Disordered complex networks: Energy optimal lattices and persistent homology
Disordered complex networks: Energy optimal lattices and persistent homology
复制标题
无序复杂网络：能量最优晶格和持久同源性
DOI：
10.1109/tit.2022.3163604
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发表时间：
2022
影响因子：
2.5
通讯作者：
Shirai Tomoyuki
中科院分区：
文献类型：
--
作者：
Ghosh Subhroshekhar; Miyoshi Naoto; Shirai Tomoyuki
无序复杂网络是统计物理学的一个基本研究方向，近年来作为无线网络信息传输的随机模型引起了人们的极大兴趣。虽然在数学上很容易处理，但基于规则泊松点过程模型的网络对网络效率提出了挑战。另一方面，强相关的替代方案，如基于随机矩阵谱的网络(Ginibre网络)，在可处理性和健壮性方面带来了巨大的挑战。在这项工作中，我们证明了基于欧几里得格子的随机扰动的网络模型在Poisson和刚性结构的网络之间进行内插，并允许我们实现两个世界的最佳：在用信号与干扰加噪声比(Abbrv)来衡量网络效率方面显著改进Poisson模型。SINR)和相关的覆盖概率概念，同时保留了相当大的数学和计算简单性以及对擦除和噪声的稳健性。我们研究了该模型中基晶格的最优选择，将其与欧几里得晶格关于Epstein Zeta函数的最优性问题联系起来，而爱泼斯坦Zeta函数又与晶格能量概念有关。这导致了我们在2D中选择三角形晶格，在3D中选择面心立方晶格，我们考虑了它们的高斯扰动。我们提供了理论分析和经验研究，证明了覆盖概率随着扰动强度的增加而减小，最终收敛到泊松网络的覆盖概率。在低无序的情况下，我们的研究表明基站附近覆盖函数的近似统计行为是对数正态分布，其参数依赖于格子的Epstein Zeta函数，以及在大阈值下控制网络覆盖概率的幂定律常数的相关近似依赖关系。在2D中，我们确定了扰动三角晶格(abbrv.Ptl)和Ginibre网络通过比较它们的网络拓扑结构，通过比较它们在总变化和对称最近邻距离上的持续图来测量最接近的距离。我们证明，在这种情况下，PTL和Ginibre网络表现出非常相似的覆盖概率分布，PTL的性能至少和Ginibre网络一样好。因此，在这种无序强度下的PTL网络可以被认为是Ginibre网络模型的有效替代，同时从理论和经验角度都提供了更好的易处理性的优点。
Disordered complex networks are of fundamental interest in statistical physics, and they have attracted recent interest as stochastic models for information transmission over wireless networks. While mathematically tractable, a network based on the regulation Poisson point process model offers challenges vis-a-vis network efficiency. Strongly correlated alternatives, such as networks based on random matrix spectra (the Ginibre network), on the other hand offer formidable challenges in terms of tractability and robustness issues. In this work, we demonstrate that network models based on random perturbations of Euclidean latticesinterpolatebetween Poisson and rigidly structured networks, and allow us to achieve thebest of both worlds: significantly improve upon the Poisson model in terms of network efficacy measured by theSignal to Interference plus Noise Ratio(abbrv. SINR) and the related concept ofcoverage 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, whose Gaussian perturbations we consider. We provide theoretical analysis and empirical investigations to 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, our studies suggest an approximate statistical behaviour of the coverage function near a base station as a log-normal distribution with parameters depending on the Epstein Zeta function of the lattice, and related approximate dependencies for a power-law constant that governs the network coverage probability at large thresholds. In 2D, we determine the disorder strength at which the perturbed triangular lattice (abbrv. PTL) and the Ginibre networks are theclosestmeasured by comparing their network topologies via a comparison of theirPersistence Diagramsin the total variation as well as the symmetrized nearest neighbour distances. We demonstrate that, at this very same disorder, the PTL and the Ginibre networks exhibit very similar coverage probability distributions, with the PTL performing at least as well as the Ginibre. Thus, the PTL network at this disorder strength can be taken to be an effective substitute for the Ginibre network model, while at the same time offering the advantages of greater tractability both from theoretical and empirical perspectives.