MultilayerGraphs.jl is a Julia package for the creation, manipulation and analysis of the structure, dynamics and functions of multilayer graphs.
A multilayer graph is a graph consisting of multiple standard subgraphs called layers which can be interconnected through bipartite graphs 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.
Formally, a multilayer graph can be defined as a triple

$V$ 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.
Each layer
Multiple theoretical frameworks have been proposed to formally subsume all instances of multilayer graphs (De Domenico et al. (2013); Kivelä et al. (2014); Boccaletti et al. (2014); Lee et al. (2015); Aleta and Moreno (2019); Bianconi (2018); Cozzo et al. (2018); Artime et al. (2022); De Domenico (2022)).
Multilayer graphs have been adopted to model the structure and dynamics of a wide spectrum of highdimensional, nonlinear, multiscale, timedependent complex systems including physical, chemical, biological, neuronal, sociotechnical, epidemiological, ecological and economic networks (Cozzo et al. (2013); Granell et al. (2013); Massaro and Bagnoli (2014); Estrada and GomezGardenes (2014); AzimiTafreshi (2016); Baggio et al. (2016); DeDomenico et al. (2016); Amato et al. (2017); DeDomenico (2017); Pilosof et al. (2017); de Arruda et al. (2017); Gosak et al. (2018); SorianoPanos et al. (2018); Timteo et al. (2018); Buldú et al. (2018); Lim et al. (2019); Mangioni et al. (2020); Aleta et al. (2020); Aleta et al. (2022)).
MultilayerGraphs.jl is an integral part of the JuliaGraphs ecosystem extending 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 integerlabelled vertex representation to a more userfriendly framework exporting all the objects an experienced practitioner would expect such as nodes (Node
), vertices (MultilayerVertex
), layers (Layer
), interlayers (Interlayer
), etc.
MultilayerGraphs.jl features multilayerspecific 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 ecosystem so that any Multilayer(Di)Graph
can be utilised as an argument to the GraphSpace
constructor in Agents.jl.
To install the latest stable release of MultilayerGraphs.jl, make sure you have installed Julia v1.8 or later and run the following command:
using Pkg
Pkg.add("MultilayerGraphs")
The development version can be installed as follows:
using Pkg
Pkg.add(url="https://github.com/JuliaGraphs/MultilayerGraphs.jl")
Let's begin by importing the necessary dependencies and setting the relevant constants.
# Import necessary dependencies
using Distributions, Graphs, SimpleValueGraphs
using MultilayerGraphs
# Set the number of nodes
const n_nodes = 100
# Create a list of nodes
const node_list = [Node("node_$i") for i in 1:n_nodes]
We will instantiate layers and interlayers with randomlyselected edges and vertices adopting a variety of techniques. Layers and Interlayers are not immutable, and mostly behave like normal graphs. The user is invited to consult the API for further details.
Here we define a layer with an underlying simple directed graph using a graph generatorlike (or "configuration model"like) constructor which allows us to specify both the indegree and the outdegree sequences. Before instantiating each layer we sample the number of its vertices and, optionally, of its edges.
# Create a simple directed layer
n_vertices = rand(1:100) # Number of vertices
layer_simple_directed = layer_simpledigraph( # Layer constructor
:layer_simple_directed, # Layer name
sample(node_list, n_vertices; replace=false), # Nodes represented in the layer
Truncated(Normal(5, 5), 0, 20), # Indegree sequence distribution
Truncated(Normal(5, 5), 0, 20) # Outdegree sequence distribution
)
Then we define a layer with an underlying simple weighted directed graph. This is another kind of constructor that allows the user to specify the number of edges to be randomly distributed among vertices.
# Create a simple directed weighted layer
n_vertices = rand(1:n_nodes) # Number of vertices
n_edges = rand(n_vertices:(n_vertices * (n_vertices  1)  1)) # Number of edges
layer_simple_directed_weighted = layer_simpleweighteddigraph( # Layer constructor
:layer_simple_directed_weighted, # Layer name
sample(node_list, n_vertices; replace=false), # Nodes represented in the layer
n_edges; # Number of randomly distributed edges
default_edge_weight=(src, dst) > rand() # Function assigning weights to edges
)
Similar constructors, more flexible at the cost of ease of use, enable a finer tuning. The constructor we use below should be necessary only in rare circumstances, e.g. if the equivalent simplified constructor layer_simplevaldigraph
is not able to infer the correct return types of default_vertex_metadata
or default_edge_metadata
, or to use and underlying graph structure that isn't currently supported.
# Create a simple directed value layer
n_vertices = rand(1:n_nodes) # Number of vertices
n_edges = rand(n_vertices:(n_vertices * (n_vertices  1)  1)) # Number of edges
default_vertex_metadata = v > ("vertex_$(v)_metadata",) # Vertex metadata
default_edge_metadata = (s, d) > (rand(),) # Edge metadata
layer_simple_directed_value = Layer( # Layer constructor
:layer_simple_directed_value, # Layer name
sample(node_list, n_vertices; replace=false), # Nodes represented in the layer
n_edges, # Number of randomly distributed edges
ValDiGraph(
SimpleDiGraph{Int64}();
vertexval_types=(String,),
vertexval_init=default_vertex_metadata,
edgeval_types=(Float64,),
edgeval_init=default_edge_metadata,
),
Float64;
default_vertex_metadata=default_vertex_metadata, # Vertex metadata
default_edge_metadata=default_edge_metadata # Edge metadata
)
# Create a list of layers
layers = [layer_simple_directed, layer_simple_directed_weighted, layer_simple_directed_value]
There are many more constructors the user is encouraged to explore in the package documentation.
The interface of interlayers is very similar to that of layers. It is very important to notice that, in order to define a Multilayer(Di)Graph
, interlayers don't need to be explicitly constructed by the user since they are automatically identified by the Multilayer(Di)Graph
constructor, but for more complex interlayers the manual instantiation is required.
Here we define an interlayer with an underlying simple directed graph.
# Create a simple directed interlayer
n_vertices_1 = nv(layer_simple_directed) # Number of vertices of layer 1
n_vertices_2 = nv(layer_simple_directed_weighted) # Number of vertices of layer 2
n_edges = rand(1:(n_vertices_1 * n_vertices_2  1)) # Number of interlayer edges
interlayer_simple_directed = interlayer_simpledigraph( # Interlayer constructor
layer_simple_directed, # Layer 1
layer_simple_directed_weighted, # Layer 2
n_edges # Number of edges
)
The interlayer exports a more flexible constructor too.
# Create a simple directed meta interlayer
n_vertices_1 = nv(layer_simple_directed_weighted) # Number of vertices of layer 1
n_vertices_2 = nv(layer_simple_directed_value) # Number of vertices of layer 2
n_edges = rand(1:(n_vertices_1 * n_vertices_2  1)) # Number of interlayer edges
interlayer_simple_directed_meta = interlayer_metadigraph( # Interlayer constructor
layer_simple_directed_weighted, # Layer 1
layer_simple_directed_value, # Layer 2
n_edges; # Number of edges
default_edge_metadata=(src, dst) > # Edge metadata
(edge_metadata="metadata_of_edge_from_$(src)_to_$(dst)",),
transfer_vertex_metadata=true # Boolean deciding layer vertex metadata inheritance
)
# Create a list of interlayers
interlayers = [interlayer_simple_directed, interlayer_simple_directed_meta]
Let's construct a directed multilayer graph (MultilayerDiGraph
).
# Create a simple directed multilayer graph
multilayerdigraph = MultilayerDiGraph( # Constructor
layers, # The (ordered) collection of layers
interlayers; # The manually specified interlayers
# The interlayers that are left unspecified
# will be automatically inserted according
# to the keyword argument below
default_interlayers_structure="multiplex"
# The automatically specified interlayers will have only diagonal couplings
)
# Layers and interlayer can be accessed as properties using their names
multilayerdigraph.layer_simple_directed_value
Then we proceed by showing how to add nodes, vertices and edges to a directed multilayer graph. The user may add vertices that do or do not represent nodes which are already present in the multilayer graph. In the latter case, we have to create a node first and then add the vertex representing such node to the multilayer graph. The vertexlevel metadata are effectively considered only if the graph underlying the relevant layer or interlayer supports them, otherwise they are discarded. The same holds for edgelevel metadata and/or weight.
# Create a node
new_node_1 = Node("new_node_1")
# Add the node to the multilayer graph
add_node!(multilayerdigraph, new_node_1)
# Create a vertex representing the node
new_vertex_1 = MV( # Constructor (alias for "MultilayerVertex")
new_node_1, # Node represented by the vertex
:layer_simple_directed_value, # Layer containing the vertex
("new_metadata",) # Vertex metadata
)
# Add the vertex
add_vertex!(
multilayerdigraph, # MultilayerDiGraph the vertex will be added to
new_vertex_1 # MultilayerVertex to add
)
# Create another node in another layer
new_node_2 = Node("new_node_2")
# Create another vertex representing the new node
new_vertex_2 = MV(new_node_2, :layer_simple_directed_value)
# Add the new vertex
add_vertex!(
multilayerdigraph,
new_vertex_2;
add_node=true # Add the associated node before adding the vertex
)
# Create an edge
new_edge = MultilayerEdge( # Constructor
new_vertex_1, # Source vertex
new_vertex_2, # Destination vertex
("some_edge_metadata",) # Edge metadata
)
# Add the edge
add_edge!(
multilayerdigraph, # MultilayerDiGraph the edge will be added to
new_edge # MultilayerVertex to add
)
Finally we illustrate how to compute a few multilayer metrics such as the global clustering coefficient, the overlay clustering coefficient, the multilayer eigenvector centrality, and the multilayer modularity as defined in De Domenico et al. (2013).
# Compute the global clustering coefficient
multilayer_global_clustering_coefficient(multilayerdigraph)
# Compute the overlay clustering coefficient
overlay_clustering_coefficient(multilayerdigraph)
# Compute the multilayer eigenvector centrality
eigenvector_centrality(multilayerdigraph)
# Compute the multilayer modularity
modularity(
multilayerdigraph,
rand([1, 2, 3, 4], length(nodes(multilayerdigraph)), length(multilayerdigraph.layers))
)
All the information regarding the future developments of MultilayerGraphs.jl can be found in the issues.
The ongoing development of this package would greatly benefit from the valuable feedback of the esteemed members of the JuliaGraph community, as well as from graph theorists, network scientists, and any users who may have general questions or suggestions.
We therefore encourage you to participate in discussions, raise issues, or submit pull requests. Your contributions are welcome!
If you utilize this package in your project, please consider citing this repository using the citation information provided in CITATION.bib
.
This will help to give appropriate credit to the contributors and support the continued development of the package.
MultilayerGraphs.jl (v0.1) and its features were announced on the following platforms:
MultilayerGraphs.jl (v1.1) and its features were announced on the following platforms:
 When: 2023/05/23  2023/05/26
 Where: The Fields Institute, Toronto, Canada
 Who: Pietro Monticone and Claudio Moroni
 What: "Multilayer Network Science in Julia with
MultilayerGraphs.jl
" (Website, Slides, Video)
 When: 2023/07/25  2023/07/29
 Where: Massachusetts Institute of Technology, Cambridge, MA, USA
 Who: Pietro Monticone and Claudio Moroni
 What: "
MultilayerGraphs.jl
: Multilayer Network Science in Julia" (Website, Short Presentation, Long Presentation, Video)
Here is a list of software packages for the creation, manipulation, analysis and visualisation of multilayer graphs implemented in the R language:
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
implements functions to import, export, create and manipulate multilayer graphs, several stateoftheart multiplex graph analysis algorithms for centrality measures, layer comparison, community detection and visualization;mully
implements functions to import, export, create, manipulate and merge multilayer graphs and utilities to visualise multilayer graphs in 2D and 3D;multinets
implements functions to import, export, create, manipulate multilayer graphs and utilities to visualise multilayer graphs.
Here is a list of software packages for the creation, manipulation, analysis and visualisation of multilayer graphs implemented in the Python language:
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
implements data structures for multilayer graphs and multiplex graphs, methods to import, export, create, manipulate multilayer graphs and for the rulebased generation and lazyevaluation of coupling edges and utilities to visualise multilayer graphs.
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 apart from MultilayerGraphs.jl itself.
 De Domenico et al. (2013) Mathematical Formulation of Multilayer Networks. Physical Review X;
 Kivelä et al. (2014) Multilayer networks. Journal of Complex Networks;
 Boccaletti et al. (2014) The structure and dynamics of multilayer networks. Physics Reports;
 Lee et al. (2015) Towards realworld complexity: an introduction to multiplex networks. The European Physical Journal B;
 Bianconi (2018) Multilayer Networks: Structure and Function. Oxford University Press;
 Cozzo et al. (2018) Multiplex Networks: Basic Formalism and Structural Properties. SpringerBriefs in Complexity;
 Aleta and Moreno (2019) Multilayer Networks in a Nutshell. Annual Review of Condensed Matter Physics;
 Artime et al. (2022) Multilayer Network Science: From Cells to Societies. Cambridge University Press;
 De Domenico (2022) Multilayer Networks: Analysis and Visualization. Springer Cham.
 De Domenico (2023) More is different in realworld multilayer networks. Nature Physics