Simplifying the homology of networks via strong collapses
There has recently been increased interest in applications of topology to areas ranging from control and sensing, to social network analysis, to high-dimensional point cloud data analysis. Here we use simplicial complexes to represent the group relationship structure in a network. We detail a novel algorithm for simplifying homology and “hole location” computations on a complex by reducing it to its core using a strong collapse. We show that the homology and hole locations are preserved and provide motivation for interest in this reduction technique with applications in sensor and social networks. Since the complexity of finding “holes” is quintic in the number of simplices, the proposed reduction leads to significant savings in complexity.