Multiplex congruence network of natural numbers

TL;DR

This paper introduces a multiplex network model based on congruence relations among natural numbers, revealing unique topological features, controllability properties, and potential cryptographic applications.

Contribution

It uncovers the topological and controllability properties of multiplex congruence networks, a novel approach linking number theory with network science.

Findings

Each layer is a sparse, scale-free network.

Layers exhibit strong controllability despite scale-free structure.

Controllability is robust to targeted attacks but vulnerable to random failures.

Abstract

Congruence theory has many applications in physical, social, biological and technological systems. Congruence arithmetic has been a fundamental tool for data security and computer algebra. However, much less attention was devoted to the topological features of congruence relations among natural numbers. Here, we explore the congruence relations in the setting of a multiplex network and unveil some unique and outstanding properties of the multiplex congruence network. Analytical results show that every layer therein is a sparse and heterogeneous subnetwork with a scale-free topology. Counterintuitively, every layer has an extremely strong controllability in spite of its scale-free structure that is usually difficult to control. Another amazing feature is that the controllability is robust against targeted attacks to critical nodes but vulnerable to random failures, which also differs from normal scale-free networks. The multi-chain structure with a small number of chain roots arising from each layer accounts for the strong controllability and the abnormal feature. The multiplex congruence network offers a graphical solution to the simultaneous congruences problem, which may have implication in cryptography based on simultaneous congruences. Our work also gains insight into the design of networks integrating advantages of both heterogeneous and homogeneous networks without inheriting their limitations.