2.1. The relevance and use of centrality measures within (student) networks

A network is a set of nodes connected by edges or ties. For instance, with regard to student networks, a students’ graph represents the students (i.e., the nodes) and their connections (i.e., the edges or ties) with other student(s) within the network. The literature recognizes the relevance of centrality measures for representing the importance of nodes in a graph [16]. The centrality concept gives information about its prestige, prominence, or involvement, how a node get access to and spreads information, and the node’s proximity to phenomena that are observed within a network [16–19]. Various network studies in many fields have used centrality measures to represent nodes and possibly the links between the nodes’ centrality and some variable(s) of interest. Among them we find studies in management and organizations [12, 20–26], economics and finance [27–31], marketing research [8, 32–35], sociology and political science [36–42], and biological networks [14, 43–51]. Also, student centrality within their network and the links with education outcome(s) have been the subjects of many studies. Those investigations concern performance and achievement [4, 24, 52–70], other aspects of learning (e.g., attitudes about the courses, sharing and construction of knowledge) [54, 71], delinquency [72, 73], sense of community [74, 75], and dropping out of school [76, 77]. Table 1 in S1 Appendix shows 63 studies conducted on centrality within networks (of which 27 were student networks), together with the centrality indices that were computed and used within those studies. In Tables 2 and 3 in S1 Appendix we computed the numbers and percentages of the centrality measures that were used within all network types (Table 2) or within student networks only (Table 3). Those percentages, which are represented in Fig 1, show that studies dedicated to networks have mostly conceptualized centrality as (1) the simplest measure of centrality, i.e., degree centrality; (2) closeness centrality; and/or (3) betweenness centrality. However, as stated earlier, many other metrics have been developed in SNA in order to assess a node’s centrality within a graph. S1 Appendix and Fig 1 show that (student) networks have rarely been represented using those alternative centrality measures.

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Fig 1. Top 10 centrality measures in networks studies.

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Then, for network studies that concerned student centrality linked to some outcome of interest, some results showed centrality to have positive effects on education outcomes while others demonstrated no or even negative impacts. For instance, with regard to student performance, while the degree and closeness centralities seemed to have a positive effect on achievement, the impact of betweenness centrality appeared to be less clear (see S2 Appendix for non-exhaustive instances of college student network studies and academic achievement). Together with other issues, such as the tie type (e.g., friendship versus strategic ties) that is investigated [4, 70], the choice of the centrality measures might explain the nature of the observed links between centrality and education outcomes, along with the inconsistencies in the findings [6]: The way centrality is computed mathematically defines centrality and establishes how individuals are represented within a network. Furthermore, studies showed that the ranks of the scores obtained for different centrality indices did not always match (e.g., a node could have high scores on some centrality measures, but average or low scores on other measures of centrality) [1, 8, 9]. As explained earlier, one objective of this paper is to highlight which centrality measures might be the most informative for friendship student networks. This research aims to select, from some chosen indices, the best-suited centrality measures for representing a friendship student network. Those centrality indices might then be used to study the impacts of student networks on education outcomes (e.g., student performance) in further research.

2.2. A theoretical taxonomy of centrality measures for student networks

Taxonomies have been defined as “a formal specification of a shared conceptualisation” [78]. Used in a variety of fields (pharmacology, engineering, physic, law, finance, etc.), they help to describe, organize, explain and predict phenomena; they yield knowledge about the relationships between different categories or objects; and they help researchers or practitioners to communicate about those phenomena [79–81]. Taxonomies may, however, “be subject to a wide range of interpretations and misunderstandings” [81]. Their appeal to a community (i.e., their sharedness) and fit with the reality they represent (i.e. their conceptualisation) therefore need to be validated [82]. Four criteria discussed in Guizzardi [83, 84] may be used in order to verify their fit with reality, namely, their soundness, completeness, lucidity, and laconicity (see Fig 2, where the constructs and/or objects colored in grey represent an absence of soundness, completeness, lucidity, or laconicity).

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Fig 2. Soundness, completeness, lucidity, and laconicity: Comparison between the theoretical classes or constructs and the reality or objects they represent.

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A taxonomy is sound when there is no construct excess and each of its constructs matches an underlying reality in an intended universe of discourse (e.g., within a student network). For example, a sound taxonomy of centrality measures for student networks contains only theoretical categories that represent centrality notions or dimensions for student networks in the real world faithfully. A complete (or exhaustive) taxonomy has no construct deficit and, hence, a construct for each aspect of the underlying reality [80]. For example, with an exhaustive list of theoretical categories, a complete taxonomy of centrality measures would reflect each centrality dimension existing within student networks, including those that have not been observed yet. A lucid taxonomy has no construct overload (homonymy) and hence only constructs that each maps to (at most) a single aspect of the underlying reality. For example, a lucid taxonomy would not contain a theoretical category that refers to several latent dimensions of student centrality. A laconic taxonomy has no construct redundancy (synonymy) and, hence, at most one construct for each aspect of the underlying reality. Classes must therefore be mutually exclusive, with no object that might belong to more than one category [80]. For example, a laconic taxonomy would contain only one theoretical category for each centrality dimension existing in student networks. In the absence of these criteria, the taxonomy could lead to ambiguous interpretations of the centrality measures pertaining to, for instance, a student network.

According to Lü et al. [15], “a valuable work is to arrange well-known centralities and classify them.” As stated before, numerous centrality indices exist in SNA, and clear theoretical classifications that are easy to use might help both experienced and novice researchers to understand and choose centrality indices from the vastness of options [32]. This work might be especially valuable for novice researchers in student networks, since taxonomies are useful to understand quickly the “essential traits of the classified object by simply knowing in which category and with which other objects it has been grouped” [80]. Moreover, centrality measures are related to the objectives linked to the use of those indices [9]. As Kozma et al. [79], who proposed a taxonomy of instructional treatments, different instructional treatment types having different impacts on learning and cognition, classifications of centrality indices might be valuable for efficiently visualizing and determining centrality measures that correspond to the goals of studies on student networks. For instance, if the purpose of one research initiative is to study the impact of a student's information control on some variable(s) of interest, such as learning or academic performance, taxonomies of centrality measures could help to select the most appropriate indices. Moreover, as stated before, the four criteria discussed in Guizzardi [83, 84] may be used to verify the validity of taxonomies that conceptualize the notion of centrality within (student networks). Now, not only has this work never been done before for student graphs, but a valid taxonomy of centrality measures tested on a student network might also be useful to generalize and validate the theoretical classifications of centrality indices proposed in other studies and for other graph types [e.g., in 12–15].

3.1. Choosing and defining the centrality measures

As stated before, our specific graph of interest is a friendship student network. We obtained a list of suitable centralities for our friendship student network by using the CINNA package [6] implemented in R©, and more precisely the function proper_centralities. The CINNA package can compute a great variety of centrality indices and is able to deal with directed and un-weighted networks, which is the case for our graph. The function output, i.e., the complete list of the suitable centralities applicable to our graph, is presented in S3 Appendix. Among those suitable centralities, the measures for representing our friendship student network were selected by means of the sequence shown in Fig 4 and composed of the following steps:

1. Who are the nodes, what is/are the network type(s)? (i.e., in our case, a directed friendship student network; see the data section for the details);
2. A deep understanding of centrality measures is a necessary condition for (1) pursuing the methodology, (2) choosing the centrality indices from a vast set of measures, and (3) enabling deeper knowledge of the considered network(s) and the mechanisms occurring within the graph(s) of interest. With regard to the definitions of the centrality indices that were proposed by the CINNA library (see S3 Appendix for the proposed measures and S4 Appendix for the thorough definitions of the centrality indices that we chose), the second step consists of selecting which centrality measure might be suitable and interesting for further studies to investigate the links between some network(s) and outcome(s) of interest (in our case, between student centrality and education outcomes such as learning, performance, and so on). Our selection of the centrality indices—made in line with the perspective of a friendship student network—are justified below, when we present the centrality measures that we chose. Finally, for this second step, indices considered as irrelevant to the network(s) of interest and future research questions related to this/these network(s) were not selected for further analyses. For instance, the current flow closeness centrality [85] is an index specific to electrical currents and was therefore not chosen since it is not suitable for social networks.
3. For the remaining indices: In the third step, we determined whether there were any measures with highly similar formulas, i.e., that differed by only a very few parameters. For instance, communicability betweenness centrality, flow betweenness centrality, load centrality, and stress centrality are all variants of betweenness centrality. We chose the centrality index figuring in the highest number of documents on Google Scholar and—as a benchmark—that was most used in network literature (see S1 Appendix): For betweenness centrality, we chose the measure that was proposed by Freeman [86].
4. For the remaining indices: In order to continue the methodological process with a reasonable number of indices, centrality measures that figured in very few documents on Google Scholar were not selected for further analyses.

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Fig 4. Choosing networks’ centrality indices: Sequence.

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Among the complete list of suitable centralities presented in S3 Appendix, the set of centrality measures chosen for our friendship student network is composed of eighteen centrality indices. Detailed definitions and explanations of these indices are given in S4 Appendix. We consider a centrality measure that takes into account the edge’s direction, i.e., that can be computed separately on the incoming and outgoing ties, as two distinct indices. We explain, for each of the eighteen centrality measures, why they might be suitable for friendship student networks, and eventually for further studies conducted on those graph types and for their links with some education outcome(s):

1. Eccentricity centrality (computed separately on the incoming and outgoing ties): Eccentricity represents proximity versus isolation, i.e., the ease versus the difficulty of being reached by or to reach others within the network [66, 67]. As this centrality measure is related to the access of valuable information disseminated within a network [59], we found it important to investigate its representativeness for friendship student networks, since it was potentially linked to education outcomes of interest.
2. Closeness centrality (Freeman) (computed separately on the incoming and outgoing ties): This index was selected as a benchmark whose representativeness needed to be tested, since it is used mostly in the literature on student networks (see S1 Appendix). Furthermore, we included this measure because the closeness centrality concerns the speed or efficiency with which information will spread between nodes [12, 56, 87]: students with high levels of closeness centrality will enjoy efficient, easier, and faster access to information, advice, resources, and (academic) benefits in the network [12, 24, 26, 62, 64, 69, 88].
3. Residual closeness centrality (computed separately on the incoming and outgoing ties): Residual closeness centrality reflects the significance of a node as a communication link for its network [12], given that its removal significantly increases the distance between other nodes. This index was used to investigate nodes’ centrality in several investigations [12, 14, 89]. However, as far as we know, no studies have yet used this measure on social networks, and we selected this index to assess its representativeness for such graph types, since being a strong communication link might be important in student networks.
4. Betweenness centrality (Freeman): We also chose this measure as a benchmark to be assessed, since it has been used widely in previous studies on student networks (see S1 Appendix). Moreover, this index is interesting for studies conducted on student networks since students with high levels of betweenness centrality connect other nodes and facilitate communication between other students in the network [26, 55, 56, 66, 67, 69]. Betweenness centrality represents the access and control that a node has over the (novel) information and resources contained in and flowing through a graph [9, 12, 15, 61, 69, 90], and therefore reflects its power or influence on other nodes [64, 87, 88].
5. Geodesic k-path centrality (computed separately on the incoming and the outgoing ties): This bounded k-betweenness index—proposed by Borgatti & Everett [91]- allows for computing the number of neighbors that are reachable by the fastest path up to length k, i.e., that are on a geodesic path less than k away. According to the authors, “long” shortest paths, which are considered when computing betweenness centrality, are not necessarily relevant for the spread of information through the network. Moreover, as stated before, the role of betweenness centrality in student network literature is still not clear. As far as we know, no research has yet studied geodesick-path centrality in student networks, and we selected this index because it might be important for students by informing about their reception and/or dissemination of local information instead of the totality of information that circulates through the entire network [13].
6. Bottleneck centrality (computed separately on the incoming and outgoing ties): According to Obadi et al. [55], bottlenecks are “central nodes that provide the only connection between different parts of a network”. Nodes with high bottleneck scores are therefore the most important ones in the network [92]. This index is used mostly in biological studies [44, 46, 50, 92, 93], but we decided to assess its importance for social graphs such as friendship student networks, since it represents the degree of confluence of links through a given node [93], i.e., through a given student in our case.
7. Eigenvector centrality: This centrality measures reflects the power, influence, or importance of a node in a network [8, 10, 61, 64]. The additional idea behind this score is that a node will be more prestigious or powerful if its neighbors are also central or well-connected [9, 18, 19, 26], an interesting centrality concept for social graphs. Few studies [e.g., 61, 64, 66, 67, 77] have investigated eigenvector centrality for student networks. We included this index since, given the advantages provided by high levels of eigenvector centrality, its representativeness for those graphs might be important and should therefore be assessed.
8. Page rank: The Page rank score [94, 95] quantifies the relative importance of a node within the network [61]. In social networks (e.g., friendship student networks), members that are cited by many individuals who have a high degree of Page rank will see their own Page rank increase [4, 19]. Only two [4, 61] of the twenty-seven studies concerning student networks that we reviewed used the Page rank score to investigate the centrality of students within their network, even though this measure might be a valid candidate for centrality and provides valuable information about the importance of a student within her/his friendship network.
9. Hub &authority scores: Related to networks, the authority score of a node reflects the importance of a node according to the number of important nodes, i.e., hubs that point towards it. Then, a node will be central if it points towards other important nodes, i.e., if it possesses a high hub score by pointing to good authorities. Here again, few studies [61, 77] have used those two distinct indices for investigating the central position of a student within her/his network, even though those measures provide different types of information, and might reflect specific and important centrality measures of a node within friendship student networks.
10. MNC–Maximum Neighborhood Component (computed separately on the incoming and outgoing ties): The MNC [46] is used mostly in the study of biological networks [e.g., 14, 47, 96, 97], even though it could also be used to identify central nodes in other graph types, such as human networks [46]. As far as we know, no study has already used the MNC to compute centrality within friendship student networks. We included this measure because the representativeness of this index could be interesting to estimate, since it concerns a student's centrality, which is related to the degrees of connectivity of her/his friends.
11. Cross-clique connectivity: The cross-clique connectivity [41] of a node represents the level of connectedness of this node to different sub-communities in a network. For a node, a high value of cross-clique connectivity represents its large influence in the graph, the spread and promotion of its ideas, its transfer of information between sub-communities in the network, and its role in the cohesiveness of its clique [13, 26, 87]. As far as we know, this way of representing centrality has yet not been the focus of research within friendship student networks, even though it could potentially highlight important information linked to (a student’s) having a cohesiveness role within her/his network.

3.2. Proposed theoretical taxonomy

Based on the definitions of the eighteen indices that we chose (S4 Appendix) and examples of taxonomies from Song et al. [12], Lü et al. [15]; Ghazzali & Ouellet [13] and Ashtiani et al. [14], who each worked on different sets of centrality measures, we propose five theoretical categories to classify our eighteen centrality measures (see Table 1). To construct our theoretical taxonomy, we took three criteria that concern centrality measures into account, namely, (1) their formulas, (2) the benefits of high levels of centrality, and (3) the topological structure of the network and consideration (or non-consideration) of neighborhood properties. As the definitions of the eighteen indices show (see S4 Appendix), the first category (distance-based) assesses a node’s proximity to the other members of the graph. The second category (geodesic path-based) is related to the geodesic paths on which nodes are located. The third category is based on connectivity, i.e., on the number of direct connections a node possesses. Finally, the fourth and five categories take the neighborhood of a node, the former neighbors’ prestige, and, lastly, the topology of the members adjacent to the node in question into account.

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Table 1. Centrality measures taxonomy: Categories.

https://doi.org/10.1371/journal.pone.0244377.t001

3.3. Computation of the centrality measures on a real network

We used the igraph package [98] implemented in R© to compute the eighteen centrality measures for each student in our specific friendship student network.

3.4. Principal Component Analysis as a methodological tool

Previous studies have used various techniques to compare several centrality indices and/or highlight the most representative centrality measures within specific networks. Among those methods, we find, for instance, the computation of correlation coefficients between centrality indices [3, 13, 32]; Principal Component Analysis [6, 14, 99]; hierarchical clustering [e.g., 14]; the comparison between published network data sets and a Erdös–Renyi random network used as a baseline [10]; and more complex techniques, such as the influence maximization problem (IMP), heuristic and greedy algorithms, message passing theory, and percolation methods (see [15]).

According to Ashtiani & Jafari [6] and Ashtiani [100], PCA allows one to determine the most informative centrality indices and which centrality measures best represent the nodes within a network, i.e., which indices identify the central nodes most accurately.

PCA is a factorial analysis method that uses the correlations, i.e., inter-dependencies, between variables (in our case the eighteen centrality indices) to reduce the p-dimensional space of these variables to a k-dimensional space (with k<p). PCA results in a minimal number of principal components, i.e., factorial axis or latent dimensions that corresponds to maximum data dispersion, with these principal components being linear combinations of the initial variables. We performed PCA on our eighteen centrality indices and worked on standardized data to neutralize the problem of centrality measures with different units. We used the Varimax procedure as it ensures a better distribution of the variables over the factors by rotating the axis and allows easier interpretation of the factorial axis [101, 102]. We used SPSS 23 to perform PCA on our eighteen centrality measures.

First, we performed PCA on the centrality indices because the method can highlight the k latent dimensions of centrality within a graph, in our case our friendship student network. Also, by computing the coordinates of each variable on each highlighted dimension, PCA enables one to identify the factor on which a variable has the highest loading, so that identifying the centrality indices that belong to the same latent dimension is possible in turn. Comparison of the PCA output—the k highlighted latent dimensions—with the proposed theoretical classification of centrality indices then enabled us to check whether this theoretical classification could be validated.

Second, we performed PCA because it also enables one to determine the most representative centrality indices (among a complete set of measures) for a network of interest such as our student graph. The first step consists of retrieving the relative contribution of each centrality index (i.e., each p variable) for each of the k dimensions (i.e., the k factorial axes) retained in the PCA. The following formula computes a variable's contribution to a factorial axis k: (1) Where ρ(X_p, F_k)² represents the quality of the variable’s representation on the factorial axis k, and is equal to the squared correlation coefficient between the variable X_p and the axis F_k; and ∑ρ(X_p, F_k)² represents the variance or inertia preserved on the factorial axis k, and is equal to the sum of the squared correlation coefficients between each variable p and the factorial axis k [102].

The second step then consists of computing each centrality measure’s average contribution to the factorial plan, i.e., the average contribution on all k factorial axes, i.e., Cont_p We compared each average contribution to a threshold of (1/18)×100 = 5.55%−, i.e., in our case a centrality measure’s theoretical contribution, since we worked on eighteen indices. For our paper, values higher or lower than 5.55% indicate a contribution that is above or below the theoretical average contribution, respectively.

4.1. Collect of the data and management of the missing ties

The data were collected at Saint-Louis University in Brussels (i.e., USL-B), Belgium, in October 2016. The juridic department of the USL-B approved the study. The survey that was dispensed contained all the required information for an informed consent by the participants, and the survey was not mandatory. The data were anonymized before further analyses. Data collection took place during academic lectures that covered all curricula proposed by the university, i.e., law; economics; management sciences; literature, philosophy & history; communication, political science, & social science; and translation & interpretation studies. A total of 574 (43.95% of the population) first-generation freshmen students (students registered in their first year of studies for the first time) completed a paper survey. They were asked about their friendship ties at university. In the survey, the student’ friends were described as the “persons with whom students spend personal time, with whom they interact on a regular basis (face to face, by phone, or on online social medias), whom they see outside classes, whom they trust, and/or with whom they share their personal issues” [52, 55, 57].The nodes graph, i.e., the friendship student network, was drawn from the collected data. 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 [103]. Thorough analysis of our graph revealed that 296 students were named as friends at least once by the 574 respondents but did not complete the survey and 651 of the 1911 designations (the total number of ties within the network) made by the respondents concern those missing students. Since SNA methods require complete recording of interactions between actors belonging to the studied network [16], we decided to impute the friendship relations for the 296 missing actors by means of Exponential Random Graph Models (ERGMs) [103] and statnet [104, 105], more specifically the ERGM package [106] implemented in R©. We tested progressively inclusive models and the final model used to simulate the ties on the missing actors included the effects of the number of edges in the network, node mixing by gender (gender was significant to predict the edge probabilities), node mixing by curriculum (the curriculum was significant to predict the edge probabilities), and a homophily effect for students in the same curriculum (i.e., nodes with the same curricula were more likely to be connected). All details (the justification for imputing the missing ties by means of ERGMs, the general formulas of the ERGMs, the terms used in our model, and its validation) are shown in S5 Appendix. The simulation enabled us to impute 639 ties to the 296 missing students. The total number of ties within the augmented network was therefore 2550. This imputation enabled us to compute centrality measures for each of the 870 students belonging to the augmented network, that is, the 574 respondents and the 296 missing actors to whom ties were imputed.

4.2. Comparison between the respondents’ network and the augmented network: Visualization and structural properties of the two graphs

Since as explained earlier, SNA methods require the complete recording of interactions between actors within the studied network, in order to compare the augmented network to the respondents' network, we had to use—for this last network—the complete cases methodological approach, i.e., we had to delete within the respondents’ network the nominations corresponding to students that did not complete the survey (i.e., we removed the 651 nominations made towards the missing actors).

Fig 5 shows the two networks: the original graph- i.e., the respondents’ network—that haves a number of nodes that is equal to 574, and the augmented network that haves a number of nodes that is equal to 870. Table 2 shows the structural properties of the original network and of the augmented network.

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Fig 5. Representation of the original network and of the augmented network.

https://doi.org/10.1371/journal.pone.0244377.g005

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Table 2. Structural properties of the respondents’ network and of the augmented network.

https://doi.org/10.1371/journal.pone.0244377.t002

The reciprocity (i.e., the percentage of dyads with mutual ties within a network: [18]) shows that, for the respondents, 49% of the links are reciprocal, and that, for the augmented network, the proportion of reciprocated ties is equal to 37%. The proportion of mutual ties is higher in the respondents’ network probably because of the deletion of nominations towards missing actors, which has for effect to artificially increase the reciprocity. The diameter—i.e., the longest geodesic distance between any two students in the network [18, 61]—is equal to 27 for the network composed of the students who responded to the survey, and equal to 18 for the augmented network. Results show that the imputation of missing ties allowed decreasing the shortest distance between the two most distant students within the graph, which seems logical since in the respondents’ network we deleted the nominations made by students towards missing actors. Then, the average shortest path length computes the mean of the geodesic distance between each pair of nodes within the network [107]. In average, the shortest path length between each dyad is equal to 9.40 in the respondents’ graph and equal to 7.13 in the augmented network. The imputation process also allowed decreasing the average geodesic distance that is needed to access other students within the network. Finally, Fig 6 shows left-skew phenomena for the degree distribution of our two networks: the two histograms and cumulative density graphs show that there is a maximum of density at low values of node degree. Left-skewed degree distributions show similarity to scale-free networks [14]. In scale-free networks, most nodes have few links and only few nodes entertain many ties [108]. Since the probability of measuring a high value of node degree varies inversely as a power of that value [109], the distribution of nodes linkages in scale free-networks follows a power law [108]. The power law appears in many fields, including the social sciences [109]. As in Ashtiani et al. [14], we compared the degree distribution of both of our networks to the power law distribution in order to assess the scale-free property of our two graphs. Both of our networks seem to follow a power law distribution (Table 2). First, we observed small scores (which denote a better fit between the power law distribution and the data) for the Kolmogorov-Smirnov test statistic (i.e., resp. 0.07 and 0.08 for the respondents’ network and for the augmented network). Second, if the resulting p-value of the Kolmogorov-Smirnov test is greater than 0.1, then the power law is a plausible hypothesis for fitting the distribution of nodes [110]. The two high p-values (i.e., resp. 0.68 and 0.93 for the respondents network and for the augmented network) show that the distribution of nodes for both our two networks do not significantly differ from the power law distribution (i.e., our two networks could have been drawn from the fitted power-law distribution). It is interesting to see that the augmented network seems to better fit the power-law distribution than the network composed only of respondents (again, probably because, for the respondents' network, we deleted the nominations that were sent towards students who did not answer to the survey).

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Fig 6. Distribution of node degree for the two networks.

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5.1. Correlations between the centrality measures

Three conditions are necessary for a PCA to be relevant. The first condition is that the variables must be correlated [111]. For the augmented network, the Pearson’s correlation coefficients between each centrality measure together with their significance levels are shown in Table 3. The results show that the correlations between centrality measures that can be computed separately on the incoming and outgoing ties (e.g., the eccentricity in- and out- centralities) are all positive and significant (p-values ≤ 0.05). Then, the results show many positive significant correlations between centrality measures (p-values ≤ 0.05), except for the bottleneck indices (in- and out-), for which few significant correlations are observed. We notice the strongest relationships between the eccentricity and closeness in- centralities (ρ = 0.96); the eccentricity and closeness out- centralities (ρ = 0.97); the residual closeness and geodesic k-path in- centralities (ρ = 0.97); the residual closeness and geodesic k-path out- centralities (ρ = 0.97); betweenness with the residual closeness in- (ρ = 0.62), geodesic k-path in- (ρ = 0.62), and geodesic k-path out- (ρ = 0.62) centralities; the eigenvector prestige and Kleinberg's authority centrality scores (ρ = 0.97); Kleinberg's hub centrality with Kleinberg's authority centrality (ρ = 0.86) and the eigenvector prestige scores (ρ = 0.85); the Page rank score with the residual closeness (ρ = 0.51) and geodesic k-path (ρ = 0.49) in- centralities; and cross-clique connectivity and the MNC: the maximum neighborhood component (in-: ρ = 0.78, and out-: ρ = 0.72).

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Table 3. Pearson correlations between the eighteen centrality measures for the augmented network.

https://doi.org/10.1371/journal.pone.0244377.t003

5.2. The centrality latent dimensions within friendship student networks

For a PCA to be relevant, two other conditions (other than the correlations between variables) must be met. First, the Bartlett's test verifies whether highly correlated variables might be correlated to the same latent factor(s) [101, 111], which is the case (χ² = 20512.07; p-value = 0.000). Second, the Kaiser-Meyer-Olkin index (KMO) tests the compressibility of information [111]. As the value of the KMO (= 0.713) is higher than 0.5 (the critical threshold), we can consider the factorization to be statistically acceptable.

Concerning the quality of a variable’s representation, 50% of the information contained in each variable must be preserved in the factorial plan [101]. This fourth condition, which was computed automatically by SPSS 23, was met for all our centrality measures (see Table 4).

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Table 4. Sum of the squared correlation coefficients between a variable and each factorial axis.

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Then, to determine the minimum number of axes that preserves the maximum percentage of information, we used the Kaiser criterion, which consists of keeping the k factorial axes possessing eigenvalues (i.e. projected inertia/variance or information’s degree preserved by an axis) that are higher than 1 [101, 111]. According to this criterion, six latent dimensions were retained. Moreover, they conserved 82.82% of the total inertia present in the original dimensional space (see Table 5). This respects an additional criterion based on a 60% threshold for the minimum percentage of conserved variance in the factorial plan [101, 111].

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Table 5. Percentages of variance retained by the first six factorial axes.

https://doi.org/10.1371/journal.pone.0244377.t005

Table 6 shows the variables and their factor loading on the components for which their saturation is the highest, i.e., the latent dimensions on which the variables have the highest loading. The closeness, residual closeness, eccentricity, and geodesic k-path out- centralities are correlated with the first factorial axis. According to these four indices’ definitions and formulas (in S4 Appendix), this first component or latent dimension might therefore reflect the ease with which a node reaches the other nodes, connects them, and transmits information throughout the network. The second dimension, which is highly correlated with Kleinberg's authority & hub centrality scores and the eigenvector prestige score, relates to centrality through the number of connections with prestigious friends. The geodesic k-path and residual closeness in- centralities, betweenness, and Page rank score load on the third factor. This dimension might therefore denote the ability to control the received information and a node’s degree of significance, especially by being located on the shortest (local) paths converging towards the node. The fourth dimension (the maximum neighborhood components and cross-clique connectivity) is linked to the degree of cross-connectivity of a student and her/his neighbors. The fifth component, which is highly correlated with the eccentricity and closeness in- centralities, relates to the ease with which a node is reached by the other nodes in the network and its ability to receive information. The sixth and last dimension reflects the degree of bottleneck, i.e., the degree of confluence through a given student.

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Table 6. Results of the Varimax rotation: Correlation of each variable on the factorial axis on which the saturation is the highest.

https://doi.org/10.1371/journal.pone.0244377.t006

5.3. Verifying the taxonomy on real data: A friendship student network

In order to verify the proposed taxonomy, we compared our theoretical classification (in Table 1) with the six centrality dimensions that emerged from the PCA that was performed on the augmented network, i.e., with the composition of the dimensions in terms of the eighteen centrality indices (in Table 6).

Within the taxonomy, four indices, namely, the eccentricity, closeness, residual closeness, and geodesic k-path centralities, are gathered within a first category (Category 1 in Table 1), which is built on the criteria of a geodesic distance-based formula and access to information as a centrality corollary. Those four indices, but computed for the outgoing ties only, saturate on the first latent dimension in the PCA (Table 6), which therefore matches Category 1 in Table 1. This theoretical category is therefore validated, but only for centralities computed on the nominations (declared friends) that are made by a node (student). Moreover, the closeness and eccentricity centralities that are computed on the incoming ties, i.e., the nominations received by a node, and which both saturate on the fifth factorial axis (Table 6), seem to form a subset within the first theoretical category in Table 1. Then, in the theoretical taxonomy, we assigned the residual closeness and geodesic k-path centralities to the first category, but also to a second family of indices (Category 2 in Table 1) based on a geodesic-path formula and that relies on information control and diffusion. As seen above, residual closeness and geodesic k-path out- centralities have been verified to be part of Theoretical Category 1. However, they are validated for Theoretical Category 2 when they are computed on the incoming ties. They form a latent construct (the third dimension in Table 6) together with the betweenness index, with the latter also being validated for Theoretical Category 2 of centrality measures. As shown for the eccentricity and closeness centralities, the residual closeness and geodesic k-path indices therefore seem to be divided into two distinct categories according to the nature of the ties (i.e., in- versus out-),.

Then, the second latent dimension resulting from the PCA (Table 6), which brings together Kleinberg's authority & hub centrality scores along with the eigenvector prestige score, validates the third and fourth theoretical categories of indices within the taxonomy (Table 1), i.e., the categories based on the degree of connectivity and the prestige of the connections, respectively, which both reflect power and influence.

The fifth theoretical category in Table 1 concerns centrality indices that take the topology properties of the neighborhood into account in their formulas and relate to information spread and cohesiveness roles. The cross-clique connectivity and maximum neighborhood components (in- and out-) were proposed as being part of this category. That was confirmed by the PCA, which showed those three indices gathering on a same latent factor (the fourth dimension in Table 6). It should be noted that since a clique is composed of three or more nodes, cross-clique connectivity was also proposed as part of the third theoretical category in Table 1, which is based on the number of connections. However, the PCA confirmed that cross-clique connectivity belonged to the same category as the MNC scores.

Finally, bottleneck (in- and out-) and Page rank scores do not behave as expected according to the taxonomy. First, as shown above with the correlations and PCA, the two bottleneck indices do not correlate significantly with most of the other sixteen centrality measures, and both saturate on a specific factorial axis (the sixth dimension in Table 6). Yet based on the shortest paths in their algorithms, bottlenecks seem therefore to measure a different centrality type than the residual closeness, geodesic k-path, and betweenness indices. Second, we expected the Page rank score to be validated within the same theoretical category as the eigenvector and Kleinberg's authority & hub scores, since the Page rank formula takes the prestige of the incoming ties into account when computing a node’s centrality. Instead, PCA showed a maximum saturation of Page rank on the same factorial axis as the geodesic k-path (in-), residual closeness (in-), and betweenness centralities.

5.4. Generalization and summary of the results

In order to generalize our findings, we computed the centrality indices on the original network (i.e., on the 574 respondents). Then, we applied the PCA to the eighteen centrality measures computed for those 574 respondents only and compared the PCA outputs with those from on the augmented sample (the 870 students). The tables related to the PCA performed on the 574 respondents are presented in S6 Appendix. The results show that even though some differences are highlighted between the two PCAs, we found roughly the same latent factors, which supports the generalizability of our results. The similarities and differences that were highlighted are as follows: (1) five factorial axes emerged for the PCA carried out on the 574 respondents instead of six latent dimensions for the augmented sample (see Table 5 in S6 Appendix); (2) the first dimension (which included the closeness out-, residual closeness out-, eccentricity out- and geodesic k-path out- centralities in the initial PCA) is similar between the two PCA, except for the fact that betweenness is added to this dimension when the PCA is performed on the 574 respondents. However, as shown in Table 3 of the S6 Appendix, betweenness does not meet the requirement of 50% of the information preserved in the factorial plan. Therefore, its proximity to the four centrality measures that compose the first dimension should be considered with caution; (3) the second dimension (which includes the Kleinberg's authority, eigenvector, and Kleinberg's hub scores) matches with the dimension in the PCA carried out on the augmented sample exactly; (4) as in the initial PCA, the third dimension includes the geodesic k-path in- and residual closeness in- centralities, but, on the one hand, the eccentricity in- and closeness in- centralities are now part of this component, and, on the other hand, betweenness and the Page rank score are no longer included in this dimension. We stated earlier that betweenness is not well represented in the factorial plan, which, as for the initial PCA, is also the case for the Page rank score (see Table 4 in S6 Appendix). We also note that this dimension highlighted for the respondents only would validate the first category in our theoretical taxonomy, but only for incoming ties. This might be due to the fact that the incoming ties no longer include the imputed ties from the non-respondents; (5) the fourth dimension (which includes the MNC in- and out- together with cross-clique connectivity) corresponds to the latent factor generated by the first PCA, except for the Page rank score, which is now added to this dimension. But as just stated, its proximity to the three other centrality measures that compose this latent factor must be considered with caution; (6) in the correlation matrix (see Table 1 of S6 Appendix), the two bottleneck scores seem to be more linked to the other centrality measures than when computed for the 870 students. However, whether the PCA is performed on the respondents only, as on the augmented sample, they continue to form a unique latent dimension.

Table 7 compares the centrality dimensions that came out of the PCA (for both networks) with the theoretical categories proposed within the taxonomy, and summarizes the above findings: For both networks, the first latent dimension validates the first theoretical category, but for indices computed on the outgoing ties only, while for the augmented network (resp. the respondents’ network), the fifth (resp. third) dimension matches, but only for incoming ties, two centrality measures that were proposed within Theoretical Category 1. For both networks, a unique dimension, i.e., Dimension 2, which includes and represents three indices, validates the membership in a unique class for those indices that were theoretically proposed as belonging to two theoretical categories, i.e., the third and fourth categories in Table 1. Then, for both networks, except for the Page rank score (for which we expected saturation on the second dimension) and the two bottleneck indices, which both saturate on the sixth (resp. fifth) dimension, the third dimension matches with Theoretical Category 2. However, for the respondents’ network, the betweenness does not belong to the third dimension as it was expected. Finally, for both networks, the fifth theoretical category of centrality measures is validated by the fourth PCA dimension.

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Table 7. The proposed theoretical classification and the centrality dimensions that came out of the PCA: Comparison.

https://doi.org/10.1371/journal.pone.0244377.t007

5.5. The most representative measures of centrality for friendship student networks

The last objective of the paper was to find the best centrality measures, i.e., the most representative and significant indices, when we investigate and represent friendship student networks. As in Ashtiani et al. [14] for biological networks, our goal was therefore to establish, from within a set of centrality indices, the measures that best categorize the central students and distinguish them from the peripheral ones.

As explained earlier in Section 3.4., we first retrieved the relative contribution of each of the eighteen centrality indices for each of the six dimensions that were retained in the PCA(performed on the augmented network). Then, we computed the average contribution of each of the eighteen centrality measures to the factorial plan (i.e., Cont_p). We compared each average contribution to the threshold of 5.55%, with higher (lower) values than this threshold indicating a contribution that is above (below) the theoretical average contribution. Table 8 shows the centrality measures and their average contributions (in descending order) to the factorial plan. The eight indices that best represent the variance of student centrality within a network are the bottleneck indices (in- &out-), eccentricity and closeness centralities computed on the incoming ties, maximum neighborhood component measures (in- &out-), and Kleinberg's authority & eigenvector prestige scores. On the contrary, other centralities (e.g., the betweenness index and eccentricity and closeness centralities computed on the outgoing ties) seem to contribute less to the factorial plan, being below the average threshold of 5.55%.

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Table 8. Average contributions of the centralities to the factorial plan.

https://doi.org/10.1371/journal.pone.0244377.t008