Predicting Who Says “Got It” in Group Chats: A Dynamic Mathematical Model

This article analyzes the reply behavior of WeChat group members to a leader’s notification by constructing a dynamic mathematical model.

Problem Analysis and Assumptions

To build a reasonable model, the following assumptions are made:

Group size : the total number of members is denoted as N.

Member activity : each member i has an activity level a_i, ranging within a specified interval.

Message importance : the importance of the notification is I, with higher values increasing reply probability.

Leader‑member relationship : the relationship strength between member i and the leader is r_i.

Initial independence : initially, each member’s reply decision is an independent event.

Time decay and peer influence : over time the importance decays, and replies from some members influence the others.

Model Construction

Initial Model: Independent Reply Probability

For member i, the initial probability of replying “got it” is defined by a formula that combines activity level a_i, importance I, and relationship strength r_i.

The expected number of replies at the initial stage is the sum of these individual probabilities.

Dynamic Model

(1) Time Decay Factor

As time t progresses, members’ attention to the message declines. This decay is modeled by a function f(t) with decay rate λ, which drops quickly at first and then stabilizes.

(2) Peer Influence Factor

Replies from already‑responded members encourage others to reply. This influence is represented by a logarithmic term with strength β, added to the base probability.

Consequently, the reply probability for member i at time t becomes a combination of the initial probability, the time‑decay factor, and the peer‑influence term.

Total Reply Count Calculation

At each discrete time step, a random number determines whether each remaining member replies; the total number of replies is the sum of individual reply states (1 for replied, 0 otherwise).

Case Study

Simulation parameters include total members, notification importance, activity distribution, relationship‑strength distribution, influence strength, decay rate, and total time steps.

Simulation results are illustrated in the following figures:

The dynamic model successfully predicts the evolution of reply numbers, and the simulations validate its effectiveness.