1. Discovering Influential Nodes From Social Trust Network
Thesis Proposal by
Sabbir Ahmed
Thesis Committee:
Chair- Dr. R. Frost [Computer Science]
Advisor – Dr. C. I. Ezeife [Computer Science]
Internal Reader – Dr. A. Ngom [Computer Science]
External Reader – Dr. E. H. Kim [Department of Physics]
University of Windsor, School of Computer Science

2. Outline 1. Introduction 2. Related Work 3. Proposed solution framework
Background
(SNA, Data Mining )
Viral Marketing
Influence Maximization
Challenges and Thesis Problem
2. Related Work
3. Proposed solution framework
Thesis Contribution
4. Thesis Plan
5. References
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3. 1.1 Data Mining
Data Mining is way of efficiently discovering interesting rules from large databases.
Increasing revenue
Cutting time and costs
Decision making
Medical Diagnosis
Techniques include:
Classification (e.g. classify ‘Cancer’ or ‘Not Cancer’)
Association Rule (e.g. discover rule - Bread => Milk)
Clustering (e.g. Cluster - ‘Risky’, ‘Not Risky’)
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4. 1.1 What is “Social Network”
A social network is a social structure
Made up of individuals / organizations / entities called "nodes“
Which are tied (connected) by one or more specific types of interdependency.
Such as friendship, kinship, common interest, financial exchange, dislike etc.
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5. 1.1 Modeling Social Network
The structure of social networks, most commonly, are modeled as directed or undirected graph G=(V,E).
Where V is the set of all nodes in the network and E is the set of directed or undirected edges between nodes.
E.g.
V = {1,2,3,4,5,6,7,8,9,10,11}
E = {(1,5), (1,3), (1,3),(2,3),(2,4),
(4,5), (5,7) (10,11)} 1 5 8 3 9 7 2 11 10 6 4
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6. 1.1 Types of Social Network Graph
Directed Graph
Undirected Graph
Signed graph
Weighted graph + - –
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7. 1.1 Social Network Mining
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8. 1.2 Viral Marketing Also known as Target Advertising
Initiate chain reaction by Word of mouth effect.
Also known as diffusion process.
Low investments, maximum gain
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9. 1.2 Social Influence
Social influence - a force that person A (i.e., the influencer) exerts on person B to introduce a change of the behavior and/or opinion of B
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10. 1.2 Problem Setting Given Goal Question
A limited budget B for initial advertising (e.g. give away free samples of product)
Estimates for influence between individuals
Goal
Trigger a large cascade of influence
e.g. further adoptions of a product
Question
Which set of individuals should B target at?
if we can try to convince a subset of individuals to adopt a new
product or innovation
But how should we choose the few key individuals to use for seeding this process?
Which blogs should one read to be most up to date?
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11. 1.2 Terms Used Active Diffusion Model [M] Influence Probability [pu,v]
We say a node (user) is active if he/she adopts a product or performs an action.
Diffusion Model [M]
model that describes the diffusion process.
Influence Probability [pu,v]
Probability of user v getting activated given u activates.
Seed Set
Initially activated set
Influence Spread [σM(A)]
# of users getting activated by seed set A with diffusion model M.
Marginal Gain
Additional # of users getting activated as a result of adding any node v with respect to A. i.e. σM(A+v) - σM(A)
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12. 1.2 Diffusion Models
A diffusion model attempts to describes the entire diffusion process
And determines which nodes will be activated due the influence spread through the social network.
Example - Linear threshold (LT) model and Independent Cascade (IC) model
Mathematical sociology.
General Threshold Model – An extension of IC and LT Model.
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13. 1.2 GT Model At any time any node u is either active or inactive.
Diffusion process starts with initial set of active nodes.
Let us consider an inactive user u and set of its active neighbors S.
According to GT model probability of user u activating given active set S is:
Probability increases as more of u’s neighbors also gets activated.
If pu(S) ≥ θu , we can conclude that u activates.
Where θu is the activation threshold of user u
θu is chosen randomly and uniformly from interval [0,1]
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14. 1.2 General Threshold Model - Example
pu(S) ≥
1.2 General Threshold Model - Example
Source: David Kempe’s slides
Inactive Node
0.6
Active Node
0.2
0.2
0.3
Threshold x
0.1
Joint Influence Probability
0.4 U
Probably skip this slide
0.3
0.5
Stop!
0.2
0.5 w v
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15. 1.2 Influence Spread function
Given a social network graph G=(V,E)
A diffusion model M
An initial set of active vertices A⊆ V,
The influence spread of set A, denoted σM(A), is the expected number of vertices to become active, under the influence of vertices in set A, once the diffusion process is over.
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16. 1.2 Influence Maximization
Problem: Given social network G with influence probabilities, budget k, find k-node set S that maximizes σM(A)
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17. 1.2 Other application area
Spread of virus in computer network.
Contamination prevention in water distribution network.
Social media recommendation
Expert finding
Etc.
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18. Adding S’ helps very little
1.2 Solving the problem
Kempe et al. showed that solving the problem exactly is NP-hard
Also proved that:
Influence Spread function is submodular and monotone
New node: S1 S1 S’ S’
Adding S’ helps very little
Adding S’ helps a lot S3 S4 S2 S2
Seed Set A={S1, S2}
Seed Set A={S1, S2, S3, S4}
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19. 1.2 Monotone and Submodularity
For all seed set it hol
Monotone:
A submodular function f is said to be monotone if
Benefit of adding a node to a small seed set
Benefit of adding a node to a large seed set
adding an element to a set cannot cause f to decrease
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20. 1.2 Greedy Algorithm
A submodular monotone function can be maximized with (1-1/e) or 63% approximation guarantee using Greedy algorithm. (Nemhauser et al. 1978)
That is if A* is an optimal solution.
The greedy algorithm guarantee the following:
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21. 1.2 Greedy Algorithm
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22. 1.2 Issues and Challenges Scalability Dynamic Social Graph Privacy
Considering Negative Influence in IM
Requires new Diffusion model
Learning Influence Probability
+ve and –ve Influence Probability
Thesis focus
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23. 1.2 Motivation
All previous works consider only positive influence among users.
In real life a user can also have some degree of negative influence on another node.
If I do not trust someone I probably will not be influenced by him.
In fact I may get negatively influenced.
Viral marketing differentiates itself because it is based on trust among individuals’ close social circle of families, friends, and coworkers.
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24. 1.2 Thesis Problem
Given a trust network graph, G(V,E) with every edge (u,v) of E is directed and labelled either positive (trust) or negative (distrust).
We need a diffusion model which consider both +ve and –ve influence.
How we can learn both positive and negative influence from patterns of actions?
How to extract influential nodes under this new model? + - –
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25. 2 Related Work Domingos & Richardson 2001 – Kempe et al. 2003 –
Finding influential users to maximize expected lift in profit.
No concrete problem formulation.
Too many assumptions.
Kempe et al –
Defined IM as discrete optimization problem.
Proved this to NP-Hard.
Influence spread is monotone and submodular.
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26. 2 Related Work Leskovec et al. 2007 Chen et al. 2009, 2010
Lazy forward optimization.
700 times faster than Greedy.
Chen et al. 2009, 2010
Scalability of greedy algorithm.
Proposed heuristic algorithms.
Goyal et al. 2010
Learn Influence probabilities.
Goyal et al. 2011
Credit Distribution Model.
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27. 3. Proposed Solution Recall General Threshold Model
According to GT model probability of user u activating given S is:
If pu(S) ≥ θu , we can conclude that u activates
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28. 3. Trust-General Threshold Model
S+ is all the trusted active user of node u
S- is the active distrusted user of node u
In T-GT model we say:
If pu(S+) > pu(S-) then u activates.
Note – No need of threshold θu
+ve joint influence probability
-ve joint influence probability
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29. 3. Example
Since pu(S+) < pu(S-) according to T-GT model node C wont activate.
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30. 3. Trust Matrix A matrix denoted as - TM(u,v)
TM(u,v) is ‘+ve’ if u trusts v, ‘-ve’ otherwise.
Users in trust network often tend not to declare distrust
Need a way to estimate the unknown ‘?’
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31. 3. Predict edge sign: A ML formulation
Machine Learning formulation by Leskovec et al.:
Predict sign of edge (u,v)
Class label:
+1: positive edge
-1: negative edge
Learning method:
Logistic regression u v + ? –
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32. 3. Frequent Action Pattern
Positive Frequent Action Pattern
Av.u - The number of actions performed by any user u after the same action were performed by a trusted user v.
Negative Frequent Action Pattern
A’v.u - The number of actions not performed by any user u after the same actions were performed by a distrusted user of u.
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33. 3. Computing Influence Probabilities
Positive influence probability:
Negative influence probability:
Where Av is the # of action performed by v.
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34. 3. Action Sequence Table
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35. 3. Action Sequence
Action sequence of an action a, Seq (a), is a sequence of users performing the action a with respect to time.
For example – Seq(a) = {u1, u2, u3, u4}
Sub sequence of any user u for action a, denoted as Seq(a,u), is the sequence of users performing action a before user u.
For example – Seq(a,u3) = {u1, u2}
Also Seq(a)-Seq(a,u3) = {u4}
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36. 3. Action Pattern Generator
Complexity of APG algorithm is O(A*2 |V|2 ) in worst case.
Where A is number of actions and |V| is number of total nodes.
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37. Running Example
Action Sequence Table
Trust Matrix
The algorithm will start with action a from the action sequence table whose sequence is {u1, u2, u3, u4}.
According to TM, u5 (did not perform a) do not trust u1,u3 and u4. So the algorithm will add 1 to A’u1.u5, A’u3.u5 and A’u4.u5
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38. Running Example (Cont.)
Action Sequence Table
Trust Matrix
According to TM, u3 (did perform a) do trusts u1 and u2. So the algorithm will add 1 to Au1.u3 and Au2.u3
APG algorithm will continue the same process for all action and return Au.v , A’u.v and Av for all u and v.
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39. Running Example (Cont.)
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40. Running Example (Cont.)
Positive influence probability of user u2 on user u5 that is:
Similarly the negative influence probability of user u2 on user u4 that is:
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41. Influence Maximization under T-GT Model
Unlike previous works influence spread is not monotone.
That is adding a node (or user) may result in influence spread to decrease.
Let S is the initially activated seed set
And ∂(S) are the nodes that were successfully activated by the seed set S.
i.e. the influence spread, σ(S), is actually the number of nodes in ∂(S) or |∂(S)|
Let a node w have a negative influence (due to distrust) on another node u which is in ∂(S).
According to T-GT Model adding w to S will cause the probability of u getting activated to decrease.
This will cause the influence spread of S+w, i.e. σ(S+w), to decrease as |∂(S)-1| < |∂(S)|.
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42. Influence Maximization under T-GT Model
Greedy or Lazy Forward Optimization is not applicable for IM under T-GT model.
However we claim that though influence spread function in T-GT model is not monotone but still sub modular. [Need to provide formal proof!!]
So IM under T-GT is a optimization of non-monotone submodular function.
Based on this we can use deterministic local search algorithm of Feige et al. [2007].
Provides 1/3 approximation guarantee.
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43. Solution Framework
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44. Thesis Contribution
First to consider negative influence and trust network for IM.
Proposed a new diffusion model, T-GT, model to incorporate –ve influence.
Proposed a pattern based approach to compute +ve and –ve probabilities.
Show that standard IM algorithms are not applicable to T-GT model.
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45. Future Work Considering dynamic network.
Needs to be scalable to very large social network.
Need more sophisticated way to assign threshold θ.
Action based influence.
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46. 4. Proposed Timeline
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47. 5. References
Chen, W., Wang, C., and Wang, Y Scalable influence maximization for prevalent viral marketing in large-scale social networks. In Proceedings of the 16th ACM SIGKDD international conference on Knowledge discovery and data mining. KDD ’10. ACM, New York, NY, USA, 1029–1038.
Chen, W., Wang, Y., and Yang, S Efficient influence maximization in social networks. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining. KDD ’09. ACM, New York, NY, USA, 199–208.
Domingos, P. and Richardson, M Mining the network value of customers. In Proceedings of the seventh ACM SIGKDD international conference on Knowledge discovery and data mining. KDD ’01. ACM, New York, NY, USA, 57–66.
Feige, U.; Mirrokni, V.S.; Vondrak, J.; , "Maximizing Non-Monotone Submodular Functions," Foundations of Computer Science, FOCS '07. 48th Annual IEEE Symposium on , vol., no., pp , Oct. 2007
Goyal, A., Bonchi, F., and Lakshmanan, L. V Learning influence probabilities in social networks. In Proceedings of the third ACM international conference on Web search and data mining. WSDM ’10. ACM, New York, NY, USA, 241–250.
Goyal, A., Bonchi, F., and Lakshmanan, L. V. S A data-based approach to social influence maximization. Proc. VLDB Endow. 5, 73–84.
Kempe, D., Kleinberg, J., and Tardos, E Maximizing the spread of influence through a social network. In Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining. KDD ’03. ACM, New York, NY, USA, 137–146.
Leskovec, J., Huttenlocher, D., and Kleinberg, J. 2010a. Predicting positive and negative links in online social networks. In Proceedings of the 19th international conference on World wide web. WWW ’10. ACM, New York, NY, USA, 641–650.
Leskovec, J., Krause, A., Guestrin, C., Faloutsos, C., VanBriesen, J., and Glance, N Cost-effective outbreak detection in networks. In Proceedings of the 13th ACM SIGKDD international conference on Knowledge discovery and data mining. KDD ’07. ACM, New York, NY, USA, 420–429.
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48. Acknowledgment
Some material of the presentation is taken from slides of:
Dr. David Kempe [Yahoo Research]
Amit Goyal [UBC]
Dr. Jure Leskovec [Stanford University]
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49. QUESTION?
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