From networked students’ centrality to student networks’ density: What really matters for student performance?

This study examines the effects of a student networks’ structural components (i.e., centrality and density) on the academic achievement of its nodes. First, several studies investigated the links between student centrality within a network and student performance. Central students are hypothesised to have access to more diverse and novel information, resources, knowledge, academic benefits and power (Thomas, 2000; Vaquera & Kao, 2008; Gašević & al., 2013; Zwolak & al., 2017), which in turn can lead to better grades, more success at college and/or higher persistence (e.g., Yang and Tang, 2003; Cho & al., 2007; Zhang & al., 2008; Hommes & al., 2012; Gašević & al., 2013; Vaughan & al., 2015; de-Marcos & al., 2016; Mushtaq & al., 2016; Zwolak & al., 2017; Liu & al., 2018; Saqr & al., 2018; Vargas & al. 2018). The centrality measures that are most often used when studying centrality in relation with student performance are the degree, the closeness and the betweenness centralities. In addition to these well-known measures, we test indices that received less attention or that have not yet been used to predict student performance (e.g., the eccentricity, the geodesic k-path centrality, the Kleinberg's authority centrality scores, the cross-clique connectivity …). This exhaustive set of measures allows for distinguishing the effects on performance of (1) the connectivity or number of direct connections a node possesses, (2) of the proximity of a node with the other members of the graph, (3) of the geodesic paths in which nodes are located, and (4) of the neighbourhood of a node (i.e., the prestige and the topology structure of the neighbourhood). Second, the density, measured as the number of observed links within a network divided by the total number of all possible links within this network, represents the ‘degree of social capital and constraint contained within a network and the speed at which information circulates within the network’ (Hanneman & Riddle, 2005). We investigate the links between the density of natural sub-communities present in the student network - i.e., of student groups that emerge based on strong ties linking those students - and academic achievement. Few studies (e.g., Eggens & al., 2008; Rizzuto & al. 2009; Saqr & al., 2018) investigated the links between network density and educational outcomes such as student persistence and student performance. Moreover, the combined effects of student centrality and of network density have been scarcely studied (e.g., Eggens & al., 2008). Then, several authors (Swan, 2001; Picciano, 2002; Wang, 2004; de Laat & al., 2007; Hommes & al., 2012; Gašević & al., 2013; Agudo-Peregrina & al., 2014; Yang & al., 2016) stressed the need to generalize the results of previous studies that were conducted in only one curriculum, one course or within students' specific groups (e.g., female students). For this research, the data have been collected in October 2016 at Saint-Louis University in Brussels, Belgium. The University proposes the following curricula: Law; Economy; Management Sciences; Literature, Philosophy & History; Communication, Political & Social Sciences; and Translation & Interpretation Studies. Consequently, the university was a valuable choice regarding the heterogeneity of information that would be gathered. 574 First generation freshmen students (i.e., students registered in their first year of studies and for the first time) were interrogated about their friendship relations at university. The nodes graph, i.e., the student network, was drawn from the collected information. Since the survey was not mandatory, students who did not participate could nevertheless be cited as ties - a case of missing or non-respondent actors (Robin & al., 2004). A thorough analysis of our graph reveals that 296 students were nominated at least one time by the 574 respondents but did not complete the survey. According to Wasserman & Faust (1994), social network analysis methods require the complete recording of interactions between actors belonging to the studied network. Using the respondent only approach (i.e., in our case deleting the nominations that correspond to the 296 students who did not participate) instead of working with the all cases approach (i.e., in our case including the 296 missing actors in further analyses), can bias the indices that are computed on the graph (e.g., the centrality measures) (Gile & Handcock, 2006; Huisman, 2009; Gile & Handcock, 2017). Due to the nature of our graph (i.e., directed), to the high proportion of missing actors, and due to the fact that those actors were not missing at random - we observed significant differences of gender and of curricula between respondents and missing actors - we used the Exponential Random Graph Models to impute friendship relations for the 296 missing actors. This imputation allowed for computing centrality measures for each of the 870 student belonging to the augmented network (the 574 respondents and the 296 missing actors for which ties were imputed). We performed principal components analysis on our set of complex centrality indices (i.e., all indices except degree centrality) to highlight the latent dimensions of centrality in our student network. Six dimensions were retained and, for further analysis, we choose the six centrality indices with maximum saturation on their respective component - i.e., that were the most representative on their respective dimension (Malhotra & al., 2007). Since one of research objectives was to test the density of sub-communities in the prediction of academic achievement, we used an agglomerative hierarchical clustering - specifically the Fast Greedy Community clustering algorithm proposed by Clauset & al. (2004) - in order to identify these sub-communities. This allowed computing the density of ties inside each sub-community in order to study the links between density and student performance. Therefore, we model performance by centrality at student level and by density at network level. This involves micro-units - the students - that are nested within macro-units - their sub-community of belonging. Most studies dealing with micro- and macro- units use classic Ordinary Least Square techniques. However, these regression techniques have not been designed for such cases since they ignore the postulate of data independence (students sharing same sub-community of belonging being more similar to each other than students belonging to different macro-units). In order to predict achievement by features belonging to different analysis levels (i.e., centrality at the student level and density at the sub-community level), we use multilevel models, which we believe to be the most appropriate modelling technique in case of such complex structure of data. The multilevel models, by recognizing the hierarchical structure of data, remove the OLS constraints that are related to data independence. The results first demonstrate a positive impact of the geodesic k-path centrality (k ≤ 3) on GPA. The geodesic k-path centrality, proposed by Borgatti & Everett (2006), counts the shortest paths up to length k emanating from/going towards a given node, and allows for computing the number of individuals or neighbours that are reachable by the fastest path up to length k – i.e., that are on a geodesic path less than k away. This k-betweenness index bounds therefore the length or distance of the geodesic paths on which nodes are located (Alahakoon & al., 2011; Pfeffer & Carley, 2012; Ghazzali & Ouellet, 2017). In some earlier studies (e.g., Obadi & al., 2010; Gašević & al., 2013; Zwolak & al., 2017), the betweenness centrality was not found to be significant in the prediction of student performance. According to Borgatti & Everett (2006), long shortest paths (that are also considered when computing the betweenness centrality) are not necessarily relevant in the spreading of information through the network. On the contrary, the geodesic k-path centrality informs about the reception and/or diffusion of local information, instead of the spreading of information through the whole network (Ghazzali & Ouellet, 2017). We might therefore argue that the geodesic k-path centrality has a positive effect on student GPA through more access to local, trusty and relevant information. As far as we know, no research has yet studied the links between the geodesic k-path centrality and student performance, and the results indicate that this centrality index might be important when studying academic achievement. Then, as in previous studies (e.g., Cho & al., 2007; Zhang & al., 2008; Gašević & al., 2013; Zwolak & al., 2017), we show positive and significant links between closeness centrality and student performance. The proximity of a student to all other students in the network - measured as the reciprocal of the total geodesic distance between a given node i and all other nodes in the network (Freeman, 1979) - seems therefore to increase student performance. Finally, results show a positive impact of density on GPA, which, however, seems bounded by a ceiling effect. We might argue that at a certain point, the constraints related to high levels of density outweighs the social capital that is brought by density, this resulting in a density that no longer contributes to student performance