Signed networks

Signed networks, consisting of both positive and negative edges, offer a powerful framework for modeling complex systems with antagonistic interactions. While this field has gained significant attention recently, key challenges remain, such as developing a reliable distance metric for signed edges, identifying factions within the network, and predicting alliances or conflicts based on the network's structure.In this study, we introduce a novel approach to address these challenges using the concept of signed communicability, defined as the exponential of the adjacency matrix. Communicability functions have been explored for the case of unsigned networks, yet no extension to signed networks had been done before. By considering all walks between two nodes, the communicability function measures the effective level of alliance or conflict between them. Additionally, it induces a metric that fulfills the axioms of an Euclidean distance even in the presence of signed edges. Finally, it can be used to get a measure of cosine similarity between nodes.These novel metrics are then combined with standard data analysis methods in order to gain insights into the structural properties of signed networks. Specifically, our framework can induce low-dimensional network embeddings, uncover hidden factions, establish alliance hierarchies, and quantify political polarization. We show the effectiveness of this framework in a variety of applications, including the analysis of political networks, tribal systems, and international relations. Overall, our results provide a rigorous mathematical foundation and a set of versatile tools for the empirical analysis of signed networks.

Beliefs shape how we interact with the world, and cognitive and social biases guide these interactions, reinforcing the dynamic interplay between our beliefs and environment. Beliefs are interconnected, often forming coherent “belief systems” through which we make sense of the world. For example, refusal of childhood vaccinations is associated with home births. Although many of such correlations between beliefs can be naturally explained by how much they coherently fit together, our understanding of nontrivial associations is still in its infancy. For instance, why do people who support strong gun rights deny climate change? Why are liberals stereotyped as “tax-hiking, government-expanding, latte-drinking, sushi-eating, Volvo-driving, New York Times-reading, body-piercing, Hollywood-loving”? In this work, we build on a previous belief system dynamics framework, by adding a spontaneous internal belief update mechanism and the mechanism where we observe others’ expressed beliefs and form our beliefs about them. We show that the addition of this simple mechanism is enough to produce the emergence of unrelated yet correlated beliefs in a population. We also show that these new beliefs can be polarized or shared among subpopulations.

Bipartite networks provide a major insight into the organisation of many real-world systems, unveiling the mechanisms that drive the interactions occurring between distinct groups of nodes. Of particular interest are two-mode networks whose edges admit a sign: examples are provided by human interactions with entities such as products, where agents either cast a positive or negative vote or abstain from voting at all. One of the most relevant issues encountered when modelling a bipartite network is that of devising a way to obtain a monopartite projection onto the layer of interest that preserves as much as possible the information encoded into the original structure. In the present contribution we propose an unsupervised algorithm to obtain statistically validated projections of bipartite, signed networks, according to which any, two nodes sharing a statistically-significant number of concordant (discordant) relationships are connected by a positive (negative) edge. More precisely, we propose two variants of it, according to the way ambivalent patterns and missing ties are treated. Since assessing the statistical significance of any, two nodes similarity requires a proper benchmark, here we consider four, different Exponential Random Graphs, with homogeneous as well as heterogeneous constraints, either leaving the topology free or keeping the topology fixed. Our algorithm outputs a matrix of link-specific p-values, from which a validated projection can be obtained upon running a multiple hypothesis testing procedure. After testing our method on a synthetic configuration output by a fully controllable generative model, we apply it to four, social real-world configurations: in all cases, non-trivial, mesoscopic structures, induced by relationships that cannot be traced back to the constraints defining the employed benchmarks, are detected.

We study a social network where each agent possesses a set of attributes. Following the homophily principle, using agents’ attributes, we denote relations as positive (friendly) or negative (unfriendly). Distinguishing the signs of relationships between pairs of agents can be performed for each attribute separately or considering all attributes together in the multidimensional space. Structural balance theory (SBT) analyses stability of groups in signed social networks. The theory concerns structures of all sizes focusing on triads - connected trios of people. SBT states that unbalanced triads are less stable than balanced ones. We apply our signed network construction definition to study the NetSense dataset containing relationships between university students and students' opinions on important social topics. We test for which conditions SBT principles can be measured in the system. To this aim, we use static and dynamical structural balance metrics, such as density of balanced triads and triad transition probabilities, respectively. We compare the measures obtained for the real network with those for three different null models and two randomized processes. Our results show that for the analyzed dataset, SBT influence is not observed in the case of signs constructed using separate attributes. Triad densities for real networks are not statistically different from densities in null models. However, when considering all attributes together, for the range of tolerance values, multidimensional triads are significantly more balanced in the real network. We also propose an agent-based model with triad dynamics causing coevolution of attributes and edge signs. This model reproduces transition probabilities better than randomized processes for a similar range of tolerance values. Summing up, multidimensional attributes are sufficient to measure SBT influence.

Human interactions in the online world are made of a combination of positive and negative exchanges. These diverse interactions can be captured using signed network representations, where the edges can take either positive or negative weights to indicate friendship or animosity between individuals. In political contexts, signed networks offer valuable insights into online social polarization by capturing antagonistic interactions and ideological divides. This study analyzes polarization on the Menéame platform, a Spanish social media site that facilitates engagement with news stories through comments and voting. Using a dual-method approach---Signed Hamiltonian Eigenvector Embedding for Proximity (SHEEP) for signed networks and Correspondence Analysis (CA) for unsigned networks---we investigate how including negative ties enhances the understanding of structural polarization. We quantify the level of positive and negative structural polarization on the platform across different conversation topics. We find that on the Menéame network, negative ties are necessary for detecting antagonism, while the unsigned network delineates ideological communities. We also show that far-left users on the platform are more likely to engage with users across ideological lines, even if through negative interactions, whereas far-right users interact primarily with users similar to themselves.