A multimodal network (MMN) is a novel mathematical construct that captures the structure of biological networks, computational network models, and relationships from biological databases. An MMN subsumes the structure of graphs and hypergraphs, either undirected or directed. Formally, an MMN is a triple (V,E,M) where V is a set of vertices, E is a set of modal hyperedges, and M is a set of modes. A modal hyperedge e=(T,H,A,m) in E is an ordered 4-tuple, in which T,H,A are subsets of V and m is an element of M. The sets T, H, and A are the tail, head, and associate of e, while m is its mode. In the context of biology, each vertex is a biological entity, each hyperedge is a relationship, and each mode is a type of relationship (e.g., 'forms complex' and 'is a'). Within the space of multimodal networks, structural operations such as union, intersection, hyperedge contraction, subnetwork selection, and graph or hypergraph projections can be performed. A denotational semantics approach is used to specify the semantics of each hyperedge in MMN in terms of interaction among its vertices. This is done by mapping each hyperedge e to a hyperedge code algo:V(e), an algorithm that details how the vertices in V(e) get used and updated. A semantic MMN-based model is a function of a given schedule of evaluation of hyperedge codes and the current state of the model, a set of vertex-value pairs.
An MMN-based computational system is implemented as a proof of concept to determine empirically the benefits of having it. This system consists of an MMN database populated by data from various biological databases, MMN operators implemented as database functions, graph operations implemented in C++ using LEDA, and mmnsh, a shell scripting language that provides a consistent interface to both data and operators. It is demonstrated that computational network models may enrich the MMN database and MMN data may be used as input to other computational tools and environments. A simulator is developed to compute from an initial state and a schedule of hyperedge codes the resulting state of a semantic MMN model.