RTG: A Recursive Realistic Graph Generator using Random Typing

RTG: A Recursive Realistic Graph Generator using Random Typing

Leman Akoglu and Christos FaloutsosCarnegie Mellon University

Outline• Motivation• Problem Definition• Related Work• A Little History• Proposed Model• Experimental Results• Conclusion

04/20/23 Akoglu, Faloutsos ECML PKDD 2009 2

Motivation - 1 Complex graphs --WWW, computer, biological, social networks, etc. exhibit many common properties:

- power laws - small and shrinking diameter - community structure - …

How can we produce synthetic but realistic graphs?


http://www.aharef.info/static/htmlgraph/

Motivation - 2

Why do we need synthetic graphs?• Simulation• Sampling/Extrapolation• Summarization/Compression• Motivation to understand pattern generating

processes


Problem Definition

Discover a graph generator that is:G1. simple: the more intuitive the better!G2. realistic: outputs graphs that obey all “laws”G3. parsimonious: requires few parametersG4. flexible: able to produce the cross-product of

un/weighted, un/directed, uni/bipartite graphsG5. fast: generation should take linear time with

the size of the output graph




Related Work

1. Graph Properties What we want to match

2. Graph Generators What has been proposed earlier


Related Work 1: Graph Properties


Related Work 2: Graph Generators

• Erdős-Rényi (ER) model [Erdős, Rényi `60]• Small-world model [Watts, Strogatz `98]• Preferential Attachment [Barabási, Albert `99]• Winners don’t take all [Pennock et al. `02]• Forest Fire model [Leskovec, Faloutsos `05]• Butterfly model [McGlohon et al. `08]



• Erdős-Rényi (ER) model [Erdős, Rényi `60]• Small-world model [Watts, Strogatz `98]• Preferential Attachment [Barabási, Albert `99]• Winners don’t take all [Pennock et al. `02]• Forest Fire model [Leskovec, Faloutsos `05]• Butterfly model [McGlohon et al. `08]


• Model some static graph property• Neglect dynamic properties • Cannot produce weighted graphs.


• Random dot-product graphs [Kraetzl, Nickel `05] [Young, Scheinerman `07]

• Utility-based models [Fabrikant et al. ’02] [Even-Bar et al. `07] [Laoutaris, `08]

• Kronecker graphs [Leskovec et al. `07] [Akoglu et al. `08]04/20/23 Akoglu, Faloutsos ECML PKDD 2009 11





• Kronecker graphs [Leskovec et al. `07] [Akoglu et al. `08]

• Produces only undirected graphs • Cannot produce weighted graphs.• Requires quadratic time



• Kronecker graphs [Leskovec et al. `07] [Akoglu et al. `08]



• Hard to analyze





• Kronecker graphs [Leskovec et al. `07] [Akoglu, `08]04/20/23 Akoglu, Faloutsos ECML PKDD 2009 14

• Multinomial/Lognormal distrib.• Fixed number of nodes

• Hard to analyze




rankco

unt

A Little History - 1

[Zipf, 1932] In many natural languages,

the rank r and the frequency fr of words follow a power law:

fr 1/r∝


A Little History - 2 [Mandelbrot, 1953] “Humans optimize avg. information per

unit transmission cost.”


A Little History - 2 [Miller, 1957] “A monkey types randomly on a keyboard:

Distribution of words follow a power-law.”


k equiprobable keys

. . . . .a b λ $ + Space

A Little History - 2

[Conrad and Mitzenmacher, 2004] “Same relation still holds when keys have

unequal probabilities.”


. . .a b λ $ + Space



Preliminary Model 1RTG-IE: RTG with Independent Equiprobable keys


Space

, where

Lemma 1. W is super-linear on N (power law):

Lemma 2. W is super-linear on E (power law):

Lemma 3. In(out)-weight Wn of node n is super-linear on in(out)-degree dn (power law):


Please find the proofs in the paper.


Graph Properties


, where

Lemma 1. W is super-linear on N (power law):

Lemma 2. W is super-linear on E (power law):

Lemma 3. In(out)-weight Wn of node n is super-linear on in(out)-degree dn (power law):


Please find the proofs in the paper.


L11. Weight PL

L05. Densification PL

L10. Snapshot PL

Advantages of the Preliminary Model 1

G1 - IntuitiveG1 - Easy to implementG2 - Realistic –provably follows several rules

G3 - Handful of parameters –k, q, W G5 - Fast –generating random sequence of char.s


Problems of the Preliminary Model 11- Multinomial degree distributions


rank

coun

t

in-degree

coun

t

Problems of the Preliminary Model 1

2- No homophily, no community structure Node i connects to any node j with prob. di*dj independently, rather than connecting to ‘similar’ nodes.


Preliminary Model 2RTG-IU: RTG with Independent Un-equiprobable keys


Solution to Problem 1:[Conrad and Mitzenmacher, 2004]

rank

coun

t

in-degree

coun

t

rank

coun

t

in-degree

coun

t

. . .ab λ $ + Space

. . . . .a b λ $ + Space

Proposed ModelRTG: Random Typing Graphs


Solution to Problem 2:“2D keyboard”• Generate source- destination labels in one shot.• Pick one of the nine keys randomly.


Solution to Problem 2:“2D keyboard”• Repeat recursively.• Terminate each label when the space key is typed on each dimension (dark blue).


pa*pa


Solution to Problem 2:“2D keyboard” How do we choose the keys? Independent model does not yield community structure!


pa*pb

pb*pa pb*pb

q*pa q*pb

pa*q

pb*q

q*q


Solution to Problem 2:“2D keyboard”• Boost probability of diagonal keys and decrease probability of off-diagonal ones (0<β<1: imbalance factor)



Solution to Problem 2:“2D keyboard”• Boost probability of diagonal keys and decrease probability of off-diagonal ones (0<β<1: imbalance factor)• Favoring of diagonal keys creates homophily.


Proposed Model


Parameters• k: Number of keys• q: Probability of hitting the space key S• W: Number of multi- edges in output graph G• β: imbalance factor

Up to this point, we discussed directed, weighted and unipartite graphs. Generalizations - Undirected graphs: Ignore edge directions; edge generation is symmetric. - Unweighted graphs: Ignore duplicate edges. - Bipartite graphs: Different key sets on source and destination; labels are different.

Proposed Model




Experimental ResultsHow does RTG model real graphs?

• Blognet: a social network of blogs based on citations undirected, unweighted and unipartite N = 27, 726; E = 126, 227; over 80 time ticks.• Com2Cand: the U.S. electoral campaign donations network from organizations to candidates directed, weighted ($ amounts) and bipartite N = 23, 191; E = 877, 721; W = 4, 383, 105, 580 over 29 time ticks.


Experimental Results Blognet RTG


degreedegreeco

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coun

t

L01. Power-law degree distribution [Faloutsos et al. `99, Kleinberg et al. `99, Chakrabarti et al. `04, Newman `04]



trianglestrianglesco

unt

coun

t

L02. Triangle Power Law (TPL) [Tsourakakis `08]

Experimental Results 1 Blognet RTG


rank rank

λ rank

λ rank

L03. Eigenvalue Power Law (EPL) [Siganos et al. `03]

Graph Properties




#nodes #nodes

#edg

es

#edg

es

L05. Densification Power Law (DPL) [Leskovec et al. `05]



time time

diam

eter

diam

eter

L06. Small and shrinking diameter [Albert and Barabási `99, Leskovec et al. `05]



time time

size

size

L07. Constant size 2nd and 3rd connected components [McGlohon et al. `08]



#edges #edges

λ 1 λ 1

L08. Principal Eigenvalue Power Law (λ1PL) [Akoglu et al. `08]



resolution resolution

entr

opy

entr

opy

L09. Bursty/self-similar edge/weight additions [Gomez and Santonja `98, Gribble et al. `98, Crovella and Bestavros `99, McGlohon et al. `08]

Graph Properties


Experimental Results 2 Com2Cand RTG


time time

time time

diam

eter

diam

eter

size

size



#edges #edges

rank rank

λ 1 λ 1

λ rank

λ rank



in-degree

coun

t

in-degree

coun

t

resolution resolution

entr

opy

entr

opy



in-degreein-degree (#checks)

in-w

eigh

t

in-w

eigh

t ($

am

ount

)

L10. Snapshot Power Law (SPL) [McGlohon et al. `08]



#edges

Tota

l wei

ght

L11. Weight Power Law (WPL) [McGlohon et al. `08]

Tota

l wei

ght

#edges

Graph Properties


Experimental ResultsOn “modularity” [Girvan and Newman `02]


No significant modularity --RTG-IE

“Modularity “decreaseswith increasing β

mor

e co

mm

unity

str

uctu

re

Graph Properties


Experimental ResultsOn complexity


Computation timegrows linearlywith increasing W

2M multi-edges in 7 sec.s

#multi-edges

time

(ms)



Conclusion 1Our model is:G1. simple and intuitive --few lines of codeG2. realistic --graphs that obey all eleven

properties in real graphsG3. parsimonious --only a handful of parametersG4. flexible --can generate

weighted/unweighted, directed/undirected, unipartite/bipartite graphs and any combination of those

G5. fast --linear on the size of the output graph04/20/23 Akoglu, Faloutsos ECML PKDD 2009 58

Conclusion 2We showed that: RTG mimics real graphs well.


Contact


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

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

mailto:[email protected]

mailto:[email protected]

A Little History - 3 The infinite monkey theorem: A monkey typing randomly on a keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.


Burstiness and Self-similarity If each step is a time tick, weight additions are uniform!

• Start with a uniform interval• Recursively subdivide weight additions to each half, quarter, and so on, according to the bias b > 0.5• b -fraction of the additions happen in one “half” and the remaining in the other.

TotalWeight

Time

Proposed Model


Related Work: Graph Properties


Documents

RTG: A Recursive Realistic Graph Generator using Random Typing