# The Intrinsic Scale of Networks is Small

## TL;DR

This paper introduces the concept of intrinsic scale in networks, revealing that many networks have a surprisingly small scale at which their unique structure becomes distinguishable, and this scale varies with network type and robustness.

## Contribution

The paper defines the intrinsic scale of networks and provides an extensive empirical study across diverse network types, highlighting its small size and relation to network structure.

## Key findings

- Intrinsic scale is small (7-20 vertices) across various networks.
- Networks designed for specific tasks tend to have smaller intrinsic scales.
- Network structure at different scales can be fragile or robust.

## Abstract

We define the intrinsic scale at which a network begins to reveal its identity as the scale at which subgraphs in the network (created by a random walk) are distinguishable from similar sized subgraphs in a perturbed copy of the network. We conduct an extensive study of intrinsic scale for several networks, ranging from structured (e.g. road networks) to ad-hoc and unstructured (e.g. crowd sourced information networks), to biological. We find: (a) The intrinsic scale is surprisingly small (7-20 vertices), even though the networks are many orders of magnitude larger. (b) The intrinsic scale quantifies ``structure'' in a network -- networks which are explicitly constructed for specific tasks have smaller intrinsic scale. (c) The structure at different scales can be fragile (easy to disrupt) or robust.