  I analyse a model of an evolving network represented as a directed graph;
each node corresponds to one molecular species and the links to catalytic
interactions between species. Over short timescales the graph remains fixed
while relative populations of the molecular species change according to a set
of coupled differential equations. Over long timescales the system is subject
to periodic perturbations, each of which adds one new node to the graph, with
random links to other nodes, and removes one node with the least relative
population. Starting from a sparse random graph, a small autocatalytic set
(ACS) inevitably forms and then grows by accreting nodes until it spans the
entire graph. The resultant fully autocatalytic graph, whose probability of
forming by pure chance is very small, nevertheless forms in this model in an
average time that grows only logarithmically with the size of the system. ACSs
can also get destroyed, often accompanied by the sudden extinction of a large
number of species. I show that the largest of the extinction events in this
model are caused by one of three mechanisms, each of which produces a specific
discontinuous change in the graph's topology. The model is analytically
tractable: two theorems are proved which determine the set of nodes with least
relative population in the attractor, for any given graph. This in turn can be
used to analytically demonstrate the inevitability of the formation and growth
of ACSs and calculate the associated timescales. Finally, I show that the
formation and growth of ACSs is robust to the relaxation of many of the
idealizations made to enhance the analytical tractability of the model.
