  In this paper we study connections between structured storage or Lyapunov
functions of a class of interconnected systems (dynamical networks) and
dissipativity properties of the individual systems. We prove that if a
dynamical network, composed as a set of linear time invariant (LTI) systems
interconnected over an acyclic graph, admits an additive quadratic Lyapunov
function, then the individual systems in the network are dissipative with
respect to a (nonempty) set of interconnection neutral supply functions. Each
supply function from this set is defined on a single interconnection link in
the network. Specific characterizations of neutral supply functions are
presented which imply robustness of network stability/dissiptivity to removal
of interconnection links.
