twographs

Twographs are a simple model for representing two distinct kinds of relationships among the same set of entities. Formally, a twograph is a triple (V, E1, E2), where V is a finite set of vertices and E1 and E2 are edge sets on V. Each edge set can be directed or undirected independently, allowing one to capture two related or complementary relations, such as friendship and professional collaboration, or transportation and communication links among the same places.

The two layers can be represented by adjacency matrices A1 and A2, where Ai[i, j] indicates the presence of an edge of type i between i and j. Twographs are a special case of multiplex networks or multi-layer graphs, focusing on exactly two layers. Analyses can treat layers separately or in a combined form, for example by projecting onto V, forming a composite adjacency M = A1 + A2, or by considering inter-layer coupling when nodes can switch types.

Common tasks include characterizing the structure of each layer (degree distributions, connected components), measuring inter-layer relationships (overlap of neighborhoods, correlation between edge sets), and performing path-based queries that may traverse edges of either type with or without layer switching. Algorithms from graph theory and network science can be adapted to operate on twographs, with attention to how results differ when two edge types interact.

Applications span social science, biology, and infrastructure, including modeling people’s social ties alongside work relationships, gene interaction networks with two modes of interaction, or regional transportation and communication networks. Twographs provide a concise, two-layer abstraction that complements broader multiplex and multi-layer network frameworks.