# Network Comparison: Embeddings and Interiors

## TL;DR

This paper introduces a method to compare networks by embedding them into a metric space, explores the concept of network interiors to improve approximation, and demonstrates practical discrimination of different network models.

## Contribution

It defines a valid network distance metric based on optimal embeddings and introduces interior-based methods to enhance comparison accuracy.

## Key findings

- Network distance is a valid metric in network space.
- Interior comparison aligns with actual network distances.
- Method effectively discriminates between different network models.

## Abstract

This paper presents methods to compare networks where relationships between pairs of nodes in a given network are defined. We define such network distance by searching for the optimal method to embed one network into another network, prove that such distance is a valid metric in the space of networks modulo permutation isomorphisms, and examine its relationship with other network metrics. The network distance defined can be approximated via multi-dimensional scaling, however, the lack of structure in networks results in poor approximations. To alleviate such problem, we consider methods to define the interiors of networks. We show that comparing interiors induced from a pair of networks yields the same result as the actual network distance between the original networks. Practical implications are explored by showing the ability to discriminate networks generated by different models.