Maintaining Large Dense Subgraphs on Dynamic Networks

1. Maintaining Large Dense Subgraphs on Dynamic Networks Atish Das Sarma Ashwin Lall Danupon Nanangkoi Amitabh Trehan

2. Density Network density is probably the most fundamental network metric for understanding how networks tick.... Third Degree Centrality (blog), June 16, 2011

3. Sparse but Dense The Internet World Wide Web Telephone call Graph Social Networks... • Globally Sparse but with dense substructures

4. Dense subgraphs • Web communities / Friend groups • Good structures to study network structure and dynamics e.g. link spam (Websites linking to each other) • Network backbones? • (Relatively) Robust over a dynamic network • Unlike most graph problems, (relatively) easy to approximate

5. Density • Density of Graph G(V,E): • Density of Subgraph S (= induced density on G):

6. Our problem • Efficient Distributed algorithms for discovering densest subgraphs/ bounded size densest subgraphs • Maintaining the subgraphs when edges change (Dynamic graphs)

7. Our Model • Initial Graph over n nodes • Edge Dynamic: At each time step, Adversary may add or removeupto r edges; Dynamic diameter D • After adversarial action, nodes communicate with direct neighbors • Communication: synchronous broadcast following CONGEST model (O(log n) size messages)

8. Our Results:At-least-k Densest Subgraph Problem • Subgraph should have at least k vertices: Discovery known to be NP-hard • Algorithm that w.h.p. the at-least-k densest subgraph at that time given the max at-least-k density is at least • Static graphs (r=0): A distributed algorithm that obtains w.h.p. a in O(D logn) rounds of the CONGEST model

9. Our Results:Densest Subgraph Problem • A distributed algorithm for any dynamic graph with dynamic diameter D and rate r that w.h.p. the densest subgraph at that time given the max density is at least • Static graphs (r=0): A distributed algorithm that obtains w.h.p. a in O(D logn) rounds of the CONGEST model (the first such distributed algorithms)

10. Additional details • Algorithms run continuously to maintain the approximations at all times • Self-awareness: Nodes are aware they are part of certain dense subgraphs • Nodes need knowledge of the dynamic diameter D

11. Outline: Algo Maintain() • Repeat forever ... • Algo computes set of nodes V, V’,.. and their induced Graph sizes (n,m), (n’,m’).... (V’ subset of V and so on...) • Uses approx-estimation algorithms to estimate number of nodes and edges (modified ELECT (AfekMatias’94) and Randomized ApproxCounting(KuhnLynchOshman’2010)

12. Outline: Algo DensestSubgraph(k) • Pick the densest subgraph of size from Maintain() • Calculate shortfall • Repeat a Padding algorithm till enough nodes: Each coin not in set flips a coin with prob. of head

13. Thank you Image: caseorganic@flickr.org , under creative commonss licence

14. Conclusions and Future Work • We provide distributed approximation algorithms for densest and at-least-k densest subgraph problems in the CONGEST model for both static and dynamic cases. • While most graph problems are hard to approximate even in the static case, density is a useful exception • Can we extend to node deletions, other density definitions, improve our upper bounds and provide a lower bound?

15. Density