$\{1,\infty\}$ and unit edge capacities. In Definition \[d:pq\] below, this is generalized as follows. Every node $u$ has capacity $q_u$ (the resistance of $u$ to failures), and an amount $p_u$ of flow-supply that $u$ can deliver to any other node (including itself) if $u$ is chosen to be in the source set $S$.

\[d:pq\] Let $G=(V,E)$ be a graph with node-capacities $\{q_u:u \in V\}$. For $S \subseteq V$ and $v \in V$ the $(S,v)$-$q$-connectivity $\lambda_G^q(S,v)$ is the maximum flow value from $S \setminus \{v\}$ to $v$ (every edge has capacity $1$). Equivalently, $\lambda_G^q(S,v)$ is the minimum capacity of a cut $C \subseteq E \cup V \setminus \{v\}$ such that $G \setminus C$ has no $(S,v)$-path. Given [*flow-supplies*]{} $\{p_u:u \in S\}$, the $(S,v)$-$(p,q)$-connectivity $\lambda_G^{p,q}(S,v)$ is the maximum $(s,v)$-flow value $\lambda_{G'}^q(s,v)$ in the network $G'$ obtained by adding to $G$ a new node $s$ and connecting it to every $u \in S$ with $p_u$ edges; hence if $p_u \geq q_u$ for every $u \in S$ then: $\lambda_G^{p,q}(S,v)=p_v+\lambda_G^q(S,v)$ if $v \in S$, and $\lambda_G^{p,q}(S,v)=\lambda_G^q(S,v)$ otherwise.

Note that $\lambda_G(S,v)=\lambda_G^{p,q}(S,v)$ for $p_u= \infty$ and $q_u=\infty$ for every $u \in V$, and that $\lambda_G^Q(s,v)=\lambda_G^q(s,v)$ for $q_u=1$ if $u \in Q \setminus \{s,v\}$ and $q_u=\infty$ otherwise. Now 