Hierarchically Modular Structure in Complex Networks Aaron Clauset - - PowerPoint PPT Presentation

Hierarchically Modular Structure in Complex Networks

Aaron Clauset Santa Fe Institute 3 November 2008 DIMACS / DyDAn “Network Models of Biological and Social Contagion”

Modular Hierarchies

Grassland species*

*thank you: Jennifer Dunne

plant

→ →

herbivore

→

parasite

Modular Hierarchies

c

Modular Hierarchies

c

The Task

How can we extract

• this hierarchical (multi-scale) structure

from complex networks?

→

network ?

chierarchy

One Approach

Model-based inference

• 1. describe how to generate hierarchies (a model)
• 2. “fit” model to empirical data
• 3. test “fitted” model
• 4. extract predictions + insight
• 5. profit!

A Model of Hierarchy

→

A Model of Hierarchy

probability assortative modules

pr

D, {pr}

“inhomogeneous” random graph

→ →

model instance

→

Pr(i, j connected) = pr i j i j = p(lowest common ancestor of i,j)

Model Features

• explicit model = explicit assumptions
• very flexible (many parameters)
• captures structure at all scales
• arbitrary mixtures of assortativity, disassortativity
• learnable directly from data

Learning From Data

a direct approach

• likelihood function

( scores quality of model)

• sample the good models

via Markov chain Monte Carlo

• technical details in arXiv : physics/0610051

L = Pr( data | model )

From Graph to Ensemble

From Graph to Ensemble

• Given graph
• run MCMC to equilibrium
• then, for each sampled , draw a resampled

graph from ensemble A test: do resampled graphs look like original?

D

G G

Grassland species* plant

→ →

herbivore

→

parasite

*thank you: Jennifer Dunne

10 10

1

10

!3

10

!2

10

!1

10

a

Degree, k Fraction of vertices with degree k

Degree Distribution

resampled

→

• riginal

→

Clustering Coefficient

resampled

→

• riginal

→

0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25

Fraction of graphs with clustering coefficient c Clustering coefficient, c

resampled

→

• riginal

→

2 4 6 8 10 10

!3

10

!2

10

!1

10

b

Distance, d Fraction of vertex!pairs at distance d

Distance Distribution

resampled

→

• riginal

→

Missing Links

A test: can model predict missing links?

Predicting is Hard

• remove edges from
• how easy to guess a missing link?

n = 75 m = 113 pguess ≈ k n2 − m + k = O(n−2) k pguess = k/(2662 + k) G

• Given incomplete graph
• run MCMC to equilibrium
• then, over sampled , compute average

for links

• predict links with high values are missing

Test idea via leave-k-out cross-validation perfect accuracy: AUC = 1 no better than chance: AUC = 1/2

(i, j) ∈ G

Predicting Missing Links

D

pr G pr

Missing Structure

0.2 0.4 0.6 0.8 1 0.4 0.5 0.6 0.7 0.8 0.9 1 Area under ROC curve Fraction of edges observed, k/m Grassland species network Pure chance Common neighbors Jaccard coeff. Degree product Shortest paths Hierarchical structure

simple predictors

→

hierarchy

→

pure chance

→

AUC

0.2 0.4 0.6 0.8 1 0.4 0.5 0.6 0.7 0.8 0.9 1 AUC Fraction of edges observed Terrorist association network

a

Pure chance Common neighbors Jaccard coefficient Degree product Shortest paths Hierarchical structure

Other Networks

0.2 0.4 0.6 0.8 1 0.4 0.5 0.6 0.7 0.8 0.9 1 AUC Fraction of edges observed

• T. pallidum

metabolic network

b

Pure chance Common neighbors Jaccard coefficient Degree product Shortest paths Hierarchical structure

Summary

• Many real networks are hierarchically modular
• Hierarchies can
• model multi-scale structure
• generalize a single network
• predict missing links
• Model-based inference is very powerful

Acknowledgments:

• C. Moore, M.E.J. Newman, C.H. Wiggins, and C.R. Shalizi

Fin