1. Distance-Constraint Reachability Computation in Uncertain Graphs
Ruoming Jin, Lin Liu Kent State University
Bolin Ding UIUC
Haixun Wang
MSRA

2. Uncertainty is ubiquitous!
Why Uncertain Graphs?
Increasing importance of graph/network data
Social Network, Biological Network, Traffic/Transportation Network, Peer-to-Peer Network
Probabilistic perspective gets more and more attention recently.
Uncertainty is ubiquitous!
Protein-Protein Interaction Networks
Social Networks
Probabilistic Trust/Influence Model
False Positive > 45%

3. Uncertain Graph Model Possible worlds (2#Edge) Edge Independence
Existence Probability
Possible worlds (2#Edge)
G1:
G2:
Weight of G2: Pr(G2) =
0.5
0.7
0.2
0.6
(1-0.5) * * * *
(1-0.4)
(1-0.9)
(1-0.1)
(1-0.3) = * * * *

4. Distance-Constraint Reachability (DCR) Problem
Given distance constraint d and two vertices s and t,
Target
Source
What is the probability that s can reach t within distance d?
A generalization of the two-terminal network reliability problem, which has no distance constraint.

5. Important Applications
Peer-to-Peer (P2P) Networks
Communication happens only when node distance is limited.
Social Networks
Trust/Influence can only be propagated only through small number of hops.
Traffic Networks
Travel distance (travel time) query
What is the probability that we can reach the airport within one hour?

6. Example: Exact Computation
d = 2, ?
First Step: Enumerate all possible worlds (29),
Pr(G1)
Pr(G2)
Pr(G3)
Pr(G4)
Second Step: Check for distance-constraint connectivity,
… +
Pr(G1)
* 0 +
Pr(G2)
* 1 +
Pr(G3)
* 0 +
Pr(G4)
* 1
+ … =

7. Approximating Distance-Constraint Reachability Computation
Hardness
Two-terminal network reliability is #P-Complete.
DCR is a generalization.
Our goal is to approximate through Sampling
Unbiased estimator
Minimal variance
Low computational cost

8. Start from the most intuitive estimators, right?

9. Direct Sampling Approach
Sampling Process
Sample n graphs
Sample each graph according to edge probability

10. Direct Sampling Approach (Cont’)
Estimator
Unbiased
Variance
= 1, s reach t within d; = 0, otherwise.
Indicator function

11. Path-Based Approach Generate Path Set
Enumerate all paths from s to t with length ≤ d
Enumeration methods
E.g., DFS

12. Path-Based Approach (Cont’)
Path set
Exactly computed by Inclusion-Exclusion principle
Approximated by Monte-Carlo Algorithm by R. M. Karp and M. G. Luby ( )
Unbiased
Variance

13. Can we do better?

14. Divide-and-Conquer Methodology
Example
+(s,a)
-(s,a)
+(a,t)
-(a,t)
+(s,b)
-(s,b) … … … … … …

15. Divide and Conquer (Cont’)
Summarize:
# of leaf nodes is smaller than 2|E| .
Each possible world exists only in one leaf node.
Reachability is the sum of the weights of blue nodes.
Leaf nodes form a nice sample space.
all possible worlds
Graphs having e1
Graphs not Having e1
s can reach t.
s can not reach t.

16. How do we sample? Unequal probability sampling
Start from here
Pri: Sample Unit Weight; Sum of possible worlds’ probabilities in the node.
qi: sampling probability, determined by properties of coins along the way.
Unequal probability sampling
Hansen-Hurwitz (HH) estimator
Horvitz-Thomson (HT) estimator
Sample Unit

17. Hansen-Hurwitz (HH) Estimator
sample size
= 1, blue node
= 0, red node
Estimator
Unbiased
Variance
Weight
Sampling probability
To minimize the variance above, we have :Pri = qi
Pri = p(e1)*p(e2)*(1-p(e3))*…
Pri: the leaf node weight
qi: the sampling probability
P(e1)
1-P(e1)
p(e1) : 1 – p(e1)
P(e2)
1-P(e2)
1-P(e4)
P(e4)
1-P(e3)
p(e2) : 1 – p(e2)
P(e3)
p(e3) : 1 – p(e3)

18. Horvitz-Thomson (HT) Estimator
# of Unique sample units
Estimator
Unbiased
Variance
To minimize vairance, we find
Pri = qi
Smaller variance than HH estimator

19. Can we further reduce the variance and computational cost?

20. Recursive Estimator Unbiased Variance: n1 + n2 = n
Sample the entire space n times
Sample the sub-space n1 times
Sample the sub-space n2 times
We can not minimize the variance without knowing τ1 and τ2. Then what can we do?

21. Sample Allocation We guess: What if We find: Variance reduced!
n1 = n*p(e)
n2 = n*(1-p(e))?
We find: Variance reduced!
HH Estimator:
HT Estimator:

22. Sample Allocation (Cont’)
Sampling Time Reduced!!
Sample size = n
Directly allocate samples
n1=n*p(e1)
n2=n*(1-p(e1))
n3=n1*p(e2)
n4=n1*(1-p(e2))
Toss coin when sample size is small

23. Experimental Setup Experiment setting Goal: System Specification
Relative Error
Variance
Computational Time
System Specification
2.0GHz Dual Core AMD Opteron CPU
4.0GB RAM
Linux

24. Experimental Results Synthetic datasets Erdös-Rényi random graphs
Vertex#: 5000, edge density: 10, Sample size: 1000
Categorized by extracted-subgraph size (#edge)
For each category, 1000 queries

25. Experimental Results Real datasets
DBLP: 226,000 vertices, 1,400,000 edges
Yeast PPIN: 5499 vertices, edges
Fly PPIN: 7518 vertices, edges
Extracted subgraphs size: 20 ~ 50 edges

26. Conclusions
We first propose a novel s-t distance-constraint reachability problem in uncertain graphs.
One efficient exact computation algorithm is developed based on a divide-and-conquer scheme.
Compared with two classic reachability estimators, two significant unequal probability sampling estimators Hansen-Hurwitz (HH) estimator and Horvitz-Thomson (HT) estimator.
Based on the enumeration tree framework, two recursive estimators Recursive HH, and Recursive HT are constructed to reduce estimation variance and time.
Experiments demonstrate the accuracy and efficiency of our estimators.

27. Thank you !
Questions?