Educational attainment plays an important role in shaping an individual’s future career opportunities and overall life trajectory, and understanding disparities in educational aspirations provides valuable insights into educational inequality.

This study investigates how socioeconomic status (SES) moderates the influence of peers on children’s educational aspirations. Past research has shown that adolescents’ educational aspirations are largely shaped by their friends’ aspirations. We employ the concept of "sticky expectations" — the tendency for high-SES children to maintain high educational aspirations irrespective of objective academic performance. In a novel approach, we integrate this framework with insights on peer influence to explore whether high-SES children are less susceptible to peer effects compared to their low-SES counterparts. Besides direct influence from friendship ties, we explore the role of parental expectations, migration background, and classroom composition in shaping educational aspirations.

We use the CILS4EU longitudinal school network data and apply Multilevel Stochastic Actor-Oriented Models (MSAOM) to study the dynamic interplay between educational aspirations, socioeconomic background, and social influence. MSAOMs allow us to separate selection and influence effects, providing a clear estimation of how peers affect educational aspirations while simultaneously accounting for endogenous network processes, the dynamic nature of the data and its multilevel structure.

The concept of centrality is one of the essential tools for analyzing complex networks. However, the notion of importance can be defined in various ways, depending on the network's nature, the characteristics of its components, and the specific processes the researcher is exploring. Over the years, an enormous number of centrality indices have been proposed to account for different aspects of a network. Some of these measures are based on the number of links to other nodes in a network (e.g., degree or spectral centrality). Others assess how closely a node is positioned to other nodes in terms of distance, or how often it appears on the shortest paths connecting pairs of nodes. There are also centrality measures, which are based on ideas from information theory, cooperative game theory, voting theory, Dempster–Shafer evidence theory, multi-criteria decision-making, signal processing, physics, biology, geometry and many other fields. Centrality measures are so vast and diverse that they resemble a zoo, with each model representing a unique species and exhibiting its own distinct characteristics and behaviors. In general, the choice of the most appropriate centrality measure depends on the type of a network and the interpretation of important elements.

Due to the large number of centrality measures available, a significant number of challenges have emerged, particularly in terms of selection, comparison, and validation of these models. First, many centrality measures remain unknown because the most extensive existing reviews are limited to just 50-70 measures. Second, access to many centrality models is limited, with even the most comprehensive libraries containing no more than 40 measures. Next, many models (including some classical centralities) are being reinvented rather than built upon, leading to unnecessary duplication. Fourth, many new measures are assigned the same name (e.g. the neighborhood centrality or the improved 'X' centrality), possibly due to a lack of awareness of existing models. Furthermore, new measures are often insufficiently validated, with comparisons typically limited to only 5-10 existing models, most of which are classical measures. Finally, the number of centrality measures is still growing exponentially, as both researchers and reviewers encounter significant challenges in effectively evaluating or comparing new models with existing ones.

Addressing this research gap, we examine 400 existing centrality measures across both artificial and real networks. First, although many centrality measures offer different interpretations and capture various aspects of network topology, a significant number of them may be highly correlated. Do we need all of these measures and how to choose k most uncorrelated measures? We perform the correlation analysis in order to identify relationships between different models. Our results demonstrate that many centrality measures of different nature are well correlated and that some comprehensive methods agree well with simple models. Next, since most real networks are partially observed, some centrality measures can be misused and lead to wrong interpretation. Therefore, we evaluate the sensitivity of centrality measures to small changes in the graph structure and identify centrality indices that are highly vulnerable to incomplete data.

For some broadcasters of association football (soccer) matches it is a routine half-time feature to present average player locations as a proxy for their team's spatial organization. This is commonly referred to as the actual formation, in contrast to the tactical formation presented before the start of the match. Locations averaged separately over the movements of individuals are routinely, but misleadingly, found to suggest arrangements that are more compact than the tactical formations.

We model player movement as a social influence process in which players adjust their distances to tactical reference positions, other players, and the ball. By constructing an influence network from observed proximity relations and viewing average locations as positions in an opinion space attained at equilibrium, we obtain tactical reference positions from an inverse problem of opinion dynamics: instead of determining equilibrium opinions from given beliefs in the Friedkin-Johnsen model, we determine unknown beliefs (tactical assignments) from observed opinions (player locations).