# Robust Clustering Oracle and Local Reconstructor of Cluster Structure of Graphs

## TL;DR

This paper introduces sublinear time algorithms for analyzing and reconstructing the cluster structure of large, noisy graphs using conductance-based definitions, enabling efficient local clustering and property testing.

## Contribution

It formalizes noisy clusterable graphs, develops a robust clustering oracle, and provides a local reconstructor, all operating in sublinear time with noisy data.

## Key findings

- Developed a sublinear time algorithm for analyzing cluster structure.
- Constructed a robust clustering oracle supporting typical cluster queries.
- Designed a local reconstructor for noisy clusterable graphs.

## Abstract

Due to the massive size of modern network data, local algorithms that run in sublinear time for analyzing the cluster structure of the graph are receiving growing interest. Two typical examples are local graph clustering algorithms that find a cluster from a seed node with running time proportional to the size of the output set, and clusterability testing algorithms that decide if a graph can be partitioned into a few clusters in the framework of property testing. In this work, we develop sublinear time algorithms for analyzing the cluster structure of graphs with noisy partial information. By using conductance based definitions for measuring the quality of clusters and the cluster structure, we formalize a definition of noisy clusterable graphs with bounded maximum degree. The algorithm is given query access to the adjacency list to such a graph. We then formalize the notion of robust clustering oracle for a noisy clusterable graph, and give an algorithm that builds such an oracle in sublinear time, which can be further used to support typical queries (e.g., IsOutlier($s$), SameCluster($s,t$)) regarding the cluster structure of the graph in sublinear time. All the answers are consistent with a partition of $G$ in which all but a small fraction of vertices belong to some good cluster. We also give a local reconstructor for a noisy clusterable graph that provides query access to a reconstructed graph that is guaranteed to be clusterable in sublinear time. All the query answers are consistent with a clusterable graph which is guaranteed to be close to the original graph.