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 FaloutsosCarnegie Mellon University

School of Computer Science

1

Motivation Graphs are popular!

Social, communication,

network traffic, call graphs…

2

…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

Outline1. Motivation

2. Related Work - Patterns - Generators - Burstiness

3. Datasets

4. Laws and Observations

5. Proposed graph generator: RTM

6. (Sketch of proofs)

7. Experiments

8. Conclusion 3

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

4

y(x) = Ax−γ, A>0, γ>0

Blog Network

time

diam

eter

Graph Patterns (II)

5

DBLP Keyword-to-Conference NetworkInter-domain Internet graph

Densification [Leskovec et al.‘05]

and Weight [McGlohon

et al.‘08] Power-laws Eigenvalues Power Law [Faloutsos et al.‘99]

Rank

Eig

enva

lue

|E|

|W|

|srcN|

|dstN|

Degree Power Law [Richardson and Domingos, ‘01]

In-degree

Cou

nt

Epinions who-trusts-whom graph

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]6

Edge and weight additions are bursty, and self-similar.

Entropy plots [Wang+’02] is a measure of burstiness.

Burstiness

Time

We

ight

s

Resolution

En

trop

y

Resolution

En

trop

y

Bursty: 0.2 < slope < 0.9

slope = 5.9


2. Related Work - Patterns

- Generators

3. Datasets



6. Sketch of proofs

7. Experiments

8. Conclusion8

Datasets

9

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.

1

10






3Datasets

11






Unipartite networks: |N| |E| time

5. BlogNet 60K, 125K, 80 days

6. NetworkTraffic 21K, 2M, 52 months

3Datasets

20MB

12






Unipartite networks: |N| |E| time

5. BlogNet 60K, 125K, 80 days

6. NetworkTraffic 21K, 2M, 52 months

3Datasets

20MB5MB

25MB



- Generators

3. Datasets



6. Sketch of proofs

7. Experiments

8. Conclusion13

Observation 1: λ1 Power Law(LPL) Q1: How does the principal eigenvalue λ1 of the

adjacency matrix change over time?

Q2: Why should we care?

14

Observation 1: λ1 Power Law(LPL) Q1: How does the principal eigenvalue λ1 of the

adjacency matrix change over time?

Q2: Why should we care?

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

A1:

15

λ1(t) E(t)∝ α,

α ≤ 0.5

λ1 Power 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

16

DBLP Author-Conference network

Observation 2:λ1,w Power Law (LWPL)

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

A:

17

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?

18

FEC Committee-to-

Candidate network

wi,j ∝ wi * wj

Wi,j

Wi Wj

ji

A:



- Generators

3. Datasets



6. Sketch of proofs

7. Experiments

8. Conclusion19

Problem Definition Generate a sequence of realistic weighted

graphs that will obey all the patterns over time.

SUGP: static un-weighted graph properties small diameter power law degree distribution

SWGP: static weighted graph properties the edge weight power law (EWPL) the snapshot power law (SPL)

20

Problem Definition DUGP: dynamic un-weighted graph properties

the densification power law (DPL) shrinking diameter bursty edge additions λ1 Power Law (LPL)

DWGP: dynamic weighted graph properties the weight power law (WPL) bursty weight additions λ1,w Power Law (LWPL)

21

2D solution: Kronecker Product

22

Idea: Recursion Intuition: Communities within

communities Self-similarity Power-laws

2D solution: Kronecker Product

23

3D solution: Recursive Tensor Multiplication(RTM)

24

4

2

3

I

X I1,1,1


25

4

2

3

I

X I1,2,1


26

4

2

3

I

X I1,3,1


27

4

2

3

I

X I1,4,1


28

4

2

3

I

X I2,1,1


29

4

2

3

I

X I3,1,1


30

4

2

3

I


31

4

2

3

I

X I1,1,2


32

4

2

3

I

X I1,2,2


33

4

2

3

I

42

32

22


34

sen

der

s

recipients

t-slices

time


35

t1 t2 t3


36

t1t2 t3

3 12

5

2

1234

1 2 3 4 1 2 3 41234

1234

1 2 3 4

2 3

1

2 34

1

2 34

1

12

4

5 2

3

21

2 342



- Generators

3. Datasets



6. (Sketch of proofs)

7. Experiments

8. Conclusion37

Experimental Results

38

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

Time

diam

eter


39





degree

coun

t


40





|N|

|E|


41





|E|

|W|


42






43

In-degree

In-w

eigh

t




DWGP: Weight PL bursty weight additions λ1,w PL Out-degree

Out

-wei

ght


44






45





|E|

λ1

|E|

λ1,w

Conclusion 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

Wi,j

Wi Wj

47

Contact usMary McGlohonwww.cs.cmu.edu/[email protected]

Christos Faloutsoswww.cs.cmu.edu/[email protected]

Leman Akogluwww.andrew.cmu.edu/[email protected]

Documents

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