# Distance-generalized Core Decomposition

## TL;DR

This paper introduces the $(k,h)$-core, a distance-based generalization of the $k$-core, preserving key properties and enabling efficient computation for large networks through novel bounds and parallelization.

## Contribution

It defines the $(k,h)$-core, analyzes its properties, and develops an efficient, parallelizable algorithm for large-scale networks.

## Key findings

- $(k,h)$-core preserves properties of classic core decomposition.
- The proposed algorithm enables scalable computation on large networks.
- The $(k,h)$-core aids in approximating dense structures like $h$-clubs.

## Abstract

The $k$-core of a graph is defined as the maximal subgraph in which every vertex is connected to at least $k$ other vertices within that subgraph. In this work we introduce a distance-based generalization of the notion of $k$-core, which we refer to as the $(k,h)$-core, i.e., the maximal subgraph in which every vertex has at least $k$ other vertices at distance $\leq h$ within that subgraph. We study the properties of the $(k,h)$-core showing that it preserves many of the nice features of the classic core decomposition (e.g., its connection with the notion of distance-generalized chromatic number) and it preserves its usefulness to speed-up or approximate distance-generalized notions of dense structures, such as $h$-club. Computing the distance-generalized core decomposition over large networks is intrinsically complex. However, by exploiting clever upper and lower bounds we can partition the computation in a set of totally independent subcomputations, opening the door to top-down exploration and to multithreading, and thus achieving an efficient algorithm.