*4.2. Global Mean-First Passage Time*

The global mean first-passage time (MFPT) is defined as the average of the first-passage time (FPT) between two nodes of a network. Mathematically, the global mean-first passage time (MFPT) for *G*(*t*) is:"

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

The global mean-first passage time for the categorical path-path product network:" Since *Et* = (2*t* − <sup>2</sup>)*n* − (2*t* − 2) and *Nt* = *nt*:

$$\begin{aligned} \epsilon < F\_N &> \quad &\quad \frac{2((2t-2)n - (2t-2))}{(nt-1)}\\ &\quad \times \quad \sum\_{i=0}^{t-1} \sum\_{j=0}^{n-1} \frac{1}{\left( (2 - 2\cos\frac{j\pi}{n})d\_{i+1} + d\_{j+1}(2 - 2\cos\frac{i\pi}{l}) - (2 - 2\cos\frac{j\pi}{n})(2 - 2\cos\frac{i\pi}{l}) \right)}\\ (i,j) &\quad \neq \quad (0,0) \end{aligned}$$

The global mean-first passage time for the categorical cycle-path product network:" Since *Et* = <sup>2</sup>*n*(*<sup>t</sup>* − 1) and *Nt* = *nt*:

$$\begin{array}{rcl} \chi < F\_N &=& \frac{2(2n(t-1))}{(nt-1)}\\ & \times & \sum\_{i=0}^{t-1} \sum\_{j=0}^{n-1} \frac{1}{\left( (2-2\cos\frac{2j\pi}{n})d\_{i+1} + d\_{j+1}(2-2\cos\frac{i\pi}{t}) - (2-2\cos\frac{2j\pi}{n})(2-2\cos\frac{i\pi}{t}) \right)} \end{array}$$

The global mean-first passage time for the categorical cycle-cycle product network: Since *Et* = <sup>2</sup>*n*(*<sup>t</sup>* + 2) − 2 and *Nt* = *n*(*<sup>t</sup>* + <sup>2</sup>):

$$\begin{array}{rcl} \chi < F\_N > & = & \frac{2(2n(t+2)-2)}{n(t+2)-1} \\ & \times & \sum\_{i=0}^{t+1} \sum\_{j=0}^{n-1} \frac{1}{\left( (2-2\cos\frac{2j\pi}{n})d\_{i+1} + d\_{j+1}(2-2\cos\frac{2j\pi}{1+2}) - (2-2\cos\frac{2j\pi}{n})(2-2\cos\frac{2i\pi}{1+2}) \right)} \end{array}$$
