Many network datasets exhibit connectivity with variance by resolution and large-scale organization that coexists with localized departures. When vertices have observed ordering or embedding, such as geography in spatial and village networks, or anatomical coordinates in connectomes, learning where and at what resolution connectivity departs from a baseline is crucial. Standard models typically emphasize a single representation, i.e. stochastic block models prioritize coarse partitions, latent space models prioritize global geometry, small-world generators capture local clustering with random shortcuts, and graphon formulations are fully general and do not solely supply a canonical multiresolution parameterization for interpretation and regularization. We introduce wavelet latent position exponential random graphs (WL-ERGs), an exchangeable logistic-graphon framework in which the log-odds connectivity kernel is represented in compactly supported orthonormal wavelet coordinates and mapped to edge probabilities through a logistic link. Wavelet coefficients are indexed by resolution and location, which allows multiscale structure to become sparse and directly interpretable. Although edges remain independent given latent coordinates, any finite truncation yields a conditional exponential family whose sufficient statistics are multiscale wavelet interaction counts and conditional laws admit a maximum-entropy characterization. These characteristics enable likelihood-based regularization and testing directly in coefficient space. The theory is naturally scale-resolved and includes universality for broad classes of logistic graphons, near-minimax estimation under multiscale sparsity, scale-indexed recovery and detection thresholds, and a band-limited regime in which canonical coefficient-space tilts are non-degenerate and satisfy a finite-dimensional large deviation principle.

翻译：许多网络数据集展现出具有分辨率方差的连通性，以及大规模组织与局部偏离共存的特征。当顶点具有观测排序或嵌入时（例如空间和村庄网络中的地理信息，或连接组中的解剖学坐标），学习连通性在何处以及何种分辨率下偏离基线至关重要。标准模型通常强调单一表示：随机块模型优先考虑粗粒度划分，潜在空间模型优先考虑全局几何结构，小世界生成器通过随机捷径捕捉局部聚类，而图函数表述则完全通用，并不单独提供用于解释和正则化的规范多分辨率参数化。我们提出了小波潜在位置指数随机图（WL-ERGs），这是一种可交换的逻辑图函数框架，其中对数优势连通性核通过紧支撑正交小波坐标表示，并通过逻辑链接映射到边概率。小波系数由分辨率和位置索引，这使得多尺度结构变得稀疏且可直接解释。尽管在给定潜在坐标时边保持独立，但任何有限截断都会产生一个条件指数族，其充分统计量是多尺度小波交互计数，且条件律允许最大熵表征。这些特性支持直接在系数空间中进行基于似然的正则化和检验。该理论天然具有尺度解析特性，包含对广泛逻辑图函数类的普适性、多尺度稀疏性下的近极小极大估计、尺度索引的恢复与检测阈值，以及一个带限机制——在该机制中规范系数空间的倾斜非退化且满足有限维大偏差原理。