Embedded-Graph Theory
This work addresses a limitation in graph theory for applications requiring nuanced relational data, such as in linguistics or complex networks, though it appears incremental as it builds on existing graph concepts.
The paper tackles the problem of expressing complex relations in graphs by proposing a new 'embedded-graph' type with distributed representations, enabling the description of linguistic and complicated relations not possible with existing edge-graphs or weighted-graphs, and introduces mathematical definitions and transformations for this graph type.
In this paper, we propose a new type of graph, denoted as "embedded-graph", and its theory, which employs a distributed representation to describe the relations on the graph edges. Embedded-graphs can express linguistic and complicated relations, which cannot be expressed by the existing edge-graphs or weighted-graphs. We introduce the mathematical definition of embedded-graph, translation, edge distance, and graph similarity. We can transform an embedded-graph into a weighted-graph and a weighted-graph into an edge-graph by the translation method and by threshold calculation, respectively. The edge distance of an embedded-graph is a distance based on the components of a target vector, and it is calculated through cosine similarity with the target vector. The graph similarity is obtained considering the relations with linguistic complexity. In addition, we provide some examples and data structures for embedded-graphs in this paper.