We (@claudio_moroni and @pitmonticone) are excited to announce the new version of **MultilayerGraphs.jl**: a Julia package for the creation, manipulation and analysis of the structure, dynamics and functions of multilayer graphs.

## Main Changes

Here we list the main changes since the previous announcement post:

- We have integrated the package within the JuliaGraphs ecosystem;
- We have integrated the package within the JuliaDynamics ecosystem so that any
`Multilayer(Di)Graph`

can be utilised as an argument to the`GraphSpace`

constructor in Agents.jl (see the related closed issue); - We have implemented a general multilayer graph removing the requirement for each layer to have the same vertices (see the related closed issue);
- We have implemented the weight tensor, the metadata tensor and the supra weight matrix (see the related closed issue);
- We have implemented a concrete
`MultilayerGraph`

and`MultilayerDiGraph`

, which are not graph wrappers in order to greatly simplify the library, since modifications to edges, vertices, etc. shall no longer be applied twice (see the related closed issue); - We have implemented multiple interfaces for layer and interlayers called "uniform", "standard" and "transparent" (see the related closed issue);
- We have implemented faster graph realization algorithms such as the Havel-Hakimi algorithm for undirected graphs (Hakimi (1962)) and the Kleitman-Wang algorithm for directed ones (Kleitman and Wang (1973)) (see the related closed issue);
- We have implemented more general graph generators ("configuration models") capable of using a different distribution for each dimension of multiplexity (see the related closed issue);
- We have implemented graph generators ("configuration models") for layers (see the related closed issue);
- We have improved the readability of console print of custom structs (see the related closed issue);
- We have implemented more user-friendly constructors for layers and interlayers (see the related closed issue).

For a more comprehensive exploration of the package features and functionalities we strongly recommend consulting the package README and documentation.

## Usage

Here we are going to synthetically showcase some of the main features of MultilayerGraphs.jl (v1.1) such as:

- how to define
**layers**and**interlayers**with a variety of constructors and underlying graphs; - how to construct a
**directed multilayer graph**with those layers and interlayers; - how to add nodes, vertices and edges to the multilayer graph;
- how to compute some multilayer metrics as presented in De Domenico et al. (2013).

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]
```

### Layers and Interlayers

We will instantiate layers and interlayers with randomly-selected 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 generator-like (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]
```

### Multilayer Graphs

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_simplevaldigraph
```

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 vertex-level metadata are effectively considered only if the graph underlying the relevant layer or interlayer supports them, otherwise they are discarded. The same holds for edge-level 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_simplevaldigraph, # 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_simpledigraph)
# 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))
)
```

## Future Developments

Implement more default multilayer graphs (e.g. multiplex graphs);

Implement configuration models / graph generators for interlayers;

Implement a fully-fledged multilayer configuration model / graph generator;

Implement multilayer graph data visualisation functionalities;

### Contacts

Author | GitHub | Discourse | Forem | |
---|---|---|---|---|

Claudio Moroni | @ClaudMor | @Claudio__Moroni | @claudio20497 | @claudio_moroni |

Pietro Monticone | @pitmonticone | @PietroMonticone | @PietroMonticone | @pitmonticone |

## Top comments (0)