Incivility refers to behaviors that violate collective norms and disrupt cooperation within the political process. Although large-scale online data and automated techniques have enabled the quantitative analysis of uncivil discourse, prior research has predominantly focused on impoliteness or toxicity, often overlooking other behaviors that undermine democratic values. To address this gap, we propose a multidimensional conceptual framework encompassing Impoliteness (IMP), Physical Harm and Violent Political Rhetoric (PHAVPR), Hate Speech and Stereotyping (HSST), and Threats to Democratic Institutions and Values (THREAT). Using this framework, we measure the spread of online political incivility in Brazil using approximately 5 million tweets posted by 2,307 political influencers during the 2022 Brazilian general election. Through statistical modeling and network analysis, we examine the dynamics of uncivil posts at different election stages, identify key disseminators and audiences, and explore the mechanisms driving the spread of uncivil information online. Our findings indicate that impoliteness is more likely to surge during election campaigns. In contrast, the other dimensions of incivility are often triggered by specific violent events. Moreover, we find that left-aligned individual influencers are the primary disseminators of online incivility in the Brazilian Twitter/X sphere and that they disseminate not only direct incivility but also indirect incivility when discussing or opposing incivility expressed by others. They relay those content from politicians, media agents, and individuals to reach broader audiences, revealing a diffusion pattern mixing the direct and two-step flows of communication theory. This study offers new insights into the multidimensional nature of incivility in Brazilian politics and provides a conceptual framework that can be extended to other political contexts.

Abstract Signed graphs allow for encoding positive and negative relations between nodes and are used to model various online activities. Node representation learning for signed graphs is a well-studied task with important applications such as sign prediction. While the size of datasets is ever-increasing, recent methods often sacrifice scalability for accuracy. We propose a novel message-passing layer architecture called graph spring network (GSN) modeled after spring forces. We combine it with a graph neural ordinary differential equations (ODEs) formalism to optimize the system dynamics in embedding space to solve a downstream prediction task. Once the dynamics is learned, embedding generation for novel datasets is done by solving the ODEs in time using a numerical integration scheme. Our GSN layer leverages the fast-to-compute edge vector directions and learnable scalar functions that only depend on nodes’ distances in latent space to compute the nodes’ positions. Conversely, graph convolution and graph attention network layers rely on learnable vector functions that require the full positions of input nodes in latent space. We propose a specific implementation called spring-neural-network using a set of small neural networks mimicking attracting and repulsing spring forces that we train for link sign prediction. Experiments show that our method achieves accuracy close to the state-of-the-art methods with node generation time speedups factor of up to 28,000 on large graphs.

Many social and economic systems can be represented as attributed networks encoding the relations between entities who are themselves described by different node attributes. Finding anomalies in these systems is crucial for detecting abuses such as credit card frauds, web spams or network intrusions. Intuitively, anomalous nodes are defined as nodes whose attributes differ starkly from the attributes of a certain set of nodes of reference, called the context of the anomaly. While some methods have proposed to spot anomalies locally, globally or within a community context, the problem remain challenging due to the multi-scale composition of real networks and the heterogeneity of node metadata. Here, we propose a principled way to uncover outlier nodes simultaneously with the context with respect to which they are anomalous, at all relevant scales of the network. We characterize anomalous nodes in terms of the concentration retained for each node after smoothing specific signals localized on the vertices of the graph. Besides, we introduce a graph signal processing formulation of the Markov stability framework used in community detection, in order to find the context of anomalies. The performance of our method is assessed on synthetic and real-world attributed networks and shows superior results concerning state of the art algorithms. Finally, we show the scalability of our approach in large networks employing Chebychev polynomial approximations.