A hypernetwork-based urn model for explaining collective dynamics

Abstract

The topological characterization of complex systems has significantly contributed to our understanding of the principles of collective dynamics. However, the representation of general complex networks is not enough for explaining certain problems, such as collective actions. Considering the effectiveness of hypernetworks on modeling real-world complex networks, in this paper, we proposed a hypernetwork-based Pólya urn model that considers the effect of group identity. The mathematical deduction and simulation experiments show that social influence provides a strong imitation environment for individuals, which can prevent the dynamics from being self-correcting. Additionally, the unpredictability of the social system increases with growing social influence, and the effect of group identity can moderate market inequality caused by individual preference and social influence. The present work provides a modeling basis for a better understanding of the logic of collective dynamics.

Fig 1. An illustration of the urn model.

Individuals will make choices (choose red or blue balls) by observing others’ actions.

Fig 2. Schematic illustration of the dynamic process on our hypernetwork-based urn model.

When time = 2, individual 5 has already made a choice at time 1, and individual 9 will make a choice based on the state of individual 11.

Fig 3. Schematic illustration of social network structure of human interaction.

Social influence allows individuals to perceive external information instead of being independent, then network analysis offers a heightened precision in delineating the intricate configurations underlying human interaction patterns. Furthermore, group effects elucidate various groups in social interaction, subsequently forming hypernetworks.

Fig 4. Proportion of the red balls with different probability of social influence p_s and probability of conformity p_c.

(a) is the result with a false start in a hypernetwork with δ₁ = 0.5; (b) is an equal start in a hypernetwork with δ₁ = 0.5; (c) is a correct start in a hypernetwork with δ₁ = 0.5; (d) is the timeline result with p_s = 0.5, p_c = 0.5.

Fig 5. Unpredictability results under different probability of social influence p_s and probability of conformity p_c.

(a) is the result with a false start in a hypernetwork; (b) is an equal start in a hypernetwork; (c) is a correct start in a hypernetwork. We fix δ₁ = 0.5.

Fig 6. Inequality results for different probability of social influence p_s and probability of conformity p_c.

(a) is the result under a false start; (b) is the results under an equal start; (c) is the results under a correct start. We fix δ₁ = 0.5.

Fig 7. Comparison of numerical and simulation results.

(a) are the results with fixed p = 0.5; (b) are the results with fixed p_c = 0.5. There are one red ball and nine blue balls at the initial.

Fig 8. Unpredictability results for different numbers of hyperedges and probability of social influence p_s.

(a) is the result with a false start; (b) is an equal start; (c) is a correct start. We fix δ₁ = 0.5, p_c = 0.5.

Fig 9. Unpredictability results at different diffusion times.

Each diffusion time represents the moment when an individual makes a choice. We set a false start and fix δ₁ = 0.5, p_c = 0.5, p_s = 0.5.

Fig 10. An illustration of diffusion paths in fully-connected networks and hypernetworks.

In the left fully-connected network, individual 6 has numerous adjacent individuals, resulting in a more intricate diffusion path for its social influence compared to the hypernetwork.