1. Energy Models for Graph Clustering Bo-Young Kim Applied Algorithm Lab, KAIST

2. Introduction  Graph clusters : groups of densely connected nodes  A common structure shared by many real-world system  Decomposed into subsystems : strong intra-subsystem interaction, weak inter-subsystem interaction. e.g. groups of friends, cohesive modules in software systems.  Graph layout : nodes → positions in low-dimensional Euclidean space  Goal: Introduce and evaluate measures(=energy models) that quantify how well a given graph layout reflects the graph clusters  Node-repulsion LinLog  Edge-repulsion LinLog

3. Introduction  Interpretability vs. Readability  The most energy based graph layout technique : produce easily readable box-and-line visualization of graphs e.g. Goal of Fruchterman and Reingold model(1991)  Interpretable layout : positions of Euclidean distances of nodes reflect certain properties of the graph The density of subgraphs (This case) Graph-theoretic distances of nodes Direction of edges in directed graphs

4. Basic Definitions  A graph G=(V,E)  Assume that connected, undirected, no edge weight  A d-dimensional layout of the graph G a vector of node positions  The distance of u and v in p :

5. Clustering Criteria  A subgraph with many internal edges and few edges to the remaining graph  a graph cluster  Coupling measures: a smaller coupling  a better clustering  The cut  The node-normalized cut  The edge-normalized cut  Shi and Malik’s normalized cut  Expansion  Conductance  Newman’s modularity

6. The Cut  The cut  for two disjoint sets of nodes V 1 and V 2  Expected cut between V 1 and V 2 :  Biased towards bipartitions : smaller with | V 1 | ≪ | V 2 | than | V 1 | = | V 2 |

7. The Node-Normalized Cut  The node-normalized cut (=density of the cut, the ratio of the cut)  Expected value:  Still biased towards bipartitions (measure of size : # edge) smaller with deg(V 1 ) ≪ deg(V 2 ) than deg(V 1 ) = deg(V 2 ) ( ∵ Expected value: )

8. The Edge-Normalized Cut  The edge-normalized cut  Expected value:

9. Other Clustering Criteria  Shi and Malik’s normalized cut   biased toward small clusters when is not fixed.

10. Other Clustering Criteria  The expansion  The conductance   Bias The expansion : biased towards similarly-sized clusters The conductance : similar bias (cluster size measure : total deg)

11. Other Clustering Criteria  Newman’s Modularity  Two sets case  If V 1 ∪ V 2 =V,  maximizing Q’(V 1,V 2 ) = minimizing

12. The LinLog Energy Models  The node-repulsion LinLog energy of a layout p  The edge-repulsion LinLog energy of a layout p  Symmetry : Edges cause both attraction and repulsion

13. Separation of Clusters  The arithmetic mean and the geometric mean  G=(V,E), F ⊆ V (2), a layout p   Degree-weighted geometric mean :

14. Separation of Clusters (Proof) Let the layout p 0 be a solution of the minimization problem: Let  ⇔ ⇔ ∙∙∙(1)

15. Separation of Clusters ⇔ ⇔ ∙∙∙(2)

16. Separation of Clusters (Proof) Similar to the Theorem 1.

17. Interpretable Distance between Clusters (Proof) p 0 : a layout in P with minimum node-repulsion LinLog energy d 0 : the distance of V 1 and V 2 in p 0. The distance between all node in V 1 (resp. V 2 ) = 0  d 0 : a minimum of

18. Interpretable Distance between Clusters (Proof) Similar to the Theorem 3.

19. Example 1. Airline Routing

20. Example 2. World Trade

21. Example 3. Dictionary

22. Thank you