RTM: Laws and a Recursive
Generator for Weighted
Time-Evolving Graphs
Leman Akoglu, Mary McGlohon, Christos Faloutsos
Carnegie Mellon University
School of Computer Science
1
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
2
Outline
1. 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]
diameter
Blog Network
 Power Laws
y(x) = Ax−γ, A>0, γ>0
time
4
Graph Patterns (II)
Count
Eigenvalue
|W|
In-degree
Epinions who-trusts-whom graph
 Degree Power Law
[Richardson and
Domingos, ‘01]
|srcN|
|dstN|
Rank
|E|
Inter-domain Internet graph DBLP Keyword-to-Conference Network
 Eigenvalues Power  Densification
Law [Faloutsos et
al.‘99]
[Leskovec et al.‘05]
and Weight [McGlohon
et al.‘08] Power-laws
5
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
Burstiness

Entropy
Entropy
Weights

Edge and weight additions are bursty, and selfsimilar.
Entropy plots [Wang+’02] is a measure of
Bursty:
burstiness.
0.2 < slope < 0.9
Time
slope = 5.9
Resolution
Resolution
Outline
1. Motivation
2. Related Work
- Patterns
- Generators
3. Datasets
4. Laws and Observations
5. Proposed graph generator: RTM
6. Sketch of proofs
7. Experiments
8. Conclusion
8
Datasets
Bipartite networks:
1. AuthorConference
2. KeywordConference
3. AuthorKeyword
4. CampaignOrg
1
|N| |E| time
17K, 22K, 25 yr.
10K, 23K, 25 yr.
27K, 189K, 25 yr.
23K, 877K, 28 yr.
9
Datasets
Bipartite networks:
1. AuthorConference
2. KeywordConference
3. AuthorKeyword
4. CampaignOrg
3
|N| |E| time
17K, 22K, 25 yr.
10K, 23K, 25 yr.
27K, 189K, 25 yr.
23K, 877K, 28 yr.
10
Datasets
3
Bipartite networks:
1. AuthorConference
2. KeywordConference
3. AuthorKeyword
4. CampaignOrg
|N| |E| time
17K, 22K, 25 yr.
10K, 23K, 25 yr.
27K, 189K, 25 yr.
23K, 877K, 28 yr.
Unipartite networks:
5. BlogNet
6. NetworkTraffic
|N|
|E| time
60K, 125K, 80 days
21K, 2M, 52 months
20MB
11
Datasets
3
Bipartite networks:
1. AuthorConference
2. KeywordConference
3. AuthorKeyword
4. CampaignOrg
|N| |E| time
17K, 22K, 25 yr.
10K, 23K, 25 yr.
27K, 189K, 25 yr.
23K, 877K, 28 yr.
Unipartite networks:
5. BlogNet
6. NetworkTraffic
|N|
|E| time
60K, 125K, 80 days
21K, 2M, 52 months
20MB
25MB
5MB
12
Outline
1. Motivation
2. Related Work
- Patterns
- Generators
3. Datasets
4. Laws and Observations
5. Proposed graph generator: RTM
6. Sketch of proofs
7. Experiments
8. Conclusion
13
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:
λ (t) ∝ E(t) α,
1
α ≤ 0.5
15
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
DBLP Author-Conference network
16
Observation 2:λ1,w Power Law (LWPL)
Q: How does the weighted principal eigenvalue
λ1,w change over time?
A:
λ (t) ∝ E(t) β
1,w
DBLP Author-Conference network
Network Traffic
17
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
i
Wi
j
Wj
FEC Committee-toCandidate network
18
Outline
1. Motivation
2. Related Work
- Patterns
- Generators
3. Datasets
4. Laws and Observations
5. Proposed graph generator: RTM
6. Sketch of proofs
7. Experiments
8. Conclusion
19
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
Idea: Recursion
Intuition:
 Communities within
communities
 Self-similarity
 Power-laws
22
2D solution: Kronecker Product
23
3D solution: Recursive Tensor
Multiplication(RTM)
I
2
3
4
X
I1,1,1
24
3D solution: Recursive Tensor
Multiplication(RTM)
I
2
3
4
X
I1,2,1
25
3D solution: Recursive Tensor
Multiplication(RTM)
I
2
3
4
X
I1,3,1
26
3D solution: Recursive Tensor
Multiplication(RTM)
I
2
3
4
X
I1,4,1
27
3D solution: Recursive Tensor
Multiplication(RTM)
I
2
3
4
X
I2,1,1
28
3D solution: Recursive Tensor
Multiplication(RTM)
I
2
3
4
X
I3,1,1
29
3D solution: Recursive Tensor
Multiplication(RTM)
I
2
3
4
30
3D solution: Recursive Tensor
Multiplication(RTM)
I
2
3
4
X
I1,1,2
31
3D solution: Recursive Tensor
Multiplication(RTM)
I
2
3
4
X
I1,2,2
32
3D solution: Recursive Tensor
Multiplication(RTM)
22
I
2
32
3
4
42
33
3D solution: Recursive Tensor
Multiplication(RTM)
senders
t-slices
recipients
34
3D solution: Recursive Tensor
Multiplication(RTM)
t1
t2
t3
35
3D solution: Recursive Tensor
Multiplication(RTM)
1 2 3 4
1
2
3
4
1 2 3 4
1
2
3
4
3
1
3
1
2
3
4
1
2
5
2
t2
t1
2
1 2 3 4
3
2
4
2
1
1
t3
3
2
1
4
2
3
5
1
2
4
36
Outline
1. Motivation
2. Related Work
- Patterns
- Generators
3. Datasets
4. Laws and Observations
5. Proposed graph generator: RTM
6. (Sketch of proofs)
7. Experiments
8. Conclusion
37
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
Experimental Results
Time
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
count
Experimental Results
degree
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
|E|
Experimental Results
|N|
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
|W|
Experimental Results
|E|
41
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
In-degree
Out-weight
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
Experimental Results
Out-degree
43
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
|E|
λ1,w
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
Experimental Results
|E|
45
Conclusion
 In real graphs, (un)weighted largest eigenvalues
are power-law related to number of edges.
 Weight Wi,j of an edge is related to the total
weights Wi and Wj of its incident nodes.
 Recursive Tensor Multiplication is a recursive
method to generate (1)weighted, (2)timeevolving, (3)self-similar, (4)power-law networks.
Future directions:
 Probabilistic version of RTM
 Fitting the initial tensor I
46
Contact us
Mary McGlohon
www.cs.cmu.edu/~mmcgloho
mmcgloho@cs.cmu.edu
Christos Faloutsos
www.cs.cmu.edu/~christos
christos@cs.cmu.edu
Leman Akoglu
www.andrew.cmu.edu/~lakoglu
lakoglu@cs.cmu.edu
47