Edge-Port Conservation in Networks

An operator
F[e, i]
defined on
n
-port networks is said to satisfy the condition of edge-port conservation if it satisfies
\Sigma_{all the \\ ports} F[e_{m}, i_{m}] = \Sigma_{all the \\ edges} F[\upsilon_{k},j_{k}]
where
e_{m}, i_{m}
are the voltage and current functions at the
m
th port and
\upsilon_{k},j_{k}
are the voltage and current functions at the
k
th edge, respectively. It is shown that an operator satisfies the condition if and only if the operator is additive.