This paper introduces Graphon Attachment Network Models (GAN-M), a novel framework for modeling evolving networks with rich structural dependencies, grounded in graphon theory. GAN-M provides a flexible and interpretable foundation for studying network formation by leveraging graphon functions to define attachment probabilities, thereby combining the strengths of graphons with a temporal perspective. A key contribution of this work is a methodology for learning structural changes in these networks over time. Our approach uses graph counts—frequencies of substructures such as triangles and stars—to capture shifts in network topology. We propose a new statistic designed to learn changes in the resulting piecewise polynomial signals and develop an efficient method for change detection, supported by theoretical guarantees. Numerical experiments demonstrate the effectiveness of our approach across various network settings, highlighting its potential for dynamic network analysis.