# Local Algorithms for Hierarchical Dense Subgraph Discovery

## TL;DR

This paper introduces local, scalable algorithms for hierarchical dense subgraph discovery, generalizing core, truss, and nucleus decompositions with convergence guarantees, enabling efficient parallel processing of large networks.

## Contribution

It generalizes iterative $h$-index methods for truss and nucleus decompositions, providing convergence bounds and a local framework for scalable, parallel dense subgraph analysis.

## Key findings

- Algorithms are highly scalable and parallelizable.
- Effective in real-world network analysis.
- Convergence bounds are established.

## Abstract

Finding the dense regions of a graph and relations among them is a fundamental problem in network analysis. Core and truss decompositions reveal dense subgraphs with hierarchical relations. The incremental nature of algorithms for computing these decompositions and the need for global information at each step of the algorithm hinders scalable parallelization and approximations since the densest regions are not revealed until the end. In a previous work, Lu et al. proposed to iteratively compute the $h$-indices of neighbor vertex degrees to obtain the core numbers and prove that the convergence is obtained after a finite number of iterations. This work generalizes the iterative $h$-index computation for truss decomposition as well as nucleus decomposition which leverages higher-order structures to generalize core and truss decompositions. In addition, we prove convergence bounds on the number of iterations. We present a framework of local algorithms to obtain the core, truss, and nucleus decompositions. Our algorithms are local, parallel, offer high scalability, and enable approximations to explore time and quality trade-offs. Our shared-memory implementation verifies the efficiency, scalability, and effectiveness of our local algorithms on real-world networks.