============================== 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. .. admonition:: Multilayer network A **multilayer network** is defined as a quadruple .. math:: G = ([n], [\ell], V, E), where: * :math:`[n] = \{1, 2, \ldots, n\}` denotes a set of **actors**, * :math:`[\ell] = \{1, 2, \ldots, \ell\}` denotes a set of **layers**, * :math:`V \subseteq [n] \times [\ell]` is the set of **nodes**, * :math:`E \subseteq \bigcup_{i \in [\ell]} (V_i \times V_i)` is the set of **edges**. Each node is represented as a pair :math:`(a, i)`, indicating appearance of the actor :math:`a` in layer :math:`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: .. math:: G_i = (V_i, E_i) where each graph :math:`G_i` is **spanned on a subset of actors present in the system** within the relationship (layer) :math:`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 :math:`[n]` represents the people active on the Internet, while :math:`[\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 :math:`(a, i)`, where :math:`a` denotes the person and :math:`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 :math:`\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 .. code-block:: python 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) .. code-block:: console ============================================ 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 - number of nodes - 77 number of edges - 254 average degree - 6.5974 clustering coefficient - 0.5731 layer 'layer_1' parameters: graph type - number of nodes - 77 number of edges - 254 average degree - 6.5974 clustering coefficient - 0.5731 layer 'layer_2' parameters: graph type - number of nodes - 77 number of edges - 254 average degree - 6.5974 clustering coefficient - 0.5731 ============================================ 2. By defining separate graphs and using concrete names for the layers .. code-block:: python 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) .. code-block:: console ============================================ 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 - number of nodes - 77 number of edges - 254 average degree - 6.5974 clustering coefficient - 0.5731 layer 'l2' parameters: graph type - number of nodes - 77 number of edges - 254 average degree - 6.5974 clustering coefficient - 0.5731 layer 'l3' parameters: graph type - number of nodes - 77 number of edges - 254 average degree - 6.5974 clustering coefficient - 0.5731 ============================================ 3. By reading directly the `mpx` file .. code-block:: python import network_diffusion as nd mpx_path = "network_diffusion/tests/data/florentine.mpx" ml_network = nd.MultilayerNetwork.from_mpx(mpx_path) print(ml_network) .. code-block:: console ============================================ 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 - number of nodes - 15 number of edges - 20 average degree - 2.6667 clustering coefficient - 0.16 layer 'business' parameters: graph type - 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.