I came across the undefined terminology nondecreasing function in an appendix about "useful remarks on probability" in a textbook, "Social and Economic Networks" by Matthew Jackson. Precisely, it is for functions from ${\mathbb R}^n$ into $\mathbb R$ and the subsection was about "domination of distributions".
Here is an example of statement using nondecreasing functions:
Consider two probability distribution $\mu$ and $\nu$ on ${\mathbb R}^n$. We say that $\mu$ dominates $\nu$ if $$E_\mu[f] \geqslant E_\nu[f]$$ for every mondecreasing function $f:{\mathbb R}^n\to{\mathbb R}$.
(with - I guess - $E_\mu[f]=\sum_{d\in{\mathbb R}^n}f(d)\mu(d)$)
What is the usual definition of nondecreasing function $f:{\mathbb R}^n\to{\mathbb R}$ in the context of probability/random graphs?
Economists tend to be really sloppy with imposing an order on $\mathbb R^n$ (and other product spaces) without being explicit about what it is they mean exactly. Apologies for that.
I suspect Jackson will have made clear what they mean for a function of several variables to be non-decreasing early in the text (probably chapters $1$ or $2$. If I remember correctly, chapter $1$ is all examples).
In any case, my best guess for what this means (and this is relatively common -- see, for instance, Mas-Colell, et al, which is the standard graduate reference in microeconomic theory) is this:
Let $x,y\in\mathbb R^n$. Say that $x \geq y$ if $x_i \geq y_i$ for all $i=1,\ldots,n$.
$f:\mathbb R^n \to \mathbb R$ is nondecreasing if $x\geq y$ implies that $f(x) \geq f(y)$.
Other order notation that may come up:
$x>y$ if $x_i \geq y_i$, for all $i=1,\ldots,n$, and $x\neq y$.
$x\gg y$ if $x_i > y_i$ for all $i = 1,\ldots,n$.