In network theory, multidimensional networks, a special type of multilayer network, are networks with multiple kinds of relations. Increasingly sophisticated attempts to model real-world systems as multidimensional networks have yielded valuable insight in the fields of social network analysis, economics, urban and international transport, ecology, psychology, medicine, biology, commerce, climatology, physics, computational neuroscience, operations management, and finance.
Terminology
The rapid exploration of complex networks in recent years has been dogged by a lack of standardized naming conventions, as various groups use overlapping and contradictory terminology to describe specific network configurations (e.g., multiplex, multilayer, multilevel, multidimensional, multirelational, interconnected). To fully leverage the dataset information on the directional nature of the communications, some authers consider only direct networks without any labels on vertices, and introduce the definition of edge-labeled multigraphs which can cover many multidimensional situations. The term "fully multidimensional" has also been used to refer to a multipartite edge-labeled multigraph. Multidimensional networks have also recently been reframed as specific instances of multilayer networks. In this case, there are as many layers as there are dimensions, and the links between nodes within each layer are simply all the links for a given dimension.
Definition
Unweighted multilayer networks
In elementary network theory, a network is represented by a graph in which is the set of nodes and the links between nodes, typically represented as a tuple of nodes . While this basic formalization is useful for analyzing many systems, real world networks often have added complexity in the form of multiple types of relations between system elements. An early formalization of this idea came through its application in the field of social network analysis (see, e.g., and papers on relational algebras in social networks) in which multiple forms of social connection between people were represented by multiple types of links.
To accommodate the presence of more than one type of link, a multidimensional network is represented by a triple , where is a set of dimensions (or layers), each member of which is a different type of link, and consists of triples with and .
Note that as in all directed graphs, the links and are distinct.
By convention, the number of links between two nodes in a given dimension is either 0 or 1 in a multidimensional network. However, the total number of links between two nodes across all dimensions is less than or equal to .
Weighted multilayer networks
In the case of a weighted network, this triplet is expanded to a quadruplet , where is the weight on the link between and in the dimension . Further, as is often useful in social network analysis, link weights may take on positive or negative values. Such signed networks can better reflect relations like amity and enmity in social networks. Alternatively, link signs may be figured as dimensions themselves, e.g. where and This approach has particular value when considering unweighted networks. This conception of dimensionality can be expanded should attributes in multiple dimensions need specification. In this instance, links are n-tuples . Such an expanded formulation, in which links may exist within multiple dimensions, is uncommon but has been used in the study of multidimensional time-varying networks.
General formulation in terms of tensors
Whereas unidimensional networks have two-dimensional adjacency matrices of size , in a multidimensional network with dimensions, the adjacency matrix becomes a multilayer adjacency tensor, a four-dimensional matrix of size . By using index notation, adjacency matrices can be indicated by , to encode connections between nodes and , whereas multilayer adjacency tensors are indicated by , to encode connections between node in layer and node in layer . As in unidimensional matrices, directed links, signed links, and weights are all easily accommodated by this framework.
In the case of multiplex networks, which are special types of multilayer networks where nodes can not be interconnected with other nodes in other layers, a three-dimensional matrix of size with entries is enough to represent the structure of the system by encoding connections between nodes and in layer .
Multidimensional network-specific definitions
Multi-layer neighbors
In a multidimensional network, the neighbors of some node are all nodes connected to across dimensions.