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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…
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
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. 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
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.
3Datasets
11
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
3Datasets
20MB
12
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
3Datasets
20MB5MB
25MB
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. 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:
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. 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
3D solution: Recursive Tensor Multiplication(RTM)
25
4
2
3
I
X I1,2,1
3D solution: Recursive Tensor Multiplication(RTM)
26
4
2
3
I
X I1,3,1
3D solution: Recursive Tensor Multiplication(RTM)
27
4
2
3
I
X I1,4,1
3D solution: Recursive Tensor Multiplication(RTM)
28
4
2
3
I
X I2,1,1
3D solution: Recursive Tensor Multiplication(RTM)
29
4
2
3
I
X I3,1,1
3D solution: Recursive Tensor Multiplication(RTM)
30
4
2
3
I
3D solution: Recursive Tensor Multiplication(RTM)
31
4
2
3
I
X I1,1,2
3D solution: Recursive Tensor Multiplication(RTM)
32
4
2
3
I
X I1,2,2
3D solution: Recursive Tensor Multiplication(RTM)
33
4
2
3
I
42
32
22
3D solution: Recursive Tensor Multiplication(RTM)
34
sen
der
s
recipients
t-slices
time
3D solution: Recursive Tensor Multiplication(RTM)
35
t1 t2 t3
3D solution: Recursive Tensor Multiplication(RTM)
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
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. 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
Experimental Results
39
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
degree
coun
t
Experimental Results
40
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
|N|
|E|
Experimental Results
41
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|
|W|
Experimental Results
42
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
Experimental Results
43
In-degree
In-w
eigh
t
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 Out-degree
Out
-wei
ght
Experimental Results
44
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
Experimental Results
45
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|
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]