Twograph

Twograph is a concept used to model systems that integrate two interacting networks. In its standard form, a twograph consists of two graphs defined over a shared set of entities, together with a mechanism that links the two layers. This structure is used when the same items participate in two different kinds of relationships, and analysis benefits from considering both networks jointly.

Formally, let G1 = (V, E1) and G2 = (V, E2) be graphs on a common vertex set V. The twograph T is given by the triple (V, E1, E2) together with an inter-layer relation C that connects elements across the two layers. Depending on the model, C may be empty, contain a one-to-one mapping between corresponding vertices, or specify arbitrary cross-layer edges between V and V (or a subset V×V).

Representations of twographs include block adjacency matrices, where the two intra-layer adjacencies appear on the diagonal blocks and the inter-layer connections form the off-diagonal blocks. Algorithms for shortest paths, centrality, and community detection can be extended to operate on twographs by aggregating metrics across layers or by designing inter-layer-aware variants.

Common use cases include social and professional networks, where one layer encodes friendship links and the other encodes collaboration, or multimodal datasets where different data modalities form separate graphs over the same entities. Twographs are also used in biology to couple gene interaction networks with metabolic or regulatory networks, and in transportation to combine different travel modes.

Twographs are related to multiplex or multilayer networks, but emphasize the explicit coupling between two layers. Related concepts include two-mode (bipartite) graphs and interdependent networks. The term appears in various papers and software libraries under different naming conventions, reflecting its role as a building block for joint analysis of dual-graph systems.