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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

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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

- 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

- Motivation
- Related Work
- Patterns

- Generators

- Burstiness

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

- 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

- 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

- 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]

Resolution

Entropy

- 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

- Motivation
- Related Work
- Patterns

- Generators

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

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

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.

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

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

- Motivation
- Related Work
- Patterns

- Generators

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

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

Q2: Why should we care?

14

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

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

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

A:

λ1,w(t) ∝ E(t) β

DBLP Author-Conference network

Network Traffic

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

Motivation

Related Work

- Patterns

- Generators

Datasets

Laws and Observations

Proposed graph generator: RTM

Sketch of proofs

Experiments

Conclusion

19

- 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

- 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)

- Idea: Recursion

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

I

2

3

4

X I1,1,1

24

I

2

3

4

X I1,2,1

25

I

2

3

4

X I1,3,1

26

I

2

3

4

X I1,4,1

27

I

2

3

4

X I2,1,1

28

I

2

3

4

X I3,1,1

29

I

2

3

4

30

I

2

3

4

X I1,1,2

31

I

2

3

4

X I1,2,2

32

22

I

2

32

3

4

42

33

t-slices

time

senders

recipients

34

t1

t2

t3

35

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

Motivation

Related Work

- Patterns

- Generators

Datasets

Laws and Observations

Proposed graph generator: RTM

(Sketch of proofs)

Experiments

Conclusion

37

- SUGP:

- DUGP:

diameter

Time

- SUGP:

- SWGP:

- DUGP:

- DWGP:

count

degree

39

- SUGP:

- SWGP:

- DUGP:

- DWGP:

|E|

|N|

40

- SUGP:

- SWGP:

- DUGP:

- DWGP:

|W|

|E|

41

- SUGP:

- SWGP:

- DUGP:

- DWGP:

42

- SUGP:

- SWGP:

- DUGP:

- DWGP:

In-weight

In-degree

Out-weight

Out-degree

43

- SUGP:

- SWGP:

- DUGP:

- DWGP:

44

- SUGP:

- SWGP:

- DUGP:

- DWGP:

λ1

|E|

λ1,w

|E|

45

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

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