An Exploratory Analysis

Matthew Fam—Dartmouth Social Systems Lab

Acknowledgement

The following research was produced by Matthew Fam with the help of Christopher Welker, a graduate student in Dartmouth’s Department of Psychological and Brain Sciences. It was completed in Dartmouth College’s Social Systems Lab, led by Dr. Thalia Wheatley. A portion of this work was made possible by a generous stipend from Dartmouth’s Undergraduate Advising and Research (UGAR), specifically the James O. Freedman Presidential Scholars Program.

Abstract

How do people become popular online, and do those same behaviors predict popularity in person? Prior work has shown that people’s centrality—a measure of popularity—differs between their online and offline social networks (Gaito et al., 2012). Here, we investigate which behaviors predict centrality in online and offline social networks. We analyzed an open-source, multi-layer social network (N = 79) with four categories of edges: (i) undirected, online social connections on Facebook; (ii) directed, self-reported (offline) friendships; (iii) undirected, face-to-face (offline) interactions measured by radio frequency identification (RFID) tags; and (iv) undirected instances of simultaneous presence (offline) in a shared space (“co-locations”) measured by RFID tags. Using correlation, regression, clustering, and structural equation modeling (SEM), we found that centrality in the colocation network predicted greater centrality in the online social network, but not in the offline friendship network. This suggests that simply being around others in a physical space translates to centrality in an online social network. However, co-location did not lead to centrality in the offline friendship network, implying that in-person connections require more complex behavioral patterns than mere co-presence.
  Keywords: social networks, friendship, social media, social interaction

Methods

Data used in this exploratory analysis were retrieved from SocioPatterns. The referenced data sets were created by recording contacts and relations between between 2nd year students in a high school in Marseille, France over the span of five days during December 2013. Four data sets contained information about students in nine classes: directed contacts between students (reported at the end of the fourth day), a directed network of reported friendships, and pairs of students for which Facebook friendship status was known (whether present or absent) respectively. A fifth file included general identification data for the study participants and later update provided a sixth file containing the colocation contacts of students in a high school, measured by radio-frequency identification (RFID) scanning devices.

This collection of data was manipulated (i.e. wrangled, analyzed, and visualized) using R. Initially, each selected network’s dataset was converted to an adjacency matrix. Centralities (betweenness, closeness [in- and out-closeness in directed networks], eigenvector, degree, and pagerank) were calculated for each ID in addition to various relational/behavioral metrics of interest. Measures calculated for separate networks were tied together by matching subjects’ general identifying characteristics (gender, class, etc.). After this setup was complete, data was filtered to only include participant IDs present across all of the data sets relevant to this exploration. This combined and filtered dataset was then used for the following analyses. Code for data preparation is available online.

Analyses Walkthrough

NOTE: A decision to define significance according to an ⍺-level of 0.05, or a 95% significance level, was chosen arbitrarily prior to running any statistical tests to minimize the potential of p-hacking.

DISCLAIMER: The “analyses” provided below are one possible way of interpreting the data and corresponding results. As an exploratory analysis—not a full-fledged scientific publication—these results, descriptions, and discussions may not quite satisfy the accepted standard for scientific rigor. Rather, this work is meant to document an approach to exploring the data at hand while highlighting points of interest, notable observations, relevant thoughts, and possible ideas for further exploration. Though a solid effort was made to properly apply the scientific method, a certain level of liberty was taken in order to examine avenues that would have been closed otherwise for lack of sufficient evidence. These instances are documented and made obvious to maintain the integrity and credibility of the study.

LIMITATIONS: The included data, particularly in the cases of the interaction and colocation networks, only account for activity during the school day. They fail to consider social activity between subjects outside of school hours and neglect other forms of social interaction (digital messaging, telephone communication, etc.). Additionally, data was not collected for each subject across all four networks. As such, comparative analyses (and corresponding visualizations) were performed after the relevant networks were filtered to include only those subjects present in all data sets. However, to ensure the most realistic portrayal of each network and each subject’s position within, centrality measures, behavioral metrics, and other parameters were calculated based on all available data.

Network Visualizations

Prior to exploring the statistics behind the given networks, it may be informative to visualize them.



The figure above depicts the various networks analyzed in this project:(A) an online, undirected network of Facebook “friend” connections; (B) an offline, undirected face-to-face interaction network measured by radio frequency identification (RFID) tags (recordings every 20s); (C) an offline, directed network of self-reported friendships; and (D) an offline, undirected simultaneous presence in a shared space (colocation), measured by RFID (recordings every 20s). In each of these networks, colors mark the classes/specializations that the corresponding students belong to. “Bio” classes focus on biology; “MP” classes—mathematics and physics; “PC” classes—physics and chemistry; “PSI” classes—engineering. Students in Bio1 are depicted in red, Bio2 in orange, Bio3 in light green, MP1 in bright green, MP3 in teal, PC1 in sky-blue, PC2 in purple, and PSI in pink; males are depicted as squares and females as circles. Edge darkness in (B) the interaction network and (D) the colocation network represents total time spent interacting or colocating respectively. Arrowheads in (C) the friendship network mark the direction of reported friendships (an arrow pointing from individual 1 to individual 2 suggests that individual 1 reported individual 2 as a friend).

Network Comparisons

Given the presence of four separate networks (Facebook friendships, in-person friendships, interactions, and colocations), understanding the relationship between these networks is a logical starting point. Namely, it is important to assess the consistency of an individual’s relative centrality across networks, within networks, and/or within specific measures.

Note that this initial test will not produce any scientifically rigorous results. Rather, the results will provide some basic information about the structure of the data and the relationships between different variables—perhaps revealing specific avenues worth exploring in more detail. Since no defined question is being asked, no significance tests will be performed to minimize risk of Type I error.



In the figure above, besides the specification of each centrality measure, the network on which that measure is based is noted in parentheses (I-Interaction Network, FN-Friendship Network, FB-Facebook Network, CN-Colocation Network). The dendograms are color-coded to show the division of the data into four groups, based on heirarchical clustering to determine how closely related the measures are to each other. The decision to classify the dendogram into four groups was done to assess whether the data would divide along the inherent network classes, of which there are four.

As one might expect, these results suggest that centrality measures within networks are closely related, clustering together with a few exceptions: Eigenvector (Colocation Network), Colocation Entropy (Colocation Network), and some Friendship network measures—which, despite clustering with the Facebook Network metrics, remain closely tied to their Friendship network counterparts, existing on some kind of blurred boundary between the respective networks. Though these instances might seem to contradict the pattern of similarity between intra-network centrality measures, colocation-eigenvector still has similarly high correlations with other colocation network measures. Colocation entropy on the other hand does not seem to correlate especially well with any other measures, within or across networks. The closest network measure relation seems to be Facebook-PageRank, although the other measure of entropy, social entropy (interaction network), is a close second. In this context, the misallignment is not much of a concern given that entropy is not a centrality measure like the other values being correlated.

Looking deeper, it seems that centrality measures within networks are most closely related in the Facebook network, followed by the colocation and interaction networks, both similarly intra-related. The friendship network measures have the most variation among themselves. The nature of the networks may explain this phenomenon. Being that the Facebook, interaction, and colocation networks are not directed (interactions, colocations, and Facebook friends are alwats two-sided/mutual), the consistency across measures makes sense. The added simplicity of the Facebook network, where there is no element of weight, might then explain why its measures are more closely connected than those within the interaction or colocation networks. In both of the latter cases, there is an element of weight in the form of time spent interacting or colocating. When considering the friendship network on the other hand, the directionality of connections allows for more complexity and thus more variation.

Of the four networks, the Facebook and friendship networks appear to be the most closely related networks. This might be intuitively explained by the nature of the networks—the Facebook and friendship networks being similar in that they are social networks in contrast to the interaction network, which measures behavioral activity, and the colocation network, which measures sharing of space. These two behavioral networks (interaction and colocation networks) seem to be more closely related to each other than to any other network.

Notably, three of the four eigenvector measures across networks clustered tightly together, with only the interaction network’s measure of eigenvector separating from its counterparts within other networks. This suggests eigenvector is likely the most consistent centrality measure across networks and may serve as the best indicator of overall social success.

Class Differences

Due to the presence of participants within a school, one might expect the infrastructure in place to have an impact on networks. Though there is no data about how students’ schedules differed based on class or how they may have been segregated during the school day, it is important to consider the possibility. Additionally, there is a chance that students’ prioritization/preference for certain subjects underlies some behavioral, personality qualities which may affect/inform their centrality. As a result, it might be useful to explore the relationship between class and social entropy—beginning with a visualization.



Visualizing social entropy by class reveals some variation across class. It is unclear whether this variation is enough to suggest that the variation in social entropy might be due to significant confounding variables, perhaps related to class rather than the psychological or behavioral factors we are more interested in deciphering.

Before exploring the impact of possible confounds that come with class, it makes sense to run a one-way ANOVA to assess whether this variation in social entropy is due to chance.

Source of Variance Sum of Squares Degrees of Freedom F-Value P-Value
Intercept 11.795 1 34.996 0.000
Class 2.759 7 1.169 0.331

The effect of class on social entropy is not significant (p = 0.331), which allows us to examine our data with confidence, knowing that class-structure and related scheduling isn’t affecting behavior too much. Nevertheless, class still warrants attention and may be included in a model to minimize confounds (however small they may be).

Before moving on, a linear regression analysis may shed more light on the effects of individual classes on social entropy.

Independent Variable Regression Coefficient Standard Error T-Value P-Value
Intercept 1.717 0.290 5.916 0.000
Bio2 0.458 0.364 1.257 0.213
Bio3 0.404 0.313 1.292 0.201
MP1 0.247 0.319 0.773 0.442
MP2 -0.04 0.389 -0.101 0.919
PC1 0.62 0.389 1.593 0.116
PC2 0.015 0.375 0.039 0.969
PSI 0.508 0.356 1.429 0.157

Linear regression analysis (using Bio1 as the standard of comparison for the other classes) reveals that no class significantly correlates with social entropy, which adds confidence in working with the data as is.


Visualizing number of friends by class reveals a decent amount of visible variation across class. Running a one-way ANOVA will reveal whether this variation is due to chance.

Source of Variance Sum of Squares Degrees of Freedom F-Value P-Value
Intercept 30.250 1 2.871 0.095
Class 217.354 7 2.947 0.009

Number of friends does in fact vary significantly (p = 0.009) by class.

A linear regression analysis will reveal the effects of individual classes.

Independent Variable Regression Coefficient Standard Error T-Value P-Value
Intercept 2.75 1.623 1.694 0.095
Bio2 4.25 2.035 2.089 0.040
Bio3 5.69 1.748 3.255 0.002
MP1 4.197 1.786 2.350 0.022
MP2 1.05 2.178 0.482 0.631
PC1 2.85 2.178 1.309 0.195
PC2 2.417 2.095 1.153 0.253
PSI 5.5 1.988 2.767 0.007

In this case, presence in Bio1, Bio2, MP1, and PSI significantly (p = 0.04, p = 0.002, p = 0.022, p = 0.007 respectively) and positively (r = 2.089, r = 3.255, r = 2.35, r = 2.767 respectively) correlated with number of friends. This difference among classes may be worrying to an extent as hinting at some underlying confound, but this effect cannot be attributed to behavioral differences between classes or forced/limited entropy given the aforementioned lack of difference in social entropy across classes. Still, this brings up a question of what about certain classes makes them associated with a greater number of offline friends. We might look at the interaction of gender within these networks and classes.

Gender Differences

One might expect gender itself to have some impact on social networks. In this vein, and with the added motivation of uncovering the qualities of classes which result in friendship differences, it seems worthwhile to search and account for any differences that may result from gender.



Plotting social entropy by gender reveals no clear variation across gender. In fact, both the means and the spread of the data seem comparable. This result is reassuring that confounds will not interfere too much with our upcoming analyses.

Lack of variation means that it is not necessary to perform any rigorous statistical test (ANOVA or otherwise) to assess the effect of gender alone on social entropy. Rather, due to the results being visually unremarkable, it may be best to avoid doing so to avoid potentially introducing Type I error.


Visualizing number of friends by gender reveals some variation across class. An one-way ANOVA will assess whether these variations are due to chance.

Source of Variance Sum of Squares Degrees of Freedom F-Value P-Value
Intercept 1864.170 1 156.015 0.000
Gender 45.496 1 3.808 0.055

Number of friends does not vary significantly (p = 0.055) by gender, which reduces worries about gender-related confounds. Nevertheless, because this relationship approaches significance, it does not hurt to include it in models to reduce any unseen effects, no matter how small. Still, due to the lack of significance, no linear regression will be run to examine the effects of each individual gender. If there were a large number of categories within the gender-class in our dataset, that may have been more pertinent, but it does not make sense here.

However, it may be interesting to examine the relationship of class and and gender together on number of friends and social entropy—something that is not clearly apparent. This will serve to remove any impact that gender may have had on the class effect observed earlier, to confirm or refute the aforementioned results.

Looking first at social entropy:
Source of Variance Sum of Squares Degrees of Freedom F-Value P-Value
Intercept 11.795 1 34.583 0.000
Class 2.774 7 1.162 0.336
Gender 0.055 1 0.162 0.689

Two-way ANOVA fails to confirm that social entropy varies significantly based on class or gender, when controlling for each other. This aligns with the previous, separate findings looking at social entropy by class and by gender separately.

Moving on to number of friends:

Source of Variance Sum of Squares Degrees of Freedom F-Value P-Value
Intercept 30.250 1 3.089 0.083
Class 234.468 7 3.42 0.003
Gender 62.610 1 6.393 0.014

As before, class contributes to number of friends significantly (p = 0.003), even when controlling for gender. However, controlling for class allows gender to emerge as a significant (p = 0.014) contributor to number of friends. Gender may not ahve been significant when viewed alone due to some opposing effects by class, which are mitigated by controlling for that category. Linear regression analysis should make the impacts of individual classes and genders more clear.

Independent Variable Regression Coefficient Standard Error T-Value P-Value
Intercept 2.75 1.565 1.757 0.083
Bio2 2.828 2.041 1.386 0.170
Bio3 5.292 1.693 3.126 0.003
MP1 3.045 1.781 1.710 0.092
MP2 -0.145 2.152 -0.067 0.947
PC1 2.054 2.123 0.967 0.337
PC2 1.753 2.037 0.861 0.392
PSI 4.504 1.956 2.302 0.024
Male 1.991 0.787 2.528 0.014

Linear regression analysis (using Bio1 as the standard of comparison for other classes and females as the standard of comparison for other genders) reveals that when controlling for individual classes and genders, being male significantly (p = 0.014) predicts more friends (r = 1.991), as does being a part of the Bio 3 (r = 5.292, p = 0.003) or PSI (r = 4.504, p = 0.024) classes.

Interestingly, controlling for gender led to a change in which classes significantly predicted number of friends, with Bio2 and MP1 no longer significant (p = 0.337, p= 0.024respectively). The positive correlation (r = 5.733) between Bio3 and number of friends remained significant (p = 0.001) and relatively unchanged. The positive correlation (r = 5.076) between PSI and number of friends also remained significant (p = 0.012) though the magnitude of the effect decreased slightly.

The change in the significance of Bio2’s and MP1’s correlations with number of friends when controlling for gender would suggest that the initial source of variation within this class was gender. Given that each of these classes was found to be significantly and positively correlated with number of friends and that being male was shown to be significantly and positively correlated to number of friends, one would expect to find that their compositions are disproportionately male. To check this, a bar graph of the gender distribution of each class (using the full data set, rather than just individuals within the filtered network) could be useful.


Checking the aforementioned theories reveals some unexpected results—Bio2 is made up of 21 females and 13 males. Contrary to expectations, the class is over 60% female. MP1 fits the expectations more clearly, with 11 females and 18 males.

In the case of Bio3, it makes sense that significance remained when accounting for gender due to the opposite of the expected phenomenon—the class is disproportionately female, with 32 females and 13 males. Thus, accounting for gender, in which case female does not predict greater number of friends (but fewer), would not cancel a gender-based advantage. In this sense, Bio3 might seem like an anomaly to the correlation between male gender and number of friends compared to female. However, there are likely other underlying factors that could explain the difference. For instance, Bio3 is one of the biggest class and the one with the most females.

The case of PSI is another interesting one. It seems logical that account for gender decreased the strength of the positive correlation with number of friends since the class is predominantly male, with 24 males and 10. Nevertheless, accounting for gender’s contribution to number of friends, is not enough to cancel the large positive relationship within the class. We are left to assume some underlying factor within this class is driving this correlation with number of friends.

These contradictory findings regarding the contributions of gender to social entropy and number of friends provides cause to assess the correlation between social entropy, scaled by gender, and number of friends. Though visualizing this social entropy across gender did not show much variation, there may be hidden patterns or differences that were lost in the simplicity of the plot.

Before that however, it makes sense to examine the composition of the classes when considering only the participants who made it through the initial filtering process to exclude those who were not present in all of the networks in question.


Taking a look at the filtered classes, the composition by gender matches the expected outcomes. Bio2 and MP1 appear to be predominantly male while Bio3 is predominantly female, explaining why Bio3 maintained a significant relationship with number of friends when accounting for gender. PSI on the other hand appears to have an equal number of males and females after filtering, explaining why the strength of the correlation decreased slightly, but not enough to render the relationship insignificant.

However, these results introduce an extra reason to remain skeptical and cautious of all findings, given that the filtered dataset looks much different than the complete one. Though centrality measures and other calculations were conducted on the full dataset prior to filtration in order to minimize such concerns, there still remains some artificial noise in the data as a result of the clean-up process.

At this point, this approach seems to have reached a roadblock. On one hand, these results alleviate some of our worries about what confounds the class structure within the dataset may be contributing as well as potentially significant confounds associated with gender. We now better understand the scale and direction of these intricacies. However, there doesn’t seem to be an obvious step forward from here. Thus, we will shift our focus to the networks at large.

Network Modelling

In order to develop a more informative model for the relationships of individuals across networks, a heatmap displaying the results of a partial correlation test controlling for each of the other included measures when assessing two variables could prove helpful. Such a test, would be similar to a multiple linear regression model incorporating each of the given measures.

The above figure is quite difficult to decipher and does not seem to indicate much. Creating random and fixed effects models incorporating one network’s degree measure against centrality measures from all other networks failed to reveal much either. When applying partial correlations, intra-network similarities between centrality measures seem to weaken quite a bit (as well as perhaps inter-network similarities). Perhaps this approach is too aggressive in that pitting different centrality measures from the same network against each other (by controlling for them) overcompensates for expected multicollinearity, effectively removing the underlying patterns which characterize centrality across different measures. In other words, many of the individual measures are too similar too each other. Thus, controlling across them removes the essence of what the data is measuring, rendering the numbers meaningless. Nevertheless, it is interesting that friendship and interaction network measures group together more in this heatmap while colocation and Facebook network measures cluster together more, unlike the patterns of similarity witnessed in the earlier network measure correlations heatmap (Figure 2). To account for different ways of measuring centrality within networks, perhaps the best approach is to create an average (or otherwise simplified) centrality score for each individual-network pair, based on normalized centrality measures. First, this suspicion about mulicollinearity will be explored by calculating variance inflation factors for each centrality measure when predicting social entropy.


Examining variance inflation factors for each network measure reveals significant collinearity among several measures (colocation entropy, colocation degree, Facebook degree, colocation eigenvector, Facebook eigenvector, colocation pagerank, and Facebook pagerank) when predicting social entropy. Interaction degree in partiuclar revealed almost no collinearility, with other measures from the friendship and interaction networks similarly lower in collinearity as measured by variance inflation factor. These results are fascinating for several reasons: not only do they imply that interaction network measures, from which social entropy is derived, are among the least collinear when modelling social entropy, but they also suggest that measures from one network tend to provide only a general reflection of popularity in other networks, not specific to or reflective of the nuances associatd with unique centrality measures. Instead, looking at centrality in one network seem useful only as a distant, “blurred” view of popularity in others.

Though insightful, these results create a dilemma: while the large collinearity among measures from different networks would support the creation of some cumulative centrality measure for each network, the low collinearity between intra-network measures suggests that such an approach would remove much of the information present within the data. This is something which must be assessed and addressed before performing such a transformation.

Standardizing

As an alternative approach to attempt to make comparisons of intra- and inter-network measures simpler, standardizing (z-scoring around a mean of 0 with a standard deviation of 1) of each measure was performed.



Though standardization should not affect general correlations (and it in fact does not, upon performing such checks), it does seem to drastically change the partial correlations between network measures. Partial correlations seem to be some kind of exception to the rule. The standardized partial correlation heatmap of network measures looks much more random than the previously shown simple correlaton heatmap (Figure 2) and the previous, unstandardized partical correlation heatmap (Figure 9) even. Still, clustering largely maintains centrality measures from the same network together with few exceptions (i.e. colocation network eienvector is clustered with Facebook measures; Facebook closeness and pagerank, interaction network [social] entropy, and colocation network [colocation] entropy are sorted between colocation network and friendship network measures). Standardized partial correlation seems to also create much more defined separation between networks, with all the networks seemingly equidistant from each other based on clustering. This was not the case with the previous simple correlation (Figure 2).

At this point, the data and its corresponding, calculated measures are beginning to seem quite overwhelming; their quantity and complexity makes it difficult to decipher the meaning of results or reach any intuition on relationships. There is too much collinearity to cleanly parse variables or measures for the purpose of isolating the important measures therein. As a result, simplifying the data while maintaining its features and robustness is key. The simplest way to do this might be averaging the different centrality scores for each network to create a single popularity value for each individual in each network, as mentioned earlier. However, this approach will likely prove too basic, reducing the data in a way that sacrifices a lot of useful information. A better technique may be running principal component analysis (PCA) or latent variable analysis, both of which should effectively reduce the dimensionality of the data and reveal more complex, hidden relationships among variables (or linear combinations thereof) than are observable otherwise.

Principal Component Analysis



Principal component analysis (PCA) of the various calculated network measures corroborates that the data is quite complex. Nevertheless, more than 40% of the model’s variance can be captured in 2 dimensions, over 50% in 3 dimensions, and more than 70% in 5 dimensions. In 4 dimensions (the same number of dimensions as there are networks involved in this model), around 63% of variance is explained. Exploring the network measures contributing to the 5 most revealing dimensions reasserts the relationship between the Facebook and friendship networks and the interaction and colocation networks, observed in the earlier correlation heatmaps (Figures 2 and 11). Furthermore, this PCA affirms the distinction among the calculated centrality measures. It is clear that “popularity” differs across the included networks, and cannot be generalized across in-person and online interactions.

In order to reduce dimensionality and achieve more interpretable results, however, it makes sense to use PCA to reduce the network measures for each individual network into a single score (or the minimum reasonable number).


Separate PCAs for measures obtained within each network reveal that the Facebook network is the most easily reduced—sufficiently represented by 1 or 2 dimensions. As one would expect, this also seems to suggest that the Facebook network is the simplest of the bunch. The interaction and colocation networks follow, revealing similar levels of complexity captured by 2 or 3 dimensions. The friendship network clearly presents as the most complex network, requiring 3 or 4 dimensions to capture comparable amounts of variance.

Exploring the contributors to each of these principal components paints the first dimension as a relatively general measurement of centrality for each network, integrating each of the forms of centrality (degree, betweenness, closeness, eigenvector, and pagerank) to a similar degree. Other dimensions reveal a much more selective image, heavily skewed by one or two centrality measures (or three in the case of the interaction network’s second principal component). Given this intuition, reducing each network’s centrality measures to a single, all-inclusive “popularity” score may be a reasonable decision, even if only accounting for 50-60% of variance in some cases.

Following this simplification of the network centrality measures, another correlation heatmap can show the relationship between popularity in each network and other, more interpretable network features.


Though this set of correlations reveals some interesting relationships between centrality and behavioral patterns, it is still slightly too convoluted to achieve clear and meaningful results. It is interesting that all the network’s cumulative centrality measures hang together with the exception of the interaction network’s. In general, the interaction network’s measures seem to lump together, despite the rest of the netwokr measures appearing fairly mixed among each other, even across networks. Still, the data is too complex to reach a solid conclusion. Latent variable analysis, or clustering more generally, should achieve the remainder of the desired simplification.

Clustering

Latent Profile Analysis (LPA)

The goal of implementing latent variable analysis (in the form of latent profile analysis [LPA]) is to separate groups of subjects based on their network centralities, detecting different versions of “popularity” and/or different patterns of social behavior in the process.

Given that there is no presumed/hypothesized operating model or any clear groupings to test with LPA, plotting Bayesian Information Criteria (BIC) for the possible models will provide us with the best path to explore. Two general approaches will be attempted: the first, grouping subjects based only on their simplified network centrality scores to isolate differences across networks, and the second, including centrality measures alongside other socio-behavioral features.

Several BIC analyses with different combinations of variables produced only one useful model, apparent when incorporating the original series of centrality and entropy measures. In this case, BIC revealed that the preeminent LPA model consists of 4 clusters in a diagonal orientation with varying volume and varying shape. Running this model created clusters of 18, 3, 23, and 35 respectively.

Integrate Completed Likelihood (ICL) criteria is another method to determine the best fitting model for a series of data. While BIC and ICL are similar, the latter imposes a penalty for models with greater entropy or uncertainty. This alternative algorithm will be run to ensure the best LPA approach from the outset.

ICL corroborates the results of BIC, suggesting that the included subjects fit into 4 clusters. This makes sense when considering that our data includes 4 networks and their corresponding centrality measures. However, the hope is that this LPA will reveal something more interesting than just that. Since people who are central in one network are likely central in other networks as well, there is a decent chance that the 4 clusters generated are more enlightening. Plotting these profiles should make any such findings clear(er).

Network/Behavioral Features



Exploring the different groups’ profiles across the features of their social behavior reveals the differences that distinguish these groups. The subjects could be separated into one group marked by normal colocations and interactions, but high number and quality of facebook and in-person friendships, particularly female friends and remarkably mutual (perhaps a very close group of female friends; 1), a group with high colocations and interactions, but low entropy in their social behaviors and slightly unpopular based on their online and in-person friendship parameters (2), another group which seems relatively unpopular in terms of in-person friendship, but average otherwise (Facebook friends, colocations, and interactions; 3), and a final group, around average regarding all metrics (4).

Looking at these groups through the lens of just their PCA-derived centrality measures, the subjects were divided into a group with low centrality in all networks, only slightly lower than average in terms of interactions and colocations, but much lower than average with respect to friendship and Facebook centrality (1); a group with above average centrality in all networks except colocation, and enormously above average with respect to interactions (2); one with around average centrality in all networks except friendship, where the group exceeded the average centrality by quite a bit (3), and a final group with above average Facebook and colocation centrality, but average friendship slighltly below average interaction centrality (4).

Centrality Measures



Examining these groups in terms of their comprehensive network centrality measures directly, rather than their PCA-derived counterparts or network features reveals one group with above average centrality across the Facebook and friendship networks and more average centralities across the interaction and colocation networks (1), another group with below average facebook centrality, average friendship ranking (except Pagerank, suggesting a tight-knit group with limited influence beyond this small circle; 2), a group with average Facebook, interaction, and colocation centralities, but below average friendship centralities (3), and a group around average across the board (4).

Structural Equation Modeling (SEM)

An alternative method would be to focus on the networks themselves rather than individuals—the relationships between the networks rather than the features that separate or connect subjects. Confirmatory factor analysis (CFA) is a form of structural equation modeling (SEM) which can be used to detect (and measure significance of) relationships between measured variables and the latent constructs they compose. In this case, different network centrality measures would be used as independent variables contributing to the dependent, latent variable that is the network from which those measures were derived.


Note: Only significant values are visible. Regression coefficients are shown between each latent variable and its contributing variables. Covariances are shown between latent variables.

To capture the relationships between the latent facebook and friendship network variables to combinations of the behavioral latent variables, regressions of these latent variables must be run:
Dependent Variable Independent Variable Estimate Standard Error Z-Value P-Value
Facebook ~ Interaction 0.148 0.094 1.575 0.115
Colocation 0.114 0.054 2.125 0.034
Friendship ~ Interaction 0.048 0.094 0.508 0.611
Colocation -0.006 0.050 -0.127 0.899
Furthermore, some interesting covariances must be observed to account for independent variable relationships when connecting independent variables to dependent variables:
Dependent Variable Independent Variable Estimate Standard Error Z-Value P-Value
Facebook ~~ Friendship 0.513 0.124 4.144 0.000
Interaction 0.193 0.109 1.774 0.076
Colocation 0.186 0.078 2.391 0.017
Friendship ~~ Interaction 0.051 0.104 0.490 0.624
Colocation 0.004 0.068 0.062 0.951
Interaction ~~ Colocation 0.261 0.083 3.137 0.002

These results are consistent with the findings we have been collecting, suggesting a relationship between the Facebook and friendship networks and another strong relationship between interaction and colocation. Notably, SEM also highlights a significant (p = 0.017) positive (r - 0.186) relationship between Facebook and colocation centrality.

K-Means Clustering

Though different clustering algorithms (BIC, ICL) were tested earlier to ensure the optimum structure and quantity of groupings, these methods all fit under the umbrella of LPA. To examine whether this classification structure is an inherent fit for the data, an unrelated method will be tested: k-means clustering.

Again, we must first discover the optimal number of groups by which to cluster the data. The go-to way of doing so is called the “elbow point method.” This method involves optimizing the amount of variation within each group by plotting the reduction in variation against the number of clusters (within group sum of squares [WSS] plot). The goal is to strike the right balance (somewhat subjective).


Typically, the optimal number of clusters is that after which there is little reduction of within-group variation with each added group. This creates a sort of “elbow” in the plot, from which the method gets its name. The final selection of the best model falls onto a human decision, which is a weakness of the approach, but it is simply being used here for comparison with the results of LPA. Since the “elbow” can often be arguable, it is best to select a few points which can reasonably be considered the “elbow” of the plot and examine each of the corresponding models to see if the results make any intuitive sense. In this case, their doesn’t seem to be a defined “elbow” to the plot. The graph does not look like the ideal plot produced by k-means clustering, perhaps because the data is relatively complex and does not lend itself to clustering well. Nevertheless, the “sweet spot,” if one can call it that, seems to be 3 or 6 clusters. Given the inconclusive nature of the results, it is best to examine more options.

In this case, alternative methods for finding the ideal number of clusters can be performed. The average silhouette method computes the ability of the clusters in each model to encapsulate the objects within it. The gap statistic method compares variation within clusters against a random distribution of data with no reasonable form of clustering. These are only a few of many possible options.

Average Silhouette Method

Gap Statistic Method

30 Extra Indices (from NBClust)

The results of different clustering validation techniques seem to vary quite a bit. Besides the individual cluster optimization results shown, the NbClust package was used to test clustering schemes using 30 different indices (Figure 21). 5 indices suggested 2 clusters, 5 suggested 3, and 8 suggested 5. As a result, looking at all of the possibilities from 2-8 seems like the safest option, covering the range of possibilities suggested by the employed techniques. To determine the best nymber of clusters, the data must be visualized according to the different ways it can be clustered. Given that the data in question does not fit into 2 dimensions, it can be plotted along the greatest 2 principle components as determined by PCA. However, this visualization technique would not explain the data in any meaningful way (given that the PCA-derived dimensions are not intuitive measures of observable network features), such that we could determine the best number of clusters. Instead, the groups will be graphed on account of their network features and centralities as was done with LPA in the previous section.



Even when manually visualizing each clustering option, choosing the optimal number of clusters is difficult. The best choice is unclear at best. Comparing the k-means clustering profiles when fitting the data into 4 groups reveals different results than those seen earlier through LPA. The two algorithms seem to be picking up on different features and patterns, hence forming completely distinct groupings with the same data.

For the sake of completeness, the groups created by 4-way k-means clustering include one group of 40 subjects (1), another with 4 (2), a group with 18 participants (3), and a cluster with 17 people (4). One group exhibits around average behavioral features, hovering slightly below average in measures related to Facebook, slightly above average in those related to friendship, and slightly below average with respect to colocation and interaction measures (1). Another group appears to be below average with respect to Facebook and frienship measures, but only slightly below average in terms of colocation and interaction measures (2). This group’s interaction patterns seem quite notable in that they seem to be the unpopular friends within the popular circles, perhaps some kind of mediators between higher centrality and lower centrality social circles. Given that the group is made up only 4 people, it is difficult to make such a presumption with any certainty. It could be that the individuals possess a few outlier variables that can’t necessarily “average out” due to the small sample size. The third group presents as average in terms of Facebook and friendship measures, but above average with respect to interaction and colocation (3). The final group is about average in terms of all the network/behavioral features except for the Facebook-related values, where it shows above average features (4). Separating these groups by their PCA-derived network measures, the separation of the groups is much clearer: one group presents with about average Facebook and colocation centrality, but above average friendship and interaction centrality (1); another with extremely low Facebook centrality, extremely high friendship and interaction centrality, and slightly below average colocation centrality (2; note the small sample size of this group may be the reason for these extreme values); a third group with around average Facebook and friendship centralities, but above average interaction and colocation centralities (3); and a final group marked by above average Facebook centrality and below average centrality in all other networks (4).

Due to the difficulty in discovering the optimal number of clusters for the k-means algorithm, a more robust approach seems necessary. Rather than the previously employed methods, which looked at the cluster sets separately, a more dynamic approach will be tested—one which factors how each individual’s position among the given groups changes with each clustering. This gives a sense of which groups are the most stable among the possibilities.



Drawing out the cluster trees for the possible results of k-means clustering seems to confirm BIC and ICL’s finding that the data fits best into 4 clusters (although the aforementioned suggestions were with respect to LPA). Though the results are open to interpretation, it appears that the clusters formed when grouping the data into 4 divisions are more stable than those formed by the other possible k-means structures. This is evinced by the fact that when moving from 4 to 5 clusters, no subjects move across clusters. Instead, the existing clusters split into several smaller groups or remain unchanged. Going from 4 groups to 5 groups is the only time when this proves to be the case, placing the 4-cluster approach unique among the options. Furthermore, the determined clusters seem to be relatively well-distanced from one another, especially in the case of Principle Component 1, suggesting that the groups are sufficiently and observably distinct.

Validation Measure Optimal Score Optimal Clustering Method Optimal No. Clusters
Connectivity 5.005 hierarchical 2
Dunn 0.535 hierarchical 2
Silhouette 0.306 hierarchical 2

It is important to note that hierarchical clustering into 2 groups (as seen in the Network Comparisons section, where the Facebook and friendship Networks were clustered together and the interaction and colocation networks were together) seems to be the optimal form of clustering. However, a K-Means and LPA approach remains valuable nonetheless due to the granularity of the results. Delving into these procedures is motivated by the desire to discover features of the clusters present within the data, rather than just general observations of correlation. This procedure is also better suited at understanding individual subjects and their differences rather than focusing on the networks themselves.

Visualizing Clusters and Further Feature Extraction

Due to the difficulty in clustering the data and extracting features from those clusters, the composition of the clusters produced by different algorithms seems worth exploring. Are the groups created by k-means clustering and LPA similar? Do they pick up on similar features or profiles even though the parallel coordinate plots viewed earlier didn’t show a clear connection? Another way of looking at the cluster profiles, aside from the parallel coordinate plots shown earlier, would be through a radar plot focused only on the PCA-derived network centralities (since more parameters would make the plot quite overwhelming). These might reveal similar/parallel profiles for certain groups across the two clustering algorithms.

K-Means Clusters

LPA Clusters

Though the LPA-based radar plot certainly shows more of the differences being picked up on by the clustering algorithm, the difference between the groups isn’t exactly intuitive.
Another way to compare the clustering results is to visualize the clusters overlaid on each other in space. This involves plotting individuals against their two greatest principle components.

K-Means Clusters

LPA Clusters

Neither of these visualizations seems particularly great at revealing the differences or separation between groups.
Yet another method of visualization that may prove interesting is repeating the initial set of network graphs, highlighting cluster rather than class.

K Means Clusters

LPA Clusters

Visualizing the LPA- and K-means-derived clusters within the available networks is reassuring—it reveals a good indication of the legitimacy of the employed clustering algorithms. The graphs above clearly present a pattern wherein individuals who have been clustered together by either algorithm appear relatively close together across networks 9and separate from other groups). Though the intricacies of the distinctions are not quite obvious or simple enough to reach meaningful conclusions from, the separation is there.

Conclusion

Throughout the exploratory analysis, the most significant and repeated finding observed seemed to be the connection within relational networks (among Facebook and in-person friendship networks) as well as that within behavioral networks (among interaction and colocation networks). Though these groups of networks significantly correlated across media (online vs. offline), only the Facebook and colocation networks did the same across relational-behavioral lines. As a result, it seems fair to conclude that while mere presence in a shared space with lots of people and/or central people is enough to predict the same of an individual in online spaces, more complex behaviors seem to underly offline friendships. It may be expected that colocation, especially when partially forced as in a school environment, does not correlate with self-reported friendship, but even interaction fails to achieve significance in this department. In other words, one’s self-reported friendship network does not predict one’s interactions just as one’s interactions do not predict friendships. This might hint at underlying biases/impressions only being exaggerated or deepened via interaction. Though this study did not delve deep enough to establish causaul relationships, it seems that one can achieve a position of centrality or importance online by placing oneself among many others of influence in person (perhaps akin to networking). Becoming central in a network of offline friends is a much more difficult task which cannot be coeerced through interactions even.

The study also revealed a few smaller, but notable observations. First, eigenvector centralities seemed to be the strongest centrality correlates across networks. These measures tended to cluster together even when that came at the cost of separating from their own networks’ other centrality metrics. In addition, gender and class seemed to have some confounding effects on centrality, though not enough to seriously unstabilize our results. Male gender significantly predicted high centrality compared to female gender while being a part of the PSI (engineering) class or the Bio2 class provided an advantage in centrality networks when compared to other classes.