]{}

[*Case (a):*]{} $C_{j^*}^{-} \geq 1/2.$\
We focus on those $w_{ij}$ with $i\in S^-$ and $j=j^*$. Indeed, since all $w_{ij}$ are nonnegative, we have $$\label{wsubset} \sum_{i \in S^{-}, ~j \in S^{+}}
w_{ij} \geq \sum_{i \in S^{-}} w_{ij^*}.$$ For each element in the sum on the right-hand [side]{}, we have $$w_{ij^*} = \sum_{k=1}^n a_{ki}\ a_{kj^*}
\geq a_{j^*i}\ a_{j^* j^*} \geq a_{j^*i}\ \eta,$$ where the inequalities follow from the facts that $A$ has nonnegative entries, its diagonal entries are positive, and its positive entries are at least $\eta$. Consequently, $$\sum_{i \in S^{-}} w_{ij^*}
\geq \eta\ \sum_{i \in S^{-}} a_{j^*i}
= \eta\, C_{j^*}^{-}. \label{wlowerbound}$$ Combining Eqs. (\[wsubset\]) and (\[wlowerbound\]), and recalling [the assumption]{} $C_{j^*}^{-} \geq 1/2$, the [result]{} follows. [An]{} illustration of [this argument]{} can be found in Figure \[cdef\][(b)]{}.

[*Case (b):*]{} $C_{j^*}^{+} \geq 1/2.$\
We focus on those $w_{ij}$ with $i=i^*$ and $j \in S^{+}$. We have $$\label{wsubset2}
\sum_{i \in S_{-}, ~j \in S^{+}} w_{ij} \geq \sum_{j \in
S^{+}} w_{i^* j},$$ since all $w_{ij}$ are nonnegative. For each element in the sum on the right-hand side, we have $$w_{i^* j} = \sum_{k=1}^n a_{ki^*}\ a_{k j}
\geq a_{j^* i^*}\ a_{j^* j} \geq \eta\, a_{j^* j},$$ where the inequalities follow [because all entries of 