The Science of Human Connection: How Graduate Students make Friends & Seek Advice

This post is based on a research paper I published with Diego Gomez-Zara and Prof. Noshir Contractor as part of Northwestern University’s SONIC Lab.

1. Introduction

It’s a question that has vexed generations of socially anxious high school students and formed the crux of sitcoms since time immemorial: who ends up becoming friends with whom and why? Who grabs a coffee with whom and whom do we turn to when we need advice? Do “birds of the same feather flock together” or do opposites attract? From a Diversity, Equity, Inclusion (DEI) perspective, a related question of greater significance might be to understand which subgroups within a student social network might be disadvantaged due to their social positioning. And finally, we might ask a more global & abstract question: how do communities & subgroups form in a social setting?

Questions such as the above have been the topic of countless books (“How to win Friends & Influence People”), T.V. shows (“Friends”), and most recently, number crunching data scientists such as myself. If you’ve found yourself asking these questions and have wondered how math can help answer the problem, then you’ll find some great insights below.

On a more serious note, a variety of professional stakeholders might ask a diversity of questions that are amenable to a data-driven approach, yet also involve a distinct human component. For example, educators might want to identify students at risk of falling behind. Project managers may want to identify strategies to build more innovative, high-performing teams. Senior executives might want to evaluate how individual personality traits can be used as an input to building effective teams & org charts. Corporate governance may seek to identify root causes holding women back as they approach senior levels of an organization. Each of these personas can find something to gain from this study.

2. Domain Background

People, social ties, and math meet in a field of data science called Social Network Analytics (SNA). True to its name, Social Network Analytics seeks to apply quantitative analysis techniques to what is often viewed as a fundamentally qualitative problem: social ties between humans. With the explosion of digital links spanning people the globe, opportunities to analyze the ensuing data have proliferated. Historically, data about social networks was either collected non-intrusively by researchers observing social interactions or explicitly requested via surveys. However, in recent decades, social media platforms such as Facebook, Instagram, and Snapchat are explicitly modeled as social networks, while “enterprise social media” such as Microsoft Teams, Slack, and Yammer generate significant “digital exhaust” that can then be mined for useful insight.

As an example of research topics commonly explored, researchers analyzing professional networks have found that men tend to have larger and more homogeneous networks than women, which give men access to additional mentorship opportunities. Others have explored how personality moderates friendship ties by analyzing Twitter networks: while extraverted users tend to make friends mainly with other extraverted users, neurotic-conscientious friend pairings tended to form more often than neurotic-neurotic and conscientious-conscientious friend pairings. 

3. Dataset

Like any data scientist, I tried to answer this question by collecting some data. I was just beginning my graduate studies at Northwestern University’s M.S. in Analytics program and and decided to obtain data from the most important social network I found myself in: my fellow Master’s degree candidates in the program’s Theta Cohort.

In my cohort, there were 44 students evenly split between female and male. All students had undergraduate degrees in STEM fields and about 50% came with at least some work experience. As part of our program, we completed a year long practicum project in a team of four to five fellow students which involved completing a data science consulting project with a real-world client. By design, each student spends a large part of their time with their practicum team, learning essential team work skills in the process.

I initially set out to collect data along five dimensions:

1. Demographics: This included a student’s race, nationality, age, years of work experience, and where they were staying.
2. Personality: I administered a Big Five Personality test quantifying each student’s personality along the dimensions of Openness, Conscientiousness, Extraversion, Agreeableness, Neuroticism.
3. Friendship: Each student was asked which other cohort members they considered a friend.
4. Advice: Each student was asked which other cohort members they turned to for advice. Advice networks are important because they depict the flow of ideas in a social network.
5. Teamwork: Each student was asked to identify which other students he was working with as part of his practicum team. The idea was to investigate to what extent an exogenous team assignment influenced friendship & advice tie formation.

4. Methodology

4.1 Descriptive Network Analysis

Descriptive Network Analysis consists of using various summary statistics to understand a network better. While some statistics are network-wide (i.e. they summarily describe the entire network), other statistics are node-level (i.e. they describe characteristics of a single node). Some network-wide statistics we evaluated included:

• Density: Ratio between number of edges observed in the network and the number of all possible ties in a network
• Centralization (graph level): To what extent is the network oriented around a single node
• Reciprocity: What proportion of ties in the network are mutually linked

Other node-level statistics we explored included:

• Network constraint: Measures the extent to which a node connects two others with complementary sources of information. The lower the network constraint, the more sources of non-complementary information the node has access to.
• In-degree centrality: How many edges are in-bound to a given node from other nodes. This is often used to estimate the importance of a node in a network.
• Hubs & Authorities: The notion of authority scores comes from the HITS (Hyperlink Induced Topic Search) algorithm, a link analysis algorithm used to rate web pages. When applied to social networks, HITS assigns two scores for each node: a hub score and an authority score. A hub score measures a node’s tendency to point to other authoritative sources of information. Meanwhile, a high authority score signifies a node that many hubs point to. Given the recursive nature of this definition, an iterative algorithm (HITS) is used to compute hub and authority scores, respectively. We applied HITS analysis to the advice network because we wanted to identify authoritative sources of knowledge (i.e., authorities) in the network.

4.2 Community Detection Analysis

Community Detection involves identifying densely connected sub-groups (“clusters”) within a graph based solely on the interconnections between nodes & edges. Our rationale behind deploying community detection algorithms was to uncover what communities existed based solely on the vertex-edge structures in the network, and then compare to the various individual attributes to see if they matched up with our intuition. We used modularity maximization to determine the communities in our graph. Modularity maximization views the community detection problem as an integer linear program, with modularity as the objective function to be maximized. We decided to use maximum modularity clustering because our graph was relatively small (44 nodes approx.), and this made the problem computationally tractable.

4.3 Exponential Random Graph Models

We utilized an Exponential Random Graph Model (ERGM) to evaluate the effect of personality traits in forming friendship ties and advice ties. ERGMs are a class of probabilistic graph models that seek to identify the key forces that drive the formation of the network. ERGMs use an exponential-family probability distribution to compute the probability of seeing a network given a set of network statistics & associated parameters. The ERGM model fitting process then involves using Maximum-Likelihood Estimation (MLE) to estimate the parameters that make seeing the observed network the most likely. An ERGM allows social network analytics researchers to better understand whether, for example, similarity in race is a factor driving friendship tie formation.

Positive and significant coefficients indicate that the corresponding independent variable is more likely to influence a tie occurring than by chance. Negative and significant coefficients indicate that the independent variable is less likely to result in a tie occurring than by chance alone. We used Markov-Chain Monte Carlo (MCMC) to identify maximum likelihood estimates (MLE) for parameter values. MCMC simulates thousands of random networks fitting the model’s quantifiable properties, rather than attempting to count the impossibly large number of possible network’s edge permutations. We selected network statistics based on driving factors previously studied in social network formation.

Network effects we included were of four main types:

1. Structural Effects: We controlled whether endogenous effects influenced the formation of ties in their friendship and advice networks: the number of ties, to what extent participants tended to reciprocate these ties, and to what extent participants tend to close triads. To measure these properties, we included the terms edges, mutual, and dgwesp from the R ergm package, respectively. A positive and significant effect means that the respective structural signature is more likely to occur in the friendship/advice network than by chance.
2. Individual Attributes: We controlled whether individuals’ attributes were likely to explain the formation of ties in their friendship and advice networks. We separated the effects by the sender (i.e., out-link) and receiver (i.e., in-link). For example, do people tend to turn to older classmates for advice rather than those of the same age; or, do people instead tend to seek advice from people of similar age?
3. Edge Covariates: Does presence or absence of an edge in a parallel network influence edge formation in the network under study? For example, if two people are already asking each other for advice, are they also likely to become friends with each other? For each one of these prior networks, we added these edge-covariate terms using edgecov from the ergm R package.
4. Categorical homophily effects: We also accounted for homophily among the categorical variables of gender, race, and identified country. Homophily describes the tendency for nodes with similar values to form ties within themselves. Its opposite is the heterophily: a phenomenon where nodes with opposite values tend to form ties between themselves. We operationalized homophily using the nodematch term from the ergm R package. This term counts how many nodes connected in the network share the same value for that categorical attribute.
5. Numerical homophily effects: Homophily can also appear between nodes with numerical covariates. In our study, this included age and the five big five personal trait scores (Openness, Conscientiousness, Extraversion, Agreeableness, Neuroticism). For example, students of similar ages may end up becoming friends with each other. Numerical homophily was operationalized using the absdiff term from the ERGM R package.

5. Findings

5.1 Descriptive Network Analysis

5.1.1 Demographic Connection Patterns

After coloring each network by nationality, race, and gender, we observed strong clustering patterns in terms of how the cohort members made friends and asked each other for advice.

  

5.1.2 Density, Centralization, and Reciprocity

Statistic Name	Network	Friendship Network Value	Advice Network Value
Density	Friendship	25.9%	38.3%
Centralization	Friendship	21.3%	43.4%
Reciprocity	Friendship	68.6%	61.6%

The density of the advice network (38.3%) was higher than that of the friendship network (25.9%). This could reflect a lower standard applied when cohort members ask each other for advice than when cohort members identify another a friend. With respect to centralization, the advice network (43.4%) was significantly more centralized than the friendship network (21.3%). Meanwhile, both the friendship & advice networks exhibited a similar levels of reciprocity (68.6% for the friendship network & 61.6% for the advice network). High levels of reciprocity in the friendship & advice network might reflect commonly held-norms that friendship & advice tend to be reciprocal relationships among equals, especially in the absence of an organized hierarchy.

5.1.3 Network Constraint, In-degree Centrality, and Authority

The most interesting insight we found during our descriptive network analysis consisted of gender disparities between network constraint, centrality, and authority. To make statistically meaningful conclusions, we used a one-sided, two-sample, (unpaired) t-test comparing male and female students.

In particular, female students tend to occupy high network constraint positions and male students tend to occupy low network constraint positions in the friendship network (m_female = 0.1851 > m_male = 0.1677; t = 1.2618; p > 0.05; df = 45.415) and advice network (m_female = 0.1294 > m_male = 0.1158; t = -2.52; p < 0:001; df = 44.625). This indicates that females report friendship ties with others who already report friendship ties with one another. In short, in both the advice and friendship networks, women appeared in high network constraint positions more frequently than did men.

Network	Male Average	Female Average	t-statistic	p-value	df
Friendship	0.1677	0.1851	1.2618	<0.05	45.415
Advice	0.1158	0.1294	-2.52	<0.001	44.625

Secondly, in the advice network, female students tend to have lower indegree centrality than male students (m_female = 16.08 < m_male = 19.96; t = 3:64; p < 0:001; df = 45.821). Thus, female students are less likely to be sought out for advice than male students.

Network	Male Average	Female Average	t-statistic	p-value	df
Advice	19.96	16.08	3.64	<0.001	45.821

Finally, for Authority scores, we find that female students have lower authority scores on average than male students in the advice network (m_female = 0:67;m_male = 0.77; t = 3:15; p < 0.001; df = 45.957). In other words, women were less likely to be seen as authoritative sources of information than were male students.

Network	Male Average	Female Average	t-statistic	p-value	df
Advice	0.77	0.67	3.15	<0.001	45.957

We did not find any meaningful differences based on race and identified-country. In other words, there was no discernable disadvantage or disadvantage that accrued to students in terms of their race and nationality.

5.2 Community Detection

We applied community detection analysis to both the friendship & advice network. In the friendship network, we detected three main communities (modularity = 0.39). We hypothesize that the communities were driven by nationality, given that students of a particular nationality dominated in each of the clusters. Figure 3 shows in red the cluster formed by students from China, followed by the blue cluster formed by students from the USA, and the green cluster formed by students from India.

In the advice network, we detected two communities driven primarily by the top-2 identified-countries among participants (modularity = 0.22). The blue cluster is formed by students who are not identified with China and the red cluster is formed by students identified with China. Figure 4 shows the clusters formed in the advice network.

In other words, friendship and advice communities were mostly explained by students’ identified countries. We did not find communities driven by race or gender.

5.3 Exponential Random Graph Modelling

We found that the magnitude and direction of effects varied for both the friendship and advice networks. In the case of the friendship network, reciprocity (coeff. = 1.911; p < 0.001) and triadic closure (coeff. = 0.524; p < 0.001) were more likely to occur than by chance. We found that working on the same practicum team (coeff. = 0.794; p < 0.001) and asking each other for advice (coeff. = 1.891; p < 0.001) were strong predictors of a friendship between two students. Female students (coeff. = -0.469; p < 0.01), students with higher levels of extraversion (coeff. = 0.009; p < 0.05) and neuroticism (coeff. = 0.008; p < 0.05) were more likely to be mentioned as friends. Male students (coeff. = 0.617; p < 0.001), older students (coeff. = 0.167; p < 0.001), and students identified with India (coeff. = 1.214; p < 0.05) were more likely to mention others as friends. Similarly, students with high levels of openness (coeff. = 0.013; p < 0.001), conscientiousness (coeff. = 0.011; p < 0.05), and neuroticism (coeff. = 0.008; p < 0.05) were likely to mention others as friends. Regarding the homophily effects, the ERGM results show that students who identified with the same country (coeff. = 1.305; p < 0.001), had similar agreeableness levels (coeff. = -0.008; p < 0.05), dissimilar openness levels (coeff. = 0.008; p < 0.05) and dissimilar extraversion levels (coeff. = 0.005; p < 0.10) were more likely to form ties. Conscientiousness and neuroticism were found to be insignificant in determining friendship ties.

The ERGM results of the advice network indicate that reciprocity (coeff. = 0.907; p < 0.001) and triadic closure (coeff. = 1.241; p < 0.001) were more likely to occur than by chance. However, while reciprocity levels of the advice network were lower than in the friendship network, the closure levels were higher in the advice network. Asking or receiving advice was more likely to occur between friends (coeff. = 1.941; p < 0.001), but not necessarily between practicum teammates (coeff. = 0.356; p < 0.10). Unlike the friendship network, male students were more likely to be asked for advice, although this coefficient was not statistically significant (coeff. = 0.228; p < 0.10). Compared to the students who identified with European countries, students identified with China were less likely to be asked for advice (coeff. = -1:416; p < 0.001). We found that students who identified with China (coeff. = 3.165; p < 0.001) and white students (coeff. = 0.708; p < 0.01) were more likely to ask for advice. Also, students who were more likely to ask for advice had lower levels of openness (coeff. = -0:011; p < 0.001) and neuroticism (coeff. = -0.021; p < 0.001), and higher levels of extraversion (  coeff. = 0.007; p < 0.05) and agreeableness (coeff. = 0.026; p < 0.001). Regarding homophily effects, we found that students who identified with the same country were more likely to establish a tie (coeff. = 0.791; p < 0.001). Finally, the results show that students with different agreeableness levels ( coeff. = 0.01; p < 0.01) and similar extraversion levels (coeff. = -0.005; p < 0.10) were more likely to establish an advice tie.

6. Impact & Implications

The impact from our study was three-fold:

• Invisible inequity with respect to gender
• Nationality-driven communities
• Assigned teamwork as a level to engineer social networks

6.1. Invisible Inequity

Our descriptive network analysis uncovered important insights into which subgroups in the student network are likely to be disadvantaged. Our results suggest that female students were likely to be more constrained than male students in friendship networks. In other words, female students were less likely to be exposed to novel information beyond what is already circulating in their closed cohesive group of friends. In contrast, male students were more likely to act as brokers and have friends in different social circles. In the advice networks, female students were also more constrained than male students. In our sample, male students were able to receive and provide advice from different students, whereas female students were more likely to receive and provide advice within closed social circles. Female students’ advice was also overall less solicited than male students. These findings reveal that even when a cohort is gender-balanced, social networks of female and male students could evolve unequally. Instead of attributing women’s disadvantage to a largely male faculty body or to their minority status, we attribute their disadvantage to the high network constraint positions that they tend to occupy in the graduate student network. As a result, they can be less likely to get access to novel information needed to produce innovative ideas. It is reasonable to expect that this phenomenon will replicate over time once students begin their career as tech professionals. Thus, our work provides one possible, early career explanation for a lack of female representation in tech leadership roles. Our findings also have implications for business leaders, sociologists, and policy-makers as they seek to develop diversity initiatives. As our analysis shows, even with perfect gender parity, network effects can create ’invisible inequity’ as early as graduate school.

6.2 Community Detection

Our rationale behind analyzing the communities in the friendship and advice networks was to investigate what roles demographics played in forming communities. Our results show that participants were likely to form friendship and advice groups based on their nationalities. Rather than gender or race, it was their self-identified country of origin that influenced how students established friendship and advice circles. For example, American students tended to form ties with fellow American students, as did Indian and Chineses students. We checked whether other attributes were likely to explain students’ social groups. Neither race nor gender was likely to explain these subdivisions among students. The community detection analysis indicates the hardwired tendencies to seek advice and establish friendships with people from the same identified-country. Similar results were reported by [30], who found high levels of homophily in nationality on research co-authorship networks. It would be reasonable to extrapolate that such nationality-based communities will continue to appear even when these students begin their careers as tech professionals. Thus, one possible solution for managers looking to foster more innovative, high-value teams would be to encourage team formation with individuals from diverse national backgrounds. One study found that CEO’s with highly heterogeneous in terms of nationality social networks had higher firm valuations, due to enhanced social learning [31]. One possible explanation for forming friendships and advice relationships with people of the same country is the low cultural dissonance. People from the same country are more likely to share the same language, social norms, and references than people from different countries [32].

6.3 Exponential Random Graph Models

Lastly, our intention behind using exponential random graph models was to evaluate whether prior group interaction and personality traits together explain the formation of friendship and advice ties. In the case of the friendship network, our results show that homophily and prior collaborations were most likely to explain a friendship tie between two students. Students who nominated another person as a friend have similar levels of extraversion and had a prior group activity together. However, people with different levels of conscientiousness and agreeableness were more likely to establish friendships. In the advice network, we found that giving and receiving advice were more likely to happen between students who identified with the same country, who lived in the same apartment complex, were from the seam practicum team, and who already identified themselves as friends. We did not find any significant effects between students’ personality traits and their advice relationships. One possible explanation is the instrumental nature of advice networks. Rather than seeking advice from people who are compatible in terms of personality, students sought advice from other students who are familiar to them in terms of prior experiences and identified countries, which ultimately promote trust among individuals. Living nearby in the same apartment building also played a role in facilitating advice ties. Additionally, ERGM results show a high multiplexity between friendship and advice networks, in the sense that one tie is likely to predict the emergence of the other. Since we cannot infer any causality with these results, future longitudinal work should explore the causal effect between friendship and advice ties.

7. Conclusion

In this paper, we studied the most likely factors that explain the formation of friendship and advice relationships in a professional STEM master’s program. Our case study has shown that students’ race, gender, identified-country, and personality are key factors determining whether friendship and advice ties form between students. However, the effect is not the same for all personality traits. In addition, having prior collaborations has a positive influence on building ties in both networks. Finally, we show that there is a strong gender effect in favor of males in terms of network position. Sociologists, policy researchers, and management scientists can use our findings at the initial graduate school level to motivate a interdisciplinary, longitudinal analysis of female attrition in tech roles over time – commonly referred to as the leaky bucket problem. Our results provide empirical evidence regarding the social bias in favor of interacting with similar people, and the social challenges that female students must face when they start a graduate program.

7.1 Next Steps

There are some limitations to our study that can serve as a foundation for future work. First, we relied on self-reported data to analyze students’ friendship and advice networks. Students could have forgotten to nominate other students as friends, as well as those from whom they sought advice. Second, students’ understanding of what constitutes friendships may not be consistent. To address the above two factors, future extensions could study whether students’ online interaction patterns on social networks such as Twitter, LinkedIn, Instagram mirror our findings in real-life. In particular, an analysis of student collaboration patterns on enterprise social media such as Slack would add a new light to our findings. Third, we did not assess whether students’ prior collaborative experience working in groups with other students was positive or not. Future studies may build on our work by evaluating the individual experiences (positive/negative) in working  with their practicum team, and how this affects friendship/advice relationships. Fourth, neither African-American nor Hispanic students were part of this sample, which are the largest minority groups in the US. Future case studies should consider a good representation of these minorities in their analyzes. Finally, our survey was conducted at a singular moment. Future studies could benefit from longitudinal analysis to study how friendship and advice networks evolve over time.

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