# Walkability Compass--A Space Syntax Solution for Comparative Studies
## 2 Method
### The Research Method
#### 2.2.2 Space Syntax Calculations Method (Steps 4-6)
Metrics mathematics and potential fields of use are described in [PERSON]'s works [60; 61]. Based on this, we decided to focus on three measures related to the walkability issue: Gravity index (Gr), Reach (Re), and Straightness (St). The space syntax metrics were completed with population density data retrieved from the worldpop.org database, recalculated to building centroids, and reported as PoP metrics. The PoP metric assumes that the presence of people fulfils the walkability [16; 17]. Four selected metrics (indicators) were used as case studies of cities' walkability explanatory indexes. Indexes were calculated at the method workflow step 6. The mathematical formulas are explained below. For practical reasons, indexes calculated on point type feature layer (building centroids) are interactively visualised using the clustering method implemented in WebMap (step 9). In order to analyse the spatial pattern drawn by walkability high attribute values, an aggregation procedure was proceeded using 4 ha hexagon tessellation (step 7). The study walkability explanatory index mathematical formulae are:  
The Gr index in the graph is calculated as the sum of walking destinations (e.g., shops, schools, cultural establishments) divided by the sum of distances to the destinations within 15 min distances. The centrality of gravity was introduced by [PERSON] (1959) and is calculated by the following, a little simplified for the presented research, [PERSON]'s [61] Formula (1):  
\[\text{G[i]r}=\Sigma\;\text{j}\in\text{G-[i]},\text{d[i,j]}\leq\text{r}\;\text{W[ j]}/\text{d[i,j]} \tag{1}\]  
where G[i] means gravity of the building i; \(\Sigma\)--sum of all buildings j, weights W, reachable on the graph G, within the radius r on the shortest distance d between i and j divided by the shortest distance from i to j. Weight in this calculation was not used, so it is equal to one in all cases, and the Gr index represents the number of reachable buildings divided by the shortest distances. The result reflects not the only number of destinations but the distances between them. For example, two buildings have three travel destinations within 15 min--in the case of simple destination counting, both buildings would have the same walkability, but, if the destinations would be located closer to the first building, then its gravity would be higher and better reflect a possibly shorter travel time. In the presented research, the weight of the travel destinations was not used. The higher gravity means it is allocated closer to a more significant number of travel destinations. The Re index is calculated as the number of buildings weighted by the perimeter of the facades reachable within 15 min. The index captures how many surrounding buildings each building reaches within a given search radius on the network [61] (Formula (2))  
\[\text{R[i]r}=\Sigma\;\text{j}\in\text{G-[i]},\text{d[i,j]}\leq\text{r}\;\text{W[ j]} \tag{2}\]  
where R[i] means the reach of the building i; \(\Sigma\)--sum all buildings j, weights W, reachable on the graph G, within the radius r on the shortest distance d between i and j. If no weight is assigned, then W is equal to 1 and Reach, in such a case, simply represents the number of reachable buildings. In the presented research, the building perimeter is used as a weight. The St index measures how the shortest path from a building to the other buildings within a 15 min radius resembles a straight line. It is calculated as ratios between linear Euclidean line distances and the shortest distances between the buildings within a 15 min radius. It is estimated by the following Formula (3) [61; 62]:  
\[\text{S[i]r}=\Sigma\;\text{j}\in\text{G-[i]},\text{d[i,j]}\leq\text{r}\;\text {\delta[i]},\text{d[i,j]}\;\text{W[j]} \tag{3}\]  
where S[i] means straightness of the building i; \(\Sigma\)--the sum of ratios between the straight-line distance \(\delta\) from i to j and the shortest path distance in the graph G multiplied by weights W of j. Building perimeter was used as a weight in the presented research.