The purpose of this study is to detect project teams in a group. A key point in considering group's relationships is the reciprocal influence, whereby group's members influence each other. There was conducted a survey based on reciprocal nomination method, and then a social network was constructed. Participants were first-year bachelor students of Tomsk Polytechnic University. Various social network analysis algorithms were used to cluster network in communities. The results of analysis were discussed with the teachers and students, and then detected community teams were adjusted within the key actors of group. The results of the study may be used to create project teams, which can make successful collective actions in educational projects.
A problem arising very frequently is how to identify new teams in an existing
group. Community detection helps to understand the distribution of key social
actors and their interrelations in the network [
Firstly, we need to define what is meant by a project team. A project team
is a group of people who are able to act in concert and to achieve collectively
the common goal. Team members have specific and unique roles, where the
performance of each role contributes to achievement of the team’s goal. In project
teams members care about the success of other team’s members because their
own goal attainment is often inextricably bound to collective achievement [
This study is a part of one-year follow-up research of social network changes among the members of students group. The purpose of this study is to detect project teams in a group. We present an approach for project teams creation based on SNA methodology. This approach includes descriptive analysis, community structure identification and key actor analysis using graph theory.
The structure of the paper is following. In Section 2, we give short overview of related works. Then, in Section 3, we introduce the dataset, on which our approach has been tested. We calculate the measures of our networks, detect key social actors. In Section 4, we examined an influence of teams’ interactions over time on the academic performance. A brief summary follows in Section 5. 2
Pijl et al. [
Rienties et al. [
Pronin et al. [
Liu et al. [
Lomi et al. [
Ertem et al. [
In contrast with modularity that was proposed by M. Newman [
Research questions. Our one-year follow-up research is aimed to explore the community organization of the network and the influence of the structural factors of the network on academic performance of students. We formulated the following questions concerning the community organization of the network: Q1. What is the network structure at the classroom level?
Q2. How to use the network structure to increase academic performance of students?
Dataset. In our experiments, a local social network was built on reciprocal nomination between 20 first-year bachelor students of Tomsk Polytechnic University during the fall semester of 2015. We used the direct-preference questions. The 20 students answered the four social network questions: 1. Name classmates with whom you spend free time. 2. Name classmates to whom you are applying for the information, related to academic activities. 3. Name classmates who could influence your academic performance. 4. Name classmates with whom you don’t want to cooperate in framework of creative project.
The students were allowed to nominate up to four classmates. Participants were 20 first-year bachelor students who were 17 to 19 years old (M = 18.5, SD = 0.35; 75% male). Data were collected on-line with the Google forms.
Data Preprocessing. Four square matrices A1, A2, A3, and A4 of size 20 were generated respectively on the basis of the questionnaire. In each adjacency matrix A1, A2, A3 the element (i, j) is equal to 1 if row student i nominated the column student j, otherwise the element (i, j) = 0. In matrix A4 the element (i, j) is equal to −1 if row student i nominated the column student j, otherwise the element (i, j) = 0. Then each matrix A1, A2, A3 was summarized with the matrix A4, and the binarization procedure was applied to the result: if the element (i, j) is less or equal than 0 then it gets set to 0, otherwise it is set to 1.
The social networks are represented by directed graphs Gk = (Vk, Ek), k = 1, 2, 3, where a set of vertices Vk includes n = 20 members of group, and a set of edges Ek ⊆ Vk × Vk presents the relation «reciprocal nomination» that corresponds to k = 1, 2, 3 questions. A key point in considering these relationships is the reciprocal influence, whereby team’s members influence each other. The graph Gk is directed, i.e. every edge (i, j) ∈ Ek links the source vertex i and the target vertex j. The direction of the edges is makes each adjacency matrix A1, A2, A3 of the each directed graph G1, G2, G3 non-symmetric because the source vertex defines nomination to the target vertex but not vice versa. The number of edges mk = |Ek| ≤ n · (n − 1), k = 1, 2, 3.
Descriptive Network Statistics. To address research question Q1 we calculated descriptive network statistics in order to define network structure at the classroom level. Figure 1 illustrates the original networks G1, G2, G3, reflecting the structure of reciprocal nominations in the group. These networks G1, G2, G3 have the identical vertex set V , |V | = n = 20, but different sets of edges E1, E2, E3: |E1| = m1 = 81, |E2| = m2 = 72, |E3| = m3 = 71 respectively, in which one vertex represents an actor, and one edge denotes the nomination between any two actors. To each actor in the student’s group labels were assigned: A01, A02, . . . , A20. In our experiment, loops and multi-edges are not allowed.
It’s seen that the diameter of graph G2 equal to 6 because the longest path between actors A15, A06, A12, A19, A11, A08 and A10 takes 6 edges, the diameter of graphs G1 and G3 equal to 5 (Figure 1, green arrows), while the lengthes of average path between any vertex pairs are {2.166, 2.236, 2.190} respectively. In our experiment, density range from 18%(0.186) to 21%(0.213) only, it can be explained by the limitation of the questionnaire, which recommended to nominate up to four actors. The transitivity range from 0.399 to 0.409 indicates the low level of intra-group interaction, the reciprocity range from 0.11 to 0.66.
A basic property of the vertices in a graph is their degree. Degree provides information on the position of actors and how they communicate. The in-degree din(v) (out-degree dout(v)) of vertex v is equal to the number of incoming (outgoing) edges. The in-, out-, total degree distribution of vertices of the graphs G1, G2, G3 are shown in Figure 2. As we can see from Figure 2, the modal interval equal to 4, which has frequency more than any other interval. where A is the adjacency matrix. In our case, clustering coefficient cl(v) is from 0.248 to 0.377.
To evaluate the significance of the network statistics (Table 1) the simulation of the random graphs was used. Based on topological properties of the each graph G1, G3, G3 1000 random networks were generated to compute their average network statistics. Table 1 gives these statistics in the last column hG2i corresponding to the network G2. The network G2 has approximately the same average path length than 1000 random graphs of the same size (2.236 and 2.262 respectively), and the network G2 has a clustering coefficient that is higher than the corresponding value of 1000 random graphs (0.3767 and 0.258 respectively).
Community Detection Algorithms. At present, many community
detection algorithms have been designed [
Denote by C = {C1, C2, . . . , Cp} a partition of a set of vertices V . We call
C a clustering of a graph G and the Ci, which is required to be nonempty,
community, i = 1, 2, . . . , p. Following to the paper [
Let us list major features of these algorithms. The edge betweenness
algorithm [
Using algorithms mentioned above each graph G1, G2, and G3 was divided into communities. Figure 3 shows results of network partitioning G2 into clusters, which are denoted by different colors. Five communities were detected with edge betweenness algorithm, while using walktrap and optimal algorithm partitioned the graph G2 into 4 communities. The numbers of actors in these communities are different: V (Cb) = {14, 1, 1, 3, 1}, V (Cw) = {3, 8, 5, 4} and V (Cop) = {7, 5, 5, 3}. We use subscripts {b, w, op} to denote the clustering algorithm that was used. Figure 4 shows distribution of number communities detected with the edge betweenness algorithm, walktrap algorithm and optimal algorithm for 1000 random graphs of the same size as the network G2. According to Figure 4, it is clear that the actual number of communities detected in the original network G2 (4 communities for walktrap and optimal algorithms) would be considered typical from the perspective of random graphs, while using the edge betweenness algorithm partitioned 1000 random graphs into number communities from 1 to 14. where m(C) = |{(u, v) ∈ E : u ∈ C, v ∈ C}|, E(m(C)) is the expected value of m(C) under some model of random edge assignment.
The edge betweenness and walktrap algorithms are the hierarchical clustering
algorithms, and the modularity score of the current clustering is stored after each
time step. In the optimal community algorithm the highest modularity is defined
after (n − 1) merges. The algorithms above use the modularity score to decide
where to stop the splitting or merging. Another way to qualify the
communities detection is to compute scoring function based on the internal connectivity,
external connectivity and combination of internal and external connectivity of
vertices set [
123 13 02 41 01 0 5 4 6 5 2 1 4
In the each matrix row and column sums belong to the number of edges
incident on a given community. In the matrixes we bold the number of intercluster
edges. Using data from matrixes we calculated the conductance [
Community Structures Comparison. After getting the communities, the
partitions were compared using various metrics, and results are presented in
Table 3. Normalized mutual information (NMI) measure is based on the fact
that if two partitions are similar to each other, then only a small amount of
additional information is needed to infer one clustering assignment from the
other [
Key actor analysis. Next to the analysis described above, we identified key social actors. We used the comparison of relative values of eigenvector centrality and betweenness centrality. The betweenness centrality gives a higher score to a vertex that sits on many shortest path of other vertex pairs, and it centrality usually refers to the access to novel information and control benefits. Eigenvector centrality gives a higher score to a vertex if it connects to many high score vertices. We calculated the linear regression model, Figure 5 shows a scatter plot of Eigenvector centrality as a function of Betweenness centrality. The equation for the line in Figure 5 is y = 0.0102x + 0.2833 (red line), this linear model was significant (F = 14.118, p-level = 0.0014 < 0.05).
Figure 5 shows each vertex’s relative value of eigenvector centrality and
betweeness, scaled by the value of the regression residuals, labels of actors scaled
by the absolute value of residuals. D. Conway [
The results of the community structure identification (Figure 3) and key actor analysis (Figure 5) were discussed with the teachers and students. A frequently mentioned disadvantage of the community structure based on the edge betweenness algorithm was providing more unbalanced partition than walktrap and optimal algorithms. The statistical characteristics of community sizes and their variations are: M = 4.0, SD = 5.65 (edge betweenness algorithm), M = 5.0, SD = 2.16 (walktrap), M = 5.0, SD = 1.36 (optimal algorithms).
Teachers recommended to separate the key actors A12 and A19 into the different teams. According to the community structure identification the following options are available: to move a key actor from C2w to C1w or from Cop to Cop 1 4 (Figure 3). The conducted SNA modelling leads to the next result: the compromise between the modularity and the conductance is to move the actor A19 from C2w to C1w. In this case, as we expected, the modularity score decreased from 0.314 to 0.304, while the conductance increased from 0.22 to 0.263. 4
As a result, four project teams were formed after the midterm (9th week of the fall semester 2015): T1 = {A08, A10, A11, A19}, T2 = {A04, A05, A07, A09, A12, A17, A18}, T3 = {A06, A13, A14, A15, A16}, and T4 = {A01, A02, A03, A20}. To address research question Q2 we examined an influence of teams’ interactions over time on the academic performance.
In our experiment, during the second period of the term (from 10th to 18th weeks) students from the experimental group (gexp, n1 = 20) additionally were meeting with the peers of their team once a week during a 2 hours tutorial in the classroom, and they worked on a project. It is notable that students cannot change teams during the experiment. A control group (gcont, n2 = 21) is not splitted into project teams and does not have any additional meetings.
Measurement of academic performance on the experimental group and the control group was collected at two time points: a) at the midterm (p1 = 9 weeks), b) at the end of term (p2 = 9 weeks). We received this information directly from the Education Office. In both groups the each student could earn 480 points during the fall semester of 2015 (50% at the midterm, 50% at the end of the term). In our experimental group, average overall performance at the midterm was M = 122.7 points, SD = 26.96, range R = [45, 163].In the control group, average overall performance at the midterm was M = 140.09 points, SD = 22.14, range R = [97, 182].
Figure 6 gives descriptive statistics of academic performance (in points) to the experimental and control groups as well as the each team. There are outlier points in the dataset, these observations correspond to the points of the actor A04, who dropped out from the experiment after 11th week.
Firstly, the Shapiro-Wilk test was applied to check whether the distribution of dependent variable came from a normally distributed population. The dependent variable is the number of points at time points (at the midterm and at the end of term). At .05 significance level the null hypothesis was rejected and there is evidence that the distribution of points in the experimental group (W = 0.87, p-value = 0.017 < 0.05) is not from a normally distributed population, while the distribution of points in the control group (W = 0.97, p-value = 0.73 > 0.05) is from a normally distributed population. Hence, we decided to use the non-parametric statistics. It is clear from Figure 6 that the median of the experimental group (gexp) at the first time point (p1) is less than the median of the control group (gcont) and vice versa at the second time point (p2). At the first data point (p1) in the experimental group the median was 128.5 points, while in the control group the median was 144 points, at the second data point (p2) the experimental median was 205, the control median was 189.
We applied the Mann-Whitney-Wilcoxon criteria to test of the null hypothesis that students from the control group tend to have the larger value of academic performance (in points) than students from the experimental group. At the .05 significance level, the null hypothesis (U = 137.5, p-value = 0.058 > 0.05) was accepted at the first time point. We conducted a randomization test of no difference in population medians (null hypothesis) against a two tailed alternative, where the difference in sample medians is the test statistic. We created 5000 randomizations of the n1 + n2 = 20 + 21 = 41 observations. The two tailed probability under the null hypothesis is p-value = 0.0376 < 0.05, and 95% confidence limits are −14.01 and 14.0. The obtained median difference was −15.5 points, which clearly falls outside the interval. Thus we can reject the null hypothesis, and conclude that the median points of the control group is significantly greater than the median points of the the experimental group. At the second data point (p2) we repeated the randomization test for comparing two medians, 95% confidence limits are −16.5 and 16.5. The obtained median difference was 16.0 points, which clearly falls inside the interval. We can not reject the null hypothesis, and we can expect that the influence of teams’ interactions on the individual academic performance is positive.
The next set of statistical tests was applied to the experimental group. Firstly, we need to test that none of k = 4 teams stochastically dominates one another. The Kruskal-Wallis test was applied to decide whether the population medians on a dependent variable are the same across all levels of a factor. The factor has four levels: 1 = T1, 2 = T2, 3 = T3, and 4 = T4. The null hypothesis is that the medians are equal across the teams. At .05 significance level, we conclude that the medians are equal across the teams (χ2 = 2.0, p-value = 0.57 > 0.05) at the midterm. Secondly, for the comparison across repeated measures at the midterm and the end of the term the Friedman’s test was used. It is used to test for differences between the two snapshot data when the dependent variable being measured is ordinal (ranks in our case). The null hypothesis that the distributions are the same across repeated measures was rejected (χ2 = 16.2, p-value = 5.7 · 10−5 < 0.05). Hence, the distributions across repeated measures are different. There is evidence of the influence of teams’ interactions on the individual academic performance. 5
Project teams are detected using various social network analysis algorithms. The key actor analysis allows us to identify individuals who have the strongest influence on other members of the group. The results of communities detection can be used in the educational process but require discussions with teachers and students. According to compromise between the SNA results and semantic recommendations of teachers and students, we have chosen the basic algorithm, and project teams were created. We found evidence of peer effects on academic performance. In the experimental group as a whole, as well as in the detected teams the academic performance increased in comparison with the control group.
The further research of our longitudinal study can be continued in the following directions. At first, it is community detection in terms of motifs, i.e. dyads, triads (two or three students are only connected to each other) as a subgraph with a fixed number of vertices and with a given topology. Such description allows us to identify complexity levels of a project to each team and different assessment methods of team performance. At second, it is an application of qualitative analysis of relations inside and outside project teams and assessment of potential predictive factors of relations.