In percolation theory, one studies the transition a structure can undergo between a connected and a fragmented phase, and its properties at and around this critical point. For an overview we refer to (Bunde and Havlin 1991; Stauffer and Aharony 1994; Isichenko 1992; Efros 1986, e.g.). For instance, randomly occupying the cells of a square grid, at the critical probability of, in this case, \(\approx\)0.59, abruptly a giant cluster emerges that extends from one side of the system to the other (Brockmann 2018). The control parameter is the variable that is being changed (e.g. the occupancy probability) and the transition is reflected in the order parameter, which is usually associated with macroscopic properties of the system (e.g. the probability that an occupied cell belongs to the giant cluster), and expected to diverge or become trivial (e.g., 1) at the transition. However, the strict theory applies only to the limit of infinite system size. In this example of occupying a grid, the cells are discrete. Another form is continuum percolation, where objects (typically circles) are placed in continuous space, eventually forming a percolating structure (Meester and Roy 2008; Balberg 2009; Mertens and Moore 2012, e.g.). While percolation theory has been developed using derivations and simulations of random processes, percolation-like transitions have also been observed in many empirical studies. Here we selectively review works carried out in the context of cities and urban systems and, more generally, in the context of landscape.

In landscape ecology, neutral models represent a sort of null hypothesis used to explain biodiversity patterns assuming all species are functionally equivalent (Gardner et al. 1987). The properties of these neutral models, including the number of clusters, size distribution, fractal dimension, separation between clusters, and correlation length (e.g. defined as mean distance between two sites on the same finite cluster, (Bunde and Havlin 1991)), can be compared with real-world data (Turner 1989; Gustafson and Parker 1992; Keitt, Urban, and Milne 1997, e.g.). Through modeling and empirical validation, such approaches are employed to explore general landscape structures (With and Crist 1995; With, Gardner, and Turner 1997; Saura and Martínez-Millán 2000; Riitters et al. 2007) to better understand processes related to land-cover patterns, habitat loss, influences on animal movement, species distribution, and the maintenance of landscape functions (McIntyre and Wiens 1999).

In Table 1 we list empirical works that treat percolation-like transitions in the urban context, providing details on the data and the variables of interest, see also Fig. 1. In the urban context, most works use the (a) clustering distance, (b) some sort of density, or (c) attributes of network edges as control parameters. As order parameter, the (i) cluster size(s), (ii) respective entropy, or (iii) number of clusters are used. For instance, an examination of the population density in UK wards reveals a transition in population and built area when population density varies (Arcaute et al. 2015). The authors suggest to consider this transition a proxy for defining cities. By varying the clustering distance, (Behnisch et al. 2019) demonstrate that the building locations in Germany form a giant cluster at the critical distance, which is related to the typical distance between neighboring buildings (Rozenfeld et al. 2008, see also). Yet, (Nam and Eom 2025) utilize population density in spatial units within cities to investigate how the number of clusters changes as the occupancy probability varies. (Bian, Yeh, and Zhang 2024) demonstrate the importance of percolating green corridors in cities. Further papers include (Molinero, Murcio, and Arcaute 2017; Piovani, Molinero, and Wilson 2017; Shreevastava, Rao, and McGrath 2019; Cao et al. 2020; Du et al. 2020, e.g.). Overall, the works represent examples in which understanding critical urban behavior can be a decisive data-driven approach to characterizing the morphology and functioning of the urban fabric.

Table 1.Selection of empirical studies investigating percolation-like transitions in the urban context. For each reference in the first column, we list the data analyzed as well as the order and control parameters. In all cases, the control parameter represents a threshold.
Authors Data Order parameter Control parameter
Arcaute et al. (2015) population density largest cluster area, population population density
Arcaute et al. (2016) road network clusters sizes clustering distance
Fluschnik et al. (2016) land-cover average cluster size clustering distance
Molinero, Murcio, and Arcaute (2017) road network entropy of cluster sizes angle between roads
Piovani, Molinero, and Wilson (2017) road network entropy of cluster sizes edge length
Huynh et al. (2018) bus stops locations largest cluster size, area clustering distance
Behnisch et al. (2019) building locations cluster sizes clustering distance
Shreevastava, Rao, and McGrath (2019) land surface temperature number of clusters temperature
Cao et al. (2020) nighttime light, population, road network entropy of cluster areas population density, road intersection density; day/night value
Du et al. (2020) air quality index fraction of the largest cluster air quality index
Tapia-McClung, Montejano-Escamilla, and Caudillo-Cos (2023) road network entropy of cluster areas clustering distance
Nam and Eom (2025) population density number of clusters occupancy probability
Figure 1
Figure 1.Hierarchical clustering (cladogram) of papers analyzing percolation-like transitions in the urban context. We processed the specific information (data, order parameter, control parameter), as detailed in Table 1, with TF-IDF vectorization and cosine similarity methods. E.g. Du et al. (2020) is more similar to Shreevastava, Rao, and McGrath (2019) than to Arcaute et al. (2015).

Percolation can also be employed for modeling of cities and their evolution (Bitner, Hołyst, and Fiałkowski 2009; Fragkias and Seto 2009; Murcio, Sosa-Herrera, and Rodriguez-Romo 2013, e.g.). The percolation front represents a special form of percolation. If the occupancy probability changes along space, then also percolation happens along this gradient – with a so-called percolation front around the critical concentration. This idea is explored for cities, where a radial gradient is combined with correlated noise, leading to clusters that resemble cities (H. Makse, Havlin, and Stanley 1995; H. A. Makse et al. 1998).

Another domain where percolation-like transitions are explored is traffic dynamics. For example, the car speed threshold that separates isolated local flows from global flows in a road network serves to characterize traffic efficiency (D. Li et al. 2015). Studies show that the volume (F. Wang et al. 2015) and the duration of traffic flow (Zeng et al. 2019) affect the characteristics of this transition, and that the relationship between the size of the residential area and the width of the roads influences traffic congestion (Utami et al. 2020). (Ambühl, Menendez, and González 2023) find correlations between traffic congestion percolation and average flow through the network. (R. Wang, Wang, and Li 2023) report that all mobility networks undergo abrupt transitions when reaching a universal critical threshold. As demonstrated by (Verbavatz and Barthelemy 2021), one-way roads can lead to a percolation-like transition belonging to a new universality class. (Cogoni and Busonera 2021) find stable traffic patterns that persist at specific times of the day during the week and on weekends. (Ebrahimabadi et al. 2023) describe that the distribution of the supply and demand sources within the urban landscape can exert a significant influence on the critical traffic rate within the city. In addition, (C. Fan, Jiang, and Mostafavi 2020) combined an epidemic model with percolation ideas to predict flooding in urban networks. Further research on traffic dynamics includes (Talebpour, Mahmassani, and Hamdar 2017; Guo et al. 2018; Olmos et al. 2018; Serok et al. 2019; Chen et al. 2025, e.g.).

Percolation ideas are also employed to analyze forests and vegetation. In particular, (Saravia, Doyle, and Bond-Lamberty 2018) and (Taubert et al. 2018) show that regions with the highest rates of deforestation are closer to a critical threshold of fragmentation, and if fragmentation continues, consequences such as species loss and degradation of ecosystem services cannot be reversed. Similarly, some vegetation patterns in saturated environments, such as rain-forest, may benefit from being close to criticality (Villegas et al. 2024). In relation to forest fires, (Perestrelo et al. 2022) propose a percolation model to predict the phase transition during fire regimes.

Studies of settlement or vegetation connectivity can be subsumed under landscape criticality. Research on percolation theory is still progressing and advances are being made in various directions (Araújo et al. 2014; Saberi 2015; J. Fan, Meng, and Saberi 2019; M. Li et al. 2021; Sun et al. 2023; Schorcht et al. 2025, e.g.). Percolation theory provides a unifying framework for analyzing phase transitions in urban and landscape systems. From traffic resilience and urban sprawl to vegetation dynamics, critical thresholds offer actionable insights. While percolation theory is primarily based on randomness, real-world systems usually exhibit more complex (spatial) properties. Future research could be dedicated to bridge these two and to a better understanding of their differences.

Acknowledgements

We thank A.-K. Brenner, J.W. Kantelhardt, H. Samaniego, and W. Xu for usefull comments. R.L. Fagundes and Y. Li acknowledge financial support from the German Research Foundation (DFG) for the UPon (#451083179) project. R.L. Fagundes and D. Rybski thank Leibniz Association for funding the project “Landscape Criticality in the Anthropocene – Biodiversity, Renewables and Settlements” (CriticaL). D. Rybski additionally appreciates financial support through the DFG project Gropius (#511568027). F. L. Ribeiro thanks CNPq (grant numbers 403139/2021-0 and 424686/2021-0) and Fapemig (grant number APQ-00829-21 and APQ-06541-24) for financial support.