Friendship paradox biases perceptions in directed networks

We rely on observations of our peers to understand social norms, assess risks, and learn behaviors. However, these observations are often systematically biased, leading to a distorted view of the world. One well-known source of this bias is the friendship paradox: on average, individuals are less popular than their friends. This paradox impacts our self-perception and comparisons with others; for example, people tend to be less happy than their friends, and researchers often have less impact than their collaborators. Any trait correlated with popularity is susceptible to this misperception. This explains why adolescents overestimate risky behaviors among their peers and why social media can exacerbate negative social comparisons. Unlike many friendships, online social networks are often directed. On platforms like Twitter, users follow others but aren't necessarily followed back. This asymmetry creates four variants of the friendship paradox: individuals tend to have fewer friends and followers than their friends and followers. This effect is significant, with a substantial portion of social media users perceiving themselves as less connected than their network neighbors. The conditions under which these variations exist haven't been fully explored. This study analyzes directed networks where nodes possess traits (gender, political affiliation, hashtag usage). The global prevalence of a trait is the overall fraction of nodes exhibiting it, while the observed prevalence is the fraction of friends possessing it. If influential nodes are more likely to have a particular trait, its observed prevalence will be significantly higher than its actual prevalence. Our analysis shows that, similarly to generalized friendship paradox in undirected networks, correlation between node traits and out-degree amplifies perception bias. This paper identifies a new paradox in directed networks: a trait will seem more popular locally among one's friends than globally. We demonstrate that this effect is amplified when high out-degree nodes (with the trait) are connected to low in-degree nodes. Despite the biased nature of individual perceptions, we propose a robust method to estimate the global prevalence of a trait.

Prior research extensively documented biases in perception stemming from social networks. Studies highlighted the “majority illusion,” where individuals perceive a majority holding a certain attribute even if it's globally rare, and the underestimation of minority group size. These phenomena result from the friendship paradox, impacting observations in directed networks as well. Previous work has explored the friendship paradox in undirected networks, showing how it biases perceptions of traits and behaviors, particularly when popular individuals are more likely to exhibit those traits. However, the nuances of this bias in directed networks, where information flow is asymmetrical, requires further investigation. The concept of inversity, which describes the correlation between the degrees of connected nodes in an undirected network, has also been explored in relation to local and global versions of the friendship paradox. This current research extends these previous investigations by considering both local and global perception biases in directed networks, providing a more nuanced understanding of the influence of network structure on individual perceptions.

This study employed a mixed-methods approach, combining theoretical analysis with empirical validation using Twitter data. The theoretical component involved defining and analyzing four variants of the friendship paradox in directed networks. The researchers formalized the concepts of random node, random friend (sampled proportionally to out-degree), and random follower (sampled proportionally to in-degree) to mathematically characterize these paradoxes. They derived expressions showing that two variants always exist, while the other two require a positive correlation between in-degree and out-degree. These findings were illustrated using a Twitter subgraph. To analyze perception bias, the study considered binary node attributes (e.g., hashtag usage). Global prevalence was defined as the expected attribute value of a random node, while perceived prevalence was the expected value among friends. Global perception bias was defined as the difference between these two prevalences. The analysis showed that a positive correlation between node attributes and out-degree amplifies this bias. A new measure, local perception bias, was introduced to capture the deviation of an individual's perception from global prevalence, accounting for how individuals divide their attention among friends. The relationships between local and global biases were explored, identifying conditions for positive and negative biases. The empirical validation utilized a Twitter dataset collected in 2014, focusing on a subgraph of users discussing ballot initiatives. The researchers examined hashtag usage as a binary attribute, measuring both global and perceived prevalence. They identified hashtags that appeared more popular than they actually were due to local perception bias, noting the impact of network structure on this bias. Specific examples of hashtags with both positive and negative biases were discussed. Finally, a novel polling algorithm was proposed to estimate global prevalence by sampling random followers and aggregating their perceptions. The algorithm's bias and variance were analyzed, demonstrating a bias-variance trade-off that leads to improved accuracy compared to existing polling methods. Synthetic polling experiments on the Twitter subgraph validated this improved performance.

The study's key findings can be summarized as follows: 1. **Four Variants of Friendship Paradox in Directed Networks:** The research formally defined and analyzed four variants of the friendship paradox in directed networks. Two variants were shown to always exist, while the other two require a positive correlation between a node's in-degree and out-degree. Empirical analysis on a Twitter subgraph confirmed the prevalence of these paradoxes. 2. **Global and Local Perception Biases:** The study introduced two measures of perception bias: global perception bias, reflecting the difference between the globally observed and perceived prevalence of an attribute, and local perception bias, reflecting the deviation of an individual's perception from the global prevalence. The analysis revealed how these biases are influenced by the correlation between a node's attribute and its out-degree, and the correlation between a friend's attribute and a follower's attention. These biases were shown to be larger in networks with heterogeneous degree distributions. 3. **Empirical Validation with Twitter Data:** An analysis of hashtag usage on Twitter provided empirical support for the theoretical findings. The study identified hashtags exhibiting significant positive and negative local perception biases, illustrating how network structure can distort perceptions of popularity. Specific examples of highly biased hashtags were provided, including those associated with social movements, memes, and current events. 4. **Follower Perception Polling (FPP) Algorithm:** A novel polling algorithm, FPP, was proposed to efficiently estimate global prevalence by leveraging the friendship paradox. The algorithm samples random followers and aggregates their perceived prevalence. Theoretical analysis showed that the FPP algorithm achieves a bias-variance trade-off, leading to more accurate estimates than traditional methods. Empirical results using the Twitter data confirmed the superior performance of the FPP algorithm compared to intent polling and node perception polling.

This research significantly advances our understanding of how network structure shapes individual perceptions and influences collective phenomena. The findings demonstrate the substantial impact of the friendship paradox in directed networks, leading to systematic biases in the perception of attributes. The conditions under which these biases are amplified – positive correlation between attribute and popularity, and positive correlation between friend's attribute and follower's attention – provide valuable insights into the mechanisms driving these distortions. The empirical validation using Twitter data reinforces the theoretical findings, showing how these biases manifest in real-world social networks. The superior performance of the proposed FPP algorithm highlights a practical approach to mitigate the effects of these biases, improving the accuracy of prevalence estimation in online social networks. This work has implications for understanding information diffusion, social contagion, and the formation of social norms, as these processes are fundamentally shaped by individual perceptions of prevalence.

This paper demonstrates that friendship paradox significantly biases perceptions in directed networks, leading to systematic over- or underestimation of attribute prevalence. The analysis identifies key conditions driving these biases and introduces a novel polling algorithm that leverages the paradox to improve estimation accuracy. The empirical results from Twitter data provide strong support for the theoretical findings. Future research could explore how these biases affect collective behavior and investigate strategies for mitigating them through network interventions, such as targeted link recommendations.

The study's primary limitation is the use of a subgraph sampled from the vast Twitter network. This sampling process may introduce biases and limit the generalizability of the findings to the entire network. The dataset was collected in 2014, and the dynamics of Twitter and hashtag usage may have changed since then. Furthermore, the focus on hashtag prevalence as a binary attribute may not fully capture the complexity of social phenomena. Future research should address these limitations by employing larger and more representative datasets and exploring alternative methods for capturing attribute prevalence.