Probabilistic Paths and Centrality in Time
description
Transcript of Probabilistic Paths and Centrality in Time
![Page 1: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/1.jpg)
Probabilistic Paths and Centrality in Time
Joseph J. Pfeiffer, III Jennifer Neville
![Page 2: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/2.jpg)
a
c
b
d
e f
![Page 3: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/3.jpg)
Betweenness Centrality
a
c
b
d
e f
![Page 4: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/4.jpg)
Time Varying Graphs
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Aggregate Time 1 Time 2 Time 3 Time 4 Time 5
=
![Page 5: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/5.jpg)
Time Varying Graphs
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Aggregate Time 1 Time 2 Time 3 Time 4 Time 5
=
Represent Current Graph
![Page 6: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/6.jpg)
Time Varying Graphs
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Aggregate Time 1 Time 2 Time 3 Time 4 Time 5
=
Represent Current GraphBetweenness Centrality
![Page 7: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/7.jpg)
Time Varying Graphs
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Aggregate Time 1 Time 2 Time 3 Time 4 Time 5
=
Represent Current GraphBetweenness Centrality
![Page 8: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/8.jpg)
Time Varying Graphs
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Aggregate Time 1 Time 2 Time 3 Time 4 Time 5
=
Messages are irregular – large changes in metric values between slices
![Page 9: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/9.jpg)
Related Work
• Betweenness centrality through time (Tang et al. SNS ’10)
• Vector clocks for determining edges with minimum time-delays (Kossinets et al. KDD ’08)
• Finding patterns of communication that occur in time intervals (Lahiri & Berger-Wolf, ICDM ’08)
![Page 10: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/10.jpg)
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
![Page 11: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/11.jpg)
Probabilistic Graphs
a
c
b
d
e f
.95
.95
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8
![Page 12: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/12.jpg)
Probabilistic Shortest Paths
a
c
b
d
e f
.95
.95
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8
![Page 13: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/13.jpg)
.95
Probabilistic Shortest Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8a-c-b: .95*.65 = 0.61
![Page 14: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/14.jpg)
.95
Probabilistic Shortest Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8a-c-b: .95*.65 = 0.61a-d-b: .80*.80 = 0.64
![Page 15: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/15.jpg)
.95
Probabilistic Shortest Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8a-c-b: .95*.65 = 0.61a-d-b: .80*.80 = 0.64a-c-d-b: .95*.95*.80 = 0.72
![Page 16: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/16.jpg)
.95
Probabilistic Shortest Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8a-c-b: .95*.65 = 0.61a-d-b: .80*.80 = 0.64a-c-d-b: .95*.95*.80 = 0.72(1-0.61)*(1-.64)*0.722
![Page 17: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/17.jpg)
.95
Probabilistic Shortest Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8a-c-b: .95*.65 = 0.61a-d-b: .80*.80 = 0.64a-c-d-b: .95*.95*.80 = 0.72(1-0.61)*(1-.64)*0.722
![Page 18: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/18.jpg)
.95
Probabilistic Shortest Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8a-c-b: .95*.65 = 0.61a-d-b: .80*.80 = 0.64a-c-d-b: .95*.95*.80 = 0.72(1-0.61)*(1-.64)*0.722Shared Edges
![Page 19: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/19.jpg)
.95
.95
Probabilistic Shortest Paths
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8a-c-b: .95*.65 = 0.61a-d-b: .80*.80 = 0.64a-c-d-b: .95*.95*.80 = 0.72(1-0.61)*(1-.64)*0.722Shared Edges
![Page 20: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/20.jpg)
.95
Probabilistic Shortest Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8Intractable to Compute Exactly
![Page 21: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/21.jpg)
.95
Probabilistic Shortest Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8Intractable to Compute Exactly
Approximate with Sampling
![Page 22: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/22.jpg)
.95
Probabilistic Shortest Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8Sample each edge independently
![Page 23: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/23.jpg)
.95
Probabilistic Shortest Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8Sample each edge independentlyDistribution of graphs
![Page 24: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/24.jpg)
.95
Probabilistic Shortest Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8Sample each edge independentlyDistribution of graphsExpected Betweenness Centrality
![Page 25: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/25.jpg)
.95
Most Likely Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8Most Likely Path
![Page 26: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/26.jpg)
.95
Most Likely Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8a-c-b: .95*.65 = 0.61a-d-b: .80*.80 = 0.64a-c-d-b: .95*.95*.80 = 0.72
![Page 27: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/27.jpg)
.95
Most Likely Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8a-c-b: .95*.65 = 0.61a-d-b: .80*.80 = 0.64a-c-d-b: .95*.95*.80 = 0.72
People with strong relationships are still unlikely to pass on all information…
![Page 28: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/28.jpg)
.95
Most Likely Handicapped (MLH) Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8a-c-b: 0.61*β2
a-d-b: 0.64*β2
a-c-d-b: 0.72*β3
Transmission Probability
![Page 29: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/29.jpg)
.95
MLH Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8a-c-b: 0.61*.52 = 0.15a-d-b: 0.64*.52 = 0.16a-c-d-b: 0.72*.53 = 0.09
Transmission Probability
![Page 30: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/30.jpg)
.95
MLH Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8a-c-b: 0.61*.52 = 0.15a-d-b: 0.64*.52 = 0.16a-c-d-b: 0.72*.53 = 0.09
Transmission Probability
![Page 31: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/31.jpg)
.95
MLH Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8a-c-b: 0.61*.52 = 0.15a-d-b: 0.64*.52 = 0.16a-c-d-b: 0.72*.53 = 0.09
Transmission ProbabilityEasy to Compute
![Page 32: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/32.jpg)
.95
MLH Paths
a
c
b
d
e f
.95a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
a
c
b
d
e f
Time 1 Time 2 Time 3 Time 4 Time 5
.8.8
.65
.65 .8.8
.8a-c-b: 0.61*.52 = 0.15a-d-b: 0.64*.52 = 0.16a-c-d-b: 0.72*.53 = 0.09
Transmission ProbabilityEasy to Compute
Use MLH Paths for Betweenness Centrality
![Page 33: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/33.jpg)
Link Probabilities: Relationship Strength
Time0
1
P(e)
![Page 34: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/34.jpg)
Link Probabilities: Relationship Strength
Probability of no message contributing to relationship
Time0
1
P(e)
![Page 35: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/35.jpg)
Link Probabilities: Relationship Strength
Probability of no message contributing to relationship
* =
0
1
P(e)
Time
![Page 36: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/36.jpg)
Link Probabilities: Relationship Strength
Probability of no message contributing to relationship
* = - =
Time0
1
P(e)
![Page 37: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/37.jpg)
0
1
P(e)
Link Probabilities: Relationship Strength
Probability of no message contributing to relationship
* = - =
Any Relationship Strength
Time
![Page 38: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/38.jpg)
Evaluation
• Enron Emails• 151 Employees – 50,572 messages over 3 years• Known dates in time
• 10,000x for Sampling Method• Time slice length was 2 weeks
• Evaluated all metrics at end of every two weeks• Aggregate, Slice, Sampling, MLH
![Page 39: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/39.jpg)
Method Correlations and Sample Size
Aggregate/Sampling
Slice/Sampling
Aggregate/Slice
SamplingAggregate
Slice
![Page 40: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/40.jpg)
Correlations – August 24th, 2001
![Page 41: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/41.jpg)
Lay and Skilling
Sampling MLH
Slice Aggregate
Lay
Lay
Lay
Lay
SkillingSkilling
SkillingSkilling
![Page 42: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/42.jpg)
Lavorato and Kitchen
Sampling MLH
Slice Aggregate
Lavorato
Lavorato
Lavorato
Lavorato
Kitchen
Kitchen
Kitchen
Kitchen
![Page 43: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/43.jpg)
Shortest Paths on Unweighted Discrete Graphs are a special case of Most Likely Handicapped Paths
![Page 44: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/44.jpg)
Shortest Paths and Most Probable Handicapped Paths
Discrete Probabilistic
1
![Page 45: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/45.jpg)
Shortest Paths and Most Probable Handicapped Paths
Discrete Probabilistic
Length: 1 Probability: β
1
![Page 46: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/46.jpg)
Shortest Paths and Most Probable Handicapped Paths
Discrete Probabilistic
Length: n Probability: βn
… …1
![Page 47: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/47.jpg)
Shortest Paths and Most Probable Handicapped Paths
Discrete Probabilistic
Length: nn < n+1
Probability: βn
βn > βn+1
… …1
![Page 48: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/48.jpg)
Shortest Paths and Most Probable Handicapped Paths
Discrete Probabilistic
Length: nn < n+1
Probability: βn
βn > βn+1
… …1Shortest Paths can be formulated as
Most Probable Handicapped Paths
![Page 49: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/49.jpg)
Computation
MLH Paths: Modify Dijkstra’s. Rather than shortest path for expansion, choose
most probable path.
MLH Betweenness Centrality: Modify Brandes’. Rather than longest path for backtracking, choose
least probable path.
![Page 50: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/50.jpg)
Conclusions
• Developed sampling approach• Developed most probable paths formulation
• Incorporated inherent transmission uncertainty
• Evaluated on Enron email dataset• Aggregate representations of time evolving graphs are unable to
detect changes with the graph• Slice samples of the graph have large variation from one slice to
the next
• Future Work: Additional metrics, such as probabilistic clustering coefficient
![Page 51: Probabilistic Paths and Centrality in Time](https://reader035.fdocuments.in/reader035/viewer/2022062323/56816049550346895dcf7044/html5/thumbnails/51.jpg)
[email protected]@cs.purdue.edu