# Seedless Graph Matching via Tail of Degree Distribution for Correlated Erdos-Renyi Graphs

## TL;DR

This paper introduces a seedless network alignment algorithm that leverages the tail of degree distributions to match nodes in correlated Erdos-Renyi graphs, outperforming previous methods on synthetic and real networks.

## Contribution

The proposed algorithm uniquely uses degree distribution tails for seedless graph matching, eliminating the need for auxiliary information.

## Key findings

- Outperforms previous methods in correct matching probability.
- Effective on both synthetic Erdos-Renyi and real networks.
- Works in sparse graph regimes where recovery is theoretically feasible.

## Abstract

The network alignment (or graph matching) problem refers to recovering the node-to-node correspondence between two correlated networks. In this paper, we propose a network alignment algorithm which works without using a seed set of pre-matched node pairs or any other auxiliary information (e.g., node or edge labels) as an input. The algorithm assigns structurally innovative features to nodes based on the tail of empirical degree distribution of their neighbor nodes. Then, it matches the nodes according to these features. We evaluate the performance of proposed algorithm on both synthetic and real networks. For synthetic networks, we generate Erdos-Renyi graphs in the regions of $\Theta(\log(n)/n)$ and $\Theta(\log^{2}(n)/n)$, where a previous work theoretically showed that recovering is feasible in sparse Erdos-Renyi graphs if and only if the probability of having an edge between a pair of nodes in one of the graphs and also between the corresponding nodes in the other graph is in the order of $\Omega(\log(n)/n)$, where $n$ is the number of nodes. Experiments on both real and synthetic networks show that it outperforms previous works in terms of probability of correct matching.