stability of influence maximization xinran he and david kempe university of southern california...

Stability of Influence Maximization

Xinran He and David KempeUniversity of Southern California

{xinranhe, dkempe}@usc.edu08/26/2014

• The adoption of new products can propagate in the social network Diffusion in the social network

0.8

0.3

0.2

0.6

0.5

0.7

0.2

He & Kempe (USC) Influence Stability KDD 2014

Diffusion In Social Networks

He & Kempe (USC) Influence Stability KDD 2014

IC Model & Influence Maximization

• Independent Cascade (IC) Model:• Each newly activated node has a single chance to activate each inactive

neighbor with probability .

• Influence Maximization: • Find people that generate the largest influence spread (i.e. expected number

of activated nodes) [KKT 2003]

Where do parameters come from?

Diffusion History

Questionnaire

Influence Maximization

0.80.5

0.30.6

0.1

0.4

0.8

Network Inference

0.50.6

0.10.6

0.3

0.4

0.9

Ground truth network

Does such instability really exist?

He & Kempe(USC) Influence Stability KDD 2014

Uncertainty in Influence Strength

= 0.0625 = 0.0625

Select one seed

0.07 0.055

He & Kempe (USC) Influence Stability KDD 2014

An Extreme Example

= 0.3 = 0.30.350.25

He & Kempe (USC) Influence Stability KDD 2014

An Extreme Example (Cont.)

0.8 0.5

0.30.6

0.1

0.4

0.8

How about this network?

Given an instance of Influence Maximization, can we

diagnose efficiently whether it is stable or unstable?

Complete answer Computing percolation threshold of any graph.

Partial solution Unstable instances Fire an alarm correctly.

Stable instances Possible false alarms.

He & Kempe (USC) Influence Stability KDD 2014

Diagnosing Instability

•Model of misestimation:𝑢 𝑣𝑝𝑢 ,𝑣

0 1

𝑝𝑢 ,𝑣𝑝𝑢 ,𝑣′

𝐼𝑢 ,𝑣

Definition (Stability of Influence Maximization): An instance is stable if the difference in influence is small for all legal and all seed sets of size .

He & Kempe (USC) Influence Stability KDD 2014

Definition of Stability

max|𝑆|=𝑘

max𝑝𝑢,𝑣′ ∈𝐼 𝑢,𝑣

¿𝜎 (𝑆 )−𝜎 ′ (𝑆)∨¿¿Optimization Problem:

Definition (Influence Difference Maximization) :Given two instances with probabilities for all , , let and be the respective influence functions. Find a set S of size maximizing .

He & Kempe (USC) Influence Stability KDD 2014

Influence Difference Maximization

• Random Greedy Algorithm [Buchbinder et al.]• Approximation guarantee: 0.266• Running time:

Main Theorem: Under the IC model, is a non-negative and submodular function of the set (but not monotone).

(number of Monte-Carlo Simulations)

Corollary : Assuming is the seed set returned by maximizing with greedy algorithm, we have , where is a constant depending on the given instance.

He & Kempe (USC) Influence Stability KDD 2014

Main Theory Result

• Noise is everywhere in social network data Þ Influence Maximization could be unstableÞ 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

Conclusion

• 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

Future work

Questions?