This appendix delves into the mathematical foundations of network representation techniques, focusing on two key areas: maximum likelihood estimation (MLE) and spectral embedding theory. It begins by exploring MLE for Erdös-Rényi (ER) and stochastic block model (SBM) networks, demonstrating the unbiasedness and consistency of estimators. The limitations of MLE for more complex models are discussed, leading to the introduction of spectral methods. The chapter then presents theoretical considerations for spectral embeddings, including the adjacency spectral embedding (ASE) and its statistical properties. It explores the concepts of consistency and asymptotic normality in the context of random dot product graphs (RDPGs). Finally, we extend these insights to multiple network models, covering graph matching for correlated networks and joint spectral embeddings like the omnibus embedding and multiple adjacency spectral embedding (MASE).