  Networks exhibiting "accelerating" growth have total link numbers growing
faster than linearly with network size and can exhibit transitions from
stationary to nonstationary statistics and from random to scale-free to regular
statistics at particular critical network sizes. However, if for any reason the
network cannot tolerate such gross structural changes then accelerating
networks are constrained to have sizes below some critical value. This is of
interest as the regulatory gene networks of single celled prokaryotes are
characterized by an accelerating quadratic growth and are size constrained to
be less than about 10,000 genes encoded in DNA sequence of less than about 10
megabases. This paper presents a probabilistic accelerating network model for
prokaryotic gene regulation which closely matches observed statistics by
employing two classes of network nodes (regulatory and non-regulatory) and
directed links whose inbound heads are exponentially distributed over all nodes
and whose outbound tails are preferentially attached to regulatory nodes and
described by a scale free distribution. This model explains the observed
quadratic growth in regulator number with gene number and predicts an upper
prokaryote size limit closely approximating the observed value.
