| Abstract: |
| The introduction of higher-order structures results in two significant changes in propagation dynamics: non-linear effects and discontinuous mechanisms. Non-linearity arises because group behaviors exhibit emergent properties that are not merely the sum of individual actions. Discontinuity refers to abrupt changes in system dynamics, where minor perturbations can lead to large-scale shifts in behavior, especially when critical thresholds are crossed.
To address these complex dynamics, this report develops a higher-order dynamical model based on simplicial complexes. Simplicial complexes allow for the representation of multi-body interactions, extending the modeling framework beyond traditional graph-based approaches. This model captures the non-linear propagation dynamics observed in complex systems and provides insights into how these interactions influence the overall spread of phenomena across networks. Additionally, to manage the discontinuous nature of higher-order interactions, the report proposes a threshold-based control model using differential complementarity systems. This model is designed to describe the discontinuous behaviors emerging from higher-order interactions, enabling the evaluation of intervention strategies aimed at controlling the spread. By incorporating threshold dynamics, the model can assess the effectiveness of various control measures and optimize intervention strategies. |
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