  The transmission of a vertex $v$ of a graph $G$ is the sum of distances from
$v$ to all the other vertices in $G$. The Wiener complexity of $G$ is the
number of different transmissions of its vertices. Similarly, the eccentric
complexity of $G$ is defined as the number of different eccentricities of its
vertices. In this paper these two complexities are compared. The complexities
are first studied on Cartesian product graphs. Transmission indivisible graphs
and arithmetic transmission graphs are introduced to demonstrate sharpness of
upper and lower bounds on the Wiener complexity, respectively. It is shown that
for almost all graphs the Wiener complexity is not smaller than the eccentric
complexity. This property is proved for trees, the equality holding precisely
for center-regular trees. Several families of graphs in which the complexities
are equal are constructed. Using the Cartesian product, it is proved that the
eccentric complexity can be arbitrarily larger than the Wiener complexity.
Additional infinite families of graphs with this property are constructed by
amalgamating universally diametrical graphs with center-regular trees.
