The impact of social networks is multifaceted, influencing information dissemination, technology adoption, and public opinion. Modern platforms like Twitter, Instagram, and TikTok differ from earlier networks like Facebook and LinkedIn; they are directed networks based on user-generated content (UGC), allowing users to follow others without reciprocal consent. The quality of UGC is crucial; high-quality content attracts more followers, leading to the rise of influencers. Existing network formation models, including stochastic actor-oriented models and strategic models, primarily focus on topological and socio-economic aspects, neglecting the impact of UGC quality. For instance, preferential attachment models, while explaining scale-free effects, don't account for the emergence of new influencers without prior fame. This paper addresses this gap by proposing a model that incorporates UGC quality into the network formation process, drawing inspiration from empirical evidence of users continuously seeking higher quality content on platforms like Twitter. The model incorporates a meritocratic principle, where users strategically form ties based on UGC quality, allowing for analytical and numerical study of network equilibria properties under different meeting probabilities.

Existing literature on network formation models, including the seminal work by Erdös and Rényi on random graphs and the preferential attachment model by Barabási and Albert, have mainly focused on topological properties and socio-economic factors. These models often assume bilateral connections and high transitivity, unlike the directed and often non-reciprocal nature of many modern social media platforms. The fitness model by Caldarelli et al., although incorporating user attributes, does not accurately capture the continuous search for better UGC quality observed in real-world data. Stochastic Actor-Oriented Models (SAOM) and strategic network formation models frequently use sociological or topological elements to explain tie formation, but often fail to incorporate the defining feature of UGC-based platforms: the quality of the content itself. This paper attempts to bridge this gap by developing a model that directly incorporates UGC quality into the network formation mechanism.

The researchers first analyzed longitudinal Twitter data from a network of complex network scientists to examine network formation dynamics. Their analysis revealed that users tend to form connections with those producing better quality UGC over time, which supports the intuition that users seek to optimize the quality of the content they consume. The model builds on this finding by proposing a sequential dynamical process starting from an empty network. Each agent in the network has a quality attribute (qi) representing the average quality of their UGC, drawn from a probability distribution. At each time step, an agent selects another agent randomly (using a uniform or in-degree-based distribution) and decides whether to create a link based on a meritocratic principle: a link is created if the selected agent's UGC quality exceeds the maximum quality among the agent's current followees. The payoff function used is the maximum quality among current followees. The model's convergence to equilibrium is proven, and the expected number of steps to reach equilibrium is calculated. Different meeting probability distributions, including uniform and in-degree-based preferential attachment, are considered to simulate the effects of recommendation systems. The model's analytical predictions of the in-degree and out-degree distributions, network diameter, average clustering coefficient, and follower overlap are then derived. Finally, the model is validated using three datasets collected from Twitch, focusing on the chess and poker categories after filtering for consistent user interest and language.

The study's key findings include: 1. The model converges to an equilibrium almost surely, with the time to reach equilibrium increasing exponentially with network size, reduced by preferential attachment. 2. The in-degree distribution at equilibrium follows Zipf's law, where the expected number of followers of a node is inversely proportional to its quality rank (E[di] = N/i). This result holds robustly even when considering a preferential attachment-based meeting process, suggesting resilience to recommendation systems' effects. 3. The out-degree distribution is non-monotonic, resembling a gamma or Poisson distribution, and exhibits a quickly vanishing tail, contrasting with power-law distributions frequently found in other social network analyses. 4. The average network diameter and average node distance grow proportionally to the logarithm of the network size, exhibiting the small-world property. 5. The average clustering coefficient remains above 10% even for large networks. 6. The overlap between followers of highly ranked users exhibits a pattern consistent with Zipf's law, showing high similarity in the followers of top-ranked nodes. Empirical data from Twitch strongly support the predicted Zipf's law for in-degree and the rapid decay of the out-degree distribution, though some differences are noted, potentially attributed to the ongoing network formation process and recommendation systems on Twitch.

The findings confirm that a simple, meritocratic model based on UGC quality can effectively explain several key characteristics of social media networks. Zipf's law's emergence from the model highlights the importance of quality in shaping network structure, explaining the rise of influencers based on merit rather than purely preferential attachment. The robustness of this result against recommendation systems suggests that intrinsic content quality is a primary driver of influence. The model's prediction of the out-degree distribution contrasts with commonly observed power laws, suggesting that a simple quality-based mechanism can explain the observed cut-off in the out-degree distribution. The results underscore the need for models incorporating content quality to accurately represent the dynamics of online social networks. The model's success in capturing both macro-scale properties like the small-world effect and micro-scale properties like follower overlap underlines its potential for applications in network analysis and prediction.

This study presents a novel, parsimonious model for network formation on UGC-based social media platforms that successfully predicts several key network properties, including the observed Zipf's law for in-degree and the rapid decay of the out-degree distribution. Future research could explore extensions including different update rules, multi-dimensional quality attributes, growing network dynamics, and the analysis of network spreading characteristics. Further empirical studies on platforms like Instagram and TikTok are needed to solidify these findings.

The model assumes a common interest among users, which may not always hold true in real-world scenarios. The choice of the payoff function (maximum quality) may affect the results, and exploring alternative functions could provide further insights. The Twitch data may suffer from sampling biases due to the limitations of data collection on live-streaming platforms. While the model captures many key characteristics, it does not fully replicate all real-world nuances, particularly subtle deviations observed in the Twitch data which may be impacted by platform-specific recommendation systems.