  Pattern-diluted associative networks were introduced recently as models for
the immune system, with nodes representing T-lymphocytes and stored patterns
representing signalling protocols between T- and B-lymphocytes. It was shown
earlier that in the regime of extreme pattern dilution, a system with $N_T$
T-lymphocytes can manage a number $N_B!=!\order(N_T^\delta)$ of B-lymphocytes
simultaneously, with $\delta!<!1$. Here we study this model in the extensive
load regime $N_B!=!\alpha N_T$, with also a high degree of pattern dilution, in
agreement with immunological findings. We use graph theory and statistical
mechanical analysis based on replica methods to show that in the
finite-connectivity regime, where each T-lymphocyte interacts with a finite
number of B-lymphocytes as $N_T\to\infty$, the T-lymphocytes can coordinate
effective immune responses to an extensive number of distinct antigen invasions
in parallel. As $\alpha$ increases, the system eventually undergoes a second
order transition to a phase with clonal cross-talk interference, where the
system's performance degrades gracefully. Mathematically, the model is
equivalent to a spin system on a finitely connected graph with many short
loops, so one would expect the available analytical methods, which all assume
locally tree-like graphs, to fail. Yet it turns out to be solvable. Our results
are supported by numerical simulations.
