Testing Cluster Structure of Graphs

1. Testing Cluster Structure of Graphs Artur Czumaj DIMAPandDepartment of Computer Science University of Warwick Joint work with Pan Peng and Christian Sohler (TU Dortmund) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAA

2. Dealing with “BigData” in Graphs • We want to process graphs quickly • Detect basic properties • Analyze their structure • For large graphs, by “quickly” we often would mean: in time constant orsublinear in the size of the graph

3. Dealing with “BigData” in Graphs One approach: • How to test basic properties of graphs in the framework of property testing

4. Fast Testing of Graph Properties • Does this graph have a clique of size 11? • Does it have a given as its subgraph? • Is this graph planar? • Is it bipartite? • Is it -colorable? • Does it have good expansion? • Does it have good clustering? from Fan Chung’s web page

5. Clustering in graphs • What is a good clustering? from Fan Chung’s web page

6. Clustering in graphs • Same cluster: points are well-connected • Different cluster: points are poorly connected from Fan Chung’s web page

7. Clustering in graphs parameterized complexity Sublinear algorithms

8. Clustering in graphs • Same cluster: points have high conductance • Different cluster: points are separated by a cut from Fan Chung’s web page

9. -clustering Graph is -clusterable if vertices of can be partitioned into at most sets such that for each : • each has large conductance • each set has low outer-conductance (small cut)

10. Conductance has maximum degree For every , conductance of S is defined as: Conductance of : (minimum conductance of any possible subset of size at most )

11. -clustering is -clusterableif vertices of can be partitioned into with such that for each : • each has large conductance • each set has low outer-conductance (small cut)

12. -clustering Notion of -clusterable graphs has been around informally for a while; formally introduced by OveisGharan and Trevisan 2014 Our goal: • we want to determine if is -clusterable • really fast • Recognize cluster structure in sublinear time using random sampling

13. Framework of property testing • We cannot quickly give 100% precise answer • We need to approximate • Distinguish graphs that have specific property from those that are far from having the property

14. Framework • Goal: Distinguish between the case when • graph has property P and • is far from having property P • one has to change at least edges of to obtain a graph with property We will consider graph of degree bounded by

15. Goal Design a sublinear-time algorithm that will distinguish between two cases: • -clusterablegraphs • graphs that are -far from being -clusterable(with as possible)

16. Basic case : Testing expansion -clusterablegraphs for : expanders • For graphs of bounded degree, we can distinguish expanders from graphs that are “far” even from poor expanders in time [C, Sohler ’07, Kale, Seshadhri’07, Nachmias, Shapira’08] • time is needed [Goldreich, Ron’02]

17. Basic case : Testing expansion • For graphs of bounded degree, we can distinguish expanders from graphs that are “far” even from poor expanders in time [C, Sohler ’07, Kale, Seshadhri’07, Nachmias, Shapira’08] • We are using basic properties of expanders: random walk of logarithmic length will mix (= will reach a random vertex) • Similar to testing uniformity of a distribution

18. Testing expansion • Choose nodes at random • For each chosen node run random walks of length • Count the number of collisions at the end-nodes • If the number of collisions is too large then Reject • Accept Idea: • If is an expander then end-nodes are random nodes e we can estimate number of collisions well • If is far from expander then we will have many more collisions • (requires non-trivial arguments)

19. Basic case : Testing expansion • For graphs of bounded degree, we can distinguish expanders from graphs that are “far” even from poor expanders in time [C, Sohler ’07, Kale, Seshadhri’07, Nachmias, Shapira’08] • We can distinguish between a graph that is an -expander and any graph that is -far from any -expander in time

20. Testing expansion and clustering Can we apply similar approach to test -clusterability for ? • We don’t know which vertex sets form expanders • We don’t know sizes of subgraphs-expanders • If we knew, we could try to test distributions of endpoints of random walks … • How to test small cuts? • We don’t know how to test distributions in time! • We know this only for uniform distributions – but since we don’t know which vertices are in each set, we won’t get it …

21. Testing expansion and clustering Can we apply similar approach to test -clusterability for ? Still, we will following the following key intuitions: • Randomly sample a constant number of points • Points in will define a “skeleton” of clusters • If two points will have same distribution of end-points of random walks of logarithmic length  are in same cluster • If two points are separated by a cut then they will have different distribution of end-points of random walks

22. Testing -clustering • We would like to have the following algorithm: • Sample set of random vertices • For any • distributions of endpoints of random walk of length starting at • For each pair : • if distributions , are close then • add edge to “cluster graph” on vertex set • If H is a union of at most k cliques then Accept • Else Reject

23. Testing -clustering • Sample set of random vertices • For any • distributions of endpoints of random walk of length starting at • For each pair : • if distributions , are close then • add edge to “cluster graph” on vertex set • If H is a union of at most k cliques then Accept • Else Reject Challenges: • Too slow (testing if distribution are closed needs time) • How to analyze it … • We understand random walks in expanders, but we need to understand them also on the rest of the graph

24. Testing -clustering • Sample set of random vertices • For any • multiset of endpoints of random walks of length starting at • number of pairwise self-collisions in • If there is with then abort and Reject • For each pair : • If -distribution-test accepts that distributions of and are close then add edge to “cluster graph” • If H is a union of at most k cliques then Accept • Else Reject

25. Testing -clustering • Sample set of random vertices • For any • multiset of endpoints of random walks of length starting at • number of pairwise self-collisions in • If there is with then abort and Reject • For each pair : • If -distribution-test accepts that distributions of and are close then add edge to “cluster graph” • If H is a union of at most k cliques then Accept • Else Reject Why do we need to test if is not too large? Because then -distribution testing runs fast!

26. Key theorem • The algorithm accepts every -clusterable graph (of maximum degree ) with probability . • The algorithm rejects every graph (of maximum degree ) that is -far from being -clusterable with probability , assuming that . • Running time is

27. Key properties (completeness) • - vertex distribution of a random walk of length starting at • If is -clusterablethen for any with and there is with such that for large enough and , for any : • For “short enough” , for any disjoint sets with , there exist , , such that for any • If is -clusterable then there is with such that for large enough , for any :

28. Key properties (completeness) How will the distribution of endpoints differ (for clusterable graphs)? For most of starting vertices : • Convergence within a cluster We prove this using higher-order Cheeger’s inequality. Key challenge: even if expect to stay inside the cluster in a large fraction of random walks, we can sometime leave the cluster and then we don’t have “easy” control over the distribution of endpoints.

29. Key properties (completeness) • - vertex distribution of a random walk of length starting at • If is -clusterablethen for any with and there is with such that for large enough and , for any : • For “short enough” , for any disjoint sets with , there exist , , such that for any • If is -clusterable then there is with such that for large enough , for any :

30. Key properties (completeness) How will the distribution of endpoints differ? For most of starting vertices : • Convergence outside a cluster

31. Key properties (completeness) • - vertex distribution of a random walk of length starting at • If is -clusterablethen for any with and there is with such that for large enough and , for any : • For “short enough” , for any disjoint sets with , there exist , , such that for any • If is -clusterable then there is with such that for large enough , for any :

32. Key properties (completeness) Can the distribution vector of endpoints have big values? For most of starting vertices : • Bound for distribution for clusterable graphs

33. Key properties (completeness) • - vertex distribution of a random walk of length starting at • If is -clusterablethen for any with and there is with such that for large enough and , for any : • For “short enough” , for any disjoint sets with , there exist , , such that for any • If is -clusterable then there is with such that for large enough , for any :

34. Key properties (completeness) With these properties: • If is clusterable then the cluster graph will consist of at most disjoint subgraphs, each forming a clique

35. Key properties (soudness) • If is -far from -clusterablewith then there are disjoint subsets of such that for each : and

36. Testing -clustering • Sample set of random vertices • For any • multiset of endpoints of random walks of length starting at • number of pairwise self-collisions in • If there is with then abort and Reject • For each pair : • If -distribution-test accepts that distributions of and are close then add edge to “cluster graph” • If H is a union of at most k cliques then Accept • Else Reject

37. Key properties (soudness) • If is -far from -clusterablewith then there are disjoint subsets of such that for each : and With this property, if is -far from clusterable then the cluster graph will have more than k components

38. Key theorem • The algorithm accepts every -clusterable graph (of maximum degree ) with probability . • The algorithm rejects every graph (of maximum degree ) that is -far from being -clusterable with probability , assuming that . • Running time is

39. Extensions Since -clustering is related to some properties of smallest eigenvalues of the relevant Laplacian matrix: • We can recognize graphs with a (large enough) gap between the kth and k+1st smallest eigenvalue.

40. Conclusions Clustering (or clusterability) can be tested fast • by comparing distributions of random walks • drawing conclusions from the distributions Tools: • Random sampling • Random walks • Spectral analysis