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The naming game (NG) is a classic model for studying the emergence and evolution of language within a population. In this article, we extend the traditional NG model to encompass multiple committed opinions and investigate the system dynamics on the complete graph with an arbitrarily large population and random networks of finite size. For the fully connected complete graph, the homogeneous mixing condition enables us to use mean-field theory to analyze the opinion evolution of the system. However, when the number of opinions increases, the number of variables describing the system grows exponentially. To mitigate this, we focus on a special scenario where the largest group of committed agents compete with a motley of committed groups, each of which is smaller than the largest one, while initially, most of uncommitted agents hold one unique opinion. This scenario is chosen for its recurrence in diverse societies and its potential for complexity reduction by unifying agents from smaller committed groups into one category. Our investigation reveals that when the size of the largest committed group reaches the critical threshold, most of uncommitted agents change their beliefs to this opinion, triggering a phase transition. Further, we derive the general formula for the multiopinion evolution using a recursive approach, enabling investigation into any scenario. Finally, we employ agent-based simulations to reveal the opinion evolution and dominance transition in random graphs. Our results provide insights into the conditions under which the dominant opinion emerges in a population and the factors that influence these conditions. 

Abstract Quantum networks describe communication networks that are based on quantum entanglement. A concurrence percolation theory has been recently developed to determine the required entanglement to enable communication between two distant stations in an arbitrary quantum network. Unfortunately, concurrence percolation has been calculated only for very small networks or large networks without loops. Here, we develop a set of mathematical tools for approximating the concurrence percolation threshold for unprecedented large-scale quantum networks by estimating the path-length distribution, under the assumption that all paths between a given pair of nodes have no overlap. We show that our approximate method agrees closely with analytical results from concurrence percolation theory. The numerical results we present include 2D square lattices of 200²nodes and complex networks of up to 10⁴nodes. The entanglement percolation threshold of a quantum network is a crucial parameter for constructing a real-world communication network based on entanglement, and our method offers a significant speed-up for the intensive computations involved. 

Abstract Many systems may switch to an undesired state due to internal failures or external perturbations, of which critical transitions toward degraded ecosystem states are prominent examples. Resilience restoration focuses on the ability of spatially-extended systems and the required time to recover to their desired states under stochastic environmental conditions. The difficulty is rooted in the lack of mathematical tools to analyze systems with high dimensionality, nonlinearity, and stochastic effects. Here we show that nucleation theory can be employed to advance resilience restoration in spatially-embedded ecological systems. We find that systems may exhibit single-cluster or multi-cluster phases depending on their sizes and noise strengths. We also discover a scaling law governing the restoration time for arbitrary system sizes and noise strengths in two-dimensional systems. This approach is not limited to ecosystems and has applications in various dynamical systems, from biology to infrastructural systems.