  We examine the connectivity fluctuations across networks obtained when the
horizontal visibility (HV) algorithm is used on trajectories generated by
nonlinear circle maps at the quasiperiodic transition to chaos. The resultant
HV graph is highly anomalous as the degrees fluctuate at all scales with
amplitude that increases with the size of the network. We determine families of
Pesin-like identities between entropy growth rates and generalized
graph-theoretical Lyapunov exponents. An irrational winding number with pure
periodic continued fraction characterizes each family. We illustrate our
results for the so-called golden, silver and bronze numbers.
