Signed graphs in data sciences via communicability geometry

TL;DR

The paper introduces signed communicability geometry, defining distance and angle measures for signed graphs via the matrix exponential $e^A$ of the signed adjacency matrix. It proves that these measures induce a hyperspherical embedding and are Euclidean/spherical, enabling practical data-analytic tasks such as partitioning, dimensionality reduction, hierarchy discovery, and polarization quantification. A flexible three-step framework uses communicability angles for embedding and clustering, demonstrated on diverse real-world networks (Gahuku-Gama, international relations, European Parliament voting, and E. coli gene regulation). The work provides a principled, general toolkit for signed-network analysis with potential extensions to edge prediction, directed/multilayer graphs, and predictive tasks.

Abstract

Signed graphs are an emergent way of representing data in a variety of contexts where antagonistic interactions exist. These include data from biological, ecological, and social systems. Here we propose the concept of communicability for signed graphs and explore in depth its mathematical properties. We also prove that the communicability induces a hyperspherical geometric embedding of the signed network, and derive communicability-based metrics that satisfy the axioms of a distance even in the presence of negative edges. We then apply these metrics to solve several problems in the data analysis of signed graphs within a unified framework. These include the partitioning of signed graphs, dimensionality reduction, finding hierarchies of alliances in signed networks, and quantifying the degree of polarization between the existing factions in social systems represented by these types of graphs.

Figures (12)

• Figure 1: Example of a balanced graph (a), its partition according to Theorem \ref{['thm:Harary']} (b), and the result of applying the switching transformation described in the text (c). Example of an unbalanced graph (i.e. a graph containing cycles with negative parity) (d), an illustration of some problems emerging when trying to apply Theorem \ref{['thm:Harary']} to this graph (e) as well as the graph resulting from applying the switching transformation mentioned before (f). Blue and green color denotes positive edges, while red dashed lines correspond to negative edges.
• Figure 2: Illustration of the three existing balanced complete graphs with 6 vertices, which are isospectral to one another.
• Figure 3: Example of five different signed partitions having the same "frustration" on the pentagon with one negative edge.
• Figure 4: Example illustrating a case where the algorithms based on the communicability distance fail to identify the expected balanced partition. (a) Communicability distance matrix of the graph shown in (b). (b) Partition obtained by hierarchical clustering on the communicability distance matrix. Any clustering algorithm based on the communicability distance will fail to identify the balanced partition, since the communicability distance between nodes 9 and 10 is smaller than between node 9 and any node in the clique. (c) Communicability angle matrix of the graph. (d) Partition obtained by hierarchical clustering on the communicability angle matrix. Angle-based clustering correctly identifies the balanced partition.
• Figure 5: Embedding of the four signed triangles in the induced three-dimensional communicability space.

Theorems & Definitions (34)

• Theorem 1: Harary
• Proposition 1
• Remark 1
• Lemma 1
• proof
• Definition 1
• Lemma 2
• proof
• Proposition 2
• proof