title: Multiplex congruence network of natural numbers.
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 ordinary 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.
authors.affiliation: Systems Science Institute, Beijing Jiaotong University, Beijing 100044, China.|Big Data Research Center, University of Electronic Science and Technology of China, Chengdu 611731, China.
authors.forename: Xiao-Yong
authors.initials: XY
authors.lastname: Yan
authors.affiliation: School of Systems Science, Beijing Normal University, Beijing 100875, China.|Business School, University of Shanghai for Science and Technology, Shanghai 200093, China.
authors.forename: Wen-Xu
authors.initials: WX
authors.lastname: Wang
authors.affiliation: Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China.
authors.forename: Guan-Rong
authors.initials: GR
authors.lastname: Chen
authors.affiliation: Department of Mathematics, Shanghai University, Shanghai 200444, China.
authors.forename: Ding-Hua
authors.initials: DH
authors.lastname: Shi