Network Measures
Non-Network Measures
Convex Hull
Calculate the total area of the region delimited by the outermost nodes.
This can be calculated by triangulating and computing the sum of the area of all the triangles in the network.
Given three side lengths $a,b$ and $c$ the area of a triangle can be calculated by:
\[A_\Delta = \sqrt{\frac p2\left(\frac p2-a\right)\left(\frac p2-b\right)\left(\frac p2-c\right)}\]Therefore the area of the convex hull is the sum of all these triangles:
\[A_{\text{CH}}=\sum_\Delta A_\Delta\]Distance
We can use distance measures (using either Manhattan distance or euclidean distance) to measure on a grid:
-
Mean Distance:
\[\bar\ell=\frac1{n^2}\sum_v\sum_u\text{dist}(u,v)\] -
Standard Deviation of the Distance:
\[\sigma_\text{dist}=\sqrt{\frac1{n^2-1}\sum_v\sum_u(\text{dist}(u,v)-\bar\ell)^2}\] -
Relative Standard Deviation:
\[\frac{\sigma_\text{dist}}{\bar\ell}\]
Network Measures
Occupancy
- Larger occupancy (more area taken up) = more numerous.
- Smaller occupancy (less area taken up) = less numerous.
Setting Up a Graph Geometrically
Given a set of points in an array, you can convert this array into a network by creating links between nodes based on a measure. Generally we can say, create a link between node $u$ and $v$ if:
\[\text{dist}(u,v)\leq s\]