Finding errors in weighted graphs
One approach is to compute the distance $d(u,v)$ between each pair $u,v$ of vertices using an all-pairs shortest paths algorithm, such as Floyd-Warshall.  Then, if $d(v,v) &lt; 0$, you know that there exists a cycle through $v$ that doesn't add to zero.  You can explicitly find one such cycle by augmenting the shortest-paths algorithm with predecessor links to keep track of an example shortest path from $v$ to $v$.  Then you know that at least one of the edges along that cycle has an error.

Unfortunately there doesn't appear to be any way to uniquely determine which edges are in error.  For instance, consider a graph with edge $a \to b$ of weight $1.01$ and edge $b \to c$ of weight $-1.48$ and edge $c \to a$ of weight $0.50$; you know that at least one of those edges has an error, but there's no way to know which one; it could even be all of them that have an error.