subgraph frequencies - stanford universitystanford.edu/~jugander/papers/ · fitting λ to subgraph...

Johan Ugander, Lars Backstrom, Jon KleinbergWorld Wide Web ConferenceMay !", #$!%

Subgraph Frequencies:The Empirical and Extremal Geographyof Large Graph Collections

▪ Neighborhoods: graph induced by friends of a single ego, excluding ego

Graph collections

All Friends

One-way Communication Mutual Communication

Maintained Relationships


▪ Groups: graph induced by members of a Facebook ‘group‘

▪ Events: graph induced by ‘Yes’ respondents to a Facebook ‘event’

Graph collections

All Friends




▪ Groups: graph induced by members of a Facebook ‘group‘

▪ Events: graph induced by ‘Yes’ respondents to a Facebook ‘event’

Graph collections

All Friends



Seeking a ‘coordinate system’ on these graphs

SubgraphsAll Friends



SubgraphsAll Friends



Compute frequencies

Subgraph Frequencies▪ Definition: The subgraph frequency s(F,G) of a k-node subgraph F in a graph G

is the fraction of k-tuples of nodes in G that induce a copy of F.

Motifs/Frequent subgraphs: Inokuchi et al. #$$$, Milo et al. #$$#, Yan-Han #$$#, Kuramochi-Karypis #$$&Triad census: Davis-Leinhardt !'(!, Wasserman-Faust !''&



▪ Subgraph frequency vectors:

s(·, G) = (y1, y2, y3, y4, y5, y6, y7, y8, y9, y10, y11)

s(·, G) = (x1, x2, x3, x4)


= ($.!), $.%(, $.!&, $.%!)



▪ Subgraph frequency vectors:


= ($.!), $.%(, $.!&, $.%!)

s(·, G) = (y1, y2, y3, y4, y5, y6, y7, y8, y9, y10, y11)

s(·, G) = (x1, x2, x3, x4)

Empirical/Extremal Questions

▪ Consider the subgraph frequencies as a ‘coordinate system’

▪ Empirical Geography:

▪ What subgraph frequencies do social graphs exhibit?

▪ Is there a good model?

▪ Extremal Geography:

▪ How much of this space is even feasible, combinatorially?

▪ Do empirical graphs fill the feasible space?

Empirical/Extremal Questions

▪ What’s a property of graphs and what’s a property of people?

▪ Consider the subgraph frequencies as a ‘coordinate system’

▪ Empirical Geography:

▪ What subgraph frequencies do social graphs exhibit?

▪ Is there a good model?

▪ Extremal Geography:

▪ How much of this space is even feasible, combinatorially?

▪ Do empirical graphs fill the feasible space?

What do we expect?

tria

dic clo

sure

triadic closure

triadic closure

What do we expect?

What do we expect?

We expect few wedges, many triangles for social networks.

The triad space

The triad space

You are here

*$ node graphsOrange - Neighborhoods Green - GroupsLavender - Events

You are here

Gn,p

The triad space


Subgraph frequency of



Gn,p




ExtremalGraph Theory

Subgraph frequency ofFrequency of the ‘forbidden triad’ is bounded at ! "/#.

Sharp for Kn/#,n/# (bipartite graph) when n is even.


Subgraph frequencies

‘Crowd-sourced’ inner bounds

Consider all social graphs and the complements of all graphs, anti-social graphs (which are also graphs!)

What graphs are missing?

▪ Square unlikely to form:

Triadic Closure and Squares

▪ Square unlikely to form:

▪ Square has very short ‘half-life’:

Triadic Closure and Squares

Continuous Time Markov Chain Model

tria

dic clo

sure

triadic closure

triadic closure

Continuous Time Markov Chain Model

Edge Formation Random Walk (EFRW)▪ Continuous-time Markov chain▪ Transitions between unlabeled, undirected graphs based in edge formation.

▪ Independent Poisson processes for all node pairs:▪ Arbitrary formation: rate ɣ > $ ▪ Arbitrary deletion: rate δ > $

▪ Triadic closure formation for each wedge: rate λ + $

Edge Formation Random Walk (EFRW)▪ Continuous-time Markov chain▪ Transitions between unlabeled, undirected graphs based in edge formation.

▪ Independent Poisson processes for all node pairs:▪ Arbitrary formation: rate ɣ > $ ▪ Arbitrary deletion: rate δ > $

▪ Triadic closure formation for each wedge: rate λ + $

▪ For &-node graphs, succinct Markov chain state transition diagram:

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Fitting λ to subgraph data▪ How well can we fit λ?

▪ Subgraph frequencies are modeled very well by triadic closure.

frequ

ency

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Neighborhoods, n=50

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Neighborhoods data, meanFit model, λ ν = 19.37

frequ

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Groups, n=50

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Groups data, meanFit model, λ ν = 7.02

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Events, n=50●

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Events data, meanFit model, λ ν = 7.38

(log-scale y-axis)

Extremal graph theory▪ Subgraph frequencies s(F,G) closely related to homomorphism density t(F,G).

▪ Frequency of cliques, lower bounds: Moon-Moser !'"#, Razborov #$$)▪ Frequency of cliques, upper bounds: Kruskal-Katona Theorem▪ Frequency of trees: Sidorenko Conjecture (‘Theorem for trees’)▪ Also linear relationships across sizes.▪ => Linear Program!

[Borgs et al. $%%&, Lovasz $%%']

▪ A proposition for all subgraphs:

Proposition. For every k, there exist constants ✏ and n0 such that the following

holds. If F is a k-node subgraph that is not a clique and not empty, and G is

any graph on n � n0 nodes, then s(F,G) < 1� ✏.

Extremal graph theory

▪ How do different audience graphs differ?

Audience graph classification

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▪ Classification challenges A) (*-node neigh. vs. (*-node events B) &$$-node neigh. vs. &$$-node groups


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▪ Classification challenges A) (*-node neigh. vs. (*-node events B) &$$-node neigh. vs. &$$-node groups

▪ Features: Quad frequencies : (", / (", accuracy Global features: "', / (", accuracy Quad frequencies + Global features: )!, / )#, accuracy


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▪ Subgraph frequencies usefully characterize social graphs, have extremal limits!

▪ Edge Formation Random Walk model of dense social graphs:

▪ Homomorphism density bounds yield subgraph density bounds:

Conclusions

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subgraph frequencies - stanford universitystanford.edu/~jugander/papers/ · fitting λ to subgraph...

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