(i.e., solutions with initially bounded support keep this feature for all $t>0$). In the setting of graphs this property strikingly fails, as shown in [@Hua:Mugnolo:2015]. As a technical but perhaps worthwile side remark, we note that all the main ingredients in the proof of finite speed of propagation (see [@Andreucci:Tedeev:1999], [@Andreucci:Tedeev:2000]) seem to be available in graphs too: embeddings as in [@Ostrovskii:2005], Caccioppoli inequalities as in Lemma \[l:cacc\] below, and of course iterative techniques as the one displayed in the proof of Proposition \[p:linf\_meas\]. The key exception in this regard is the fact that full localization via an infinite sequence of nested shrinking balls is clearly prohibited by the discrete metric at hand. This is a point of marked difference with the continuous setting.

Still we can prove sharpness of our $\ell^{1}$–$\ell^{\infty}$ bound by means of the following result of confinement of mass. By the same argument we can estimate also a suitable moment of the solution, which is also a new result for nonlinear diffusion in graphs, see Section \[s:bbl\].

\[p:bbl\] Let ${u}_{0}\ge 0$ be finitely supported. Then for every $1>{\varepsilon}>0$ there exists a $\varGamma>0$ such that $$\label{eq:bbl_n}
    {{\lVert{u}(t)\rVert}_{\ell^{1}(B_{R})}}
    \ge
    (1-{\varepsilon})
    {{\lVert{u}_{0}\rVert}_{\ell^{1}(V)}}
    \,,
    \qquad
    t>0
    \,,$$ provided $B_{{\lfloorR/2\rfloor}}$ contains 