This paper contributes to the theoretical literature by analyzing the relationship between changes in sparsity and their impacts on financial networks with incomplete and random core-periphery structures, which are widely studied in finance. Sparsity, which measures edge density, reflects the level of connectivity: high sparsity indicates fewer connections between agents, while low sparsity signifies a denser web of interactions. Changes in sparsity result in variations in network impacts. Building on a linear network model inspired by spatial econometrics, we find that reducing sparsity amplifies network impacts in incomplete core-periphery structures through two strategies: 1) increasing the number of core agents and 2) merging two or more core-periphery components. For networks with specific incomplete core-periphery configurations, we derive theoretical results for the average total impact and validate other impact measures through simulations. Furthermore, our analysis extends to networks with randomly generated core-periphery structures, affirming the robustness of our findings.
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