Distance-Constraint Reachability Computation in Uncertain Graphs

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Distance-Constraint Reachability Computation in
Uncertain Graphs

• Ruoming Jin, Lin Liu Kent
  State University

Bolin Ding UIUC
Haixun Wang MSRA
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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 gt 45
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Uncertain Graph Model
Edge Independence
Existence Probability

• Possible worlds (2Edge)

G2
G1
Weight of G2 Pr(G2)
0.5
(1-0.5)
0.2
0.6
0.7




(1-0.3)
(1-0.1)
(1-0.4)
(1-0.9)
0.0007938




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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.

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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?

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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)
Pr(G3)
Pr(G4)
1
0
1




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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

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• Start from the most intuitive estimators, right?

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Direct Sampling Approach

• Sampling Process
• Sample n graphs
• Sample each graph according to edge probability

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Direct Sampling Approach (Cont)

• Estimator
• Unbiased
• Variance

1, s reach t within d 0, otherwise.
Indicator function
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Path-Based Approach

• Generate Path Set
• Enumerate all paths from s to t with length d
• Enumeration methods
• E.g., DFS

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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

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• Can we do better?

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Divide-and-Conquer Methodology

• Example

(s,a)
-(s,a)
-(s,b)
(s,b)
-(a,t)
(a,t)






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Divide and Conquer (Cont)
Summarize

1. of leaf nodes is smaller than 2E .
2. Each possible world exists only in one leaf
   node.
3. Reachability is the sum of the weights of blue
   nodes.
4. 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.
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How do we sample?
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
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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(e3)
p(e2) 1 p(e2)
p(e3) 1 p(e3)
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Horvitz-Thomson (HT) Estimator
of Unique sample units

• Estimator
• Unbiased
• Variance
• To minimize vairance, we find
• Pri qi
• Smaller variance than HH estimator

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• Can we further reduce the variance and
  computational cost?

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Recursive Estimator

1. Unbiased
2. 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
t1 and t2. Then what can we do?
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Sample Allocation

• We guess What if
• n1 np(e)
• n2 n(1-p(e))?
• We find Variance reduced!
• HH Estimator
• HT Estimator

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Sample Allocation (Cont)

• Sampling Time Reduced!!

Sample size n
Directly allocate samples
n1np(e1)
n2n(1-p(e1))
n3n1p(e2)
n4n1(1-p(e2))
Toss coin when sample size is small
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Experimental Setup

• Experiment setting
• Goal
• Relative Error
• Variance
• Computational Time
• System Specification
• 2.0GHz Dual Core AMD Opteron CPU
• 4.0GB RAM
• Linux

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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

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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

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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.

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• Thank you !
• Questions?