The fate of cities under natural hazards depends not only on hazard intensity but also on the coupling of structural damage, a collective process that remains poorly understood. Here we show that urban structural damage exhibits phase-transition phenomena. As hazard intensity increases, the system can shift abruptly from a largely safe to a largely damaged state, analogous to a first-order phase transition in statistical physics. Higher diversity in the building portfolio smooths this transition, but multiscale damage clustering traps the system in an extended critical-like regime (analogous to a Griffiths phase), suppressing the emergence of a more predictable disordered (Gaussian) phase. These phenomenological patterns are characterized by a random-field Ising model, with the external field, disorder strength, and temperature interpreted as the effective hazard demand, structural diversity, and modeling uncertainty, respectively. Applying this framework to real urban inventories reveals that widely used engineering modeling practices can shift urban damage patterns between synchronized and volatile regimes, systematically biasing exceedance-based risk metrics by up to 50% under moderate earthquakes ($M_w \approx 5.5$--$6.0$), equivalent to a several-fold gap in repair costs. This phase-aware description turns the collective behavior of civil infrastructure damage into actionable diagnostics for urban risk assessment and planning.
Cities are increasingly exposed to natural hazards whose impacts extend far beyond individual buildings. The 2023 Kahramanmaraş earthquakes (M w 7.8 and 7.5) severely damaged over 280,000 buildings and displaced millions, causing losses exceeding US$34 billion [10,36,41]. Similarly, the 2024 Category 4 Hurricane
Milpitas, California, a residential city in the San Francisco Bay Area, serves as the study region, with earthquakes considered as the representative hazard for the city (Fig. 1a). The analysis focuses on multistory buildings (two or more stories), which dominate regional seismic risk due to their greater vulnerability and damage consequences. Building-level data are integrated into multi-degree-of-freedom structural models, whose nonlinear time-history analyses yield statistical distributions of structural capacities. These capacities are evaluated against spatially correlated ground-motion fields to generate damage realizations, which are then aggregated into ensembles of damage fractions (see Methods and Supplementary Note 1.).
As hazard intensity increases, the response shifts from a collectively safe to a collectively damaged state (Fig. 1c). At low magnitudes, most buildings remain undamaged; at high magnitudes, nearly all are damaged. While this trend is expected, the abruptness of the transition is striking: around M w = 5-6, the collective state switches suddenly from safe to damaged. Within this range, the damage-fraction distributions become bimodal, peaking at the two extremes, indicating that the region tends to be either largely safe or largely damaged rather than partially so. This synchronization emerges under moderate-intensity hazards, where engineering decisions are most consequential yet most uncertain. Unlike weak or extreme events, where appropriate actions are relatively straightforward (ignore or fully mobilize), moderate events require calibrated preparedness, but their polarized outcomes make such decisions inherently complex. Extended Data Figs. 1 and2 show the results for the full building portfolio, including single-story buildings.
Different cities may exhibit distinct collective behaviors under similar hazard conditions. To examine how the nature of the transition depends on the composition of the building portfolio, we vary structural diversity systematically. Specifically, we adjust the dispersion in structural capacities to represent cities with different levels of portfolio heterogeneity (Fig. 1b). This heterogeneity is parameterized by σ, which quantifies the additional dispersion in structural capacities across the region (see Methods), with σ = 0 denoting the realinventory baseline. This allows us to trace how the collective response evolves with increasing structural diversity.
At high structural diversity, the bimodality observed at moderate magnitudes (M w = 5.6 in Fig. 1d) is suppressed, smoothing the transition from collectively safe to collectively damaged states (Extended Data Fig. 3). Synchronization vanishes around σ ≈ 0.5, where the damage-fraction distribution flattens and collective damage becomes highly volatile. Notably, increasing diversity does not drive the system immediately into a fully disordered phase; instead, it remains in a volatile regime with persistently high variability and non-Gaussian statistics.
To map the collective response across a wide range of earthquake magnitudes and structural diversities, we construct a heatmap of the most probable regional damage fraction (the mode) (Fig. 2b). Two synchronized states emerge: a collectively safe state (near-zero damage) and a collectively damaged state (near-complete damage). At low σ, the transition between them is sharp, marked by a narrow phase-coexistence band where the damage-fraction distribution is bimodal. Beyond a critical diversity σ c ≈ 0.5, the transition smooths into a broad volatile regime characterized by high-variance, non-Gaussian statistics. Analysis of the northeastern San Francisco portfolio, which spans diverse districts and is therefore more heterogeneous, corroborates this trend (Extended Data Figs. 2 and4), demonstrating that greater heterogeneity weakens synchronization and broadens the transition. Specific quantitative details, such as the exact shape of the transition boundary or the extent of the volatile regime, vary with modeling assumptions, but the core phenomenological features remain robust. Supplementary sensitivity analyses confirm that these collective behaviors consistently emerge across a spectrum of model fidelities, ranging from alternative probabilistic models to physics-based simulations involving finite-element analysis with seismic wave propagation (Supplementary Figs. 1234). This consistency indicates that the observed regimes are not artifacts of specific modeling choices, but rather inherent regularities governing the interplay between hazards and structural systems. These observed shifts in collective behaviors mirror phase transitions
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