Table 1 Factors of the built environment, Individual indicators, dimensional Indicators and Synthetic Index of the latent vitality of the urban environment

Urban density

The density of built-up areas

Ground Surface Index (GSI)

GSI = Apb_i /Ac_i

Apb_i = Area of the building projection In the i-th cell;

Ac_i = Area of the i-th cell

Concentration of Population

Population Density (DPOP),

\(DPOP = \frac{{\mathop \sum \nolimits_{j = 1}^{n} \left[ {\left( {\frac{{Nab_{j} }}{{Act_{i} }}} \right)*Aint_{i} } \right]}}{{Ac_{i} }}\)

Nab_j = Number of residents in the j-th census tract;

Act_j = Area of the j-th census tract;

Aint_i = Area of the intersection of the j-th census tract and the i-th cell;

Ac_i = Area of the i-th cell

Index of Urban Density (I_UD)

I_UD = (GSI * w1) + (DPOP * w2)

W₁ = 0.5

W₂ = 0.5

Accessibility

Closeness centrality

Average Normalized angular integration in the 400 m radius (NAIN)

____________

NAIN = log (AI I + 2)

\(AI = \frac{1}{n}\mathop \sum \nolimits_{i = 1}^{n} d\theta \left( {\pi X, i} \right)\)

n = number of segments

dθ = angle between any two segments on the shortest path on segment x (van Nes and Yamu 2021)

Betweenness centrality

Average Normalized angular Choice in the 400 m radius (NACH)

______________

NACH = log (ACH I + 2)

\(ACH = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \mathop \sum \nolimits_{j = 1}^{n} \sigma \left( {i,x,j} \right)}}{{\left( {n - 1} \right)\left( {n - 2} \right)}}\)

n = number of segments

σ (i,x,j) = 1 if the shortest path from i to j comprises segment x and 0 otherwise (Al-Sayed et al. 2014)

Transit nodes spacing

Density of transit nodes (TRN_D),

TRN_D = Ntn_i

Ntn_i = Number of transit nodes in the i-th cell

Road intersection spacing

Road Intersection Density (RDI)

RDI = Nri_i

Nri_i = Number of road intersections in the i-th cell

Index of Accessibility (I_ACC)

I_ACC = (NAIN * w₁) + (NACH * w₂) + (RDI * w₃) + (TRN_D * w₄)

w₁ = 0.10

w₂ = 0.10

w₃ = 0.28

w₄ = 0.52

Availability of amenities

Intensity/quantity of uses

Density of points of interest (D_POI)

D_POI = Npoi_i = Number of points of interest in the i-th cell

Diversity of uses

Simpson’s Diversity index (DIV)

\(DIV = 1/\mathop \sum \nolimits_{i = 1}^{n} \left( {p_{ij} } \right)^{2}\)

p_ji = Proportion (n/N) of POIs of the i-th category (n) divided by the total number of Points of interest in the j-th cell(N)

Index of Availability of amenities (I_LU)

I_LU = (D_POI * w₁) + (DIV * w₂)

w₁ = 0.25

w₂ = 0.75

Environmental quality

Location and size of natural and semi-natural areas

Distance from Urban Green Areas (UGA_D)

UGA_D = D_ij

D_ij = Euclidean Distance of the i-th cell from the center of the nearest cell comprised in the j-th target (natural and semi-natural) area

Location of industrial facilities and large transport infrastructures

Distance from threat sources (THR_D)

THR_D = D_ij

D_ij = Euclidean Distance of the i-th cell from the center of the nearest cell comprised in the j-th target (threat source) area

Index of Environmental Quality (I_EQ)

I_LU = (UGA_D * w₁) + (THR_D * w₂)

w₁ = 0.50

w₂ = 0.50

Index of Latent Vitality (I_VIBE)

I_VIBE = (I_ACC * w₁) + (I_UD * w₂) + (I_LU * w₃) + (I_EQ * w₄)

w₁ = 0.22

w₂ = 0.22

w₃ = 0.50

w₄ = 0.06