Multiplex congruence network of natural numbers

Area of Science:

• Number Theory
• Network Science
• Complex Systems

Background:

• Congruence theory is vital in various scientific and technological fields, including data security and computer algebra.
• Topological features of congruence relations on natural numbers have been underexplored.
• Multiplex networks offer a framework to study complex interrelations.

Purpose of the Study:

• To explore congruence relations within a multiplex network framework.
• To analyze the topological properties and controllability of multiplex congruence networks.
• To investigate the implications for cryptography and network design.

Main Methods:

• Analysis of congruence relations structured as multiplex networks.
• Characterization of network topology, including sparsity, heterogeneity, and scale-free properties.
• Evaluation of network controllability and robustness against targeted and random failures.

Main Results:

• Each layer of the multiplex congruence network is a sparse, heterogeneous, scale-free subnetwork.
• Networks demonstrate exceptionally strong controllability, counterintuitive for scale-free structures.
• Controllability is robust to targeted attacks but vulnerable to random failures, differing from typical scale-free networks.
• A multi-chain structure with few chain roots explains the observed controllability and robustness.

Conclusions:

• The multiplex congruence network provides a graphical solution for simultaneous congruences, with potential cryptographic applications.
• The findings offer insights into designing hybrid networks that combine heterogeneous and homogeneous network advantages.
• This research bridges number theory and network science, revealing unique network dynamics and control properties.