Here we present the first largescale longitudinal study of happiness and popularity levels for a network of 39,110 Twitter users that are connected by ‘friendship’ relations.
To generate a friendship network among Twitter users we start with an initial set of 4,844,430 randomly chosen users (years 20082009), for which we downloaded the full list of users that they ‘follow’ or that they are ‘followed’ by [17]. Reciprocal ‘Follow’ and ‘Following’ ties are taken as an indication of a friendship and mutual interaction relation between the two individuals [18]. We selected the Largest Connected Component of the resulting network. The resulting network consists of 102,009 individuals that share 2,361,547 edges.
We automatically assess each individual’s Subjective WellBeing (SWB), on a scale of \([1,+1]\) according to a procedure outlined in [17]. For each user we collected the 3,200 most recently submitted tweets, serving as a comprehensive longitudinal record pertaining to the individual user. We subject this entire timeline, not its individual tweets, to a sentiment analysis [17] based on the OpinionFinder (OF) subjectivity lexicon [19]. The procedure consists of counting the number of words that are members of either the set of highly positive or highly negative words in the OF subjectivity lexicon, and calculating their fractional difference (number of positive wordsnegative words over all OF words in the timeline). Aggregating this information for all Tweets in an individual Twitter record, we determine the individual’s overall Subjective WellBeing.
The OF toolkit, although different from the above described application of its subjectivity lexicon, was ranked 11th out of 24 tools in a largescale survey of 2 and 3way sentiment classification tasks against a variety of data sets [20]. Its accuracy in scoring individual tweets ranged from 57.60 to 80.77, mostly approximating, and sometimes exceeding, that of top rated tools such as VADER or AFINN, with coverage levels from 41.23 to 54.98, depending on the specific data set against which OF was tested. We stress that our procedure uses the OF Subjectivity Lexicon against an entire 6 month timeline of up to 3,200 tweets by the same individual Twitter user, not individual tweets. Hence the reported coverage values are expected to be a significant underestimation relative to our application. In addition, we exclude users with SWB values of exactly zero.
Finally, we restrict our analysis to individuals with more than 15 friends in order to exclude individuals with excessively low social activity and those that have nonzero SWB, i.e. that have shared some subjective information. This reduces our final cohort to 39,110 subjects that are connected by reciprocal friendship relations, have at least 15 friends, and have nonzero SWB values.
We quantify an individuals ‘Happiness’ as their SWB value and quantify their ‘Popularity’ as their number of network friends counted in our bidirectional social network of reciprocal connections. Simply put, we deem an individual ‘Happy’ when they have high SWB values and ‘Popular’ when they have many friends. The Happiness and Popularity values of all subjects are then used to determine:

1.
the fraction of individuals that has lower popularity than their friends on average (P);

2.
the fraction of individuals that has lower happiness than their friends on average (H);

3.
the correlation between individual happiness and popularity, R (Happiness, Popularity).
We assess the magnitude of the Friendship Paradox in our network by calculating the fraction of the number of users \(u_{i} \in U\) whose Popularity, denoted \(D(u_{i})\) (degree of \(u_{i}\)) is lower than the average Popularity D̄ of their nearest neighbors (or ‘friends’) \(\mathcal{N}_{u_{i}} \subset U\) vs. the total number of individuals in the network \(\U\\). This yields the magnitude of the Friendship Paradox as:
$$ P = \frac{ \ \{ u_{i} \in U: D(u_{i}) < \bar{D}(\mathcal{N}_{u_{i}}) \} \ }{\U\}. $$
(1)
The magnitude of the Happiness Paradox can be obtained in a similar way to how we measure the Friendship Paradox. We simply calculate the fraction of users \(u_{i} \in U\) whose Happiness, denoted \(H(u_{i})\) (SWB value of \(u_{i}\)), is lower than the average Happiness H̄ of their nearest neighbors, \(\mathcal{N}_{u_{i}}\), vs. the total number of individuals in the network \(\U\\):
$$ H = \frac{\ \{ u_{i} \in U: H(u_{i}) < \bar{H}(\mathcal{H}_{u_{i}}) \} \}{\U\}. $$
(2)
A Friendship or Happiness Paradox for our sample is indicated by P and H values larger than 50%, i.e. a majority of individuals have lower Popularity or Happiness than their friends, on average.
To assess the correlation between Happiness and Popularity we calculate Pearson’s R between the SWB values and \(\log(\mathrm{degree})\) of all subjects in our cohort. The use of \(\log(\mathrm{degree})\) is meant to compensate for the very skewed distribution of degree values in our network.
We assess the robustness of our results by performing a bootstrapping procedure in which we randomly sample 10% of subjects and their network connections with replacement 5,000 times to assess the distribution of our paradox indicators for different samples of our network. Furthermore, we validate the statistical significance of our results by comparing them to a nullmodel where we reshuffle the SWB values across all individuals in our network. In this way, we are able to maintain the same identical distribution of SWB values and network structure, while completely eliminating any possible correlation that might be present. The nullmodel was bootstrapped 20,000 times and, as expected, it eliminated the Happiness paradox.
Finally, we determine the sensitivity of our results to reductions of the sample due to the ‘minimum friends threshold’ by recalculating all Happiness Paradox magnitudes for values of the threshold ranging from 1 to 200.