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

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

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### 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…

- Social, communication,

- …and interesting
- surprising common
properties for static

and un-weighted graphs

- surprising common
- How about weighted graphs?
- …and their dynamic properties?

- How can we model such graphs?
- for simulation studies, what-if scenarios, future prediction, sampling

Outline

- Motivation
- Related Work
- Patterns

- Generators

- Burstiness

- Datasets
- Laws and Observations
- Proposed graph generator: RTM
- (Sketch of proofs)
- Experiments
- Conclusion

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)

- 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

- 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]

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

Outline

- Motivation
- Related Work
- Patterns

- Generators

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

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

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.

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

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

Outline

- Motivation
- Related Work
- Patterns

- Generators

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

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: 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

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: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)

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

Outline

Motivation

Related Work

- Patterns

- Generators

Datasets

Laws and Observations

Proposed graph generator: RTM

Sketch of proofs

Experiments

Conclusion

19

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)

- SUGP: staticun-weighted graph properties

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

- Idea: Recursion

- Intuition:
- Communities within communities
- Self-similarity
- Power-laws

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

Outline

Motivation

Related Work

- Patterns

- Generators

Datasets

Laws and Observations

Proposed graph generator: RTM

(Sketch of proofs)

Experiments

Conclusion

37

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

- SUGP:

- DUGP:

diameter

Time

Experimental Results small diameter PL Degree Distribution Edge Weights PL Snaphot PL Densification PL shrinking diameter bursty edge additions λ1 PL Weight PL bursty weight additions λ1,w PL

- SUGP:

- SWGP:

- DUGP:

- DWGP:

count

degree

39

Experimental Results small diameter PL Degree Distribution Edge Weights PL Snaphot PL Densification PL shrinking diameter bursty edge additions λ1 PL Weight PL bursty weight additions λ1,w PL

- SUGP:

- SWGP:

- DUGP:

- DWGP:

|E|

|N|

40

Experimental Results small diameter PL Degree Distribution Edge Weights PL Snaphot PL Densification PL shrinking diameter bursty edge additions λ1 PL Weight PL bursty weight additions λ1,w PL

- SUGP:

- SWGP:

- DUGP:

- DWGP:

|W|

|E|

41

Experimental Results small diameter PL Degree Distribution Edge Weights PL Snaphot PL Densification PL shrinking diameter bursty edge additions λ1 PL Weight PL bursty weight additions λ1,w PL

- SUGP:

- SWGP:

- DUGP:

- DWGP:

42

Experimental Results small diameter PL Degree Distribution Edge Weights PL Snaphot PL Densification PL shrinking diameter bursty edge additions λ1 PL Weight PL bursty weight additions λ1,w PL

- SUGP:

- SWGP:

- DUGP:

- DWGP:

In-weight

In-degree

Out-weight

Out-degree

43

Experimental Results small diameter PL Degree Distribution Edge Weights PL Snaphot PL Densification PL shrinking diameter bursty edge additions λ1 PL Weight PL bursty weight additions λ1,w PL

- SUGP:

- SWGP:

- DUGP:

- DWGP:

44

Experimental Results small diameter PL Degree Distribution Edge Weights PL Snaphot PL Densification PL shrinking diameter bursty edge additions λ1 PL Weight PL bursty weight additions λ1,w PL

- SUGP:

- SWGP:

- DUGP:

- DWGP:

λ1

|E|

λ1,w

|E|

45

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

Mary McGlohon

www.cs.cmu.edu/~mmcgloho

Christos Faloutsos

www.cs.cmu.edu/~christos

Leman Akoglu

www.andrew.cmu.edu/~lakoglu

47

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