# Graph Product Multilayer Networks: Spectral Properties and Applications

@article{Sayama2018GraphPM, title={Graph Product Multilayer Networks: Spectral Properties and Applications}, author={Hiroki Sayama}, journal={J. Complex Networks}, year={2018}, volume={6}, pages={430-447} }

This paper aims to establish theoretical foundations of graph product multilayer networks (GPMNs), a family of multilayer networks that can be obtained as a graph product of two or more factor networks. Cartesian, direct (tensor), and strong product operators are considered, and then generalized. We first describe mathematical relationships between GPMNs and their factor networks regarding their degree/strength, adjacency, and Laplacian spectra, and then show that those relationships can still…

## Figures and Tables from this paper

## 4 Citations

### Tensor-based Spectral Analysis of Cascading Failures over Multilayer Complex Systems

- Computer Science2018 56th Annual Allerton Conference on Communication, Control, and Computing (Allerton)
- 2018

This paper proposes a scalable tensor-based framework to represent the interdependent multilayer network, and uses this framework to analyze the failure propagation based on a susceptible-infectious-susceptible (SIS) epidemic model.

### Graph Product Representation of Organism-Environment Couplings in Evolution

- Computer ScienceArtificial Life Conference Proceedings
- 2019

We present a theoretical framework that mathematically formulates the evolutionary dynamics of organism-environment couplings using graph product multilayer networks, i.e., networks obtained by…

### Adjacency and Laplacian spectra of variants of neighborhood corona of graphs constrained by vertex subsets

- MathematicsDiscret. Math. Algorithms Appl.
- 2019

Some variants of corona of graphs namely, subdivision, neighborhood corona, R-graph (respectively, Q-graph, total) semi-edge neighbor, and subdivision and subdivision of graphs are defined.

### Fully solvable lower dimensional dynamics of Cartesian product of Kuramoto models

- Computer ScienceNew Journal of Physics
- 2019

This work proposes to construct complex synchronization dynamics by applying the Cartesian product of two Kuramoto models on two star networks, and obtains coupling regimes where cluster synchronization states are often present on the product graph and the number of clusters is fully controlled.

## References

SHOWING 1-10 OF 35 REFERENCES

### Estimation of Laplacian spectra of direct and strong product graphs

- MathematicsDiscret. Appl. Math.
- 2016

### Multilayer networks

- Computer ScienceJ. Complex Networks
- 2014

This chapter shows how interconnected multilayer topology describes such networks more accurately than edge coloring does and introduces the tensor formalism used to construct them.

### Mathematical Formulation of Multilayer Networks

- Computer Science
- 2013

This paper introduces a tensorial framework to study multilayer networks, and discusses the generalization of several important network descriptors and dynamical processes—including degree centrality, clustering coefficients, eigenvectorcentrality, modularity, von Neumann entropy, and diffusion—for this framework.

### Kronecker Graphs: An Approach to Modeling Networks

- Computer ScienceJ. Mach. Learn. Res.
- 2010

Kronecker graphs naturally obey common network properties and it is rigorously proved that they do so, and KRONFIT, a fast and scalable algorithm for fitting the Kronecker graph generation model to large real networks, is presented.

### Spectral Graph Theory of the Hypercube

- Mathematics, Computer Science
- 2008

Spectral Graph Theory focuses on the set of eigenvalues and eigenvectors of these matrices of adjacency or Laplacian matrices, and provides several interesting areas of study, including the inverse eigenvalue problem of a graph.

### Eigenvalue spectra of modular networks.

- MathematicsPhysical review letters
- 2013

This work obtains in a unified fashion the spectrum of a large family of operators, including the adjacency, Laplacian, and normalized LaPLacian matrices, for networks with generic modular structure, in the limit of large degrees.

### Spectra of "real-world" graphs: beyond the semicircle law.

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2001

Methods to determine the eigenvalues of networks comparable in size to real systems are developed, obtaining several surprising results on the spectra of adjacency matrices corresponding to models of real-world graphs.

### Graph spectra and the detectability of community structure in networks

- Computer SciencePhysical review letters
- 2012

Using methods from random matrix theory, the spectra of networks that display community structure are calculated, and it is shown that spectral modularity maximization is an optimal detection method in the sense that no other method will succeed in the regime where the modularity method fails.

### The physics of spreading processes in multilayer networks

- Computer Science
- 2016

Progress is surveyed towards attaining a deeper understanding of spreading processes on multilayer networks, and some of the physical phenomena related to spreading processes that emerge from multilayered structure are highlighted.