1. Identifying local structure in large networks Reid Andersen Joint work with Fan Chung and Lincoln Lu

2. Many small world models presume a trivial underlying geometry. We address the problem of identifying local structure in an arbitrary network.

3. 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:

4. C × C nn is a (3,3)-local graph and a (4,5)-local graph C × C nn Example: Local graphs

6. GAVIN: A protein-protein interaction network |V|=1264 |E|=3294

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

12. 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:

13. A performance guarantee for EXTRACT ➘➘

14. ➘ ➘ local L global G hybrid graph H= L U G L’ recovered local graph EXTRACT applied to hybrid graphs

15. Recovery theorem: and G(w) satisfies then with probability If L is an (f, )-local graph with bounded degree (1) (II)

17. Spectral drawings of a grid and random edges Standard drawing Drawing that ignores edges removed by EXTRACT