BuildSupraTransitionMatrixFromSupraAdjacencyMatrix: Build the supra-transition matrix for a multilayer network

In manlius/muxViz: MuxViz - Visualization and Analysis of Multilayer Networks

Build the supra-transition matrix for a multilayer network

Description

Given a supra-adjacency matrix, build supra-transition the matrix corresponding to different type of random walks (discrete time Markov chians). The types of RW considered here are the same as in Ref.2-Supplementary Material: classical, diffusive, PageRank, physical, physical with relaxation and maximum-entropy.

The maximum entropy random walk (MERW) choses the stochastic matrix which maximises H(S), so that the walker can explore every walk of the same length with equal probability. Let λ_N, φ be the leading eigenvalue and corresponding right eigenvector of the adjacency matrix A. Then T_{ij} = \frac{A_{ij}}{λ_N}\frac{φ_j}{φ_i}.

Usage

BuildSupraTransitionMatrixFromSupraAdjacencyMatrix(
  SupraAdjacencyMatrix,
  Layers,
  Nodes,
  Type = "pagerank",
  r = NULL
)

Arguments

SupraAdjacencyMatrix	the supra-adjacency matrix, which contains the information in intra-layer connectivity \{W^{α}_{ij}\} and on inter-layer coupling D^{α β}_{i}\}_i.
Layers	integer, number of layers
Nodes	integer, number of nodes
Type	the type of the random walk, default "pagerank". Other types: c("classical", "diffusive", "maxent", "physical", "relaxed-physical"). For details on the transition rules see Refs.2-3.
r	relaxation parameter, should be a float in the range [0, 1]. Default is NULL, which implies r=0.5 for the physical random walk with relaxation (Type = "relaxed-physical").

Value

the supra-transition matrix of probabilities for a discrete-time random walk on the given multi-layer network.

References

1. De Domenico, M., Set al. (2013). Mathematical formulation of multilayer networks. Physical Review X, 3(4), 041022. doi 10.1103/PhysRevX.3.041022

2. De Domenico, M., et al. (2014). Navigability of interconnected networks under random failures. PNAS, 111(23). doi 10.1073/pnas.1318469111

3. Bertagnolli, G., & De Domenico, M. (2020). Diffusion Geometry of Multiplex and Interdependent Systems. arxiv preprint arxiv:2006.13032

manlius/muxViz documentation built on March 1, 2023, 10:28 a.m.