  Statistical network models are useful for understanding the underlying
formation mechanism and characteristics of complex networks. However,
statistical models for \textit{signed networks} have been largely unexplored.
In signed networks, there exist both positive (e.g., like, trust) and negative
(e.g., dislike, distrust) edges, which are commonly seen in real-world
scenarios. The positive and negative edges in signed networks lead to unique
structural patterns, which pose challenges for statistical modeling. In this
paper, we introduce a statistically principled latent space approach for
modeling signed networks and accommodating the well-known \textit{balance
theory}, i.e., ``the enemy of my enemy is my friend'' and ``the friend of my
friend is my friend''. The proposed approach treats both edges and their signs
as random variables, and characterizes the balance theory with a novel and
natural notion of population-level balance. This approach guides us towards
building a class of balanced inner-product models, and towards developing
scalable algorithms via projected gradient descent to estimate the latent
variables. We also establish non-asymptotic error rates for the estimates,
which are further verified through simulation studies. In addition, we apply
the proposed approach to an international relation network, which provides an
informative and interpretable model-based visualization of countries during
World War II.
