• Performance Evaluation: Demonstrated that the proposed GNN model outperforms traditional baselines in forecasting accuracy on the developed dataset.This article presents a novel competitive information propagation model designed for multi-layer interconnected networks, addressing the complex dynamics of information spread in online social networks (OSNs). It highlights how both positive and negative information compete for attention and influence, with individual behaviors shaped by local and global information trends. The study mathematically analyzes the model's dynamics, calculates the basic reproduction number, and establishes stability conditions for different equilibria. It also formulates an optimal control strategy to manage the spread of negative information while minimizing costs.• Competitive Information Propagation Model: Developed a model that captures the coexistence of positive and negative information within multi-layer interconnected networks, reflecting real-world complexity.• Incorporation of Individual Herd Behavior: Introduced a compartmental model where individual tendencies to spread information are influenced by both local and global prevalence, enhancing understanding of adaptive behaviors in information dynamics.• Stability Analysis: Provided a rigorous mathematical foundation by calculating the basic reproduction number and using Lyapunov theory to discuss the stability of informationfree and endemic equilibria.• Optimal Control Framework: Formulated an optimization problem to effectively suppress negative information propagation, balancing control costs with propagation dynamics, and derived solutions to enhance practical decision-making in resource management.This study introduces the DNIRep model, a novel dynamic network representation framework designed to enhance the modeling of rumor propagation in social networks. Acknowledging the limitations of existing static representation methods, DNIRep effectively captures the temporal dynamics and the complex interplay between explicit and implicit relationships among users. The model incorporates a feedback mechanism to improve the performance of node representations and demonstrates how trust levels and interactive behaviors shape public opinion dynamics.• Dynamic Network Representation Model (DNIRep): Developed a new framework that integrates explicit and implicit relationships to more accurately represent information propagation in dynamic networks.• Feedback Mechanism: Introduced a novel feedback system that enhances the updating of node representations, improving overall network representation performance.• Alignment with Real-World Dynamics: Established a model that aligns more closely with real-world social networks, addressing the unique characteristics of rumor propagation and public opinion dynamics.• Experimental Validation: Extensive simulations were conducted demonstrating that higher trust levels stabilize group opinions and that certain user interactions, including novelty of topics and opinion leaders, significantly influence public sentiment evolution.This study explores the dynamics of epidemic spread on networks characterized by second-order properties, specifically focusing on assortative mixing-where nodes connect preferentially based on their degrees. The authors propose a stochastic algorithm to construct networks that reflect real human connections, enhancing the understanding of how these structures influence the spread of infectious diseases. The research examines the impact of these networks on epidemic trajectories and evaluates whether the spread of disease can be predicted using readily observable data.• Stochastic Network Construction: Developed a flexible algorithm for constructing networks based on individual preferences for connection, allowing for the creation of diverse assortative network profiles.• Marginal Degree Preference: Introduced a method to derive the marginal degree preference from a general preference function, accommodating multiple external factors influencing connections.• Epidemic Dynamics Simulation: Conducted simulations to analyze epidemic curves and overall epidemic sizes across various network types, supporting the hypothesis that the effective reproductive number can be predicted as a function of the susceptible population fraction over time.• Guidance for Future Research: Provided a framework for further theoretical investigations into the effects of network structures on epidemic spread, emphasizing the relevance of second-order network properties in epidemiological modeling.From a theoretical standpoint, the primary contribution of the article is the derivation of an infinite exponential, Dirichlet, series for the model variables, which are interrelated by logarithmic and exponential transformations to the R-variable. The finite truncation of the series results in a Prony approximation, which can be interpreted as a sequence of coupled exponential relaxation processes, each with a distinct timescale.• A numerical Newton-Raphson approximation scheme for the R-variable is also derived and compared to the parametric solution.• The proposed numerical method is compared to a double exponential (DE) nonlinear approximate analytic solution, which reveals two coupled timescales: a relaxation timescale, determined by the ratio of the model's time constants, and an excitation timescale, dictated by the population size.• The DE solution is applied to estimate model parameters for a well-known epidemiological dataset-the boarding school flu outbreak.This special issue presents valuable insights into the application of compartment models beyond traditional epidemiology, focusing on their role in understanding social interactions and information dynamics. The featured articles explore various innovative approaches, including spatiotemporal graph neural networks and...