Distance-Constraint Reachability
Computation in Uncertain Graphs
Ruoming Jin, Lin Liu
Kent State University
Bolin Ding
UIUC
Haixun Wang
MSRA
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
False Positive > 45%
Social Networks
Probabilistic Trust/Influence
Model
Uncertain Graph Model
a
0.5
0.3
0.2
s
Edge
Independence
0.5
0.6
b
0.1
0.7
0.4
c
t
0.9
Existence Probability
• Possible worlds (2#Edge)
a
G1:
s
a
b
c
t
G2:
s
b
t
c
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) = 0.0007938
Distance-Constraint Reachability (DCR)
Problem
Given distance constraint d and two vertices s and t,
Source
a
0.5
Target
0.3
0.2
s
0.5
0.1
0.7
c
b
0.4
0.6
t
0.9
• 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.
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?
Example: Exact Computation
a
0.5
• d = 2,
0.5
0.2
s
?
0.3
0.1
0.7
c
b
0.4
0.6
t
0.9
First Step: Enumerate all possible worlds (29),
a
s
b
c
t
s
b
c
a
a
a
t
s
b
c
t
s
b
t
c
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 + …
Approximating Distance-Constraint
Reachability Computation
• Hardness
– Two-terminal network reliability is #PComplete.
– DCR is a generalization.
• Our goal is to approximate through
Sampling
– Unbiased estimator
– Minimal variance
– Low computational cost
Start from the most intuitive
estimators, right?
Direct Sampling Approach
• Sampling Process
– Sample n graphs
– Sample each graph according to edge
probability
a
0.5
0.3
0.2
s
0.5
0.1
0.7
c
b
0.4
0.6
t
0.9
a
s
b
c
t
Direct Sampling Approach (Cont’)
• Estimator
• Unbiased
• Variance
= 1, s reach t within d;
= 0, otherwise.
Indicator function
E( Rˆ B ) 
Path-Based Approach
• Generate Path Set
– Enumerate all paths from s to t with length ≤
d
a
0.3
0.2
s
0.1
0.7
c
b
0.4
– Enumeration methods
• E.g., DFS
0.6
0.9
t
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
Can we do better?
Divide-and-Conquer Methodology
• Example
a
s
b
t
+(s,a)
c
-(s,a)
a
s
b
a
t
s
c
b
a
s
+(a,t)
…
t
-(a,t) +(s,b)
-(s,b)
c
b
t
c
a
s
a
s
b
b
…
t
c
t
c
…
a
s
b
…
t
c
…
a
s
b
c
t
…
Divide and Conquer (Cont’)
Summarize:
1. # of
leaf
all possible
worlds
nodes is
smaller than 2|E| .
Graphs having e1
Graphs not Having e1
2. Each possible world
exists only in one leaf
node.
…
… ...
s can reach t.
3. Reachability is the sum
of the weights of blue
nodes.
… ...
4. Leaf nodes form a nice
sample space.
s can not reach t.
How do we sample?
Start from here
Pri: Sample Unit
Weight; Sum of
possible worlds’
probabilities in
the node.
…
… ...
Sample Unit
• Unequal probability sampling
qi: sampling
probability,
determined
by
… ...
properties of
coins along the
way.
– Hansen-Hurwitz (HH) estimator
– Horvitz-Thomson (HT) estimator
Hansen-Hurwitz (HH) Estimator
sample size
• Estimator
• Unbiased
• Variance
= 1, blue node
Weight
= 0, red node
Sampling probability
To minimize the variance above, we have :Pri = qi
Pri = p(e1)*p(e2)*(1-p(e
Pri: the
leaf node weight
3))*…
P(e1)
1-P(e1)
p(e1) : 1 – p(e
qi: the
sampling
1)
P(e2) probability
1-P(e2)
1-P(e4)
P(e4)
1-P(e
)
3
p(e ) : 1 – p(e ) P(e )
2
2
p(e3) : 1 – p(e3)
3
…
… ...
… ...
Horvitz-Thomson (HT) Estimator
• Estimator
# of Unique sample units
• Unbiased
• Variance
– To minimize vairance, we find
Pri = qi
– Smaller variance than HH estimator
Can we further reduce the variance
and computational cost?
Recursive Estimator
1. Unbiased
2. Variance:
Sample the subspace n1 times
n1 +Sample
n2 = n the entire space n times
Sample the subspace n2 times
…
… ...
We can not
minimize the
variance without
knowing τ1 and τ2.
Then what can we
… do?
...
Sample Allocation
• We guess: What if
– n1 = n*p(e)
– n2 = n*(1-p(e))?
• We find: Variance reduced!
– HH Estimator:
– HT Estimator:
Sample Allocation (Cont’)
• Sampling Time Reduced!!
Sample size = n
Directly
allocate samples
n1=n*p(e1)
n3=n1*p(e2)
Toss coin
when sample
size is small
n4=n1*(1-p(e2))
n2=n*(1-p(e1))
Experimental Setup
• Experiment setting
– Goal:
• Relative Error
• Variance
• Computational Time
– System Specification
• 2.0GHz Dual Core AMD Opteron CPU
• 4.0GB RAM
• Linux
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
Experimental Results
• Real datasets
–
–
–
–
DBLP: 226,000 vertices, 1,400,000 edges
Yeast PPIN: 5499 vertices, 63796 edges
Fly PPIN: 7518 vertices, 51660 edges
Extracted subgraphs size: 20 ~ 50 edges
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.
Thank you !
Questions?