  Consider a graph $G=(V,E)$ without isolated edges and with maximum degree
$\Delta$. Given a colouring $c:E\to\{1,2,\ldots,k\}$, the weighted degree of a
vertex $v\in V$ is the sum of its incident colours, i.e., $\sum_{e\ni v}c(e)$.
For any integer $r\geq 2$, the least $k$ admitting the existence of such $c$
attributing distinct weighted degrees to any two different vertices at distance
at most $r$ in $G$ is called the $r$-distant irregularity strength of $G$ and
denoted by $s_r(G)$. This graph invariant provides a natural link between the
well known 1--2--3 Conjecture and irregularity strength of graphs. In this
paper we apply the probabilistic method in order to prove an upper bound
$s_r(G)\leq (4+o(1))\Delta^{r-1}$ for graphs with minimum degree $\delta\geq
\ln^8\Delta$, improving thus far best upper bound $s_r(G)\leq 6\Delta^{r-1}$.
