*4.3. Average Path Length*

The average path length is an idea in network topology that can be defined as the average number of steps along with the shortest paths for all possible pairs of vertices of a network. The average path length in terms of the Laplacian eigenvalues is defined as:"

$$D\_t = \frac{N\_t}{N\_t - 1} \sum\_{k=2}^{N\_t} \frac{1}{\nu\_k} \quad (i, j) \neq (0, 0)$$

The average path length for the categorical path-path product network:

$$D\_{l} = \ \ \ \frac{nt}{nt-1} \sum\_{i=0}^{t-1} \sum\_{j=0}^{n-1} \frac{1}{\left( (2-2\cos\frac{j\pi}{n})d\_{i+1} + d\_{j+1}(2-2\cos\frac{i\pi}{t}) - (2-2\cos\frac{j\pi}{n})(2-2\cos\frac{i\pi}{t}) \right)}$$

The average path length for the categorical cycle-path product network:

$$D\_{l} = \ \frac{nt}{nt-1} \sum\_{i=0}^{t-1} \sum\_{j=0}^{n-1} \frac{1}{\left( (2 - 2\cos\frac{2j\pi}{n})d\_{i+1} + d\_{j+1}(2 - 2\cos\frac{i\pi}{l}) - (2 - 2\cos\frac{2j\pi}{n})(2 - 2\cos\frac{i\pi}{l}) \right)}$$

The average path length for the categorical cycle-cycle product network:

$$\begin{array}{rcl} D\_t &=& \frac{n(t+2)}{n(t+2)-1} \\ & \times \quad \sum\_{i=0}^{t+1} \sum\_{j=0}^{n-1} \frac{1}{\left( (2-2\cos\frac{2j\pi}{n})d\_{i+1} + d\_{j+1}(2-2\cos\frac{2i\pi}{1+2}) - (2-2\cos\frac{2j\pi}{n})(2-2\cos\frac{2i\pi}{1+2}) \right)} \end{array}$$
