# Universal vulnerability in strong modular networks with various degree distributions between inequality and equality

## TL;DR

This paper reveals a universal vulnerability in modular networks with various degree distributions, showing that adding community structures can drastically reduce network robustness against failures.

## Contribution

It uncovers a previously unknown vulnerability in well-structured modular networks, highlighting the impact of community structures on network robustness.

## Key findings

- Equal degree networks are more tolerant to node failures.
- Adding strong community structures increases network vulnerability.
- Vulnerability is universal across different degree distributions.

## Abstract

Generally, networks are classified into two sides of inequality and equality with respect to the number of links at nodes by the types of degree distributions. One side includes many social, technological, and biological networks which consist of a few nodes with many links, and many nodes with a few links, whereas the other side consists of all nodes with an equal number of links. In comprehensive investigations between them, we have found that, as a more equal network, the tolerance of whole connectivity is stronger without fragmentation against the malfunction of nodes in a wide class of randomized networks. However, we newly find that all networks which include typical well-known network structures between them become extremely vulnerable, if a strong modular (or community) structure is added with commonalities of areas, interests, religions, purpose, and so on. These results will encourage avoiding too dense unions by connecting nodes and taking into account the balanced resource allocation between intra- and inter-links of weak communities. We must reconsider not only efficiency but also tolerance against attacks or disasters, unless no community that is really impossible.