In this paper, we investigate the relationship between the coupling strengths and the extensive behaviour of the sum of the positive Lyapunov exponents of multiplex networks formed by coupled dynamical units. Considering networks where the dynamics of the nodes is given by the shift map, we do not only demonstrate which are the relevant parameters leading to extensivity, but also provide exact formulas how they are related. A distinct result was to show that it is always possible to construct infinitely large extensive networks by attaching, with rescaled inter-connections, infinitely many smaller networks. These smaller networks are effectively the building blocks of the large network. This is because these building blocks can have arbitrary topology and the strength of connections among nodes only depends on the block size, and not on the size of the whole network.
Bibliographical noteAcknowledgements: We thank Alessandro Gillies for the productive discussions.
Funding: MAA acknowledges the University of Aberdeen for her Elphinstone PhD scholarship.
Contributions: This article is a product of MAA’s PhD supervisioned by MSB. MAA developed the theoretical formalism, the analytical calculations, performed the numerical simulations and wrote the manuscript. MSB proposed the study and contributed towards the execution and the writing of this work. Both authors read and approved the final manuscript.
- Lyapunov exponents
- Multiplex networks