Analysis of conflicted nodes in three-state threshold dynamics on signed networks

Abstract

The spread of opinions and behaviors on networks is studied as complex contagion, where individual attitudes change through cumulative influence from multiple neighbors, primarily within the framework of the linear threshold model. Analytical methods including mean-field approximation, pair approximation, and approximate master equations (AMEs) have been developed for this purpose. Models on signed networks and three-state models distinguishing adoption, opposition, and indecision have also been investigated, but the former have mostly been limited to two-state dynamics, while the latter have relied mainly on simulations. The absence of an analytical framework for models combining both features has meant that individuals who withhold decisions due to conflicting influences and those who simply lack information have been conflated as a uniform inactive population. In this paper, we derive AMEs for a three-state linear threshold model on signed networks with positively active, negatively active, and inactive states. Using this framework, we define nodes that receive sufficient positive and negative influence yet fail to activate due to cancellation as conflicted nodes, and construct a method to compute their density directly from the AME state variables. Our analysis reveals that in the regime where cascades propagate widely, the majority of remaining inactive nodes are in a conflicted state rather than lacking information. Furthermore, we show that the conflicted fraction exhibits a transient peak during the cascade and use AME class-boundary fluxes to demonstrate that this peak is generated by the temporal balance between the total inflow into and total outflow from the conflicted class. Because inactive nodes that satisfy an activation condition can still move to other inactive classes as their neighborhood composition changes before their own activation event occurs, the peak is described as a population-level flux balance that aggregates multiple microscopic pathways. The framework presented here, which separates and quantifies the inactive population into “conflicted nodes” and “insufficient nodes,” provides a dynamical description of a broad class of social phenomena in which decisions are withheld under opposing pressures such as cross-pressures.