Projecting signed two-mode networks

ABSTRACT

Signed two-mode networks have so far predominantly been analyzed using blockmodeling techniques. In this work, we put forward the idea of projecting such networks onto its modes. Two projection methods are introduced which allow the application of known dichotomization tool for weighted networks to obtain a simple signed network. It turns out, however, that resulting networks may contain ambivalent ties, defined as conjunctions of positive and negative ties. We show that this requires the reformulation of matrices related to the network and introduce the complex adjacency and Laplacian matrix. These matrices are used to prove some properties related to balance theory including ambivalence.

KEYWORDS:

• Signed networks
• balance theory
• projections

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

¹ The presented results can be generalized to disconnected networks by considering each component separately.

² Note that this also implies an underlying model of structural balance.

³ diagonal entries are set to zero by default to avoid loops.

⁴ The authors define two models with different representations for ambivalent ties. the diamond model with $(1,1)$ and the triangle model with $(0.5,0.5)$. The triangle model is chosen here since the positive and negative parts sum up to one.

⁵ This was already observed by Hou, Li, and Pan (2003).