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

Bharathi-Kempe-Salek Conjecture

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

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.

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. . 4/4/2019 5

Example 6 2 1 5 3 4 Step 2: 4--5. 4/4/2019

A Property of Optimal Solution

Naïve Dynamic Programming

Naïve Dynamic Programming

Running Time It is not a polynomial-time!

Counting

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

Weight

Naïve Dynamic Programming

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)

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

A property

Equivalent Networks

At seed v

At non-seed v

At non-seed v

At non-seed v

At non-seed v

At seed v

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.

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!

At non-seed v

Another Dynamic Programming

Proof of NP-hardness

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

Special Case This is still an NP-complete problem!

Subsum Problem This is still an NP-complete problem!

Key Fact 1

<1？

h=?

Key Fact 2

References

THANK YOU!