Update README and docs

Co-Authored-By: Claudio Moroni <43729990+ClaudMor@users.noreply.github.com>
Co-Authored-By: Pietro Monticone <38562595+pitmonticone@users.noreply.github.com>

Author: 3 people

README.md

@@ -13,17 +13,29 @@
 
 <img align="right" width="220" height="220" src="https://github.com/JuliaGraphs/MultilayerGraphs.jl/blob/main/docs/src/assets/logo.png?raw=true">
 
-**MultilayerGraphs.jl** is a Julia package for the creation, manipulation and analysis of the structure, dynamics and functions of multilayer graphs [extending Graphs.jl](https://juliagraphs.org/Graphs.jl/dev/ecosystem/interface/).
+**MultilayerGraphs.jl** is a Julia package for the creation, manipulation and analysis of the structure, dynamics and functions of multilayer graphs. 
 
 ## Overview
 
-**MultilayerGraphs.jl** provides an implementation of the mathematical formulation of multilayer graphs as proposed by [De Domenico et al. (2013)](https://doi.org/10.1103/PhysRevX.3.041022) and incorporates insights from [Kivelä et al. (2014)](https://doi.org/10.1093/comnet/cnu016) and [Bianconi (2018)](https://global.oup.com/academic/product/multilayer-networks-9780192865540). The package focuses on two custom types, [`MultilayerGraph`](@ref) and [`MultilayerDiGraph`](@ref), which represent undirected and directed multilayer graphs, respectively.
+A multilayer graph is a graph consisting of multiple standard subgraphs called *layers* which can be interconnected through [bipartite graphs](https://en.wikipedia.org/wiki/Bipartite_graph) called *interlayers* composed of the vertex sets of two different layers and the edges between them. The vertices in each layer represent a single set of nodes, although not all nodes have to be represented in every layer. 
 
-A multilayer graph is composed of ***layers***, i.e. graphs whose vertices represent the same set of nodes (not all nodes need to be represented in every layer), and ***interlayers***, i.e. the [bipartite graphs](https://en.wikipedia.org/wiki/Bipartite_graph) that connect vertices in two different layers. Vertices in a multilayer graph are represented using the [`MultilayerVertex`](@ref) struct, while nodes are represented using the [`Node`](@ref) struct.
+Formally, a multilayer graph can be defined as a triple $G=(V,E,L)$, where:
 
-[`MultilayerGraph`](@ref) and [`MultilayerDiGraph`](@ref) are fully-fledged [Graphs.jl](https://github.com/JuliaGraphs/Graphs.jl) extensions. Both structs are designed to allow for layers and interlayers of any type (as long as they are Graphs.jl extensions themselves) and to permit layers and interlayers of different types. However, it is required that all layers and interlayers in [`MultilayerGraph`](@ref) are undirected, and all layers and interlayers in [`MultilayerDiGraph`](@ref) are directed.
+- $V$ is the set of vertices;
+- $E$ is the set of edges, pairs of nodes $(u, v)$ representing a connection, relationship or interaction between the nodes $u$ and $v$;
+- $L$ is a set of layers, which are subsets of $V$ and $E$ encoding the nodes and edges within each layer.
 
-[`MultilayerGraph`](@ref) and [`MultilayerDiGraph`](@ref) support the specification of vertex and edge metadata, provided that the underlying layer or interlayer also supports metadata.
+Each layer $\ell$ in $L$ is a tuple $(V_\ell, E_\ell)$, where $V_\ell$ is a subset of $V$ that represents the vertices within that layer, and $E_\ell$ is a subset of $E$ that represents the edges within that layer.
+
+Multiple theoretical frameworks have been proposed to formally subsume all instances of multilayer graphs ([De Domenico  et al. (2013)](https://doi.org/10.1103/physrevx.3.041022); [Kivelä et al. (2014)](https://doi.org/10.1093/comnet/cnu016); [Boccaletti et al. (2014)](https://doi.org/10.1016/j.physrep.2014.07.001); [Lee et al. (2015)](https://doi.org/10.1140/epjb/e2015-50742-1); [Aleta and Moreno (2019)](https://doi.org/10.1146/annurev-conmatphys-031218-013259); [Bianconi (2018)](https://doi.org/10.1093/oso/9780198753919.001.0001); [Cozzo et al. (2018)](https://doi.org/10.1007/978-3-319-92255-3); [Artime et al. (2022)](https://doi.org/10.1017/9781009085809); [De Domenico (2022)](https://doi.org/10.1007/978-3-030-75718-2)). 
+
+Multilayer graphs have been adopted to model the structure and dynamics of a wide spectrum of high-dimensional, non-linear, multi-scale, time-dependent complex systems including physical, chemical, biological, neuronal, socio-technical, epidemiological, ecological and economic networks ([Cozzo et al. (2013)](https://doi.org/10.1103/physreve.88.050801); [Granell et al. (2013)](https://doi.org/10.1103/physrevlett.111.128701); [Massaro and Bagnoli (2014)](https://doi.org/10.1103/physreve.90.052817); [Estrada and Gomez-Gardenes (2014)](https://doi.org/10.1103/physreve.89.042819); [Azimi-Tafreshi (2016)](https://doi.org/10.1103/physreve.93.042303); [Baggio et al. (2016)](https://doi.org/10.1073/pnas.1604401113); [DeDomenico et al. (2016)](https://doi.org/10.1038/nphys3865); [Amato et al. (2017)](https://doi.org/10.1038/s41598-017-06933-2); [DeDomenico (2017)](https://doi.org/10.1093/gigascience/gix004); [Pilosof et al. (2017)](https://doi.org/10.1038/s41559-017-0101); [de Arruda et al. (2017)](https://doi.org/10.1103/physrevx.7.011014); [Gosak et al. (2018)](https://doi.org/10.1016/j.plrev.2017.11.003); [Soriano-Panos et al. (2018)](https://doi.org/10.1103/physrevx.8.031039); [Timteo et al. (2018)](https://doi.org/10.1038/s41467-017-02658-y); [Buldú et al. (2018)](https://doi.org/10.1162/netn_a_00033); [Lim et al. (2019)](https://doi.org/10.1038/s41598-019-39243-w); [Mangioni et al. (2020)](https://doi.org/10.1109/tnse.2018.2871726); [Aleta et al. (2020)](https://doi.org/10.1038/s41562-020-0931-9); [Aleta et al. (2022)](https://doi.org/10.1073/pnas.2112182119)). 
+
+MultilayerGraphs.jl is an integral part of the [JuliaGraphs](https://github.com/JuliaGraphs) ecosystem extending [Graphs.jl](https://github.com/JuliaGraphs/Graphs.jl) so all the methods and metrics exported by Graphs.jl work for multilayer graphs, but due to the special nature of multilayer graphs the package features a peculiar implementation that maps a standard integer-labelled vertex representation to a more user-friendly framework exporting all the objects an experienced practitioner would expect such as nodes (`Node`), vertices (`MultilayerVertex`), layers (`Layer`), interlayer (`Interlayer`), etc.
+
+MultilayerGraphs.jl features multilayer-specific methods and metrics including the global clustering coefficient, the overlay clustering coefficient, the multilayer eigenvector centrality, the multilayer modularity and the Von Neumann entropy.
+
+Finally, MultilayerGraphs.jl has been integrated within the [JuliaDynamics](https://github.com/JuliaDynamics) ecosystem so that any `Multilayer(Di)Graph` can be utilised as an argument to the `GraphSpace` constructor in [Agents.jl](https://github.com/JuliaDynamics/Agents.jl). 
 
 ## Installation
 
@@ -62,9 +74,36 @@ The package and its features were announced on the following platforms:
 - [Forem](https://forem.julialang.org/inphyt/ann-multilayergraphsjl-a-package-to-construct-handle-and-analyse-multilayer-graphs-3k22)
 - [Twitter](https://twitter.com/In_Phy_T/status/1560594513189638146)
 
+## Related Packages 
+
+### R 
+
+Here is a list of software packages for the creation, manipulation, analysis and visualisation of multilayer graphs implemented in the [R language](https://www.r-project.org): 
+
+- [`muxViz`](https://github.com/manlius/muxViz) implements functions to perform multilayer correlation analysis, multilayer centrality analysis, multilayer community structure detection, multilayer structural reducibility, multilayer motifs analysis and utilities to statically and dynamically visualise multilayer graphs;
+- [`multinet`](https://github.com/cran/multinet) implements functions to import, export, create and manipulate multilayer graphs, several state-of-the-art multiplex graph analysis algorithms for centrality measures, layer comparison, community detection and visualization;
+- [`mully`](https://github.com/frankkramer-lab/mully) implements functions to import, export, create, manipulate and merge multilayer graphs and utilities to visualise multilayer graphs in 2D and 3D;
+- [`multinets`](https://github.com/neylsoncrepalde/multinets) implements functions to import, export, create, manipulate multilayer graphs and utilities to visualise multilayer graphs.
+
+### Python
+
+Here is a list of software packages for the creation, manipulation, analysis and visualisation of multilayer graphs implemented in the [Python language](https://www.python.org): 
+
+- [`MultiNetX`](https://github.com/nkoub/multinetx) implements methods to create undirected networks with weighted or unweighted links, to analyse the spectral properties of adjacency or Laplacian matrices and to visualise multilayer graphs and dynamical processes by coloring the nodes and links accordingly;
+- [`PyMNet`](https://github.com/bolozna/Multilayer-networks-library) implements data structures for multilayer graphs and multiplex graphs, methods to import, export, create, manipulate multilayer graphs and for the rule-based generation and lazy-evaluation of coupling edges and utilities to visualise multilayer graphs.
+
+### Julia 
+
+At the best of our knowledge there are currently no software packages dedicated to the creation, manipulation and analysis of multilayer graphs implemented in the [Julia language](https://julialang.org) apart from MultilayerGraphs.jl itself.
+
 ## References
 
 1. De Domenico et al. (2013) [Mathematical Formulation of Multilayer Networks](https://doi.org/10.1103/PhysRevX.3.041022). *Physical Review X*; 
 2. Kivelä et al. (2014) [Multilayer networks](https://doi.org/10.1093/comnet/cnu016). *Journal of Complex Networks*; 
-3. Bianconi (2018) [Multilayer Networks: Structure and Function](https://global.oup.com/academic/product/multilayer-networks-9780192865540). *Oxford University Press*.
- 
+3. Boccaletti et al. (2014) [The structure and dynamics of multilayer networks](https://doi.org/10.1016/j.physrep.2014.07.001). *Physics Reports*; 
+4. Lee et al. (2015) [Towards real-world complexity: an introduction to multiplex networks](https://doi.org/10.1140/epjb/e2015-50742-1). *The European Physical Journal B*; 
+5. Bianconi (2018) [Multilayer Networks: Structure and Function](https://global.oup.com/academic/product/multilayer-networks-9780192865540). *Oxford University Press*;
+6. Cozzo et al. (2018) [Multiplex Networks: Basic Formalism and Structural Properties](https://doi.org/10.1007/978-3-319-92255-3). *SpringerBriefs in Complexity*; 
+7. Aleta and Moreno (2019) [Multilayer Networks in a Nutshell](https://doi.org/10.1146/annurev-conmatphys-031218-013259). *Annual Review of Condensed Matter Physics*; 
+8. Artime et al. (2022) [Multilayer Network Science: From Cells to Societies](https://doi.org/10.1017/9781009085809). *Cambridge University Press*; 
+9. De Domenico (2022) [Multilayer Networks: Analysis and Visualization](https://doi.org/10.1007/978-3-030-75718-2). *Springer Cham*.