Annealing approximation in master-node network model

TL;DR

The paper develops a chiral master-node opinion dynamics model analyzed via an annealed mean-field approximation, reducing a complex parameter space to four key quantities $(R,L,\overline{A^{(1)}},A^{(4)})$ and revealing three absorbing-state regimes. It shows that the system can reach complete disagreement ($m^{*}=0$), complete consensus ($m^{*}=1$), or bistability where the outcome depends on initial conditions, with transitions that are either continuous or discontinuous. A central finding is the emergence of a saddle-node bifurcation and a tricritical point at $(L,R)=(0,0)$, driven by chiral asymmetry and master-node influence. The work connects to sociophysics by illustrating how asymmetric centralized influence can induce abrupt polarization, while offering analytic phase diagrams and suggesting avenues for extending to spatial networks and empirical parameter estimation. All mathematical expressions are cast in a precise $\LaTeX$-style format for clarity and reproducibility.

Abstract

This paper investigates absorbing-state phase transitions in opinion dynamics through a master-node network model analyzed using annealing approximation. We develop a theoretical framework examining three fundamental regimes: systems converging to complete disagreement, complete consensus, or both states depending on initial conditions. The phase behavior is governed by two key chiral parameters: $R$ measuring right-oriented influence and $L$ measuring left-oriented influence in the network interactions. Our analysis reveals a rich phase diagram featuring both continuous and discontinuous transitions between disordered and ordered phases. The discontinuous transition emerges in systems with two absorbing states, where the final configuration depends critically on initial opinion distributions. The annealing approximation provides fundamental insights into how asymmetric social influences (chirality) shape collective opinion formation, acting as a symmetry-breaking element that drives the system toward polarization or consensus.

Figures (16)

• Figure 1: Sketch of the system.
• Figure 2: Diagram $R\times L$ for case I, where $m^{\ast}=0$ is an absorbing state, while $m=1$ is not.
• Figure 3: Diagram $R\times L$ for case II, where $m^{\ast}=1$ is an absorbing state, while $m=0$ is not.
• Figure 4: Diagram $R\times L$ for case III, where $m^{\ast}=0$ and $m^{\ast}=1$ are absorbing states.
• Figure 5: Flux diagram corresponding to the cases I.1, I.2 and I.3.