  This paper studies critical nodes or "middlemen" that intermediate flows in a
directed network. The contestability of a node is introduced as a network
topological concept of competitiveness meaning that an intermediary's role in
the brokering of flows in the network can be substituted by a group of other
nodes. We establish the equivalence of uncontested intermediaries and
middlemen.
  The notion of node contestability gives rise to a measure that quantifies the
control exercised by a middleman in a network. We present a comparison of this
middleman centrality measures with relevant, established network centrality
measures. Furthermore, we provide concepts and measures expressing the
robustness of a middleman as the number of links or nodes that have to be added
to or removed from the network to nullify the middleman's power.
  We use these concepts to identify and measure middleman power and robustness
in two empirical networks: Krackhardt's advice network of managers in a medium
sized corporation and the well-known Florentine marriage network as a proxy of
power brokerage between houses in Renaissance Florence.
