to make $t$ larger in order to see paths with a “reasonable” weight. Even more remarkable is the impact of heterogeneous degrees on the cutoff width $\omega_\star$, which satisfies $\omega_\star>>\log\log N$ against $\omega_\star=\Theta(1)$ in the regular case. Finally, the gaussian limit shape $\Phi$ itself is specific to the non-regular case and is directly related to the fluctuations of degrees along a typical trajectory of the <span style="font-variant:small-caps;">nbrw</span>.

A classical result by @janson2009probability asserts that the graph produced by the configuration model is simple (no loops or multiple edges) with probability asymptotically bounded away from $0$, as long as $$\label{eq:simple-graph}
\sum_{v\in V}\deg(v)^2={\ensuremath{\mathcal{O}}}(N)\, .$$ Moreover, conditionally on being simple, it is uniformly distributed over all simple graphs with degrees $(\deg(v))_{v\in V}$. Thus, every property which holds [**whp**]{} under the configuration model also holds [**whp**]{} for the uniform simple graph model. In particular, under (\[eq:simple-graph\]), the conclusion of Theorem \[thm:main\] extends to simple graphs.

A common setting consists in generating an infinite <span style="font-variant:small-caps;">iid</span> degree sequence $(\deg(v))_{v\in{\ensuremath{\mathbb{N}}}}$ from some fixed degree distribution $Q$ and then restricting it to the index set $V=\{1,\ldots,n\}$ for each $n\geq 1$. Let $D$ denote a random integer with distribution $Q$. Assuming that $$\begin{array}{ccc}
{\ensuremath{\mathbb{P}}}\left(D\leq 2\right)=0, & 
\mathrm{Var}(D)>0, \textrm{ 