The similarity between neural and immune networks has been known for decades,
but so far we did not understand the mechanism that allows the immune system,
unlike associative neural networks, to recall and execute a large number of
memorized defense strategies {\em in parallel}. The explanation turns out to
lie in the network topology. Neurons interact typically with a large number of
other neurons, whereas interactions among lymphocytes in immune networks are
very specific, and described by graphs with finite connectivity. In this paper
we use replica techniques to solve a statistical mechanical immune network
model with `coordinator branches' (T-cells) and `effector branches' (B-cells),
and show how the finite connectivity enables the system to manage an extensive
number of immune clones simultaneously, even above the percolation threshold.
The system exhibits only weak ergodicity breaking, so that both multiple
antigen defense and homeostasis can be accomplished.Comment: Editor's Choice 201