1. Bharathi-Kempe-Salek Conjecture
Lecture 2-3
Bharathi-Kempe-Salek Conjecture
Ding-Zhu Du
University of Texas at Dallas

2. Bharathi-Kempe-Salek Conjecture

3. Solution Deterministic diffusion model -polynomial-time.
Linear Threshold (LT) – polynomial-time.
Independent Cascade (IC) – NP-hard.

4. Deterministic Diffusion Model
When a node becomes active (infected or protected), it activates all of its currently inactive (not infected and not protected) neighbors.
The activation attempts succeed with a probability 1.

5. 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. .
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6. Example 6 2 1 5 3 4
Step 2: 4--5.
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7. A Property of Optimal Solution

8. Naïve Dynamic Programming

9. Naïve Dynamic Programming

10. Running Time
It is not a polynomial-time!

11. Counting

12. Virtual Nodes Change arborescence to binary arborescence
At most n virtual nodes can be introduced.

13. Weight

14. Naïve Dynamic Programming

15. Linear Threshold (LT) Model
A node v has random threshold ~ U[0,1]
A node v is influenced by each neighbor w according to a weight bw,v such that
A node v becomes active when at least
(weighted) fraction of its neighbors are active
Given a random choice of thresholds, and an initial set of
active nodes A0 (with all other nodes inactive), the diffusion process
unfolds deterministically in discrete steps: in step t, all nodes
that were active in step t-1 remain active, and we activate any
node v for which the total weight of its active neighbors is at least
Theta(v)

16. Example Stop! w v Y Inactive Node 0.6 Active Node 0.2 Threshold 0.2
0.3
Active neighbors X
0.1
0.4 U
0.3
0.5
Stop!
0.2
0.5 w v

17. A property

18. Equivalent Networks

21. At seed v

22. At non-seed v

23. At non-seed v

24. At non-seed v

25. At non-seed v

26. At seed v

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. 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!

29. At non-seed v

30. Another Dynamic Programming

31. Proof of NP-hardness

32. Partition Problem
This is a well-known NP-complete problem!

33. Special Case
This is still an NP-complete problem!

34. Subsum Problem
This is still an NP-complete problem!

36. Key Fact 1

38. <1？

40. h=?

41. Key Fact 2

42. References

43. THANK YOU!