  We propose that the coronal magnetic field, linking concentrations on the
photosphere through an interwoven web of flux, embodies a scale-free network.
It arises from a self-organized critical dynamics including flux emergence, the
diffusion and merging of magnetic concentrations, as well as avalanches of
reconnecting flux tubes. Magnetic concentrations such as fragments, pores and
sunspots, are `nodes' joined by flux tubes or `links'. The number of links
emanating from a node is scale-free. We reanalyze the quiet-Sun data of Close
et al and show that the distribution of magnetic concentration strengths is a
power law with an index $\gamma= 1.7 \pm 0.3$, over the entire range of the
measurement, about $(2-500)\times 10^{17}$ Mx. This distribution is compatible
with that for the sizes of active regions reported by Harvey and Schwaan. Thus
magnetic concentrations may be scale-free from the smallest measurable
fragments to the large active regions. Numerical simulations of a
self-organized critical model give the same index $\gamma$, within statistical
uncertainty. The exponential distribution of flux tube lengths also agrees
quantitatively with results from the model. Calibration with the measured
diffusion constant of magnetic concentrations allows us to calculate a flux
turnover time in the model to be of order 10 hours and the total solar flux to
be of order $10^{23}$Mx, agreeing with observations. We introduce two other
statistical quantities to characterize scale-free networks. The probability
distribution for the amount of flux connecting a pair of concentrations, and
the number of distinct concentrations linked to a given one are predicted to be
scale-free, with different indices. Our approach unifies the observation of
scale free flare energies with the coronal magnetic field structure.
