# Incremental *k*-core decomposition: algorithms and evaluation

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## Abstract

A *k*-core of a graph is a maximal connected subgraph in which every vertex is connected to at least *k* vertices in the subgraph. *k*-core decomposition is often used in large-scale network analysis, such as community detection, protein function prediction, visualization, and solving NP-hard problems on real networks efficiently, like maximal clique finding. In many real-world applications, networks change over time. As a result, it is essential to develop efficient incremental algorithms for dynamic graph data. In this paper, we propose a suite of incremental *k*-core decomposition algorithms for dynamic graph data. These algorithms locate a small subgraph that is guaranteed to contain the list of vertices whose maximum *k*-core values have changed and efficiently process this subgraph to update the *k*-core decomposition. We present incremental algorithms for both insertion and deletion operations, and propose auxiliary vertex state maintenance techniques that can further accelerate these operations. Our results show a significant reduction in runtime compared to non-incremental alternatives. We illustrate the efficiency of our algorithms on different types of real and synthetic graphs, at varying scales. For a graph of 16 million vertices, we observe relative throughputs reaching a million times, relative to the non-incremental algorithms.

## Keywords

*k*-Core Streaming graph algorithms Dense subgraph discovery Incremental graph algorithms

## Notes

### Acknowledgments

This work is partially sponsored by the US Defense Advanced Research Projects Agency (DARPA) under the Social Media in Strategic Communication (SMISC) program (Agreement No. W911NF-12-C-0028). The views and conclusions contained in this document are those of the author(s) and should not be interpreted as representing the official policies, either expressed or implied, of DARPA or the US Government.This work is also partially sponsored by The Scientific and Technological Research Council of Turkey (TÜBİTAK) under Grant EEEAG #112E271.

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