Signed Networks: theory, methods, and applications

TL;DR

This work surveys signed networks, a formalism that encodes cooperative and antagonistic relations as a principled extension of traditional graphs. It systematically builds the mathematical foundations, adapts core metrics, and introduces balanced and frustrated structures through Harary and Heider theories, while also detailing null models, embeddings, and predictive tasks. The text integrates dynamics, data collection, and empirical analyses across social, political, neural, and ecological domains, highlighting challenges in scalability, data quality, and interpretability. By unifying theory, methods, and cross-domain examples, it provides a reference framework for researchers to study how positive and negative interactions jointly shape complex systems and signals future directions for theory and data-driven insight.

Abstract

Signed networks provide a principled framework for representing systems in which interactions are not merely present or absent but qualitatively distinct: friendly or antagonistic, supportive or conflicting, excitatory or inhibitory. This polarity reshapes how we think about structure and dynamics in complex systems: a negative tie is not simply a missing positive one but a constraint that generates tension, and possibly asymmetry. Across disciplines, from sociology to neuroscience and machine learning, signed networks provide a shared language to formalise duality, balance, and opposition as integral components of system behaviour. This review provides a comprehensive and foundational summary of signed network theory. It formalises the mathematical principles of signed graphs and surveys signed-network-specific measures, including signed degree distributions, clustering, centralities, motifs, and Laplacians. It revisits balance theory, tracing its cognitive and structural formulations and their connections to frustration. Structural aspects of signed networks are examined, analysing key topics such as null models, node embeddings, sign prediction, and community detection. Subsequent sections address dynamical processes on and of signed networks, such as opinion dynamics, contagion models, and data-driven approaches for studying evolving networks. Practical challenges in constructing, inferring and validating signed data from real-world systems are also highlighted, and we offer an overview of currently available datasets. We also address common pitfalls and challenges that arise when modelling or analysing signed data. Overall, this review integrates theoretical foundations, methodological approaches, and cross-domain examples, providing a structured entry point and a reference framework for researchers interested in the study of signed networks in complex systems.

Figures (17)

• Figure 1: Examples of signed networks across different domains. In social systems, nodes represent individuals and edges denote friendship/enmity or trust/distrust relationships. In ecological systems, edges capture competitive, mutualistic, or predator–prey dynamics. In economic systems, assets are connected based on correlations between their price time series. In psychological systems, nodes represent beliefs or attitudes, and signed edges encode whether they reinforce or contradict each other. In biological systems, interactions between genes and proteins can be represented through activation/inhibition links. In neuroscience, signed networks can represent either excitatory/inhibitory synapses between individual neurons or functional relationships between brain regions. Finally, in international relations, countries may be connected by alliances or conflicts. Figures adapted from dorresteijnIncorporatingAnthropogenicEffects2015aiyappaEmergenceSimpleComplex2024markuTimeseriesTranscriptomicsGene2023whitfield-gabrieliDefaultModeNetwork2012diaz-diazMathematicalModelingLocal2024.
• Figure 2: Real systems as signed networks.Top panel: A step-by-step abstraction of a real-world system. We begin with a group of individuals, each with specific interests and opinions about others, forming a complex web of interpersonal relations involving liking and disliking. Its complexity is then simplified into a signed network, where nodes represent individuals and links encode positive or negative relationships. Finally, the network is translated into an adjacency matrix, where rows and columns correspond to nodes, and the entries take values of $+1$, $0$, or $-1$ depending on whether the relationship is positive, neutral, or negative. Bottom panel: Illustrative examples of how the adjacency matrix is used to compute structural properties of the network. Specifically, we show the Katz centrality calculation and the count of closed walks, alongside visual representations of their meaning.
• Figure 3: Degree, closeness, Katz, and eigenvector centrality in signed networks. Node size is proportional to the absolute value of the corresponding centrality, while colour indicates its sign (blue for positive values, red for negative ones). We omit PageRank because the graph is undirected.
• Figure 4: Heiderian balance at play. The figure illustrates how cognitive balance operates in a real system, emphasising the local perspective and the interaction between unit relations and attitudes. We consider two individuals living in the same household, hence connected by a unit relation. From the viewpoint of the leftmost person (depicted in yellow), two situations are evaluated in terms of Heiderian balance. In the first, Dilemma 1, the other person (depicted in green) expresses liking toward the focal individual; according to Heiderian balance, this configuration is balanced only if the feeling is reciprocated. In the second situation, Dilemma 2, the housemate introduces a pet into the home. Given the structure of unit relations and attitudes in this system, balance is achieved if the focal person also likes the pet.
• Figure 5: Cognitive vs structural balance. The leftmost diagram illustrates cognitive balance, which is characterised by a local perspective: whether a situation is balanced depends on the viewpoint of a specific individual rather than a global assessment. Links in this framework are multi-dimensional, representing both attitudes (positive or negative) and unit relations that can reflect various types of connections. As a result, the overall balance of the system is inherently ambiguous, as it depends on the aggregation of all local perspectives, which may or may not agree. In contrast, the rightmost diagram depicts structural balance, which adopts a global perspective: the system is either balanced or unbalanced as a whole, independently of individual viewpoints. Here, links represent only attitudes (positive or negative), and the status of balance is unambiguous due to this global, uniform interpretation. The central panel illustrates this conceptual contrast by showing a signed triad analysed under both frameworks. Although the underlying structure seems equivalent, both notions are fundamentally different.