From Fragmentation to Spanning: Connectivity Regime Transitions in Equal-Probability Geohazard Fields

Abstract. Scalar hazard metrics such as exceedance ratio, damaged-area fraction, or marginal failure probability quantify how much of a spatial domain is activated, but they do not determine how that activation is organized. In correlated threshold systems, fields with identical occupation fractions may form fundamentally different connected structures, ranging from dispersed local clusters to dominant domains and system-spanning configurations. This study develops a connectivity-centered framework to examine this non-uniqueness under exact equal-probability conditions. An observed liquefaction manifestation field is used as an empirical reference and transformed into a latent spatial template for controlled coherence experiments. Correlated realizations are generated under prescribed smoothing levels, thresholded to identical occupation fractions, and evaluated using connected-component count, dominant-cluster ratio, fragmentation, cluster concentration, spanning probability, cluster-size hierarchy, and fluctuation-based indicators. The results show that uncorrelated equal-probability fields remain predominantly fragmented, whereas correlated fields can reorganize into clustered, dominant-connected, and spanning states under the same occupation constraint. Regime-occupancy analysis further demonstrates that equal probability does not correspond to a single connected state, but to a coherence-dependent distribution of accessible configurations. Second-largest-cluster and fluctuation analyses indicate that the fragmentation-to-spanning evolution proceeds through mesoscale aggregation and extended transition bands, rather than through a single sharp boundary. Finite-size analyses show that these organizational trends remain structurally interpretable across alternative raster scales. The framework provides a nonlinear geoscience interpretation of thresholded correlated hazard fields, in which spatial coherence acts as a control parameter and connectivity regimes emerge through pattern reorganization under fixed occupation constraints. The study therefore demonstrates that activation extent and spatial organization are distinct system properties, and that connectedness provides an essential descriptor beyond scalar probability alone. More broadly, the framework links correlated random fields, thresholded spatial patterning, null-model contrast, and connectivity-regime transitions for complexity-oriented hazard analysis.