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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 - PowerPoint PPT Presentation
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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…
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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
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
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
15
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;
20
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
21
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.
27
Outline1. Motivation
2. Related Work - Patterns
- Generators
3. Laws and Observations
4. Proposed graph generator: RTM
5. Sketch of proofs
6. Experiments
7. Conclusion
28
Experimental Results (I)
29
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.
31
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
32
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