learning influence probabilities in social networks 1 2 amit goyal 1 francesco bonchi 2 laks v. s....
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Learning Influence Probabilities in Social
Networks
1 2
Amit Goyal1
Francesco Bonchi2
Laks V. S. Lakshmanan1
U. of British ColumbiaYahoo! ResearchU. of British Columbia
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Word of Mouth and Viral Marketing We are more
influenced by our friends than strangers
68% of consumers consult friends and family before purchasing home electronics (Burke 2003)
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Viral Marketing Also known as
Target Advertising Initiate chain
reaction by Word of mouth effect
Low investments, maximum gain
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Viral Marketing as an Optimization Problem Given: Network with
influence probabilities Problem: Select top-k
users such that by targeting them, the spread of influence is maximized
Domingos et al 2001, Richardson et al 2002, Kempe et al 2003
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How to calculate true influence probabilities?
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Some Questions Where do those influence probabilities come
from? Available real world datasets don’t have prob.!
Can we learn those probabilities from available data?
Previous Viral Marketing studies ignore the effect of time. How can we take time into account?
Do probabilities change over time?
Can we predict time at which user is most likely to perform an action.
What users/actions are more prone to influence?http://people.cs.ubc.ca/~goyal 5University of British Columbia,
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Input Data We focus on actions. Input:
Social Graph: P and Q become friends at time 4.
Action log: User P performs actions a1 at time unit 5.
User Action Time
P a1 5
Q a1 10
R a1 15
Q a2 12
R a2 14
R a3 6
P a3 14
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Our contributions (1/2) Propose several probabilistic influence models
between users. Consistent with existing propagation models.
Develop efficient algorithms to learn the parameters of the models.
Able to predict whether a user perform an action or not.
Predict the time at which she will perform it.7
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Our Contributions (2/2) Introduce metrics of users and actions
influenceability. High values => genuine influence.
Validated our models on Flickr.
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Overview Input:
Social Graph: P and Q become friends at time 4.
Action log: User P performs actions a1 at time unit 5.
User Action Time
P a1 5
Q a1 10
R a1 15
Q a2 12
R a2 14
R a3 6
P a3 14
Influence Models
Q R
P
0.330
0
0.5
0.5
0.2
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Background
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General Threshold (Propagation) Model At any point of time, each node is either active or
inactive.
More active neighbors => u more likely to get active.
Notations: S = {active neighbors of u}. pu(S) : Joint influence probability of S on u.
Θu: Activation threshold of user u.
When pu(S) >= Θu, u becomes active.
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General Threshold Model - Example
Inactive Node
Active Node
Threshold
Joint Influence Probability
Source: David Kempe’s slides
vw 0.5
0.30.2
0.5
0.10.4
0.3 0.2
0.6
0.2
Stop!
U
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x
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Our Framework
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Solution Framework Assuming independence, we define
pv,u : influence probability of user v on user u
Consistent with the existing propagation models – monotonocity, submodularity.
It is incremental. i.e. can be updated incrementally using
Our aim is to learn pv,u for all edges.
)1(1)( ,uvSv
u pSp
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}){( wSpu uwu pSp , and )(
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Influence Models Static Models
Assume that influence probabilities are static and do not change over time.
Continuous Time (CT) Models Influence probabilities are continuous functions of time. Not incremental, hence very expensive to apply on large
datasets.
Discrete Time (DT) Models Approximation of CT models. Incremental, hence efficient.
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Static Models 4 variants
Bernoulli as running example. Incremental hence most efficient. We omit details here
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Time Conscious Models Do influence
probabilities remain constant independently of time?
We propose Continuous Time (CT) Model Based on exponential
decay distribution
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Continuous Time Models Best model. Capable of predicting time at which user is
most likely to perform the action. Not incremental
Discrete Time Model Based on step time functions Incremental
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Evaluation Strategy (1/2) Split the action log data into training
(80%) and testing (20%). User “James” have joined “Whistler Mountain”
community at time 5.
In testing phase, we ask the model to predict whether user will become active or not Given all the neighbors who are active Binary Classification
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Evaluation Strategy (2/2) We ignore all the cases
when none of the user’s friends is active
As then the model is inapplicable.
We use ROC (Receiver Operating Characteristics) curves True Positive Rate (TPR) vs
False Positive Rate (FPR). TPR = TP/P FPR = FP/N
Reality
Prediction
Active Inactive
Active TP FP
Inactive FN TN
Total P N
Operating Point
Ideal Point
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Algorithms Special emphasis on efficiency of applying/testing
the models. Incremental Property
In practice, action logs tend to be huge, so we optimize our algorithms to minimize the number of scans over the action log. Training: 2 scans to learn all models simultaneously. Testing: 1 scan to test one model at a time.
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Experimental Evaluation
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Dataset Yahoo! Flickr dataset “Joining a group” is considered as action
User “James” joined “Whistler Mountains” at time 5.
#users ~ 1.3 million #edges ~ 40.4 million Degree: 61.31 #groups/actions ~ 300K #tuples in action log ~ 35.8 million
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Comparison of Static, CT and DT models
Time conscious Models are better than Static Models.
CT and DT models perform equally well.http://people.cs.ubc.ca/~goyal 24University of British Columbia,
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Runtime
Static and DT models are far more efficient compared to CT models because of their incremental nature.
Testing
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Predicting Time – Distribution of Error Operating Point is chosen corresponding to
TPR: 82.5%, FPR: 17.5%.
X-axis: error in predicting time (in weeks)
Y-axis: frequency of that error
Most of the time, error in the prediction is very small
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Predicting Time – Coverage vs Error Operating Point is chosen corresponding to
TPR: 82.5%, FPR: 17.5%.
A point (x,y) here means for y% of cases, the error is within
In particular, for 95% of the cases, the error is within 20 weeks.
x
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User Influenceability
Users with high influenceability => easier prediction of influence => more prone to viral marketing campaigns.
Some users are more prone to influence propagation than others.
Learn from Training data
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Action Influenceability
Some actions are more prone to influence propagation than others.
Actions with high user influenceability => easier prediction of influence => more suitable to viral marketing campaigns.
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Related Work Independently, Saito et al (KES 2008) have
studied the same problem Focus on Independent Cascade Model of
propagation. Apply Expectation Maximization (EM)
algorithm. Not scalable to huge datasets like the one we
are dealing in this work.
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Other applications of Influence Propagations Personalized Recommender Systems
Song et al 2006, 2007 Feed Ranking
Samper et al 2006 Trust Propagation
Guha et al 2004, Ziegler et al 2005, Golbeck et al 2006, Taherian et al 2008
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Conclusions (1/2) Previous works typically assume influence
probabilities are given as input.
Studied the problem of learning such probabilities from a log of past propagations.
Proposed both static and time-conscious models of influence.
We also proposed efficient algorithms to learn and apply the models.
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Conclusions (2/2) Using CT models, it is possible to predict
even the time at which a user will perform it with a good accuracy.
Introduce metrics of users and actions influenceability. High values => easier prediction of influence. Can be utilized in Viral Marketing decisions.
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Future Work Learning optimal user activation
thresholds. Considering users and actions
influenceability in the theory of Viral Marketing.
Role of time in Viral Marketing.
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Thanks!!0.13
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0.30.1
0.41
0.8
0.27
0.2
0.9
0.01
0.7
0.6
0.54
0.1
0.110
0.20.7
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Predicting Time CT models can predict the time interval
[b,e] in which she is most likely to perform the action.
0
Joint influence Probability of u getting active
)2ln(,uvvte vtb
Θu
Time ->
is half life period
Tightness of lower bounds not critical in Viral Marketing Applications.
Experiments on the upper bound e.
)2ln(,uv
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Predicting Time - RMSE vs Accuracy CT models can predict the time interval [b,e] in which user is most likely to perform the action.
Experiments only on upper bound e.
Accuracy =
RMSE = root mean square error in days
cases total#
correct is boundupper of prediction when thecases #
RMSE ~ 70-80 days
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Static Models – Jaccard Index Jaccard Index is often used to measure
similarity b/w sample sets.
We adapt it to estimate pv,u
uor by v performed actions of#
u to vfrom propagated actions of#, uvp
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Partial Credits (PC) Let, for an action,
D is influenced by 3 of its neighbors.
Then, 1/3 credit is given to each one of these neighbors.
A B
D
1/3
C
1/31/3
performed vactions ofnumber total
daccumulate credits total, uvp
performedu or vactions ofnumber total
daccumulate credits total, uvp
PC Bernoulli
PC Jaccardhttp://people.cs.ubc.ca/~goyal 39University of British Columbia,
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Learning the Models Parameters to learn:
#actions performed by each user – Au
#actions propagated via each edge – Av2u
Mean life time –
P a1 5
Q a1 10
R a1 15
Q a2 12
R a2 14
R a3 6
P a3 14
u Au
P
Q
R
P Q R
P X
Q 0,0 X
R 0,0 X
Input
01
01
01
0,01,5 0,01,10
2
2
0,01,2
3
2
0,01,8
uv,
uv,uv, ,A
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Propagation Models Threshold Models
Linear Threshold Model General Threshold Model
Cascade Models Independent Cascade Model Decreasing Cascade Model
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Properties of Diffusion Models Monotonocity
Submodularity – Law of marginal Gain
Incrementality (Optional) can be updated incrementally
using
TSTpSp uu whenever )()(
TS
TpwTpSpwSp uuuu
whenever
)(}){()(}){(
}){( wSpu uwu pSp , and )(
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Comparison of 4 variants
ROC comparison of 4 variants of Static Models
ROC comparison of 4 variants of Discrete Time (DT) Models
Bernoulli is slightly better than Jaccard Among two Bernoulli variants, Partial Credits (PC) wins by a small margin.
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Discrete Time Models Approximation of
CT Models Incremental, hence
efficient 4-variants
corresponding to 4 Static Models
0uvvt ,vt
0
Influence prob. of v on u
uvvt ,vt
DT Model
0,uvp
0,uvp
Time ->
CT Model
Influence prob. of v on u
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Overview Context and Motivation Background Our Framework Algorithms Experiments Related Work Conclusions
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Continuous Time Models Joint influence probability
Individual probabilities – exponential decay
: maximum influence probability of v on u : the mean life time.
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Algorithms Training – All models simultaneously in no more
than 2 scans of training sub-set (80% of total) of action log table.
Testing – One model requires only one scan of testing sub-set (20% of total) of action log table.
Due to the lack of time, we omit the details of the algorithms.
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