# Notes: Vaquero, L. M., & Cebrian, M. (2013). The rich club phenomenon in the classroom

### 2017-06-27

## References

**Citekey**: @Vaquero2013-ld

Vaquero, L. M., & Cebrian, M. (2013). The rich club phenomenon in the classroom. Scientific Reports, 3, 1174. https://doi.org/10.1038/srep01174

## Notes

*Summarize*:

*Assess*:

*Reflect*:

## Highlights

social interactions generally imply better score for students engaging in them. We find that thesestudents, which is created during the first weeks of the course. Low performing students try to engage in theclub after it has been initially formed, and fail to produce reciprocity in their interactions, displaying more (p. 1)

NA (p. 1)

not true. (p. 1)

analysis has mainly tried to determine static structural features of the social learning network formed by the (p. 1)

While the relevance of the social network structure and interactions has been widely recognised in theeducational context 10ingness to communicate 11. In general, it is not just about knowing ‘‘who’’ the students interact with, but ‘‘how’’and ‘‘when’’ they do it and, importantly, what is the result of these interactions with regards to the educational12 (p. 1)

NA (p. 2)

Acquiring full knowledge on ‘‘how’’ students interact would befacilitated by having access to dynam ic interactions and their changes (p. 2)

Results (p. 2)

Diversity and assortativity analysis . Our first finding is that, in thisenvironment, social diversity is negatively correlated with (p. 2)

We analysed a record of college student interactions and compared (p. 2)

Figure 1 | Diversity and Assortativity Analysis. (A) shows a graph of one of the analysed courses including 82 students at the end of the last week of thecourse. Continuous thick blue edges indicate persistent interactions while dotted thin grey edges indicate transient interactions. High performi ngstudents are shown in dark blue, mid performing ones in red and low performing ones in green. As can be observed, high performance students form a‘‘core’’ where the highest density of persistent interactions can be observed. Low performance students remain in the periphery of the graph, mainlyholding transient interactions. (B) Scatter plot and linear regression for one of the variables analysed (number of interactions) vs. scoring in one of theclasses (R2 5 0.72). © Scatter plot and linear regression for social diversity vs. scoring in one of the classes ( R2 5 0.12). (D): Ratio of transient to persistentinteractions obtained for different groups of students with different levels of interaction (LOW, MID, HIGH). (p. 2)

group) per week and per student group (low, (B); mid, ©; and high, (D)) relative to the total number of interactions per group per week. Continuouslines represent the fit of a curve to the points as indicated in Methods. As can be observed, the % of persistent interaction increases as the course prog ressesfor all groups of students. High performing students achieved a higher % of persistent interactions than mid and low performing ones. (p. 3)

NA (p. 3)

Specifically, social diversity is defined as Shannon’s entropy associated with individual communication behaviour, normalised to the total number of interactions (see Methods in SI for more details). (p. 3)

Indeed, low performance students tend to initiate many transientinteractions regardless of the performance of the students theyinteract with. These interactions held by low performance studentsstart late in the course, allowing high performers to establish a closelyknitted group. In the following, we give details of these findings. (p. 3)

The number of connections (students that a student has interactedwith) and number of interactions (times a student has contacted orbeen contacted with/by other students), (see Methods in SI) were allpositively correlated with the final score of the student (Pearson’scorrelations of 0.81, 0.85, respectively; p , (p. 3)

NA (p. 3)

established persistent interactions before mid and low performance (p. 5)

To further analyse the effects on score, students were grouped intohigh (. 6.5), mid (between 6.5 and 3.5) and low (, 3.5) scoring(scores in Spain are typically given in a 0–10 scale, being 10 the topscore). To verify the suggested existence of less effective interactions, (p. 5)

we also classified the type of interactions in two types: 1) persistent,those sustained over time, and 2) transient, those not reciprocated (p. 5)

vague; are they truly opposite? (p. 5)

actions together with the assortativity analysis (students preferred tosuggested that at some point reciprocity R i,j (measured as the fractionof times a student i in any given group responds to a student j outsideof times a student i in any given group responds to a student j outsideher same group) should start to drop. However, reciprocity remainedher same group) should start to drop. However, reciprocity remainedunchanged with time and was similar between groups ( ^0: (p. 5)

lysis17 on these persistent interactions with regards to score indicatedthe existence of preferential interaction initiation (r 5 0.5, p , 0.05by using the Jackknife method, see Methods in SI). In other words,similarly scoring students tended to keep persistent interactions only (p. 5)

groups are highly symmetric (having almost even initiations startingfrom both ends). On the contrary, transient interactions betweendent with lower performance (with 0.87 probability). In addition, thetiming of responses was different. While persistent interactions areresponded in 8.1 6 0.3 hours on average, the response time fortransient interactions is delayed 7.21 6 0.46 days. (p. 5)

This assortative behaviour with regards to scoring is highly suggestive of a ‘‘rich club’’ phenomenon (see Methods in SI and 18,19). A‘‘rich club’’ is defined as a set of nodes with degree larger than k thattend to be more densely connected among themselves than the nodeswith degree smaller than k (p. 5)

all the types of interaction into account, we could observe no ‘‘richclub’’ effect (w rðÞ^3 for the students with more links, indicating theyalso interacted with students outside the ‘‘rich club’’). However,when only persistent interactions were taken into account, weobtained w rðÞ^1, which is in line with the idea of high scoringstudents keeping persistent interactions between themselves as indicated by our assortativity analysis. The ‘‘rich club’’ phenomenoncould not be observed during the first weeks, w(r ) =apparent only after week 4–5 for the top performing students,apparent only after week 4–5 for the top performing students,remaining stable afterwards. (p. 5)

Information cascades . Information cascades reveal spreadmechanisms in which an action or idea becomes adopted due tothe influence of others, typically, neighbours in some network. Awell-known example are cascades in the context of large productrecommendation networks 21–24. (p. 5)

Temporal analysis . One interesting finding is that the total numberof interactions per week (normalised to the maximum value in allweeks) for all groups increased over time and it saturated aroundweek 6 for mid performing students and around week 4 for high (p. 5)

to analyse another source of information: file exchange of students in (p. 5)

NA (p. 5)

We defined as trivial cascades those implying a single transfer (asingle originating source and a single destination) of informationabout the course, and non-trivial cascades, those with more complexpatterns. We found a total of 845 cascades, and 53.37% of which were trivial cascades (T1 in Figure 4), 25% were non-trivial cascades involally more complex. (p. 6)

The total number of cascades was significantly different across allthree groups 51%, 35.97% and 13.03% for high, mid and low performance students, respectively (see Table 1). (p. 7)

chronous transfers) gradually increased as the average score of thestudents involved in the cascade increased. This is also supported bythe fact that among non trivial cascades, the most common patternfor low performance students was star-like (T2 and T397.8%), while chained cascades (T4, T5 and T6 in Figure 4) weremore common for mid (53.82%) and high (76.29%) performingstudents. (p. 7)

Discussion (p. 7)

The major finding is that a higher number of online interactions(independently of the number of distinct students involved) is usually an indicator of higher score. (p. 7)

The results also show that the higher the score of the students, the (p. 7)

Supplementary Information for: The rich-club phenomenon in the classroom (p. 9)

Interaction Classification An interaction between students i and j was classified as persistent if the contact occurred at least twice per week. This definition tries to convey the importance of reciprocity as an indicator of mutual interest and value extracted from the interaction. (p. 9)

Grouping Metrics The correlations between properties of adjacent network nodes are known in the ecology and epidemiology literature as “assortative mixing”. (p. 10)

“Rich club” Connectivity Following [2, 3], students were sorted by decreasing number of links (connections). The node rank r denotes the position of a node on this ordered list. The “rich club” was defined, for the purpose of this study, as nodes with rank less than r max (e.g. 7 %). The “rich club” connectivity (r ) is defined as the ratio of the total actual number of links to the maximum possible number of links between members of the top scoring “rich club”. (p. 11)