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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 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. 

Abstract Engineering the crystal structure of Pt–M (M = transition metal) nanoalloys to chemically ordered ones has drawn increasing attention in oxygen reduction reaction (ORR) electrocatalysis due to their high resistance against M etching in acid. Although Pt–Ni alloy nanoparticles (NPs) have demonstrated respectable initial ORR activity in acid, their stability remains a big challenge due to the fast etching of Ni. In this work, sub‐6 nm monodisperse chemically orderedL1₀‐Pt–Ni–Co NPs are synthesized for the first time by employing a bifunctional core/shell Pt/NiCoO_xprecursor, which could provide abundant O‐vacancies for facilitated Pt/Ni/Co atom diffusion and prevent NP sintering during thermal annealing. Further, Co doping is found to remarkably enhance the ferromagnetism (room temperature coercivity reaching 2.1 kOe) and the consequent chemical ordering ofL1₀‐Pt–Ni NPs. As a result, the best‐performing carbon supportedL1₀‐PtNi_0.8Co_0.2catalyst reveals a half‐wave potential (E_1/2) of 0.951 V versus reversible hydrogen electrode in 0.1mHClO₄with 23‐times enhancement in mass activity over the commercial Pt/C catalyst along with much improved stability. Density functional theory (DFT) calculations suggest that theL1₀‐PtNi_0.8Co_0.2core could tune the surface strain of the Pt shell toward optimized Pt–O binding energy and facilitated reaction rate, thereby improving the ORR electrocatalysis.