As shown in Table 1, we find that \(P=94.3\), 95% CI \([93.7,94.9]\), indicating a very significant Friendship Paradox across all subjects, meaning that the great majority of users are less popular than their friends are on average. We also find a modest but robust value of \(H=58.5\%\), 95% CI \([57.2,59.8]\), indicating the presence of a Happiness Paradox. Hence a majority of subjects is indeed less happy than their friends on average. Our null-model indicates the absence of a Happiness paradox when the effects of network structure on Happiness levels are removed by random re-assignment. The lower magnitude of the Happiness Paradox could result from the rather low, yet robust, correlation between Happiness and Popularity (Pearson’s \(R=0.100\), 95% CI \([0.076,0.140]\)).

Table 1 Magnitude of Friendship Paradox, Happiness Paradox (compared to null-model produced by randomly re-assigning SBW values across all subjects), and Happiness-Popularity correlation coefficient (Pearson’s R ) for all subjects ( \(\pmb{N=39\text{,}110}\) ) Full size table

As shown in Figure 2 the joint distribution of individual Happiness levels and mean neighbor happiness in our sample is distinctly bi-modal. There exists some evidence that Subjective Well-Being itself can be distributed in a bimodal fashion across several cultures and nations [21] matching previous observations by [17]. In this case, bi-modality also occurs at the level of our friendship network which separates subjects into 2 distinct groups: Happy subjects with Happy friends (the ‘Happy’ group) and Unhappy subjects with Unhappy friends (the ‘Unhappy’ group). This result follows earlier reports of happiness being homophilic or assortative in social networks [17, 22, 23]. Note that this phenomenon is not the result of the distribution of our SWB values, but is determined by network connections; i.e. unhappy people tend to have unhappy friends and happy people tend to have happy friends.

Figure 2 Overview of the magnitude of the Happiness and Friendship paradox for the sample of Twitter subjects; individuals positioned above the diagonal experience a Happiness or Friendship paradox. (B1) Happiness Paradox: Distribution of individual Happiness (x-axis) vs. average Happiness of one’s friend’s average (y-axis). Happiness is measured in terms of longitudinal Subjective Well-Being (SWB) scores. Subjects above the red paradox line experience lower happiness (SWB) than their friends’ average. The distribution of SWB scores places a majority of subjects well above the diagonal Paradox line. Ellipses indicate the boundaries of 2 Gaussian Mixture Model components used to demarcate a Happy (red) and Unhappy (blue) groups of subjects. Paradox magnitudes are expressed in terms of the percentage of users who experience lower happiness than their friends. The 95% confidence intervals are calculated by a 5,000-fold bootstrapping of a 10% sample to determine the sensitivity of our results to random network sampling variations. (B2) and (B3) Friendship Paradox: Distribution of individual Popularity (x-axis) vs. average Popularity of one’s Friends (y-axis). Popularity is measured in terms of \(\operatorname{log}(\mathrm{degree})\) in the Friendship network. Subjects above the red paradox line experience lower popularity than their friends on average. As shown, we find significant Happiness and Friendship Paradoxes for all users, but Happy users experience a stronger Friendship Paradox whereas Unhappy users experience a stronger Happiness Paradox. Full size image

Since a Happiness Paradox specifically compares individual happiness to the average happiness of one’s friends, this homophilic bi-modality must be factored into our analysis. By performing a separate analysis for Happy and Unhappy groups of users, we attempt to equalize the effects of neighbor happiness across the two groups.

As shown in Figure 2 we use a Gaussian Mixture Model (GMM) to demarcate our Happy and Unhappy groups. We determine the location and distribution of two separate Gaussian components in the distribution of individual happiness vs. mean friend happiness and demarcate both groups by simply determining whether the SWB value of a subject and the mean SWB values of their neighbors fall within 2 standard deviations from the center of either one of the components (illustrated by the 2 ellipses in Figure 2). We thereby split our sample in 2 groups of subjects: a group of Happy individuals with Happy friends (\(N=29\text{,}033\)) and a group of Unhappy individuals with Unhappy friends (8,018). This procedure assumes a Gaussian density distribution which roughly matches the quantiles of the empirical density as shown by the contour lines of Figure 2, so that both Gaussian components capture 95% of our sample. These groups are well-separated in the underlying social network. Of all edges adjacent to individuals in the Happy group only 0.682% connect to individuals in the Unhappy group. Vice versa, of all edges adjacent to individuals in the Unhappy group only 3.103% connect to individuals in the Happy group.

We re-run our analysis for the Happy and Unhappy groups separately. The results are summarized in Figures 2 and 3.

Figure 3 Bootstrapped estimates of the correlation between Happiness and Popularity, and the magnitude of the Friendship and Happiness Paradox for Happy and Unhappy subjects. Top: Estimated Pearson’s R correlation coefficients (95% Confidence Intervals in brackets) between individual Happiness (Subjective Well-Being) vs. individual Popularity (log degree) for All subjects: 0.109 \([0.077, 0.140]\), Happy group: 0.126 \([0.081, 0.171]\), and unhappy group: -0.047 \([-0.08, -0.013]\). Middle: Distribution of Friendship Paradox values for all subjects 0.943 \([0.937, 0.949]\), happy group: 0.958 \([0.951, 0.964]\), and unhappy group 0.888 \([0.869, 0.906]\). Bottom: Distribution of Happiness Paradox values for all subjects: 0.585 \([0.581, 0.589]\), happy group: 0.578 \([0.573, 0.582]\), and unhappy group 0.666 \([0.657, 0.674]\). Full size image

These results reveal that the Happy group experiences a strong Friendship Paradox but a weak, yet very robust Happiness Paradox. The Unhappy group experiences a weaker Friendship Paradox, but a significantly stronger Happiness Paradox than the Happy group, in spite of subjects being surrounded by less Happy friends.

To determine whether the strong Happiness Paradox for the unhappy group, in spite of its lower correlation between Popularity and Happiness, may be related to interpersonal effects, we examine the relation between individual happiness and the average happiness of ones neighbors. As visually indicated by the distribution of individuals in Figure 2, the strength of the relationship between a subjects’ Happiness and the average Happiness of their friends may differ between the Happy and Unhappy group. The results of a linear regression predicting mean neighbor SWB from a user’s own SWB values match the visual tilt of the separate GMM components. For the Happy group we find \(F(1,29\text{,}031)=6\text{,}580\), \(p<0.001\) with an adjusted \(R^{2}=0.185\). The resulting regression equation is Mean Neighbor \(\mathrm{SWB} = 0.1768 + 0.1815\) (own SWB). For the Unhappy Group we find \(F(1,8\text{,}018)=3\text{,}274\), \(p<0.001\) with an adjusted \(R^{2}=0.290\). Here, the resulting regression equation is Mean Neighbor \(\mathrm{SWB} = 0.0135 + 0.5716\) (own SWB). An ANCOVA analysis to predict Average Neighbor SWB from the interaction between a user’s own SWB and their membership of the Happy or Unhappy group results in \(F(3,39\text{,}106) = 70\text{,}040\), \(p<0.001\) with an adjusted \(R^{2}=0.843\). On the basis of this last result we reject the null-hypothesis that the regression slopes between own SWB and mean neighbor SWB are equal between the Happy and Unhappy group. This outcome suggests that unhappy users may be more strongly affected by the lower happiness of their friends, possibly explaining why this group exhibits a stronger Happiness Paradox in the absence of a strong correlation between Popularity and Happiness. However, we caution that further investigation is warranted on this matter.

Finally, to determine the sensitivity of our analysis to the choice of our minimum neighbors threshold (the minimum number of neighbors a user must have to be included in our sample), we recalculate Happiness paradox values for all threshold values between 1 and 200. We use the same GMM component locations for every calculation to provide an equal basis for comparison. As shown in Table 2 and Figure 4, we find that requiring a minimum threshold of 15 neighbors for each individual in our data constitutes a significant reduction of our sample, but we still retain more than 45% of the entire sample. In exchange for this reduction, we attempt to increase the chances of removing noise caused by socially inactive, and possibly automated or defunct accounts. The value of this threshold is notably much lower than recent estimations of the online Dunbar number [18].

Figure 4 Minimum neighbor threshold vs. percentage of sample retained. In our calculation of the Happiness Paradox we apply a minimum neighbor threshold for each user, i.e. a minimum number of neighbors the individual must have to be included in our sample, to (1) reduce the chances of including defunct, automated, or particularly socially inactive users and (2) ensure that a mean neighbor degree or SWB is calculated on the basis of no less than 15 data points. This threshold leads to a reduction of our sample. This graph visualizes the magnitude of the reduction at chosen threshold value of 15 and all other values between 1 and 200. Values provided in Table 2. Full size image

Table 2 Percentage of sample remaining after applying a minimum number of neighbors threshold Full size table

The results of our sensitivity analysis, shown in Figure 5, indicate that the magnitude of the happiness paradox is largely independent from our choice of threshold, with the exception of extremely large values, i.e. where less than 10% of the original sample remains and only individuals with more than 150 friends are included, thus approaching the Dunbar limit [18]. We find a significant happiness paradox for the Unhappy group for all values of the minimum neighbor threshold indicating that this result is largely independent from our choice of threshold.