Class nd.MultilayerNetwork

Class nd.MultilayerNetwork#

In many real-world systems, the same set of entities can interact in different contexts or through different types of relationships. A multilayer network provides a way to represent such heterogeneous systems by organising interactions into separate layers while keeping the underlying actors shared.

Formally, a multilayer network can be defined as follows.

Multilayer network

A multilayer network is defined as a quadruple

\[G = ([n], [\ell], V, E),\]

where:

  • \([n] = \{1, 2, \ldots, n\}\) denotes a set of actors,

  • \([\ell] = \{1, 2, \ldots, \ell\}\) denotes a set of layers,

  • \(V \subseteq [n] \times [\ell]\) is the set of nodes,

  • \(E \subseteq \bigcup_{i \in [\ell]} (V_i \times V_i)\) is the set of edges.

Each node is represented as a pair \((a, i)\), indicating appearance of the actor \(a\) in layer \(i\). Also in such a system, edges connect nodes belonging to the same layer.

From this perspective, a multilayer network can be also understood as a collection of graphs:

\[G_i = (V_i, E_i)\]

where each graph \(G_i\) is spanned on a subset of actors present in the system within the relationship (layer) \(i \in [\ell]\).

Example#

Consider a system describing interactions across several social media platforms. Suppose we observe a group of people who maintain accounts on different services such as LinkedIn, Twitter, or Facebook. In this setting, the set \([n]\) represents the people active on the Internet, while \([\ell]\) represents the social platforms.

Each user account corresponds to a node in the multilayer network. For instance, a LinkedIn profile belonging to a particular person would be represented as the node \((a, i)\), where \(a\) denotes the person and \(i\) corresponds to the LinkedIn layer.

Edges capture interactions between users within the same platform, such as connections, follows, or messages. Consequently, each platform forms a separate graph describing interactions specific to that service.

From this perspective, the multilayer network can be decomposed into \(\ell\) interdependent graphs, each corresponding to a different social platform.

Using multilayer networks in network_diffusion#

nd.MultilayerNetwork extends the functionality of the networkx.Graph class to store and manipulate networks consisting of multiple relations grouped into distinct layers. This is a fundamental structure in network_diffusion.

Available data#

Here is an example repository containing multilayer network datasets that can be read from network_diffusion.

Example of usage#

Create multilayer networks in several ways.

  1. By defining separate graphs and layer names

import networkx as nx
import network_diffusion as nd

layers = [
    nx.les_miserables_graph(),
    nx.les_miserables_graph(),
    nx.les_miserables_graph(),
]
ml_network = nd.MultilayerNetwork.from_nx_layers(layers)

print(ml_network)

============================================
network parameters
--------------------------------------------
general parameters:
    number of layers: 3
    number of actors: 77
    number of nodes: 231
    number of edges: 762

layer 'layer_0' parameters:
    graph type - <class 'networkx.classes.graph.Graph'>
    number of nodes - 77
    number of edges - 254
    average degree - 6.5974
    clustering coefficient - 0.5731

layer 'layer_1' parameters:
    graph type - <class 'networkx.classes.graph.Graph'>
    number of nodes - 77
    number of edges - 254
    average degree - 6.5974
    clustering coefficient - 0.5731

layer 'layer_2' parameters:
    graph type - <class 'networkx.classes.graph.Graph'>
    number of nodes - 77
    number of edges - 254
    average degree - 6.5974
    clustering coefficient - 0.5731
============================================

  1. By defining separate graphs and using concrete names for the layers

import networkx as nx
import network_diffusion as nd

layers = [
    nx.les_miserables_graph(),
    nx.les_miserables_graph(),
    nx.les_miserables_graph(),
]
ml_network = nd.MultilayerNetwork.from_nx_layers(layers, ["l1", "l2", "l3"])

print(ml_network)

============================================
network parameters
--------------------------------------------
general parameters:
    number of layers: 3
    number of actors: 77
    number of nodes: 231
    number of edges: 762

layer 'l1' parameters:
    graph type - <class 'networkx.classes.graph.Graph'>
    number of nodes - 77
    number of edges - 254
    average degree - 6.5974
    clustering coefficient - 0.5731

layer 'l2' parameters:
    graph type - <class 'networkx.classes.graph.Graph'>
    number of nodes - 77
    number of edges - 254
    average degree - 6.5974
    clustering coefficient - 0.5731

layer 'l3' parameters:
    graph type - <class 'networkx.classes.graph.Graph'>
    number of nodes - 77
    number of edges - 254
    average degree - 6.5974
    clustering coefficient - 0.5731
============================================

  1. By reading directly the mpx file

import network_diffusion as nd

mpx_path = "network_diffusion/tests/data/florentine.mpx"
ml_network = nd.MultilayerNetwork.from_mpx(mpx_path)

print(ml_network)

============================================
network parameters
--------------------------------------------
general parameters:
    number of layers: 2
    number of actors: 15
    number of nodes: 26
    number of edges: 35

layer 'marriage' parameters:
    graph type - <class 'networkx.classes.graph.Graph'>
    number of nodes - 15
    number of edges - 20
    average degree - 2.6667
    clustering coefficient - 0.16

layer 'business' parameters:
    graph type - <class 'networkx.classes.graph.Graph'>
    number of nodes - 11
    number of edges - 15
    average degree - 2.7273
    clustering coefficient - 0.4333
============================================

Other functionalities#

network_diffusion provides functions to read properties of multilayer networks and centrality metrics tailored for such models (e.g., VoteRank, PageRank, community detection routines). See the reference guide for further information.

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