Geometric Multiplex Model

Introduction

This repository can be used to generate an instance of the Geometric Multiplex Model (GMM). This is an extention of the geometric $\mathbb{S}^1$-model with $L$ layers where the correlations at the level of the hidden coordinates between the layers can be tuned. The full model can be found in Kleineberg2016.

Installation

Requirements

• A C++11 (or newer) compliant compiler
• Boost C++ libraries
  • boost/math/special_functions/lambert_w.hpp
  • boost/math/special_functions/erf.hpp

# Unix (Linux / MAC OS)
g++ -O3 -std=c++11 include_else/hyp2f1.cpp include_me/S1_realization.cpp main.cpp -o GENMULTIPLEX

Usage

Format of input files

There are two input files containing S1 parameters and interlayer correlations respectively.

The file parameters.txt contains $L$ lines, one for each layer. It must have the following structure:

[gamma layer 1] [beta layer 1] [average degree layer 1] [random seed layer 1]
[gamma layer 2] [beta layer 2] [average degree layer 2] [random seed layer 2]
...
[gamma layer L] [beta layer L] [average degree layer L] [random seed layer L]

Note: This version of the GMM supports two degree distributions:

• Pareto distribution with cut-off $$\rho(\kappa)=\frac{(\gamma-1)\kappa_0^{\gamma-1}}{1-\left(\frac{\kappa_c}{\kappa_0}\right)^{1-\gamma}}\kappa^{-\gamma}.$$ This distribution is chosen automatically if $\gamma>2$.
• Uniform distribution $\rho(\kappa) = \delta\left(\kappa-\langle k\rangle\right)$. This distribution is chosen automatically if $\gamma\leq 2$.

The file correlations.txt contains $L-1$ lines, one for each consecutive pair of layers. It must have the following structure:

[nu layers 1,2] [g layers 1,2]
[nu layers 2,3] [g layers 2,3]
...
[nu layers L-1,L] [g layers L-1,L]

Running the code

Running GENMULTIPLEX can be done with or without flags

# Command line with flags
./GENMULTIPLEX -N <network size> -pf <parameter file name> -cf <correlation file name>

# Command line without flags
./GENMULTIPLEX <network size> <parameter file name> <correlation file name>

Output files

The program outputs the multiplex using a set of files. For each layer i, a hidden coordinate file layer<i>.coord is generated which contains two columns. The first contains the angular coordinates and the second the hidden degrees. Additionally, for each layer i an edge file layer<i>.edge is created.

Publications

Please cite:

Hidden geometric correlations in real multiplex networks
Kaj Kolja Kleineberg, Marián Boguñá, M. Ángeles Serrano and Fragkiskos Papadopoulos
Nature Physics 12, 1076-1081 (2016)
Full text | arXiv

Multiplexity amplifies geometry in networks
Jasper van der Kolk, Dmitri Krioukov, Marián Boguñá and M. Ángeles Serrano
arXiv preprint arXiv:2505.17688
arXiv