDD 2014
𝒎𝒂𝒙 𝑺 𝜹 𝑺
Approximation guarantee 𝒄

15. Conclusion Noise is everywhere in social network data
He & Kempe (USC)
Influence Stability
KDD 2014
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
transition

16. Future work Generalization to other diffusion models.
He & Kempe (USC)
Influence Stability
KDD 2014
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?

17. Questions?