1. Influence Maximization
Lecture 4
Influence Maximization
Ding-Zhu Du
University of Texas at Dallas
First, I want to thank you for you presence. ********In this presentation I will try to introduce The social network which is a theoretical structure to study relationships between individuals, groups, organizations, or even entire societies. 
It is related to a wide range of disciplines. These disciplines include, but are not limited to information science, biology, economics, geography, communication studies, and so on..
The study of social networks begins with the late eighteenth century, two sociologists (Émile [ei'mi:l] Durkheim and Ferdinand ['fɝdənænd] Fer迪南de Tönnies) foreshadowed the idea of social networks in their theories and research of social groups.
Nowadays, we study social networks using network analysis to identify social communities, pick influential person, and design good software.

2. Outline Kate Middleton effect Submodular Function Max
Independent Cascade
Brief overview of social networks
How to build applications on top of the social network
–  Think about a social network being MS Windows, We can build applications on it.

3. Kate Middleton effect “Kate Middleton effect
The trend effect that Kate, Duchess of Cambridge has on others, from cosmetic surgery for brides, to sales of coral-colored jeans.”
“Kate Middleton effect
Duchess of Cambridge, Kate, is a fashion icon who leads in fashion circles.
she seems to become more and more beautiful after she married Prince William. Many articles referred her nose as “adorable,” “perky,” and “feminine.”
Women in the New York and Long Island areas are rushing out to seek a plastic surgeon. There are approximately 100 young women a month so far that are requesting cosmetic surgery on their nose to get an exact duplicate of Kate’s.

4. Hike in Sales of Special Products
According to Newsweek, "The Kate Effect may be worth £1 billion to the UK fashion industry."
Tony DiMasso, L. K. Bennett’s US president, stated in 2012, "...when she does wear something, it always seems to go on a waiting list."
The fashion choices of the Duchess of Cambridge, Kate Middleton, have already brought $1 billion into the British economy.
Reiss, the High Street store that Kate loves, is planning to open 13 outposts in the U.S. within this year.

5. How to Find Kate? Influential persons often have many friends.
Kate is one of the persons that have many friends in this social network.
For more Kates, it’s not as easy as you might think!

6. Influence Maximization
Given a digraph and
k>0,
Find k seeds (Kates)
to maximize the number of influenced persons.
Market salesman

7. Theorem
Proof

8. Outline Kate Middleton effect Submodular Function Max
Independent Cascade
Brief overview of social networks
How to build applications on top of the social network
–  Think about a social network being MS Windows, We can build applications on it.

9. What is a submodular function?
Consider a function f on all subsets of a set E.
f is submodular if

10. Max Coverage
Given a collection C of subsets of a set E, find a subcollection C’ of C, with |C’|<k, to maximize the number of elements covered by C’ .
Influence Maximization is a special case
of Max Coverage.

11. Example of Submodular Function

12. Monotone Increasing
f is monotone increasing if

13. Property 1 2

14. Proof of 1

16. Proof of 2

17. Max Coverage
Given a collection C of subsets of a set E, find a subcollection C’ of C, with |C’|<k, to maximize the number of elements covered by C’ .

18. Submadular Function Max

19. Greedy Algorithm

20. Performance Ratio
Theorem (Nemhauser et al. 1978)
Proof

21. Proof
Monotone increasing
Why?

22. Theorem

23. Outline Kate Middleton effect Submodular Function Max
Independent Cascade
Brief overview of social networks
How to build applications on top of the social network
–  Think about a social network being MS Windows, We can build applications on it.

24. Deterministic Model 6 2 1 5
Two kinds of influence cascades: rumors and protectors.
Each individual has three status: inactive, rumored, protected.
The active individual activates all of its neighbors successfully.
When rumors and protectors influence an individual at the same time, then the individual is protected.
Each individual only has one chance to influence their neighbors.
A node will never change its status if it has been activated. 3 4
both 1 and 6 are source nodes.
Step 1: 1--2,3; 6--2,4. .
4/17/2017 24

25. Example 6 2 1 5 3 4
Step 2: 4--5.
4/17/2017

26. Models of Influence Diffusion
Two basic classes of probabilistic diffusion models:
threshold and cascade
General operational view:
A social network is represented as a directed graph, with each person (customer) as a node.
Nodes start either active or inactive.
An active node may trigger activation of neighboring nodes
Monotonicity assumption: active nodes never deactivate.
we focus on more operational models from mathematical sociology [15, 28] and interacting particle systems
[11, 17] that explicitly represent the step-by-step dynamics of adoption.
Assumption: node can switch to active from inactive, but does not switch in the other direction

27. Independent Cascade (IC) Model
When node v becomes active, it has a single chance of activating each currently inactive neighbor w.
The activation attempt succeeds with probability pvw .
The deterministic model is a special case of IC model. In this case, pvw =1 for all (v,w).
We again start with an initial set of active nodes A0, and the process unfolds in discrete steps according to
the following randomized rule. When node v first becomes active
in step t, it is given a single chance to activate each currently inactive
neighbor w; it succeeds with a probability pv;w —a parameter
of the system — independently of the history thus far. (If w has
multiple newly activated neighbors, their attempts are sequenced in
an arbitrary order.) If v succeeds, then w will become active in step
t+1; but whether or not v succeeds, it cannot make any further attempts
to activate w in subsequent rounds. Again, the process runs
until no more activations are possible.

28. Important understanding

29. Example Y
0.6
Inactive Node
0.2
0.2
0.3
Active Node
Newly active
node X U
0.1
0.4
Successful
attempt
0.5
0.3
0.2
Unsuccessful
attempt
0.5 w v
Stop!

30. Influence Maximization Problem
Influence spread of node set S: σ(S)
expected number of active nodes at the end of diffusion process, if set S is the initial active set.
Problem Definition (by Kempe et al., 2003):
(Influence Maximization). Given a directed and edge-weighted social graph G = (V,E, p) , a diffusion model m, and an integer k ≤ |V |, find a set S ⊆ V , |S| = k, such that the expected influence spread σm(S) is maximum.
the influence of a set of nodes A: the expected number of active nodes at the end of the process.

31. Known Results
Bad news: NP-hard optimization problem for both IC and LT models.
Good news:
σm(S) is monotone and submodular.
We can use Greedy algorithm!
Theorem: The resulting set S activates at least (1-1/e) (>63%) of the number of nodes that any size-k set could activate .

32. Proof of Submodularity

34. Decision Version of InfMax in IC
Is it in NP?
Theorem
Corollary

35. Theorem (Chen et al., 2010)
Proof

37. Disadvantage Lack of efficiency.
Computing σm(S) is # P-hard under both IC and LT models.
Selecting a new vertex u that provides the largest marginal gain σm(S+u) - σm(S), which can only be approximated by Monte-Carlo simulations (10,000 trials).
Assume a weighted social graph as input.
How to learn influence probabilities from history?
( Step 3 of the Greedy algorithm above)

38. Monte-Carlo Method
Buffon's needle

39. Research done by our group in UTD

40. Zaixin Lu, Wei Zhang, Weili Wu, Bin Fu, Ding-Zhu Du: Approximation and Inapproximation for the Influence Maximization Problem in Social Networks under Deterministic Linear Threshold Model. ICDCS Workshops 2011:

41. Zaixin Lu, Lidan Fan, Weili Wu, Bhavani Thuraisingham and Kai Yang, Efficient influence spread estimation for influence maximization under the linear threshold model, Computational Social Networks, 1 (2014)

42. Wen Xu, Zaixin Lu, Weili Wu, Zhiming Chen: A novel approach to online social influence maximization. Social Netw. Analys. Mining 4(1) (2014)

43. Welcome to Submit Papers
A New Springer Journal
Computational Social Networks
Editor-in-Chief:
Ding-Zhu Du
My T. Thai
Social network analysis [SNA] is the mapping and measuring of relationships and flows between people, groups, organizations, computers or other information/knowledge processing entities.
The nodes in the network are the people and groups while the links show relationships or flows between the nodes.
The advantage of social network analysis is that, unlike many other methods, it focuses on interaction (rather than on individual behavior).
Network analysis allows us to examine how the configuration of networks influences how individuals and groups, organizations, or systems function.
Welcome to Submit Papers

44. Yuqing Zhu, Zaixin Lu, Yuanjun Bi, Weili Wu, Yiwei Jiang, Deying Li: Influence and Profit: Two Sides of the Coin. ICDM 2013:

45. Lidan Fan, Zaixin Lu, Weili Wu, Yuanjun Bi, Ailian Wang: A New Model for Product Adoption over Social Networks. COCOON 2013:

46. Songsong Li, Yuqing Zhu, Deying Li, Donghyun Kim, Huan Ma, Hejiao Huang: Influence maximization in social networks with user attitude modification. ICC 2014:

47. THANK YOU!