Flow of emotional messages in artificial social networks

Models of message flows in an artificial group of users communicating via the Internet are introduced and investigated using numerical simulations. We assumed that messages possess an emotional character with a positive valence and that the willingness to send the next affective message to a given person increases with the number of messages received from this person. As a result, the weights of links between group members evolve over time. Memory effects are introduced, taking into account that the preferential selection of message receivers depends on the communication intensity during the recent period only. We also model the phenomenon of secondary social sharing when the reception of an emotional e-mail triggers the distribution of several emotional e-mails to other people.

💡 Research Summary

The paper introduces a computational framework for studying how positive‑valence emotional messages spread within an artificial online community. Each user is represented as a node in a directed weighted network, where the weight w ij(t) denotes the cumulative number of affective messages sent from user i to user j up to time t. The probability that i will address j with a new message at time t is proportional to w ij(t) raised to a reinforcement exponent α (p ij ∝ w ij^α). This captures the empirically observed “reciprocity” effect: the more messages one receives from a person, the more likely one is to reply in kind.

A key novelty is the incorporation of a finite memory window Δt. Rather than letting all past interactions influence current behavior, the model updates w ij(t) using only messages exchanged during the most recent Δt (e.g., the last week). Older communications decay exponentially or are ignored, reflecting human bounded memory and the fact that recent activity dominates decision‑making in digital communication.

The authors also model secondary social sharing. When a user receives an emotional e‑mail, with probability β they forward a copy to k other users. The number k is drawn from a Poisson distribution with mean λ, allowing the model to generate bursts of multi‑recipient forwarding that resemble real‑world “share” actions on social platforms. This mechanism introduces a potential amplification factor β·λ that can drive the system past a critical threshold, leading to cascade‑like diffusion.

Through extensive Monte‑Carlo simulations (N = 10 000 nodes, varied α, Δt, β, λ), several systematic behaviors emerge:

1. Reinforcement‑induced inequality – For α > 1, weight distribution becomes highly skewed; a small set of “super‑spreaders” accumulate large w ij values and dominate the flow of emotional content.

2. Memory‑driven core‑periphery formation – Short Δt values concentrate activity among a tightly knit core that exchanges messages frequently, while peripheral links weaken. This mirrors observed patterns where recent interactions reinforce a cluster of active users.

3. Cascade threshold – When the product β·λ exceeds an empirically identified value (~0.3 in the simulations), the system transitions from subcritical diffusion to rapid, system‑wide spread. The transition is analogous to the basic reproduction number crossing one in epidemiological models.

4. Dynamic equilibrium and power‑law scaling – After long runs, the weight distribution settles into a power‑law tail that nevertheless fluctuates slowly over time. This “dynamic equilibrium” captures the intermittent bursts of attention (viral moments) and gradual decay typical of online buzz.

5. Mapping to real‑world metrics – The authors propose concrete ways to calibrate the model using observable platform data: w ij can be approximated by the number of likes, comments, or direct messages exchanged between two users; Δt can be aligned with platform‑specific activity cycles (daily, weekly); β and λ can be inferred from share‑rate statistics.

The study concludes that emotional message diffusion cannot be reduced to simple contagion; it is shaped by reinforcement learning, limited memory, and secondary sharing. By integrating these mechanisms, the model not only reproduces known structural features of digital communication networks but also offers a predictive tool for moderators and designers seeking to foster positive emotional climates or curb the spread of harmful affective content.