I am reading the famous paper "Consensus and Cooperation in Networked Multi-Agent Systems" by Olfati Saber, please see "http://ieeexplore.ieee.org/document/4118472/".
I know the following discretization rule: $$\dot{x} = \frac{x(t+dt)-x(t)}{\epsilon}, \ \ \ \ \epsilon>0$$ where $\epsilon$ can be thought as a step size.
So given the continuous network system: $$\dot{x} = -\mathcal{L}(\mathcal{G})x,$$ after plugging in, we obtain the following $$x(t+dt)=x(t)+\epsilon[-\mathcal{L}(\mathcal{G})x(t)]$$ or the equivalent form: $$x_i(t+dt) = x_i(t)+\epsilon\sum_{j\in \mathcal{N}_i} (x_j(t)-x_i(t)).$$
Note: $\mathcal{L}(\mathcal{G})$ is the Laplacian matrix of the undirected network $\mathcal{G}$.
My question is how to derive the final discrete dynamic network:
$$x(k+1) = Px(k)$$ where $P = I-\epsilon \mathcal{L}(\mathcal{G})$, i.e, equation (16) in the paper presented above. In particular, how to define $k$, $k+1$?
Since you don't get to have different step sizes in the numerator and denominator, for $\varepsilon > 0$, $$\dot{x} = \frac{x(t+\varepsilon)-x(t)}{\varepsilon} \text{.} $$ (That is, you are comparing the heights of the graph of $x(t)$ at two times separated by $\varepsilon$, to get the slope of the line between the two points on the graph at those two times.)
Then you get \begin{align*} x(t+\varepsilon) &= x(t)+\epsilon[-\mathcal{L}(\mathcal{G})x(t)] \\ &= Ix(t) -\epsilon\mathcal{L}(\mathcal{G})x(t) \\ &= (I -\epsilon\mathcal{L}(\mathcal{G}))x(t) \\ &= P\,x(t) \text{.} \end{align*} Specializing to $t \mapsto k$, $\varepsilon \mapsto 1$, $x(k+1) = P \, x(k)$. Note that this specialization makes $P = I - \mathcal{L}(\mathcal{G})$.
If you choose to randomly throw symbols around as if they don't mean anything, you can start with $\dot{x} = \frac{x(t+\mathrm{d}t)-x(t)}{\varepsilon}$ and perform the same manipulations with the specializations $t \mapsto k$, $\mathrm{d}t \mapsto 1$, and get $x(k+1) = P \, x(k)$ with $P = I -\epsilon\mathcal{L}(\mathcal{G})$.