The multiscale simplicial flat norm (MSFN) of a d-cycle is a family of optimal homology problems indexed by a scale parameter {\lambda} >= 0. Each instance (mSFN) optimizes the total weight of a homologous d-cycle and a bounded (d + 1)-chain, with one of the components being scaled by {\lambda}.We propose a min-cost flow formulation for solving instances of mSFN at a given scale {\lambda} in polynomial time in the case of (d + 1)-dimensional simplicial complexes embedded in {R^(d + 1)} and homology over Z. Furthermore, we establish the weak and strong dualities for mSFN, as well as the complementary slackness conditions. Additionally, we prove optimality conditions for directed flow formulations with cohomology over Z+. Next, we propose an approach based on the multiscale flat norm, a notion of distance between objects defined in the field of geometric measure theory, to compute the distance between a pair of planar geometric networks. Using a triangulation of the domain containing the input networks, the flat norm distance between two networks at a given scale can be computed by solving a linear program. In addition, this computation automatically identifies the 2D regions (patches) that capture where the two networks are different. We demonstrate through 2D examples that the flat norm distance can capture the variations of inputs more accurately than the commonly used Hausdorff distance. As a notion of stability, we also derive upper bounds on the flat norm distance between a simple 1D curve and its perturbed version as a function of the radius of perturbation for a restricted class of perturbations. We demonstrate our approach on a set of actual power networks from a county in the USA. Our approach can be extended to validate synthetic networks created for multiple infrastructures such as transportation, communication, water, and gas networks.

翻译：多尺度单纯平坦范数（MSFN）是针对d维环族的一类最优同调问题，由尺度参数λ≥0索引。每个实例（mSFN）通过λ缩放其中一个分量，优化同调d维环与有界(d+1)维链的总权重。针对嵌入R^(d+1)的(d+1)维单纯复形及Z系数同调的情形，我们提出一种最小成本流公式，可在多项式时间内求解给定尺度λ下的mSFN实例。此外，我们建立了mSFN的弱对偶性与强对偶性，以及互补松弛条件。进一步证明了Z+系数上同调有向流公式的最优性条件。随后，基于几何测度论中定义的多尺度平坦范数——一种度量对象间距离的概念，我们提出计算平面几何网络对之间距离的方法。通过包含输入网络的区域三角剖分，特定尺度下两网络间的平坦范数距离可通过求解线性规划得到。该计算还能自动识别捕获网络差异的二维区域（补丁）。通过二维实例证明，相较于常用的豪斯多夫距离，平坦范数距离能更精确捕捉输入数据的变异特征。作为稳定性度量，我们还推导了受限扰动类别中，简单一维曲线与其扰动版本间平坦范数距离的上界，该上界是扰动半径的函数。最后，我们将该方法应用于美国某县的实际电力网络数据集。本方法可扩展用于验证交通、通信、供水及燃气等多类基础设施合成网络的准确性。