rtm laws and a recursive generator for weighted time evolving graphs n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
RTM: Laws and a Recursive Generator for Weighted Time-Evolving Graphs PowerPoint Presentation
Download Presentation
RTM: Laws and a Recursive Generator for Weighted Time-Evolving Graphs

Loading in 2 Seconds...

play fullscreen
1 / 47

RTM: Laws and a Recursive Generator for Weighted Time-Evolving Graphs - PowerPoint PPT Presentation


  • 110 Views
  • Uploaded on

RTM: Laws and a Recursive Generator for Weighted Time-Evolving Graphs. Leman Akoglu , Mary McGlohon, Christos Faloutsos Carnegie Mellon University School of Computer Science. Motivation. Graphs are popular! Social, communication, network traffic, call graphs…. …and interesting

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'RTM: Laws and a Recursive Generator for Weighted Time-Evolving Graphs' - tia


Download Now An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
rtm laws and a recursive generator for weighted time evolving graphs

RTM: Laws and a Recursive Generator for Weighted Time-Evolving Graphs

Leman Akoglu,Mary McGlohon, Christos Faloutsos

Carnegie Mellon University

School of Computer Science

motivation
Motivation
  • Graphs are popular!
    • Social, communication,

network traffic, call graphs…

  • …and interesting
    • surprising common

properties for static

and un-weighted graphs

  • How about weighted graphs?
  • …and their dynamic properties?
  • How can we model such graphs?
    • for simulation studies, what-if scenarios, future prediction, sampling
outline
Outline
  • Motivation
  • Related Work

- Patterns

- Generators

- Burstiness

  • Datasets
  • Laws and Observations
  • Proposed graph generator: RTM
  • (Sketch of proofs)
  • Experiments
  • Conclusion
graph patterns i
Graph Patterns (I)
  • Small diameter
  • 19 for the web [Albert and Barabási, 1999]
  • 5-6 for the Internet AS topology graph [Faloutsos, Faloutsos, Faloutsos, 1999]
  • Shrinking diameter

[Leskovec et al.‘05]

  • Power Laws

Blog Network

diameter

y(x) = Ax−γ, A>0, γ>0

time

graph patterns ii
Graph Patterns (II)
  • Densification [Leskovec et al.‘05]

and Weight [McGlohon

et al.‘08] Power-laws

  • Eigenvalues Power Law [Faloutsos et al.‘99]
  • Degree Power Law [Richardson and Domingos, ‘01]

|W|

|srcN|

Eigenvalue

Count

|dstN|

In-degree

Rank

|E|

Epinions who-trusts-whom graph

Inter-domain Internet graph

DBLP Keyword-to-Conference Network

graph generators
Graph Generators
  • Erdős-Rényi (ER)model [Erdős, Rényi ‘60]
  • Small-world model [Watts, Strogatz ‘98]
  • Preferential Attachment [Barabási, Albert ‘99]
  • Edge Copying models [Kumar et al.’99], [Kleinberg et al.’99],
  • Forest Fire model [Leskovec, Faloutsos ‘05]
  • Kronecker graphs [Leskovec, Chakrabarti, Kleinberg, Faloutsos ‘07]
  • Optimization-based models [Carlson,Doyle,’00] [Fabrikant et al. ’02]
burstiness

Resolution

Entropy

Burstiness
  • Edge and weight additions are bursty, and self-similar.
  • Entropy plots [Wang+’02] is a measure of burstiness.

Bursty:

0.2 < slope < 0.9

Weights

Entropy

slope = 5.9

Time

Resolution

outline1
Outline
  • Motivation
  • Related Work

- Patterns

- Generators

  • Datasets
  • Laws and Observations
  • Proposed graph generator: RTM
  • Sketch of proofs
  • Experiments
  • Conclusion
datasets
Datasets

1

Bipartite networks: |N| |E| time

1. AuthorConference 17K, 22K, 25 yr.

2. KeywordConference 10K, 23K, 25 yr.

3. AuthorKeyword 27K, 189K, 25 yr.

4. CampaignOrg 23K, 877K, 28 yr.

9

slide10

Datasets

3

Bipartite networks: |N| |E| time

1. AuthorConference 17K, 22K, 25 yr.

2. KeywordConference 10K, 23K, 25 yr.

3. AuthorKeyword 27K, 189K, 25 yr.

4. CampaignOrg 23K, 877K, 28 yr.

slide11

Datasets

3

Bipartite networks: |N| |E| time

1. AuthorConference 17K, 22K, 25 yr.

2. KeywordConference 10K, 23K, 25 yr.

3. AuthorKeyword 27K, 189K, 25 yr.

4. CampaignOrg 23K, 877K, 28 yr.

Unipartite networks: |N| |E| time

5. BlogNet 60K, 125K, 80 days

6. NetworkTraffic 21K, 2M, 52 months

20MB

11

slide12

Datasets

3

Bipartite networks: |N| |E| time

1. AuthorConference 17K, 22K, 25 yr.

2. KeywordConference 10K, 23K, 25 yr.

3. AuthorKeyword 27K, 189K, 25 yr.

4. CampaignOrg 23K, 877K, 28 yr.

Unipartite networks: |N| |E| time

5. BlogNet 60K, 125K, 80 days

6. NetworkTraffic 21K, 2M, 52 months

20MB

25MB

5MB

12

outline2
Outline
  • Motivation
  • Related Work

- Patterns

- Generators

  • Datasets
  • Laws and Observations
  • Proposed graph generator: RTM
  • Sketch of proofs
  • Experiments
  • Conclusion
observation 1 1 power law lpl
Observation 1: λ1Power Law(LPL)

Q1: How does the principal eigenvalue λ1 of the adjacency matrixchange over time?

Q2: Why should we care?

14

observation 1 1 power law lpl1
Observation 1: λ1Power Law(LPL)

Q1: How does the principal eigenvalue λ1 of the adjacency matrixchange over time?

Q2: Why should we care?

A2: λ1 is closely linked to density and maximumdegree, also relates to epidemic threshold.

A1:

λ1(t) ∝ E(t) α,

α ≤ 0.5

1 power law lpl cont
λ1Power Law (LPL) cont.

Theorem:

For a connected, undirected graph G with N nodes and E edges, without self-loops and multiple edges;

λ1(G) ≤ {2 (1 – 1/N) E}1/2

For large N,

1/N  0 and

λ1(G) ≤ cE1/2

DBLP Author-Conference network

observation 2 1 w power law lwpl
Observation 2:λ1,wPower Law (LWPL)

Q: How does the weighted principal eigenvalue λ1,wchange over time?

A:

λ1,w(t) ∝ E(t) β

DBLP Author-Conference network

Network Traffic

observation 3 edge weights pl ewpl
Observation 3:Edge Weights PL(EWPL)

Q: How does the weight of an edge relate to “popularity” if its adjacent nodes?

A:

wi,j ∝ wi * wj

Wi,j

j

i

FEC Committee-to-

Candidate network

Wi

Wj

outline3
Outline

Motivation

Related Work

- Patterns

- Generators

Datasets

Laws and Observations

Proposed graph generator: RTM

Sketch of proofs

Experiments

Conclusion

19

problem definition
Problem Definition
  • Generate a sequence of realistic weighted graphs that will obey all the patterns over time.
    • SUGP: staticun-weighted graph properties
      • small diameter
      • power law degree distribution
    • SWGP: staticweighted graph properties
      • the edge weight power law (EWPL)
      • the snapshot power law (SPL)
problem definition1
Problem Definition
  • DUGP: dynamicun-weighted graph properties
    • the densification power law (DPL)
    • shrinking diameter
    • bursty edge additions
    • λ1 Power Law (LPL)
  • DWGP: dynamicweighted graph properties
    • the weight power law (WPL)
    • bursty weight additions
    • λ1,w Power Law (LWPL)
2d solution kronecker product
2D solution: Kronecker Product
  • Idea: Recursion
  • Intuition:
    • Communities within communities
    • Self-similarity
    • Power-laws
3d solution recursive tensor multiplication rtm
3D solution: Recursive Tensor Multiplication(RTM)

I

2

3

4

X I1,1,1

24

3d solution recursive tensor multiplication rtm1
3D solution: Recursive Tensor Multiplication(RTM)

I

2

3

4

X I1,2,1

25

3d solution recursive tensor multiplication rtm2
3D solution: Recursive Tensor Multiplication(RTM)

I

2

3

4

X I1,3,1

26

3d solution recursive tensor multiplication rtm3
3D solution: Recursive Tensor Multiplication(RTM)

I

2

3

4

X I1,4,1

27

3d solution recursive tensor multiplication rtm4
3D solution: Recursive Tensor Multiplication(RTM)

I

2

3

4

X I2,1,1

28

3d solution recursive tensor multiplication rtm5
3D solution: Recursive Tensor Multiplication(RTM)

I

2

3

4

X I3,1,1

29

3d solution recursive tensor multiplication rtm6
3D solution: Recursive Tensor Multiplication(RTM)

I

2

3

4

30

3d solution recursive tensor multiplication rtm7
3D solution: Recursive Tensor Multiplication(RTM)

I

2

3

4

X I1,1,2

31

3d solution recursive tensor multiplication rtm8
3D solution: Recursive Tensor Multiplication(RTM)

I

2

3

4

X I1,2,2

32

3d solution recursive tensor multiplication rtm9
3D solution: Recursive Tensor Multiplication(RTM)

22

I

2

32

3

4

42

33

3d solution recursive tensor multiplication rtm10
3D solution: Recursive Tensor Multiplication(RTM)

t-slices

time

senders

recipients

34

3d solution recursive tensor multiplication rtm11
3D solution: Recursive Tensor Multiplication(RTM)

t1

t2

t3

35

3d solution recursive tensor multiplication rtm12
3D solution: Recursive Tensor Multiplication(RTM)

2

3

4

2

3

4

2

3

4

1

1

1

1

1

1

2

3

2

1

1

2

5

3

3

2

3

2

4

4

4

t2

t3

t1

3

4

4

4

3

2

3

2

2

2

3

3

2

2

2

1

1

5

2

1

4

1

36

outline4
Outline

Motivation

Related Work

- Patterns

- Generators

Datasets

Laws and Observations

Proposed graph generator: RTM

(Sketch of proofs)

Experiments

Conclusion

37

experimental results
Experimental Results
    • SUGP:
  • small diameter
  • PL Degree Distribution
  • SWGP:
  • Edge Weights PL
  • Snaphot PL
    • DUGP:
  • Densification PL
  • shrinking diameter
  • bursty edge additions
  • λ1 PL
  • DWGP:
  • Weight PL
  • bursty weight additions
  • λ1,w PL

diameter

Time

experimental results1
Experimental Results
    • SUGP:
  • small diameter
  • PL Degree Distribution
    • SWGP:
  • Edge Weights PL
  • Snaphot PL
    • DUGP:
  • Densification PL
  • shrinking diameter
  • bursty edge additions
  • λ1 PL
    • DWGP:
  • Weight PL
  • bursty weight additions
  • λ1,w PL

count

degree

39

experimental results2
Experimental Results
    • SUGP:
  • small diameter
  • PL Degree Distribution
    • SWGP:
  • Edge Weights PL
  • Snaphot PL
    • DUGP:
  • Densification PL
  • shrinking diameter
  • bursty edge additions
  • λ1 PL
    • DWGP:
  • Weight PL
  • bursty weight additions
  • λ1,w PL

|E|

|N|

40

experimental results3
Experimental Results
    • SUGP:
  • small diameter
  • PL Degree Distribution
    • SWGP:
  • Edge Weights PL
  • Snaphot PL
    • DUGP:
  • Densification PL
  • shrinking diameter
  • bursty edge additions
  • λ1 PL
    • DWGP:
  • Weight PL
  • bursty weight additions
  • λ1,w PL

|W|

|E|

41

experimental results4
Experimental Results
    • SUGP:
  • small diameter
  • PL Degree Distribution
    • SWGP:
  • Edge Weights PL
  • Snaphot PL
    • DUGP:
  • Densification PL
  • shrinking diameter
  • bursty edge additions
  • λ1 PL
    • DWGP:
  • Weight PL
  • bursty weight additions
  • λ1,w PL

42

experimental results5
Experimental Results
    • SUGP:
  • small diameter
  • PL Degree Distribution
    • SWGP:
  • Edge Weights PL
  • Snaphot PL
    • DUGP:
  • Densification PL
  • shrinking diameter
  • bursty edge additions
  • λ1 PL
    • DWGP:
  • Weight PL
  • bursty weight additions
  • λ1,w PL

In-weight

In-degree

Out-weight

Out-degree

43

experimental results6
Experimental Results
    • SUGP:
  • small diameter
  • PL Degree Distribution
    • SWGP:
  • Edge Weights PL
  • Snaphot PL
    • DUGP:
  • Densification PL
  • shrinking diameter
  • bursty edge additions
  • λ1 PL
    • DWGP:
  • Weight PL
  • bursty weight additions
  • λ1,w PL

44

experimental results7
Experimental Results
    • SUGP:
  • small diameter
  • PL Degree Distribution
    • SWGP:
  • Edge Weights PL
  • Snaphot PL
    • DUGP:
  • Densification PL
  • shrinking diameter
  • bursty edge additions
  • λ1 PL
    • DWGP:
  • Weight PL
  • bursty weight additions
  • λ1,w PL

λ1

|E|

λ1,w

|E|

45

conclusion
Conclusion

Wi,j

Wj

Wi

In real graphs, (un)weighted largest eigenvalues are power-law related to number of edges.

Weight of an edge is related to the total weights and of its incident nodes.

Recursive Tensor Multiplication is a recursive method to generate (1)weighted, (2)time-evolving, (3)self-similar, (4)power-law networks.

Future directions:

  • Probabilistic version of RTM
  • Fitting the initial tensor I

46

contact us
Contact us

Mary McGlohon

www.cs.cmu.edu/~mmcgloho

mmcgloho@cs.cmu.edu

Christos Faloutsos

www.cs.cmu.edu/~christos

christos@cs.cmu.edu

Leman Akoglu

www.andrew.cmu.edu/~lakoglu

lakoglu@cs.cmu.edu

47