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

Leman Akoglu, Mary McGlohon, Christos FaloutsosCarnegie Mellon University

School of Computer Science

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Motivation Graphs are popular!

Social, communication,

network traffic, call graphs…

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…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

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y(x) = Ax−γ, A>0, γ>0

Blog Network

time

diam

eter

Graph Patterns (II)

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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

D W

eig

hts

Resolution

En

trop

y

From time series data, begin with resolution T/2. Record entropy HR.

Entropy plots

Time

D W

eig

hts

Resolution

En

trop

y

From time series data, begin with resolution T/2. Record entropy HR. Recursively take finer resolutions.

Entropy plots

Time

D W

eig

hts

Resolution

En

trop

y

Entropy Plots Self-similarity Linear plot

Resolution

En

trop

y

●

slope = 5.9

Entropy Plots Self-similarity Linear plot

Resolution

En

trop

y

time

Uniform: slope=1

slope = 5.9

Entropy Plots Self-similarity Linear plot

Resolution

En

trop

y

timetime

Uniform: slope=1 Point mass: slope=0

slope = 5.9

13 McGlohon, Akoglu, Faloutsos KDD08

Entropy Plots

Resolution

En

trop

yBursty:

0.2 < slope < 0.9

Self-similarity Linear plot

timetime

Uniform: slope=1 Point mass: slope=0

slope = 5.9

Outline1. Motivation

2. Related Work - Patterns

- Generators

3. Datasets

4. Laws and Observations

5. Proposed graph generator: RTM

6. Sketch of proofs

7. Experiments

8. Conclusion14

Datasets

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Outline1. Motivation

2. Related Work - Patterns

- Generators

3. Datasets

4. Laws and Observations

5. Proposed graph generator: RTM

6. Sketch of proofs

7. Experiments

8. Conclusion16

Observation 1:λ1 Power Law (LPL)

Q: How does the principal eigenvalue λ1

change over time

A: λ1 (t) and the number of edges E(t) over time follow a power law with exponent less than 0.5,

especially after the ‘gelling point’.

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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;

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DBLP Author-Conference network

Observation 2:λ1,w Power Law (LWPL)

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

A:

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λ1,w(t) E(t)∝ β

DBLP Author-Conference network Network Traffic

Observation 3: Edge Weights PLQ: How does the weight of an

edge relate to “popularity” if its adjacent nodes

A: Weight of the link wi,j between two given nodes i and j in a given graph G has a power law relation with the weights wi and wj of the nodes;

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FEC Committee-to-

Candidate network

Outline1. Motivation

2. Related Work - Patterns

- Generators

3. Laws and Observations

4. Proposed graph generator: RTM

5. Sketch of proofs

6. Experiments

7. Conclusion

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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)

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Problem Definition cont. 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)

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One solution: Kronecker Product

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Intuition: Self-similarity! Communities within

communities Recursion yields modular

network behavior

One solution: Kronecker Product

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Recursive Tensor Product(RTM)

Use of tensors: 3rd mode is time Initial tensor I is a realistic graph itself

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RTM of a (3x3x3) tensor by itself

RTM cont.

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Outline1. Motivation

2. Related Work - Patterns

- Generators

3. Laws and Observations

4. Proposed graph generator: RTM

5. Sketch of proofs

6. Experiments

7. Conclusion

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Experimental Results (I)

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BLOG NETWORK

RTM MODEL

Experimental Results (II)

30RTM MODEL

BLOG NETWORK

Conclusion Largest (un)weighted principal eigenvalues are

power-law related to the number of edges in real graphs.

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

Recursive Tensor Multiplication is a recursive method to generate weighted, time-evolving, self-similar, modular networks.

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Future Directions Largest eigenvalues of the Laplacian matrices Second largest eigenvalue – related to global

connectivity – conductance – mixing rate of random walk on graph

Probabilistic version of RTM Fitting graphs

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