This paper theoretically investigates network sparsity and its impact. We focus on a specific topology of the network, i.e., the network with core-periphery structures, which is a widely studied financial network structure. It is crucial to understand how changes in sparsity influence network impact. For example, reducing the sparsity of a banking network can offer significant advantages. Upgrading strategically a local bank branch to a regional headquarters improves the connectedness among regional branches. Similarly, reducing the overall sparsity of a banking network by strategically merging two financial networks into one can create substantial potential impacts on banking operations. We derive closed-form solutions that quantify links between sparsity and its impact on financial networks with complete core-periphery structures. We prove that reducing sparsity increases impacts on networks with complete core-periphery components through two sparsity reduction strategies. We further validate other impact measures through simulations.
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