论文标题
无序复杂网络：能量最佳晶格和持续的同源性
Disordered complex networks: energy optimal lattices and persistent homology
论文作者
论文摘要
无序的复杂网络是无线网络信息传输的随机模型的基本兴趣。基于Poisson点过程模型的知名网络具有有关网络效率的局限性，而与基于随机矩阵光谱（RMT）的网络效率相关的替代方案（例如基于随机矩阵）的替代方案具有障碍性和鲁棒性问题。 In this work, we demonstrate that network models based on random perturbations of Euclidean lattices interpolate between Poisson and rigidly structured networks, and allow us to achieve the best of both worlds : significantly improve upon the Poisson model in terms of network efficacy measured by the Signal to Interference plus Noise Ratio (abbrv. SINR) and the related concept of coverage probabilities, at the same time retaining a considerable measure of mathematical and computational对擦除和噪音的简单性和鲁棒性。 我们研究了该模型中基本晶格的最佳选择，将其连接到Epstein Zeta功能的欧几里得晶格的著名问题最佳性，这反过来又与晶格能的概念有关。这使我们在2D中选择了三角晶格，并在3D中以面部为中心的立方晶格。我们证明，覆盖率概率随着扰动强度的增加而降低，最终会融合到泊松网络的强度。在低疾病的制度中，我们大致表征了覆盖函数的统计定律。 在2D中，我们确定了通过比较其持久图的比较它们的网络拓扑来测量的PTL和RMT网络的混乱强度。我们证明，可以将这种疾病强度的PTL网络视为RMT网络模型的有效替代品，同时提供了更大的易处理性的优势。
Disordered complex networks are of fundamental interest as stochastic models for information transmission over wireless networks. Well-known networks based on the Poisson point process model have limitations vis-a-vis network efficiency, whereas strongly correlated alternatives, such as those based on random matrix spectra (RMT), have tractability and robustness issues. In this work, we demonstrate that network models based on random perturbations of Euclidean lattices interpolate between Poisson and rigidly structured networks, and allow us to achieve the best of both worlds : significantly improve upon the Poisson model in terms of network efficacy measured by the Signal to Interference plus Noise Ratio (abbrv. SINR) and the related concept of coverage probabilities, at the same time retaining a considerable measure of mathematical and computational simplicity and robustness to erasure and noise. We investigate the optimal choice of the base lattice in this model, connecting it to the celebrated problem optimality of Euclidean lattices with respect to the Epstein Zeta function, which is in turn related to notions of lattice energy. This leads us to the choice of the triangular lattice in 2D and face centered cubic lattice in 3D. We demonstrate that the coverage probability decreases with increasing strength of perturbation, eventually converging to that of the Poisson network. In the regime of low disorder, we approximately characterize the statistical law of the coverage function. In 2D, we determine the disorder strength at which the PTL and the RMT networks are the closest measured by comparing their network topologies via a comparison of their Persistence Diagrams . We demonstrate that the PTL network at this disorder strength can be taken to be an effective substitute for the RMT network model, while at the same time offering the advantages of greater tractability.
