## 1 Introduction  
With the swift advancement of globalization and urbanization, human societies are undergoing profound changes in spatial patterns [1]. In the suburbs of major cities, urban factors are incrementally dominant, whereas in the peripheries of these urban areas, rural attributes are diminishing [2, 3]. While these peripheral areas have not achieved full urban status in terms of urbanization scale and quality, it is no longer apt to simply categorize them as rural [4]. In this context, the traditional urban-rural dichotomy fails to encapsulate the nuances of urban spatial structure, giving rise to the concept of the \"urban-rural fringe\" [5, 6]. The urgency to study urban-rural fringes stem from two critical challenges facing these regions. First, as cities continue to expand, these areas face immense developmental pressures. The dual processes of urbanization and industrialization mean these zones increasingly grapple with the contradictions between population and land use [7, 8]. Second, being dynamic, integrated, and transitional zones, urban-rural fringes have emerged as focal points in environmental conservation [9; 10]. As ecological buffers, they represent vulnerable zones in urban management, with escalating social, economic, and environmental conflicts originating and being magnified therein. The conceptual understanding and identification of urban-rural fringes have evolved significantly over time. To properly contextualize this evolution, it is essential to first distinguish the urban-rural fringe from related concepts. The urban sphere of influence refers to the area where a city's economic, social, and cultural influence is exerted [11; 12; 13; 14]. The urban-rural fringe is the transitional zone at the edge of this sphere, characterized by a mix of urban and rural land uses and socioeconomic characteristics. The urban-rural fringe and peri-urban areas both represent the transitional zones between urban and rural landscapes, with the former emphasizing the spatial and functional intermixing of both realms, and the latter focusing more on the urbanization processes and the specific roles of rural areas at the city's edge [15; 16; 17; 18]. Additionally, the urban-rural fringe is separate from suburban areas, which are residential and commercial areas adjacent to but outside the central urban area [19; 20; 21]. This unique geographical region not only amalgamates traits of both urban and rural areas but also exhibits complexities and distinct characteristics in aspects of space, function, and population dynamics. The urban-rural fringe is characterized by its transitional nature, high dynamics, and spatial heterogeneity [13]. It is a \"messy\" yet opportunistic space in policy and decision-making processes, but planning policies tend to overlook this specific peri-environment, resulting in the gradual development of an unhealthy urban form [22]. To comprehensively study the social structure, landscape patterns, and the impact on urban ecological environments within the urban-rural fringe, it is imperative to quantitatively identify them [23]. Such recognition is not only crucial for understanding the essence and functions of the urban-rural fringe but also indispensable for fostering more sustainable urban development. Precise identification of the urban-rural fringe is instrumental for better comprehension, planning, and management of this special zone, ensuring its sustainability and laying a robust foundation for future endeavors. Historically, the identification of urban-rural fringes has predominantly been anchored in qualitative methodologies. These initial approaches often relied on empirically derived metrics and administrative boundaries as the principal mechanisms for delineating these transitional spaces [24]. However, with the proliferation and improvement of urban datasets, there has been a paradigmatic shift in research methodologies. Researchers have increasingly transitioned from a focus on qualitative techniques to a more robust incorporation of quantitative analyses [4; 25].