Identifying local structure in large networks

Reid Andersen

Joint work with Fan Chung and Lincoln Lu

• Many small world models presume a trivial underlying geometry.

• We address the problem of identifying local structure in an arbitrary network.

• We introduce hybrid graphs, a random graph model with a power law

degree dist. and planted local structure represented by a local graph.

• Our algorithm Extract computes the largest local graph in a given network

with specified parameters.

• Existing clustering algorithms aren’t designed to identify local structure.• Existing performance guarantees are not suitable for this problem.

• We show that Extract approximately recovers planted local structure in

hybrid graphs• Improved approximation algorithm for max short flow.• New bounds for short versions of the max flow - min cut theorem.• Bounds on neighborhood growth in the hybrid graph model.

Also:

How can we identify local structure?

• We use short network flow to identify subgraphs with high local connectivity, which we call local graphs.

Outline:

C × Cn n

is a (3,3)-local graph

and a (4,5)-local graph

C × Cn n

Example:

Local graphs

GAVIN: A protein-protein interaction network

|V|=1264|E|=3294

Extract can be combined with various graph drawing algorithms to produce improved

drawings

Approximation algorithms for Max Short Flow

[Garg, Könemann 97]:• Algorithms for multicommodity flow, fractional packing.• Ratio

[Shahrokhi, Matula 89] and [Plotkin, Shmoys, Tardos 91]:• Introduced exponential length function technique for multi-flows

History:

[Fleischer, Skutella 02]:• Used Max Short Flow to approximate Quickest Multicommodity Flow.• Solve Max Short Flow using:

Ellipsoid method,Fractional packing.

• Adapted from multicommodity flow algorithm of [GK97].• Same approximation ratio.• Improved running time for testing local connectivity:

Our algorithm:

A performance guarantee for EXTRACT

➘➘

➘➘local

L

global G

hybrid graphH= L U G

L’

recovered local graph

EXTRACT applied to hybrid graphs

Recovery theorem:

and G(w) satisfies

then with probability

If L is an (f, )-local graph with bounded degree

(1)

(II)

Spectral drawings of a grid and random edges

Standard drawing

Drawing that ignores

edges removed by EXTRACT