Graph clustering has many important applications in computing, but due to the increasing sizes of graphs, even traditionally fast clustering methods can be computationally expensive for real-world graphs of interest. Scalability problems led to the development of local graph clustering algorithms that come with a variety of theoretical guarantees. Rather than return a global clustering of the entire graph, local clustering algorithms return a single cluster around a given seed node or set of seed nodes. These algorithms improve scalability because they use time and memory resources that depend only on the size of the cluster returned, instead of the size of the input graph. Indeed, for many of them, their running time grows linearly with the size of the output. In addition to scalability arguments, local graph clustering algorithms have proven to be very useful for identifying and interpreting small-scale and meso-scale structure in large-scale graphs. As opposed to heuristic operational procedures, this class of algorithms comes with strong algorithmic and statistical theory. These include statistical guarantees that prove they have implicit regularization properties. One of the challenges with the existing literature on these approaches is that they are published in a wide variety of areas, including theoretical computer science, statistics, data science, and mathematics. This has made it difficult to relate the various algorithms and ideas together into a cohesive whole. We have recently been working on unifying these diverse perspectives through the lens of optimization as well as providing software to perform these computations in a cohesive fashion. In this note, we provide a brief introduction to local graph clustering, we provide some representative examples of our perspective, and we introduce our software named Local Graph Clustering (LGC).
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