Spatial heterogeneity of Luoyang graded heritage networks revealed by dual-path TOPSIS–RSR analysis and MGWR

Abstract

Planning cultural-heritage corridors requires transparent links from indicators to spatial priorities. We propose an auditable grading–connectivity workflow for Ancient Capital Cultural Heritage (ACCH) in Luoyang, China. Eleven indicators are weighted via AHP–CRITIC; TOPSIS and rank-sum ratio (RSR) provide dual grading routes whose agreement delineates robust high/low tiers and boundary-sensitive mid-tiers. Reclassified tiers parameterize a minimum-cost resistance surface to map corridors, trunks and bottlenecks. MGWR identifies land-use conditions as the most spatially stable risk correlate, with heritage density, elevation and water barriers acting as local modifiers. Outputs support protection prioritization, corridor design, and tiered intervention thresholds in territorial spatial planning.

Introduction

Ancient Capital Cultural Heritage (ACCH) refers to the tangible heritage assets that can reflect the cultural value of the ancient capital in various historical periods, and can reflect the historical and cultural information of the capital city in all aspects. It is a relevant material cultural heritage of great value and far-reaching influence. It has extremely high value potential and needs to be protected. In the context of urbanization development reaching the stage of stock development in China^1,2, the focus of ACCH protection has shifted to the direction of value activation and inheritance that serves the public cultural life of the masses³. The protection of ACCH should focus on cultural-heritage spaces within the framework of territorial spatial planning (operationalised in China through the ‘Three Zones and Three Lines’ control system). and adopt a regional heritage perspective⁴. Meanwhile, comprehensive assessments indicate that cultural heritage risk assessment is evolving from a list of disaster elements towards “scenario-based, multi-source data, and quantitative thresholds,” and there is an urgent need to provide actionable spatial evidence at the city scale^5,6,7,8.

In recent years, heritage conservation has shifted from isolated monument-based practices to landscape- and network-oriented approaches that integrate cultural, ecological and urban processes at regional scales^9,10,11,12. The convergence of GIS, remote sensing and 3D/temporal data now underpins spatial pattern detection, risk assessment and dynamic regulation in heritage management¹³. Recent studies on cultural-heritage corridors move from simple line-based connections to multi-source, data-driven network reconstruction, advancing corridor identification, hierarchical structuring and performance evaluation through cross-disciplinary methods from landscape ecology and spatial planning^14,15,16,17. For connectivity modeling, circuit theory, Minimum Cumulative Resistance (MCR), graph-based indices and landscape connectivity metrics offer complementary ‘optimal-path’ and ‘multi-path diffusion’ perspectives to support coordinated protection and management of nodes–corridors–areas^18,19,20. However, existing research on cultural heritage corridors mostly focuses on the construction of individual corridors or the reference of ecological models. Although the MCR and gravity model have been introduced, there is insufficient identification of the hierarchical differences and spatial heterogeneity of heritage networks^4,21.

Given pronounced spatial nonstationarity and interaction effects in heritage–urban systems, Multiscale Geographically Weighted Regression (MGWR) and GeoDetector are increasingly used to reveal scale-dependent drivers and factor interactions, informing indicator design, zonal weighting and scenario tuning^{22,23,24,25,26,27}. In Multi-Criteria Decision Analysis (MCDA)²⁸, AHP remains a key tool for encoding expert/stakeholder preferences^29,30. Objective weighting via CRITIC captures contrast intensity and conflict to mitigate subjectivity^31,32, while hybrid objective–subjective schemes improve robustness under uncertainty³³. For ranking and prioritization, TOPSIS is widely used for its interpretability, yet non or weakly compensatory outranking approaches are increasingly employed for cross-checking and sensitivity control^34,35,36. The combination of TOPSIS and RSR is more suitable for dealing with multi-index comprehensive evaluation problems. Recent studies show this combo improves robustness and interpretability across domains^{37,38,39,40,41}. In the field of heritage protection pattern, the TOPSIS-RSR method is still in the development stage and can be used to deal with complex problems involving multiple evaluation criteria and decision makers, especially in the protection of cultural heritage, tourism planning, resource allocation, etc.

Anchored in these advances, we focus on Luoyang’s ACCH and develop an integrated pipeline that links indicator design, dual-path grading, graded connectivity and spatial diagnosis. Methodologically, we (1) build an MCDA indicator system (C1–C11) and compute composite scores with fused AHP–CRITIC weights; (2) generate three-/five-tier grades via two complementary routes—a rank-anchored route (RSR–Probit) and a distribution-adaptive route based on a composite score—and reconcile them to stabilize extremes and flag boundary cases for planning decisions; (3) construct graded heritage networks using the MCR model; and (4) quantify spatial heterogeneity with MGWR, supported by BH–FDR multiple-testing control and k-nearest-neighbor (KNN)-based Moran’s I diagnostics. These steps form a single workflow from attribute evaluation to spatio-structural interpretation of the graded ACCH system.

We pursue three tightly linked objectives. First, to construct a graded ACCH system for Luoyang and evaluate the agreement and complementarity between Method A and Method B under three- and five-tier schemes, thereby providing an auditable basis for heritage grading. Second, to derive graded MCR networks that translate discrete classes into movement potential, revealing multi-level corridors, nodes and bottlenecks. Third, to diagnose the scale–sign–texture dimensions of spatial heterogeneity for the socio-spatial drivers (C4–C11) using MGWR, and to identify a hierarchy of more consistently associated, modifying and background factors under rigorous significance and spatial-independence checks, so as to support zonal threshold calibration and corridor-oriented protection strategies. In this study, risk is operationalised as a relative loss-risk index of ACCH value under current socio-spatial pressures and management constraints; vulnerability denotes susceptibility to such loss given heritage sensitivity and exposure; priority refers to an actionable ordering for protection and intervention based on the graded loss-risk index. This framing supports HUL-informed corridor planning and territorial heritage governance by linking graded priorities to spatially varying drivers and network positions.

Methods

Study area and data

Luoyang is located in the Central Plains region of China, roughly between 33°35′–35°05′N and 111°08′–112°59′E, with a total area of about 15,230 km² (Fig. 1). As one of China’s most renowned ancient capital cities, Luoyang has a capital history of more than 1500 years and a rich cultural tradition, which has resulted in a large number of Ancient Capital Cultural Heritage (ACCH) sites.

Fig. 1

Study area.

The ACCH dataset used in this study is mainly derived from the results of the third national census of immovable cultural relics in Luoyang. It includes cultural relics protection units at all administrative levels, historic towns, villages and streets, traditional villages, World Cultural Heritage sites, scenic and historic interest areas, national archeological parks, and large sites that embody the cultural value of the ancient capitals. The detailed typology of these heritage resources is summarized in Table 1. Based on the classification of capital-heritage types, we identified the specific capital-heritage resources of the four major ancient capitals in Henan Province, and in total compiled 1190 ACCH sites. Other datasets involved in the analysis are listed in Table 2.

Table 1 Combing of the types of ACCH and assigning tables of value levels and legal status

Table 2 Data types and source tables

Study workflow

We establish a transparent, reproducible pipeline that links indicator design, hybrid weighting, dual-route grading, heritage network construction and spatial heterogeneity analysis, forming an auditable loop from data to decisions. It is specifically divided into seven steps (Fig. 2):

Fig. 2

Conceptual framework and Technical workflow.

(1) ACCH selection: temporal scope for dynastic capital periods; spatial scope for the contemporary Luoyang boundary.

(2) Indicators and normalization: An MCDA-based evaluation system (Table 3) includes value-socio-spatial factors C1–C11, followed by min–max range normalization.

Table 3 Evaluation index system for the protection pattern of ACCH

(3) Weighting and fusion: Subjective AHP (improved scale, consistency test) and objective CRITIC (variance and inter-indicator conflict) are fused to balance interpretability and data-driven sensitivity.

(4) Dual grading paths (two resolution schemes): Compute TOPSIS closeness C and weighted RSR. Method A (RSR–Probit anchoring) maps cumulative ranks to the Probit axis and projects theoretical quantiles back to TOPSIS for three/five tiers; Method B (Beta-on-F) forms F=αC+(1−α)WRSR on [0,1] with default α = 0.5.

(5) Confusion-matrix reclassification: Build A/B confusion matrices (three/five tiers), compute Po, Pe, and Cohen’s κ, and reclassify cross-labels to stabilize ambiguous mid-tiers.

(6) Graded heritage networks: Run the MCR model on socio-spatial resistances (C4-C11) to extract network for different ACCH tiers.

(7) Spatial heterogeneity by MGWR: Fit MGWR with C4–C11 as predictors to obtain coefficient surfaces β_j (u_i, ν_i); compare neighborhood sizes k to interpret local-vs-regional processes and evaluate group-wise robustness across regions and protection levels. Then, significance and spatial robustness are explicitly assessed on top of MGWR.

Pre-processing and normalization

All indicators are aligned in direction. Based on the initial data matrix X$=\left[\begin{array}{ccc}{x}_{11} & \ldots & {x}_{1p}\\ \ldots & \ldots & \ldots \\ {x}_{n1} & \ldots & {x}_{{np}}\end{array}\right]$, we normalize its values to obtain the matrix Z=$\left[\begin{array}{ccc}{z}_{11} & \ldots & {z}_{1p}\\ \ldots & \ldots & \ldots \\ {z}_{n1} & \ldots & {z}_{{np}}\end{array}\right]$. The formula for data normalization is:

Positive indicators: ${Z}_{{ij}}=\frac{{x}_{{ij}}-min({x}_{j})}{max\left({x}_{j}\right)-min({x}_{j})}$; Negative indicators: ${Z}_{{ij}}=\frac{max\left({x}_{j}\right)-{x}_{{ij}}}{max\left({x}_{j}\right)-min({x}_{j})}$.

MCDA: indicators and structure

Cultural-heritage corridor planning inevitably involves value, spatial-pattern, and management-condition criteria that are heterogeneous in units and scales. MCDA provides a transparent way to structure the problem (goal to criteria to indicators), normalize and align indicators, elicit/compute weights, and aggregate and rank alternatives in a reproducible manner. In our framework, MCDA sits between data preprocessing (Section 3.2) and composite scoring (Section 3.5), acting as the organizing logic that links indicator design, weighting, and subsequent TOPSIS–RSR grading. This design follows recent practice in GIS-enabled heritage planning and MCDA research, where AHP encode expert preferences, CRITIC supply objective contrasts, and hybrid schemes improve robustness under uncertainty. The indicator system is shown in Table 3 below.

Weighting and fusion

Subjective weights based on AHP

We derived subjective weights w_1j for indicators C1–C11 using AHP^42,43,44 with an improved methods of scale^45,46 (Table 4). A judgement matrix K = [u_ij] was constructed and the principal eigenvector was normalized to obtain w_1j. Matrix consistency was checked using the standard CI/CR criteria.

Table 4 Improved index scale tables

Objective weights based on CRITIC

Objective weights w_2j were computed using CRITIC^47,48, which combines indicator contrast and inter-indicator conflict. Contrast was measured by the standard deviation S_j and conflict was summarized using correlations r_jk between indicators:

$${S}_{j}=\sqrt{\frac{\mathop{\sum }\nolimits_{i=1}^{i}{({x}_{{ij}}-{\bar{x}}_{j})}^{2}}{n-1}},{R}_{j}=\mathop{\sum }\limits_{i=1}^{i}\left(1-{r}_{i{i}^{{\prime} }}\right)$$

(1)

Information content was defined as C_j = S_jR_j, and weights were obtained by normalization:

$${w}_{2j}=\frac{{C}_{j}}{\mathop{\sum }\nolimits_{j=1}^{j}{C}_{j}}$$

(2)

Fusion based on improved variance maximization

Based on the above analysis, the subjective and objective weights of each indicator have been determined^49,50. The improved variance maximization combined weighting method^51,52,53 is used to calculate the comprehensive weight. We compute the comprehensive weights using an improved variance-maximization fusion method, which maximizes score variance while accounting for inter-indicator conflict (correlations). The comprehensive weight is determined through the following research:

(1) A variance maximization combined weighted optimization model is constructed, with the maximum variance of the scores of the evaluated objects under the combined weights as the objective function, the priority of the evaluation indicators as the focus, and the constants n and p that have no effect on the maximization model are removed. The expression is:

$$\begin{array}{c}{W}_{j}=\lambda {w}_{1j}+(1-\lambda ){w}_{2j}\\ {MAX}{S}^{2}=\mathop{\sum }\limits_{i=1}^{i}\mathop{\sum }\limits_{j=1}^{j}[{R}_{j}{({x}_{{ij}}-{\bar{x}}_{{ij}})}^{2}]{c}_{j}=\mathop{\sum }\limits_{i=1}^{i}\mathop{\sum }\limits_{j=1}^{j}[{R}_{j}{({x}_{{ij}}-{\bar{x}}_{{ij}})}^{2}](\alpha {w}_{1j}+\beta {w}_{2j})\end{array}$$

(3)

(2) Sorted out: $\lambda =\sqrt{\frac{{\left[\mathop{\sum }\nolimits_{{\rm{j}}=1}^{{\rm{j}}}\mathop{\sum }\nolimits_{{\rm{i}}=1}^{{\rm{i}}}\left[{{\rm{R}}}_{{\rm{j}}}* {\left({{\rm{x}}}_{{\rm{ij}}}-{\bar{x}}_{{ij}}\right)}^{2}\right]* {{\rm{w}}}_{1{\rm{j}}}\right]}^{2}}{{\left[\mathop{\sum }\nolimits_{{\rm{j}}=1}^{{\rm{jn}}}\mathop{\sum }\nolimits_{{\rm{i}}=1}^{{\rm{i}}}\left[{{\rm{R}}}_{{\rm{j}}}* {\left({{\rm{x}}}_{{\rm{ij}}}-{\bar{x}}_{{ij}}\right)}^{2}\right]* {{\rm{w}}}_{1{\rm{j}}}\right]}^{2}+{\left[\mathop{\sum }\nolimits_{{\rm{j}}=1}^{{\rm{j}}}\mathop{\sum }\nolimits_{{\rm{i}}=1}^{{\rm{i}}}\left[{{\rm{R}}}_{{\rm{j}}}* {\left({{\rm{x}}}_{{\rm{ij}}}-{\bar{x}}_{{ij}}\right)}^{2}\right]* {{\rm{w}}}_{2{\rm{j}}}\right]}^{2}}}$.

Sensitivity check

To examine whether the grading results are sensitive to normative choices in indicator weighting and composition, we re-computed the integrated score F and the resulting priority pattern under three weighting schemes (AHP-only, CRITIC-only, and the fused weights in Table 5). We further repeated the calculation after dropping C1 (heritage value) and C5 (land-use condition) one at a time, with the remaining weights re-normalized. Robustness was assessed by the overlap of top 10% priority units between the baseline and each perturbed scenario.

Table 5 Subjective, objective and combined weights for C1–C11

TOPSIS–RSR scoring and grading

To assess the spatial characteristics of the ACCH in Luoyang City, we integrate TOPSIS—providing a continuous proximity-to-ideal score—with the rank-based RSR that supplies a robust ordinal backbone.

A limitation of TOPSIS is its sensitivity to outliers and the implicit full compensability among criteria, which may distort rankings under highly skewed indicator distributions. Combining TOPSIS with RSR compensates for the sensitivity of TOPSIS to extreme values while avoiding the information loss that may occur in a pure rank-based RSR approach^54,55.

TOPSIS (continuous scores)

The TOPSIS method (Technique for Order Preference by Similarity to Ideal Solution) is also called the ideal solution method^56,57,58. It is a commonly used method in MCDA. Its basic idea is to identify the best and worst solutions among a limited number of solutions by normalizing the original data matrix. We normalize the decision matrix, identify positive/negative ideals, and compute the distances to both ideals. We then rank alternatives by the closeness coefficient. The calculation method is:

Given Z =[x_ij]=$\left[\begin{array}{ccc}{X}_{11} & \ldots & {X}_{1p}\\ \ldots & \ldots & \ldots \\ {X}_{n1} & \ldots & {X}_{{np}}\end{array}\right]$, the positive and negative ideals are ${A}^{+}=max\{{X}_{\begin{array}{l}{ij}\end{array}}\}$, ${A}^{-}=min\{{X}_{\begin{array}{l}{ij}\end{array}}\}$； Then the optimal and worst solution distance are ${D}^{+}=\sqrt{\sum {{(Z}_{\begin{array}{c}{ij}\end{array}}-{A}^{+})}^{2}}$, ${D}^{-}=\sqrt{\sum {{(Z}_{\begin{array}{c}{ij}\end{array}}-{A}^{-})}^{2}}$。 Finally, we compute the degree of proximity C_i to A⁺ and sort the alternatives by C_i. The larger the value of ${\rm{C}}{\rm{i}}=\frac{{D}^{-}}{{D}^{-}+{D}^{+}}$, the better the evaluation result.

RSR (discrete classes)

The RSR method is the rank-sum ratio method, which refers to the relative average of the ranks of each evaluation object in the table in the comprehensive evaluation of multiple indicators (if the weights of the evaluation factors are different, the indicator needs to be multiplied by the weight). It is a non-parametric measurement with the characteristics of a continuous variable in the range of 0–1^59,60. Its calculation method is:

(1) The rank conversion is performed on the basis of the initial data matrix, and the rank is sorted according to the size of the index value. The rank R is obtained by the integer method, and the rank R is used to replace the original evaluation index value to obtain the rank data matrix $R=\left[\begin{array}{ccc}{R}_{11} & \ldots & {R}_{1p}\\ \ldots & \ldots & \ldots \\ {R}_{n1} & \ldots & {R}_{{np}}\end{array}\right]$;

(2) The weighted rank sum ratio WRSR value and its ranking are calculated based on the rank data matrix. The WRSR value is calculated as follows: ${\text{WRSR}}=\frac{\sum {{W}_{J}R}_{ij}}{n}$ ($\sum {R}_{{ij}}$ is the sum of the rank values of each indicator, ${\text{W}}_{\text{J}}$ is the weight of the evaluation factor);

(3) Arrange the WRSR values from small to large, list the frequency and cumulative frequency, determine the rank R and average rank $\bar{R}$, and calculate the downward cumulative frequency P. The calculation method is: $P=\frac{\bar{R}}{n}\times 100 \%$ ；$P=1-\frac{1}{4n}\times 100 \%$ (The last P value correction formula);

(4) According to the cumulative frequency, query the “Percentage and Probability Unit Comparison Table”, convert the percentage P to the probability unit Probit, calculate the regression equation, and sort by grading according to the commonly used grading number table:$\,{WRSR}=a+b\times {Probit}$。

TOPSIS–RSR integration

The study of ACCH grading faces a longstanding challenge of reconciling ordinal ranks with metric values. Therefore, to jointly exploit the geometric “ideal-solution” distance of TOPSIS and the robust, scale-free ordering of RSR, we implement two complementary thresholding paths: Method A: RSR–Probit anchoring, which maps cumulative ranks to a near-normal Probit axis to obtain robust quantile cut-points; and Method B: Beta-on-F, which defines thresholds directly on the [0,1] metric domain of a composite score F. Together, the two methods balance an ordinal backbone with distribution-adaptive cut-points and provide cross-validation of grading stability.

(1) Method A (RSR–Probit anchoring) discrete grading is anchored by the RSR–Probit model and projected to the TOPSIS space: First, the Probit score (Z) of WRSR is calculated:

$${\rm{Z}}={\Phi }^{-1}({\rm{P}})$$

(4)

And then Regress TOPSIS closeness C’ on Z:

$${\rm{C}}\mbox{'}={{\rm{\beta }}}_{0}+{{\rm{\beta }}}_{1}Z+\varepsilon$$

(5)

Taking the theoretical quantiles on the Probit axis (five-tier: q = {0.2, 0.4, 0.6, 0.8}; three- tier: q = {1/3, 2/3}), map them to the TOPSIS space by

$${{\rm{C}}}_{\mathrm{cut}({\rm{q}})}={\beta }_{0}^{{\prime} }+{\beta }_{1}^{{\prime} }{\varPhi }^{-1}(q)$$

(6)

and classify C from high to low into five (or three) ordered grades.

(2) Method B (Beta-on-F) within-grade ranking is performed by a composite index of TOPSIS and RSR: Defining a composite score F from TOPSIS and RSR (both on [0, 1]):

$${\rm{F}}={\rm{\alpha }}{\rm{C}}+(1-{\rm{\alpha }}){\rm{WRSR}},{\rm{\alpha }}\in [0,1],({\rm{default}}\,{\rm{\alpha }}=0.5)$$

(7)

We set α = 0.5 as a neutral balance between TOPSIS closeness and RSR robustness, and use F to derive three-/five-tier grades via Beta-quantile cut points.

(3) Agreement between Method A and Method B is quantified using a confusion matrix and Cohen’s κ. We use the cross-label table to produce an AB-combined grading that preserves consensus at the extremes while assigning discordant mid-tier cases to the more conservative/intersection side (Fig. 3).

Fig. 3

The confusion matrix reclassification method diagram.

MCR-based corridor/network modeling

The MCR model constructs a composite resistance surface that accounts for both natural and anthropogenic factors, accumulating cost from these sources to identify least-cost corridors and cost fields. Originating from biological dispersal pathway simulations, the MCR model serves as an effective tool for analyzing spatial heterogeneity^61,62. By calculating the minimal energy expenditure needed for spatial agents to overcome environmental resistance, the model forms potential networks for spatial diffusion. Mathematically, it is expressed as:

$$MCR=min\mathop{\sum }\limits_{i=1}^{n}{D}_{i}\times {R}_{i}$$

(8)

where D_i denotes diffusion distance and R_i represents the resistance coefficient of spatial units.

While MCR offers high spatial resolution in the identification of ecological corridors, its application to cultural heritage faces specific methodological challenges, primarily due to the difficulty of translating historical data: the palimpsest effect—which arises from the layered land use of multiple historical periods—introduces dynamic spatial resistance that cannot be accurately captured by traditional static parameters^63,64.

To overcome this limitation, Knaapen et al. introduced a GIS-based MCR framework⁶⁵, rethinking the construction of cultural heritage corridors as a diffusion process of cultural elements. The novelty of this approach lies in the development of a “spatial resistance-cultural potential” coupling model. This model defines the cultural radiation intensity of heritage nodes as the diffusion kinetic energy, while considering terrain barriers and land use renewal as dissipation factors. Consequently, a composite analytical framework is created, integrating both natural base resistance and human-induced dynamic resistance. By calculating the minimum cost path for the transmission of cultural potential across heterogeneous spaces, this model effectively reconstructs diachronic cultural communication routes.

To provide an external plausibility check for the MCR-derived corridors, we compared the spatial configuration of primary trunks with policy-identified historical–cultural heritage corridors emphasized in the Luoyang Territorial Spatial Master Plan (2021–2035), which highlights the protection of national heritage-route systems such as the Silk Road, the Grand Canal and historic post-road routes, and calls for multi-corridor cultural-heritage governance; and widely recognized route narratives and key heritage clusters in Luoyang (e.g., Silk Road-related heritage nodes and the “five-capital” site belt along the Luo River). In addition, we used major visitor-oriented heritage destinations (officially promoted itineraries and key attractions) as a weak proxy for use intensity, to check whether the modeled trunks preferentially link high-demand sites.

MGWR for spatial heterogeneity

In explaining spatial heterogeneity, MGWR allows different drivers to act at different spatial scales, and its advantages over traditional GWR have been verified in issues such as air pollution and urban thermal environment^66,67.

To characterize the spatially varying relationship between the explanatory variables and the ACCH loss-risk index described above, this study employs MGWR. Unlike the traditional GWR assumption that “all variables vary at the same spatial scale”, MGWR allows each coefficient β to be estimated at its own optimal spatial scale. For observation i, the response is y_i and predictors ${x}_{i}=\left[1,{x}_{i1},\ldots ,{x}_{{ip}}\right]$. The model form is as follows:

$${y}_{i}={\beta }_{0}\left({u}_{i},{v}_{i}\right)+\mathop{\sum }\limits_{j=1}^{p}{\beta }_{j}\left({u}_{i},{v}_{i}\right){x}_{{ij}}+\varepsilon$$

(9)

Where $\left({u}_{i},{v}_{i}\right)$ are spatial coordinates; β is the location-dependent local coefficient; β_j$\left({u}_{i},{v}_{i}\right)$ represents the marginal effect of the j-th factor at position; ε∼ N (0,δ²).

The kernel function employs an adaptive bi-square kernel, using the number of neighbors k of a sample point as the bandwidth metric; for each coefficient, the optimal bandwidth k is selected by minimizing the AICc adjusted for small samples.

We report three bandwidth metrics: the absolute neighbor count k, the relative scale k/n (the proportion of observations used to estimate a coefficient), and a standardized bandwidth z_k. The latter is computed as$\,{z}_{k}=(k-\bar{k})/{s}_{k}$, where $\bar{k}$ and s_k are the mean and sample standard deviation of bandwidths across the intercept and all covariates within the same MGWR model. Positive z_k indicates a more global process relative to the model average, while negative z_k indicates a more local process. We also provide the local R² (local coefficient of determination, used to measure the model’s explanatory power for the variation in y at location spatial coordinates $\left({u}_{i},{v}_{i}\right)$) and the significance of the coefficients.

To control for multiple testing and assess spatial independence, two diagnostic steps are embedded in the MGWR workflow. First, we apply the Benjamini–Hochberg false-discovery-rate (BH–FDR) procedure (q = 0.05) to pixelwise coefficient p-values, generating binary significance masks (sig-BH) and the share of significant locations (sig-share) for each predictor. All reported coefficient maps and summary statistics are restricted to sig-BH = TRUE pixels to suppress chance findings. Second, we compute global Moran’s I of MGWR residuals under KNN spatial weights (default k = 8, with sensitivity checks at k = 6 and 12) and obtain permutation-based two-sided p-values. Combined with Moran scatterplots, these diagnostics evaluate whether remaining spatial structure in the residuals is negligible, ensuring that the spatial heterogeneity observed in β-surfaces is driven by explanatory variables rather than unmodelled error dependence.

Results

Weighting outcomes

Based on the MCDA and AHP method, we established an evaluation system for the ACCH resources. The target layer was the Protection pattern of ACCH, and the criterion layer was Heritage Value, Protection Regional Pattern and Protection Conditions (Table 3). We calculated subjective and objective weights using AHP and CRITIC, respectively.

We determine the weights of the eleven criteria (C1–C11) using the Analytic Hierarchy Process with an improved exponential scale to enhance continuity and discrimination. For any pair of criteria C_i and C_j, experts assign an importance level $n\in \left\{\mathrm{0,1},\ldots ,8\right\}$ which is mapped to a quantitative judgment ${a}_{{ij}}={1.316}^{n}$; the anchors n = 0, 2, 4, 6, 8 represent equal, slightly, obviously, strongly, and extremely more important, and compromise levels use n = 1, 3, 5, 7.

Pairwise comparison matrices satisfy reciprocity and reflexivity, i.e., ${a}_{{ij}}=1/{a}_{{ij}}$ and ${a}_{{ii}}=1$. To combine multiple opinions, we adopt group-AHP by aggregating 50 individual matrices element-wise via the geometric mean, yielding the group matrix $\bar{A}=\left[{\bar{a}}_{{ij}}\right]$ with ${\bar{a}}_{{ij}}={\left(\mathop{\prod }\limits_{e=1}^{50}{a}_{ij}^{(e)}\right)}^{1/50}$. We then extract the priority vector by the principal right-eigenvector method: the normalized eigenvector associated with the maximum eigenvalue of $\bar{A}$ provides $w=({w}_{1},{w}_{2},\ldots ,{w}_{11})$ with ${w}_{k}\ge 0$ and ${\sum }_{k}{w}_{k}=1$. Consistency is assessed using $\mathrm{CI}=({\lambda }_{\max }-n)/(n-1)$ and $\mathrm{CR}={CI}/{RI}$, where n = 11 and RI = 1.51; the conventional acceptance threshold is CR<0.10. In our application, the aggregated matrix achieved λ_max ≈ 11.0082, givin; CI ≈ 0.000821 and CR ≈ 0.000544, which indicates satisfactory group consistency. The AHP weight vector for {C1, …, C11} is (0.17, 0.14, 0.13, 0.04, 0.12, 0.09, 0.03, 0.10, 0.04, 0.06, 0.08).

Objective importance is estimated with CRITIC, which quantifies an indicator’s information as the product of its contrast intensity (standard deviation S_j) and its conflict with the remaining indicators R_j. The objective weight is ${w}_{2j}={C}_{j}/{\sum }_{j=1}^{p}{C}_{j}$. Finally, the CRITIC weight vector for {C1, …, C11} is (0.06, 0.02, 0.11, 0.15, 0.13, 0.17, 0.07, 0.06, 0.06, 0.10, 0.07).

Interpretively, C6 (heritage-density, 0.17) and C4 (regional economy, 0.15) carry the largest objective information (high variability with low redundancy), followed by C5 (0.13) and C10 (0.10). Legal and status factors (C1–C3) retain non-negligible but comparatively smaller objective weights, consistent with their more categorical nature.

Based on the improved variance maximization combined weighting method, we calculated that α = 0.31, β = 0.69 and the combined weight vector for {C1, …, C11} is (0.09, 0.06, 0.11, 0.12, 0.12, 0.15, 0.06, 0.08, 0.05, 0.09, 0.07).

The hybrid scheme (α = 0.31, β = 0.69) places greater emphasis on data-revealed contrast/conflict (CRITIC) while retaining expert-encoded priorities (AHP). Spatial-pattern and management-condition factors (C4–C6, C10) rise in influence relative to a purely subjective scheme, which is consistent with the cross-scale patterning required for corridor identification and with the robustness aims of the subsequent TOPSIS–RSR scoring. Specific weights are shown in Table 5.

Sensitivity tests indicate that the priority pattern remains stable under reasonable perturbations. Under the fused scheme, the top-10% priority set shows substantial agreement with the baseline after dropping C1 (overlap = 77.31%) and after dropping C5 (overlap = 74.79%). The corresponding high-priority clusters remain broadly consistent in space when comparing the baseline fused pattern with the drop-C1 and drop-C5 scenarios (Fig. 4). Across weighting schemes, dropping C1 yields overlaps of 50.42% (AHP), 87.39% (CRITIC) and 77.31% (fused), while dropping C5 yields consistently similar overlaps of 73.95% (AHP), 73.95% (CRITIC) and 74.79% (fused), which are summarized visually in Fig. 5. Together, these results support that the main spatial messages are not artefacts of a single indicator or a particular weighting choice.

Fig. 4

Sensitivity maps of high-priority ACCH units under the fused weights (Top 10%): baseline, drop-C1 and drop-C5.

Fig. 5

Top-10% and 25% overlap with the baseline across weighting schemes under indicator removal (drop C1 and C5).

ACCH grading and agreement

TOPSIS–RSR alignment for method A/B

For n=1190 alternatives and p=11 indicators, TOPSIS closeness C and RSR are strongly and positively associated (Spearman ρ = 0.642, p = 4.67 × 10⁻¹³⁹), indicating aligned ordering yet non-redundant information content. The data can be seen in Fig. 6.

Fig. 6: TOPSIS vs. RSR results: bar charts and scatter plot.

a Bar chart of RSR scores; b bar chart of TOPSIS scores; c scatter plot of TOPSIS versus RSR with the fitted regression line and summary statistics.

According to method A, regressing C on the Probit-transformed cumulative ranks $\text{Z}={\Phi }^{-1}(\text{P})$ yields ${\text{C}}_{\text{cut}(\text{q})}={{\rm{\beta }}}_{0}^{{\prime} }+{{\rm{\beta }}}_{{1}{\prime}} {\Phi }^{-1}({\text{q}})$, ${{\rm{\beta }}}_{{0}{\prime}}$=0.2515, ${{\rm{\beta }}}_{{1}{\prime}}$= 0.0206, R²= 0.1765, DW = 1.959 (DW denotes the Durbin–Watson statistic). Theoretical Probit cuts projected to the TOPSIS space define five-tier thresholds [0.2341, 0.2463, 0.2567, 0.2688] and three-tier thresholds [0.2426,0.2604] (Fig. 7).

Fig. 7: TOPSIS–Z (Probit) scatter plots with five-tier and three-tier thresholds.

a Scatter plot with five-tier thresholds; b scatter plot with three-tier thresholds.

Agreements under the Beta-on-F quantile cuts are summarized in Table 6. The three-tier grading is less sensitive to α, and variations within 0.4–0.6 barely affect tier membership. Balancing robustness, stability and interpretability, we adopt α = 0.5 as the default.

Table 6 Balance and stability tests of α within the range of {0.3, 0.4, 0.5, 0.6, 0.7}

Figure 8 illustrates how the composite score F supports operational tiering under Method B (α = 0.5). The histograms (Figs. 8a, b) show that the quantile-based cut points partition the F distribution into ordered tiers, enabling consistent three-tier and five-tier grading for management use. The count summaries (Figs. 8c, d) indicate that the five-tier scheme mainly increases resolution within the middle band (Average/Fair/Good), whereas the three-tier aggregation produces a dominant “Medium” class with relatively fewer units at the extremes. This pattern supports using the three-tier output for external communication and cross-sector coordination, and the five-tier output for internal scheduling and differentiated intervention thresholds, without changing the overall priority ordering implied by the TOPSIS–RSR synthesis.

Fig. 8: The composite score F is operationally tiered under Method B (α = 0.5).

a Histogram of F with five-tier quantile thresholds (10%, 30%, 70%, 90%; dashed lines) used to derive Poor–Fair–Average–Good–Excellent grades. b Histogram of F with three-tier quantile thresholds (25%, 75%; dashed lines) used to derive Low–Medium–High grades. c Grade counts under the five-tier scheme. d Grade counts under the three-tier scheme. The distribution shows that five-tier grading primarily refines differentiation within the middle band, while three-tier aggregation yields a naturally dominant “Medium” class suitable for communication and coordination.

Reclassification of ACCH

Using the TOPSIS thresholds from RSR–Probit anchored grading (Method A) and the Beta-on-F scheme (Method B, F = 0.5C+0.5RSR) as a parametric comparator, the agreement is 43.11% with Cohen’s κ = 0.291 (five tiers) and 61.01% with κ = 0.414 (three tiers). With five tiers, exact matches occur in 43.11% of cases and Cohen’s κ = 0.291 indicates fair agreement beyond chance. When tiers are collapsed to three, matches rise to 61.01% with κ = 0.414 (moderate) because fewer boundaries reduce boundary-crossing. Practically, the disagreement concentrates near the middle cut points; extremes are more stable (Table 7). This pattern says the methods share the same directional signal but capture complementary structure.

Table 7 Confusion matrix and Cohen’s kappa coefficient calculation table based on methods A and B

Then, the confusion matrix above is reclassified according to Fig. 3. For five-tier, we define them as Excellent’, Good’, Average’, Fair’, Poor’; and for three-tier are High’, Medium’, Low’. The distribution of ACCH levels is shown in Fig. 9 below.

Fig. 9: Spatial grading patterns from two routes and their AB-combined reclassification.

a, b Three-tier results for Method A (RSR–Probit anchoring) and Method B (Beta-on-F); c, d five-tier results for Method A (RSR–Probit anchoring) and Method B (Beta-on-F); e, f AB-combined maps following the reclassification matrix. Agreement between A and B: 61.01% (κ = 0.414) for three tiers; 43.11% (κ = 0.291) for five tiers.

Under the three-tier scheme, both approaches reveal a consistent macro-pattern (Fig. 9a, b): high grades cluster in the Luoyang urban core and along the NE–SE development corridor; peripheral counties are dominated by medium–low grades, and mountainous areas to the SW/S (Nanyang–Funiu Mountains) are largely low. Discrepancies concentrate near the “High/Medium” and “Medium/Low” boundaries at the urban fringe.

To make this corridor pattern operationally interpretable, the NE–SE high-grade cluster is anchored by emblematic ACCH nodes including Longmen Grottoes, the Han–Wei Luoyang City Site, and the Erlitou Site (together with other capital-era city-wall/ruin nodes). Their concentration along the metropolitan expansion axis creates a typical urban-rural interface governance problem: protective buffers, rapid land-use conversion, and visitor-support facilities increasingly overlap near the urban fringe, which elevates coordination costs across jurisdictions and requires differentiated control intensity along the corridor. In contrast, peripheral counties are dominated by medium-low grades because heritage nodes are more dispersed and corridor continuity is weaker, shifting planning emphasis from “corridor consolidation” to selective node protection and local link strengthening.

With five tiers, the Method A produces a sharper “Excellent/Good” ring over the core and a clearer “Average/Fair” belt at the urban–rural edge (Fig. 9c). The Method B is more conservative in the same areas (larger “Average”, smaller “Good/Excellent”), reflecting distribution-adaptive quantiles given the empirical shape of F∈ (0,1) (Fig. 9d). The observed agreement declines to 43.11% with κ = 0.291(fair), driven by crossings among the mid-tiers (“Good/Average/Fair”), while extremes (“Excellent/Poor”) remain relatively robust. This aligns with the methodological expectation that rank-anchored and value-distribution thresholds emphasize different structures.

Using the reclassification matrix, A×B cross-labels are mapped to three- and five-tier A/B outputs. The combined scheme preserves extreme consensus (e.g., A=1/B=1 and A=5/B=5 become Excellent’/Poor’) and damps mid-tier oscillation by assigning ambiguous cross-pairs toward the conservative/intersection side. The resulting footprint is jointly interpretable by both methods: a stable Excellent’/Good’ core and a surrounding Average’/Fair’/Poor’ gradient, suitable as a compromise product for communication and internal management (Fig. 9e, f).

Three tiers are recommended for external communication and first-order prioritization, whereas five tiers are valuable for fine-grained internal management (boundary cases and near-optimal areas). Given the agreement levels (61.01% vs 43.11%) and κ values, we suggest using three tiers outwardly and complementing any five-tier use with a boundary-sensitivity report that flags A–B discordant points for targeted verification.

Compared with a single-rule grading scheme, the dual-route design improves planning usability by producing three concrete decision outputs. First, it yields a stable high-priority set where both routes agree, which can be directly used to delineate “immediate-action” zones and key nodes/corridors for protection and funding allocation. Second, it explicitly identifies boundary cases where the two routes disagree; these units are treated as “verification-required” candidates, enabling targeted field checks, stakeholder consultation, or data refinement instead of uniformly upgrading large areas. Third, the combination of three-tier and five-tier results supports multi-level governance: a coarse three-tier map for communication and statutory reporting, and a finer five-tier ranking for internal scheduling, monitoring intensity, and phased interventions. In this way, the added methodological structure reduces misclassification risk at decision-critical extremes while providing actionable nuance for mid-tier management.

Graded Heritage Networks

After obtaining the reclassified three/five tiers, we translate discrete grades into a continuous field of movement potential. Following Fig. 2, this subsection builds the composite resistance surface for MCR from B2 (protection regional pattern) and B3 (protection conditions)

The composite resistance surface (Fig. 10i) exhibits a clear gradient between basins, valleys, mountainous and water-barrier zones. Plains and valleys form continuous low-resistance belts due to higher land-use compatibility and lower slope/elevation penalties, whereas piedmont and river–water mosaics create high-resistance patches and narrow bottlenecks that suppress connectivity. This physical structure governs where MCR yields trunks, branches, and costly bridging segments.

Fig. 10: The reclassified heritage networks of the five-tier and three-tier based on the comprehensive resistance surface.

a-e The reclassified heritage networks of the five-tier; f-h the reclassified heritage networks of the three-tier; i the comprehensive resistance surface.

Then based on the comprehensive resistance surface, the reclassified heritage networks of the five-tier and three-tier are drawn respectively (Fig. 10).

For five-tier, the network shows sharp hierarchy (Fig. 10a~e). Excellent’ and Good’ form a low-cost, straight trunk–subtrunk backbone across the metropolitan core and the NE–SE corridor, aligning with low-resistance belts.

At the macro scale, the trunk follows the low-resistance passages of the Luo–Yi basin, whose valley/plain orientations are structurally constrained and have persisted through the Han–Tang period, providing a physical basis for long-lived movement corridors. In historical terms, Luoyang’s eastward connectivity was strengthened by imperial communication infrastructures, including the Grand Canal, which was conceived as a unified empire-wide communication system in the Sui dynasty and functioned as a long-term transport backbone. In addition, historical-transport studies in the Central Plains repeatedly highlight capital-centred post-road structures and persistent road belts around the Luoyang basin; this provides an independent historical context for the corridor-like alignment captured by MCR.

Average’ and Fair’ provide second-order parallel routes that stitch suburban and county clusters yet respond to local resistance with alternatives and splits. Poor’ spreads over high-resistance zones and connects via costly bottlenecks or narrow co-linear corridors, delineating weak links. Overall, the five-tier output supports engineering alternatives and fine-grained management, while its mid-tier boundaries are more sensitive.

The three-tier network is compact and coherent (Fig. 10f–h). High’ builds a continuous backbone along the core basin and valley corridors with higher local redundancy—ideal for primary conservation and presentation corridors. Medium’ supplies secondary feeders and stitches between core, suburbs and counties, often as valley-aligned long arcs. Low’ forms a coarse coverage mesh at the periphery with many single-chain links, where bottlenecks and crossings should be prioritized. This scheme facilitates external communication and prioritization, keeps trunks consistent with the five-tier map, and exhibits smaller boundary jitter.

Both schemes agree on trunk locations, whereas differences arise across mid-tiers (Good’, Average’, Fair’ vs. Medium’). The five-tier scheme produces more alternative/parallel branches in response to local resistance, while the three-tier scheme tends to merge them into single corridors—exactly mirroring the previously reported agreement gap (61.01% vs. 43.11%). We therefore recommend three tiers for external use, and five tiers for internal design or verification with an explicit boundary-sensitivity report.

To translate the tiered corridor outputs into operational planning actions, we summarize a tier-specific management strategy matrix (Table 8). The intent is to move from “network geometry” to “planning instruments”: high tiers identify corridors where strict controls and impact assessment are necessary to prevent fragmentation; mid tiers indicate areas suitable for adaptive, heritage-sensitive use and corridor enhancement; and low tiers highlight high-resistance segments where low-cost safeguarding, monitoring, or strategic de-prioritization is more appropriate. In China’s territorial spatial planning, these tiered strategies can be implemented by overlaying the corridor grades with the Three Zones and Three Lines (ecological, agricultural, urban spaces; ecological protection redline, permanent basic farmland, and urban development boundary) to assign differentiated control intensity and permissible interventions.

Table 8 Tiered management actions and planning controls

The tier-aware corridors translate graded assets into an intervention hierarchy. Excellent’/Good’ (or High’) units define candidate core conservation–presentation corridors and anchor points for corridor continuity; Medium/Average/Fair indicate feeder links where modest land-use or access adjustments can disproportionately improve connectivity; Poor/Low highlight peripheral areas where selective safeguards may be preferable to uniform upgrading. Bottleneck locations provide a shortlist for conflict-resolution and micro-alignment actions (e.g., river crossings, valley mouths, and inter-county edges).

The MCR-derived corridor trunks are consistent with external planning narratives and route-oriented governance in Luoyang. The Luoyang Territorial Spatial Master Plan (2021–2035) explicitly emphasizes safeguarding national historical-cultural heritage corridors (e.g., the Silk Road system, the Grand Canal and historic route systems) as part of a multi-corridor heritage governance structure. Against this policy background, the primary trunks identified in our three/five-tier networks form an interpretable backbone that preferentially links the main heritage clusters and internationally/nationally recognized nodes associated with Luoyang’s role in the Silk Road and the Luo River “five-capital” belt, rather than creating fragmented or counter-intuitive detours.

From a use-intensity perspective, officially promoted visitor-oriented itineraries in Luoyang are anchored by flagship heritage destinations (e.g., temple, grotto and museum nodes) that are repeatedly highlighted in cultural-tourism guidance; these destinations are predominantly situated on or near the low-resistance belts that organize the modeled trunk corridors. Bottlenecks concentrate at valley–terrace transitions, river-crossing constraints and inter-county edge zones, which aligns with the expectation that corridor continuity is most sensitive where terrain/water barriers and land-use discontinuities impose sharp resistance changes. Taken together, these external comparisons provide a plausibility check that the reconstructed corridors/bottlenecks reflect policy-relevant and practice-recognizable spatial structure, while we acknowledge that richer validation using mobility traces or structured expert elicitation would further strengthen external validity.

To align with the planning-support aim stated in the Abstract and Conclusion, we make two direct, plan-ready uses of the tiered network when revising the Luoyang Historical and Cultural City Conservation Plan. (1) Buffer calibration by tier and resistance context: higher-tier corridors warrant stronger protection intensity, while buffer extent can be adjusted using the resistance surface (larger buffers where land-use change is rapid and resistance is low; more targeted controls where resistance is high). (2) Corridor “no-break” safeguards: along primary trunks and at bottlenecks, we recommend a safeguard overlay where major expansion and linear infrastructure should undergo stricter review, because small interruptions can disproportionately reduce network continuity. These actions operationalize the “priority–corridor–threshold” logic and are consistent with the planning implications summarized in the Discussion.

MGWR spatial heterogeneity

Using the MGWR specification (adaptive bi-square kernels; sample size n = 1190), we first summarize the variable-specific bandwidths (Table 9): neighbor counts k, the relative scale k/n and the normalized scale index z_k, and then evaluate model performance and spatial patterns. Large k (high z_k) indicates a near-global, region-scale effect; small k (low z_k) indicates a strongly local, neighborhood-scale effect.

Table 9 Bandwidth parameters of MGWR predictors

We use the small-sample corrected Akaike Information Criterion (AICc) to select bandwidths and compare models; lower AICc indicates better information efficiency. R² measures in-sample explanatory power, while AdjR² adjusts for model size and is therefore a more conservative goodness-of-fit metric. With these scales fixed by AICc-based selection, we report out-of-sample errors, residual structure, and coefficient surfaces.

Ultimately, the MGWR achieved AICc = −3497.14, R² = 0.3007, and AdjR² = 0.2594, indicating that the model explains a modest share of variance in the ACCH loss-risk index. Therefore, MGWR is used here as an associational, spatially explicit diagnostic to characterize non-stationary relationships and multi-scale patterns, rather than to infer causal effects.

Fit and Tier-wise Bias

To contextualize the MGWR goodness-of-fit, we first calibrate the empirical mapping between observed values and model outputs. The empirical relation between observations (actual values) and predictions (MGWR output) is y(observations)=a+ by’ (predictions), and after calculation: a=0.11208 (±0.08834), b=0.33529 (±0.49179). The parameter a captures the baseline offset (systematic shift of y at given y’); b captures scale calibration (b=1: perfect scale; b<1: range compression, pulling extremes toward the center).

Figure 9-(1) plots predictions versus observations (purple dashed line: ideal 1:1; red dashed line: empirical fit). This regression equivalently shows that, given the predicted value y’, the measured baseline shifts upward overall (positive intercept), while the amplitude is compressed (slope significantly less than 1), a phenomenon known as range compression. At the image level, this manifests as: point clouds in the low-to-median range are slightly above the 1:1 line (prediction slightly overestimated); point clouds in the high-value range are clearly below the 1:1 line (high values are more significantly underestimated). This indicates that the model has a correct monotonic relationship overall, but there is a systematic scaling bias at both ends (underestimation of high values and slight overestimation of low values), which belongs to the category of “predictable errors” that can be corrected through posterior calibration.

The Fig. 11b reports Residuals (Mean)=y-y’ by prediction quintiles (Low to Extreme): +0.00469, +0.00151, −0.00629, −0.01084, +0.01064. The pattern is mildly U-shaped: the lowest and the extreme groups show underestimation (positive residuals), whereas the mid-high groups show overestimation (negative residuals). This aligns with the scale discrepancy seen in Fig. 11a, revealing a clear directionality of bias. Note that these means are on the order of 10⁻², markedly smaller than the overall RMSE (≈0.0377), implying calibratable rather than structural errors.

Fig. 11: Predicted vs. observed (MGWR) and mean residuals by prediction quintiles.

a Scatter plot of predicted vs. observed values; b bar chart of Residuals (Mean) by prediction quintiles (Low to Extreme).

Coefficient Surfaces

With coefficient surfaces allowed to vary by location, MGWR reveals a coherent spatial grammar of effects. Location-varying MGWR β_j (u_i, v_i) coefficients reveal pronounced spatial heterogeneity in the distribution of ACCH (Fig. 12). First, multiple predictors form co-directed, contiguous high-value belts along the metropolitan core–eastern plain corridor, co-locating with dense heritage clusters—evidence of stronger marginal effects and path dependence within the corridor. Second, peripheral counties lack region-wide effects; significant coefficients occur as nodal patches aligned with transport and settlement chains, matching the sparse yet serial heritage clusters and indicating context-dependent mechanisms. Third, several predictors display sign reversals (negative to positive) along a SW–NE transition zone that crosses terrain–accessibility thresholds and coincides with shifts in heritage types, implying mechanism switching across geo-network contexts. Texture differences are scale-consistent: smaller bandwidths produce fine, locally sensitive patches, whereas larger bandwidths yield smooth regional belts—together delineating a multi-scale structure of “corridor continuity–peripheral nodes–transition reversals.”

Fig. 12: Spatial heterogeneity of MGWR coefficients (β surfaces) from C4 to C11.

a C4 regional economic pattern; b C5 land-use conditions; c C6 kernel density of heritage; d C7 population density; e C8 road-network distance; f C9 elevation resistance; g C10 slope resistance; h C11 waterbody resistance.

Group-wise Robustness (Region and Level)

To ensure transferability of findings, we evaluate group-wise robustness across regions and protection levels. Across regions and protection levels, most groups exhibit ΔRMSE≤0 (ΔRMSE is defined as group-wise RMSE minus the global RMSE.), indicating errors at or below the global baseline and thus good spatial transferability and categorical stability. A clear “core–transition–periphery” gradient emerges at the regional scale (Fig. 13a~c): a few districts within the urban corridor display both larger RMSE and greater residual dispersion, suggesting multi-factor interactions and scale mismatch in the core–transition belt; peripheral mountainous counties, by contrast, have markedly lower errors, consistent with simpler processes and more stable signals. For example, Chanhe District shows RMSE ≈ 0.0718 (RMSE>0) but near-zero median residual (≈ −0.0035), implying that its weakness is driven mainly by variance inflation rather than systematic bias and may be mitigated by localized bandwidth tuning or zonal modeling. Luanchuan and Ruyang, in contrast, achieve RMSE ≈ 0.0103–0.0108 (RMSE<0), representing clear advantages.

Fig. 13: Group-wise diagnostics by region and protection level.

a RMSE and ΔRMSE for regions with median/standard overlays; b RMSE and ΔRMSE for protection levels; c Region-wise mean residual vs. RMSE; d Protection level-wise mean residual vs. ΔRMSE. Negative ΔRMSE indicates better-than-global performance; residual r = y−y’. R1, Yanshi District; R2, Mengjin District; R3, Luolong District; R4, Jianxi District; R5, Chanhe District; R6, Laocheng District; R7, Xigong District; R8, Yichuan County; R9, Yiyang County; R10, Song County; R11, Xin’an County; R12, Luanchuan County; R13, Ruyang County; R14, Luoning County. L1, National priority protected site; L2, National archeological park of China; L3, Provincial-level cultural relics protection unit; L4, Municipal/county-level cultural relics protection units; L5, Traditional Chinese Villages; L6, Henan Province 2A-level tourist attractions; L7, Henan Province 3A-level tourist attractions; L8, Henan Province 4A-level tourist attractions; L9, Provincial-level historical and cultural town; L10, Provincial-level historical and cultural village; L11, Unclassified immovable cultural relics.

At the protection-level dimension (L1–L11; Figs 13b, d), most labels also have RMSE close to the global baseline or lower. Several well-represented statutory labels show relatively stable performance, supporting the use of the graded results for the main heritage population (e.g., L4 county and city-level protected sites, L3 provincial protected sites, and L11 unclassified immovable cultural relics). In contrast, a few mixed or small-sample labels show inflated RMSE and dispersion (e.g., “4A scenic area” and “national archeological park”), which is expected when a label contains very few cases or bundles heterogeneous site types and management contexts. These categories should therefore be interpreted with explicit uncertainty notes, and where practical, combined with related labels or assessed with additional expert review. Importantly, very small groups may appear “good” simply because they contain too few observations; such results should not be over-interpreted as strong robustness.

In summary, joint examination of group mean residuals and RMSE places most groups near the origin, indicating that overall bias is small and weakly directional; a simple post-hoc linear calibration further reduces the mean bias and yields modest yet consistent improvements across groups. In summary, the spatial heterogeneity of Luoyang is reflected in three practical contrasts: (1) the core/transition belt is more complex, while the periphery is more predictable; (2) common, well-represented labels are more stable, whereas small-sample or mixed labels are less stable; (3) the remaining bias is generally small and correctable, while higher variability concentrates in a limited set of regions and labels. For operational use, we recommend keeping a cautious “decision buffer” for high-error areas and attaching uncertainty notes to small-sample categories.

Diagnostics and Residual Structure

To guarantee statistical and spatial reliability, we implement two complementary checks. (1) BH–FDR multiple-testing control adjusts pixelwise coefficient p-values (q = 0.05) to produce significance masks (sig-BH) and shares (sig-share); all coefficient mapping and summary statistics are restricted to sig-BH = TRUE pixels, thereby suppressing chance findings. (2) Moran’s I (KNN) on residuals evaluates remaining spatial dependence: global Moran’s I is computed for KNN weights with k = 6, 8, 12, and its significance is assessed by permutation-based two-sided p-values and Moran scatterplots (standardized residuals on the x-axis, spatial lag on the y-axis). This combination allows us to distinguish genuine non-stationarity in the β-surfaces from artefacts of spatially autocorrelated errors.

Using BH–FDR correction (q = 0.05), the MGWR coefficient surfaces exhibit a highly uneven pattern of statistical significance (Table 10 and Fig. 14a). Among the eight predictors, Land-use conditions (C5) is significant at all 1190 locations (sig-share = 1.000), indicating a spatially stable and strong marginal effect and indicating that C5 is the most spatially stable and consistently significant predictor (after BH–FDR) associated with the ACCH loss-risk index in Luoyang. Kernel density of heritage (C6) is significant at about 41% of the locations, forming belt-shaped clusters along the metropolitan–eastern plain corridor; heritage clustering therefore amplifies the ACCH loss-risk within this corridor but becomes much weaker in peripheral areas. Elevation resistance (C9, sig-share ≈ 0.195) and Waterbody resistance (C11, sig-share ≈ 0.142) are significant only in specific nodal and edge zones, acting as local modifiers of the loss gradient along inter-county boundaries and valley–tableland transitions. In contrast, Regional economic pattern C4, Population density (C7), Road-network distance (C8) and Slope resistance (C10) all have sig-share = 0. Once land use, heritage density and terrain–waterbody resistance are controlled for, their local coefficients no longer pass the FDR threshold, implying that their independent contributions are either absorbed by stronger predictors or spatially unstable.

Fig. 14: Robustness checks for MGWR coefficient significance and residual spatial.

a Share of locations with significant local coefficients after BH–FDR correction (bars) and corresponding significant counts (red dotted line) for each explanatory variable; b Global Moran’s I of MGWR residuals (left axis) and permutation-based two-sided p-values (right axis) under different KNN neighborhood sizes k; c Moran Scatterplot of Standardized Residuals (KNN = 8).

Table 10 Significance of MGWR coefficient surfaces

Residual spatial dependence was further assessed using KNN-based Moran’s I. The global Moran’s I values for the MGWR residuals are −0.0164, −0.0136 and −0.0074 for k = 6, 8, 12, respectively—very close to zero and slightly negative—suggesting no meaningful global spatial clustering or checkerboard pattern in the residuals. The permutation-based two-sided p-values of Moran’s I for k = 6, 8, 12 are 0.255, 0.280, and 0.500 (Fig. 14b), respectively, all far above the 0.05 significance level. Hence, the slightly negative Moran’s I values are statistically non-significant, and the null hypothesis of spatial randomness cannot be rejected. The MGWR residuals can therefore be regarded as approximately spatially independent at the global scale. The corresponding Moran scatterplot (Fig. 14c), with standardized residuals z-resid from LISA-local (local indicators of spatial association) on the x-axis and the spatial-lag (k = 8) on the y-axis, shows an almost circular cloud of points around the origin; the four quadrants (high–high, low–low, high–low, low–high) are roughly balanced, and the fitted regression line—whose slope equals the global Moran’s I—has only a very slight negative inclination.

Overall, the MGWR results indicate a consistent pattern of spatial associations behind the spatial variation of the ACCH loss-risk index in Luoyang. Land-use conditions (C5) show the most spatially stable association: coefficients are significant everywhere and vary smoothly in space, aligning with a baseline gradient from highly transformed urban–industrial corridors to relatively stable rural–mountainous sectors. Heritage kernel density (C6) and elevation and waterbody resistance (C9, C11) exhibit spatially selective associations. C6 is more strongly associated with higher loss-risk along the metropolitan–eastern plain corridor, whereas C9 and C11 are associated with localized variations in valley mouths, inter-county edges and river–terrace transitions. By contrast, regional economic pattern, population density, road-network distance and slope (C4, C7, C8, C10) do not retain significance after FDR correction, suggesting that their associations are not robust once land use and heritage aggregation are included in this specification. Taken together, these results support an interpretable, spatially heterogeneous association structure, while the MGWR framework should be interpreted as descriptive and diagnostic rather than causal.

Discussion

The dual-path grading and MGWR diagnosis jointly reveal a corridor-centred yet multi-scale heterogeneous structure of ACCH in Luoyang. At the grading stage, Method A (RSR–Probit anchoring) and Method B (Beta-on-F) provide two complementary lenses on the same evaluation space. Their agreement is moderate for three tiers (61.01%, κ = 0.414) and fair for five tiers (43.11%, κ = 0.291), with discrepancies concentrated in mid-tiers. This pattern indicates that extremes (High/Low; Excellent/Poor) are robust to the choice of grading route, whereas mid-tier boundaries are sensitive to whether thresholds are anchored to rank-based Probit quantiles or to the empirical distribution of a composite [0,1] score. The confusion-matrix-based reclassification consolidates these insights: it preserves consensus at the extremes while assigning ambiguous cross-labels toward the conservative/intersection side, producing a compromise product that is suitable for communication and internal management.

The graded MCR networks translate discrete ACCH classes into a continuous field of movement potential, making the corridor structure of Luoyang’s heritage system visible. Under the five-tier scheme, Excellent’ and Good’ assets organize into a low-resistance backbone linking the metropolitan core to the eastern plain, while Average’ and Fair’ form parallel and branching routes that stitch suburban and county-level clusters around this backbone. Poor’ sites occupy high-resistance zones and connect through narrow bottlenecks, indicating vulnerable links in the network. The three-tier scheme condenses this structure into a compact High’–Medium’–Low’ system: High’ defines primary heritage corridors suitable for flagship conservation and presentation, Medium’ provides regional feeders and cross-county stitching, and Low’ highlights peripheral areas where weak connectivity and sparse heritage call for selective intervention rather than uniform upgrading. The strong agreement in trunk locations between the two schemes suggests that the graded network is structurally stable, even though fine-grained mid-tier boundaries remain flexible.

Implications for heritage conservation planning are threefold. First, consensus extremes (High/Low; Excellent/Poor) can be translated into defensible priority tiers for zoning and investment, while discordant mid-tier units form a manageable verification list for iterative plan updates. Second, the reclassified MCR network provides a corridor hierarchy: trunk corridors guide the primary conservation–presentation system, feeder branches support cross-county stitching, and bottlenecks pinpoint locations where targeted mitigation is more efficient than corridor-wide upgrading. Third, the MGWR diagnosis helps distinguish where city-wide regulatory levers (land-use compatibility) are most relevant versus where localized corridor calibration (terrain/water constraints and clustered heritage) is needed. Together, these outputs operationalize differentiated intervention thresholds (e.g., buffer strictness and monitoring intensity) within territorial spatial planning.

The MGWR results are best understood by linking the key drivers to concrete on-the-ground settings. Land-use conditions (C5) mainly distinguishes whether a site sits in built-up land (higher development pressure), is next to water, or lies in cropland. Using the 30 m land-cover dataset, we summarized the land-cover context at ACCH locations by heritage type. For archeological/city-ruin sites (including major capital-city remains), about 35.3% of sites fall in built-up surroundings, 62.9% are water-adjacent, and only ~1.8% lie in mainly cropland surroundings; tombs show a similar profile (35.4% built-up; 61.1% water-adjacent; 3.5% cropland). This helps explain why C5 appears as the most spatially stable MGWR factor: it captures a broad urban-rural contrast in land-use pressure that applies across the city. Importantly, this contrast becomes very concrete at key corridor anchors—for example, the Yanshi Shang City Site is classified in a built-up context, consistent with the need to control construction land expansion and traffic/service intrusion around major archeological sites, while the Sui-Tang Luoyang City Site is classified as water-adjacent, highlighting the practical importance of managing waterfront redevelopment and visitor-support facilities near core capital remains.

Waterbody resistance (C11), by contrast, should be interpreted as a “gate” effect rather than a city-wide constraint: it matters most where the corridor must cross major rivers. In the corridor structure, the linkage between Longmen Grottoes and the urban core necessarily involves crossing the Yi River, so the river-crossing reach functions as a bottleneck where small changes in access infrastructure and mobility demand can disproportionately affect corridor continuity. From a planning perspective, these crossing reaches can be treated as priority corridor-control segments (e.g., strict impact assessment for bridge or embankment works, traffic calming, and visual and landscape controls). Likewise, where the corridor concentrates at crossings around the Luoyi Ancient City reach, existing bottlenecks can be designated as managed corridor connectors—not because they “are” heritage themselves, but because they are critical links that keep the heritage corridor continuous.

The robustness checks support the interpretability of this driver hierarchy. Global Moran’s I of MGWR residuals is very close to zero and slightly negative across KNN neighborhood sizes (k = 6, 8, 12), with permutation-based two-sided p-values well above conventional thresholds. Moran scatterplots show an almost circular distribution of standardized residuals around the origin, with a nearly horizontal regression line. These diagnostics indicate that residuals are approximately spatially random and that the pronounced spatial heterogeneity observed in the β-surfaces reflects genuine non-stationary effects of C5, C6, C9 and C11 instead of unmodelled spatial autocorrelation. At the same time, predicted–observed comparisons reveal a correct monotonic relation but some range compression, with modest underestimation of high-risk cases and slight overestimation of low-risk cases.

Substantively, these associational findings suggest practical entry points for graded heritage management in Luoyang. The spatially stable association of land-use conditions (C5) with the loss-risk index indicates that higher-risk areas tend to coincide with more intensively transformed land-use contexts; corridor-level strategies may therefore consider land-use compatibility zoning as a pragmatic lever. At the same time, MGWR does not establish causality, and these implications should be treated as planning hypotheses to be tested with additional evidence. Corridor-level strategies should therefore prioritize land-use compatibility zoning along the metropolitan–plain backbone, where dense heritage (high C6) and favorable access combine to generate both high risk and high potential for public presentation. In peripheral and transition zones shaped by terrain and waterbody resistance (C9, C11), protection should focus on safeguarding key valley mouths, river crossings and inter-county thresholds that function as network gates. Background variables (C4, C7, C8, C10) remain important as contextual information for policy design, but the MGWR evidence suggests that they can be addressed indirectly through land-use and corridor interventions rather than as standalone levers. Overall, these patterns suggest a two-level strategy—city-wide land-use compatibility control complemented by locally calibrated corridor alignment and buffer thresholds in terrain- and water-constrained transition zones.

Despite these strengths, several limitations should be acknowledged. First, the analysis is cross-sectional and constrained by the current ACCH inventory and contemporary socio-spatial data; it does not explicitly capture diachronic shifts in capital functions or long-term land-use change. Second, the MGWR specification is linear and additive, and factor interactions are only indirectly inferred through scale and sign patterns; explicit modeling of interactions could refine our understanding of compound drivers in the core–transition belt. Third, the construction of resistance surfaces and the assignment of indicator weights, although grounded in AHP–CRITIC fusion and literature, still entail normative choices that may vary under alternative expert panels or planning scenarios. Fourth, some protection-level categories and regional groups have small sample sizes, which can inflate uncertainty in group-wise diagnostics despite the use of post-hoc calibration. Future work could address these limitations by incorporating temporal datasets (e.g., historical land-use reconstructions), exploring nonlinear and interaction-rich local models, and embedding the present framework into participatory planning workflows that iteratively update weights, thresholds and corridor scenarios.

The findings of this study indicate that Luoyang’s ancient-capital cultural heritage (ACCH) is neither randomly scattered nor uniformly protected, but organized into a strongly differentiated graded network. High-grade heritage tends to align with a limited number of structurally important corridors, while lower-grade and isolated sites are more often located in fragmented transition zones and peripheral areas. In relation to our initial aims, this means that grading succeeds not only in ranking individual sites, but also in revealing where the current protection pattern reinforces or undermines the intended hierarchy of ACCH in space.

Because ACCH spans diverse protection levels and heritage types (L1–L11), the graded network should be interpreted in a type-sensitive way. World Cultural Heritage and national key protected sites (L1) should function as non-substitutable anchor nodes: their priority remains high regardless of resistance, while resistance and corridor hierarchy calibrate buffer strictness and corridor safeguards around them. Traditional Chinese Villages (L5) are living heritage systems, so a low grade should trigger land-use compatibility management and community resilience measures, complemented by soft connectivity rather than universal hard corridors. Finally, for small-sample or policy-mixed labels, grade-based recommendations should carry explicit uncertainty notes and targeted verification to avoid overconfident prescriptions near mid-tier boundaries.

At the level of underlying mechanisms, the analysis shows that a small subset of factors consistently shapes this graded pattern, while others play a more contextual or background role. The evidence that land-use conditions provide a city-wide risk backbone, and that a limited group of heritage and physical variables modulate this backbone in specific corridors and transition belts, is central to our second objective: to interpret spatial heterogeneity in terms that can be translated into differentiated protection thresholds, zoning rules and corridor strategies. In other words, the graded network is consistent with a pattern in which a small subset of predictors shows more stable associations with the loss-risk index, while others play more contextual roles in this specification.

For Luoyang, the immediate task is to test how far the graded network and driver hierarchy can be incorporated into real planning decisions, for example by linking heritage grades to zoning categories, control indices and corridor-based implementation projects. More broadly, the approach invites comparative work in other historic capitals and heritage-rich regions, the integration of temporal and behavioral data, and closer dialogue with practitioners and communities. Only through such iterations can graded, mechanism-aware analyses of ACCH move from technical exercises to a stable basis for long-term territorial spatial governance.