| Abstract: |
| Dynamics on networks arise in diverse contexts such as information spreading, disease transmission, epidemiology, opinion dynamics, and statistical physics. In these systems, agents (nodes) interact with their neighbors through network edges, and their states evolve in time depending on the states of their neighbors. The network structure, therefore, plays an essential role in shaping the overall dynamics. When networks are large or highly heterogeneous, direct computations become infeasible and mean-field approximations often fail to capture essential correlations. Pair approximation offers a powerful reduction framework by incorporating network heterogeneity through degree-based correlations, but most existing analyses focus on systems with discrete-valued states, where agents take values from a finite set. In this work, we extend the framework to continuous-valued dynamics. We introduce a family of degree-based density functions to characterize the state distributions of nodes with the same degree and derive the corresponding pair approximation for these density functions. |
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