Testing Cluster Structure of Graphs

We study the problem of recognizing the cluster structure of a graph in the framework of property testing in the bounded degree model. Given a parameter $\varepsilon$, a $d$-bounded degree graph is defined to be $(k, \phi)$-clusterable, if it can be partitioned into no more than $k$ parts, such that the (inner) conductance of the induced subgraph on each part is at least $\phi$ and the (outer) conductance of each part is at most $c_{d,k}\varepsilon^4\phi^2$, where $c_{d,k}$ depends only on $d,k$. Our main result is a sublinear algorithm with the running time $\widetilde{O}(\sqrt{n}\cdot\mathrm{poly}(\phi,k,1/\varepsilon))$ that takes as input a graph with maximum degree bounded by $d$, parameters $k$, $\phi$, $\varepsilon$, and with probability at least $\frac23$, accepts the graph if it is $(k,\phi)$-clusterable and rejects the graph if it is $\varepsilon$-far from $(k, \phi^*)$-clusterable for $\phi^* = c'_{d,k}\frac{\phi^2 \varepsilon^4}{\log n}$, where $c'_{d,k}$ depends only on $d,k$. By the lower bound of $\Omega(\sqrt{n})$ on the number of queries needed for testing graph expansion, which corresponds to $k=1$ in our problem, our algorithm is asymptotically optimal up to polylogarithmic factors.Comment: Full version of STOC 201

Partitioning Well-clustered Graphs with k-Means and Heat Kernel

We study a suitable class of well-clustered graphs that admit good k-way partitions and present the first almost-linear time algorithm for with almost-optimal approximation guarantees partitioning such graphs. A good k-way partition is a partition of the vertices of a graph into disjoint clusters (subsets) $\{S_i\}_{i=1}^k$, such that each cluster is better connected on the inside than towards the outside. This problem is a key building block in algorithm design, and has wide applications in community detection and network analysis. Key to our result is a theorem on the multi-cut and eigenvector structure of the graph Laplacians of these well-clustered graphs. Based on this theorem, we give the first rigorous guarantees on the approximation ratios of the widely used k-means clustering algorithms. We also give an almost-linear time algorithm based on heat kernel embeddings and approximate nearest neighbor data structures

Spectral concentration, robust k-center, and simple clustering

Non UBCUnreviewedAuthor affiliation: The Ohio State UniversityFacult

Constant factor approximation for balanced cut in the PIE model Tasos Sidiropoulos: Spectral concentration, robust k-center, and simple clustering

Non UBCUnreviewedAuthor affiliation: Microsoft ResearchOthe