  Many complex biological, social, and economical networks show topologies
drastically differing from random graphs. But, what is a complex network, i.e.\
how can one quantify the complexity of a graph? Here the Offdiagonal Complexity
(OdC), a new, and computationally cheap, measure of complexity is defined,
based on the node-node link cross-distribution, whose nondiagonal elements
characterize the graph structure beyond link distribution, cluster coefficient
and average path length. The OdC apporach is applied to the {\sl Helicobacter
pylori} protein interaction network and randomly rewired surrogates thereof. In
addition, OdC is used to characterize the spatial complexity of cell
aggregates. We investigate the earliest embryo development states of
Caenorhabditis elegans. The development states of the premorphogenetic phase
are represented by symmetric binary-valued cell connection matrices with
dimension growing from 4 to 385. These matrices can be interpreted as adjacency
matrix of an undirected graph, or network. The OdC approach allows to describe
quantitatively the complexity of the cell aggregate geometry.
