We study a network of connected nodes where each node holds an opinion - a binary state that may update over time under the influence of a node's neighbors. Nodes have biased affinities, which logically partition the network into distinct parties. Nodes in the same party tend to have a positive influence on each other, but the extent to which this holds varies across nodes and depends on the chosen affinity model. This paper considers two variations on an Ising spin-glass network model that investigate opinion formation in such biased affinity systems. These models differ in how they determine the pairwise influence between nodes. The first of these in what we dub the random interactions model randomly selects the influence two nodes exert on each other based on their respective party affiliation. The second, a profile-based model, relies on a profile, a κ-bit vector of ±1 entries based on the node's known positions regarding each of κ independent topics. In this model the similarity of the profiles of two nodes determines whether they have a positive or negative influence on each other's opinions. We investigate the formation of opinions under both models and characterize their equilibria. We show that while these systems always converge to an equilibrium, they differ in their number and types of equilibria. These differences manifest themselves in the level of influence of initial opinions, and in the likelihood of polarized outcomes across party lines.
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