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…. …and interesting

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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


Rtm laws and a recursive generator for weighted time evolving graphs

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.


Rtm laws and a recursive generator for weighted time evolving graphs

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


Rtm laws and a recursive generator for weighted time evolving graphs

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


2d solution kronecker product1

2D solution: Kronecker Product


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

    [email protected]

    Christos Faloutsos

    www.cs.cmu.edu/~christos

    [email protected]

    Leman Akoglu

    www.andrew.cmu.edu/~lakoglu

    [email protected]

    47


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