  Triadic relationships are accepted to play a key role in the dynamics of
social and political networks. Building on insights gleaned from balance theory
in social network studies and from Boltzmann-Gibbs statistical physics, we
propose a model to quantitatively capture the dynamics of the four types of
triadic relationships in a network. Central to our model are the triads'
incidence rates and the idea that those can be modeled by assigning a specific
triadic energy to each type of triadic relation. We emphasize the role of the
degeneracy of the different triads and how it impacts the degree of frustration
in the political network. In order to account for a persistent form of disorder
in the formation of the triadic relationships, we introduce the systemic
variable temperature. In order to learn about the dynamics and motives, we
propose a generic Hamiltonian with three terms to model the triadic energies.
One term is connected with a three-body interaction that captures balance
theory. The other terms take into account the impact of heterogeneity and of
negative edges in the triads. The validity of our model is tested on four
datasets including the time series of triadic relationships for the standings
between two classes of alliances in a massively multiplayer online game (MMOG).
We also analyze real-world data for the relationships between the "agents"
involved in the Syrian civil war, and in the relations between countries during
the Cold War era. We find emerging properties in the triadic relationships in a
political network, for example reflecting itself in a persistent hierarchy
between the four triadic energies, and in the consistency of the extracted
parameters from comparing the model Hamiltonian to the data.
