  Let ${\mathcal G}$ be an infinite family of connected graphs and let $k$ be a
positive integer. We say that $k$ is ${\it forcing}$ for ${\mathcal G}$ if for
all $G \in {\mathcal G}$ but finitely many, the following holds. Any
$\{-1,1\}$-weighing of the edges of $G$ for which all connected subgraphs on
$k$ edges are positively weighted implies that $G$ is positively weighted.
Otherwise, we say that it is ${\it weakly~forcing}$ for ${\mathcal G}$ if any
such weighing implies that the weight of $G$ is bounded from below by a
constant. Otherwise we say that $k$ ${\it collapses}$ for ${\mathcal G}$.
  We classify $k$ for some of the most prominent classes of graphs, such as all
connected graphs, all connected graphs with a given maximum degree and all
connected graphs with a given average degree.
