Friendship paradox biases perceptions in directed networks

Abstract

Social networks shape perceptions by exposing people to the actions and opinions of their peers. However, the perceived popularity of a trait or an opinion may be very different from its actual popularity. We attribute this perception bias to friendship paradox and identify conditions under which it appears. We validate the findings empirically using Twitter data. Within posts made by users in our sample, we identify topics that appear more often within users' social feeds than they do globally among all posts. We also present a polling algorithm that leverages the friendship paradox to obtain a statistically efficient estimate of a topic's global prevalence from biased individual perceptions. We characterize the polling estimate and validate it through synthetic polling experiments on Twitter data. Our paper elucidates the non-intuitive ways in which the structure of directed networks can distort perceptions and presents approaches to mitigate this bias.

Fig. 1. Illustration of the effects of the four versions of the friendship paradox using Twitter dataset described in the “Methods” section.

The sub-figures display the fraction of nodes (empirical probability of the paradox) of a particular degree whose a friends have more followers, b followers have more friends, c friends have more friends, and d followers have more followers, on average.

Fig. 2. Global prevalence and local bias of popular hashtags.

Histogram of the distribution of a global prevalence $\mathbb{E}\left\{f\left(X\right)\right\}$ and b local perception bias B_local of popular hashtags in the Twitter data. Local perception bias B_local (overestimating the prevalence) exists for most hashtags.

Fig. 3. The ranking of popular Twitter hashtags based on Local Bias.

Top-20 and bottom-10 are included in the ranking. The bars compare $\mathbb{E}\left\{f\left(X\right)\right\}$ (global prevalence) and $\mathbb{E}\left\{q_{\mathrm{f}}\left(X\right)\right\}$ (local perception) and include 95% confidence intervals. The hashtags can appear to be much more popular than they actually are (e.g. #ferguson) or, they can appear to be less popular (e.g. #oscars) due to local perception bias. Definition of some hashtags: #mike(/michael)brown and #ferguson (an 18-year-old African American male killed by police), #tbt (Throwback Thursday—for posting an old picture on Thursdays), #ff (Follow Friday—introducing account worth following), #tcot (Top Conservatives On Twitter), #rt (Retweet).

Fig. 4. Comparison of estimates of the prevalence of Twitter hashtags produced by the polling algorithms.

Variation of a squared bias (Bias{T}²), b variance (Var{T}), and c mean squared error (Bias{T}² +  Var{T}) of the polling estimate T as a function of a hashtag's global prevalence $\mathbb{E}\left\{f\left(X\right)\right\}$. Each point represents a different hashtag and a fixed sampling budget b = 25. The polling algorithms used are intent polling (IP), node perception polling (NPP) and the proposed follower perception polling (FPP). d Fraction of hashtags for which the FPP algorithm outperforms the other two in terms of MSE. The fraction for NPP approaches 0.5, and for IP approaches 0.8 as sampling budget b increases. These figures illustrate that the proposed FPP algorithm achieves a bias-variance trade-off by coupling perception polling with friendship paradox to reduce the mean squared error.