  Associative network models featuring multi-tasking properties have been
introduced recently and studied in the low load regime, where the number $P$ of
simultaneously retrievable patterns scales with the number $N$ of nodes as
$P\sim \log N$. In addition to their relevance in artificial intelligence,
these models are increasingly important in immunology, where stored patterns
represent strategies to fight pathogens and nodes represent lymphocyte clones.
They allow us to understand the crucial ability of the immune system to respond
simultaneously to multiple distinct antigen invasions. Here we develop further
the statistical mechanical analysis of such systems, by studying the medium
load regime, $P \sim N^{\delta}$ with $\delta \in (0,1]$. We derive three main
results. First, we reveal the nontrivial architecture of these networks: they
exhibit a high degree of modularity and clustering, which is linked to their
retrieval abilities. Second, by solving the model we demonstrate for $\delta<1$
the existence of large regions in the phase diagram where the network can
retrieve all stored patterns simultaneously. Finally, in the high load regime
$\delta=1$ we find that the system behaves as a spin glass, suggesting that
finite-connectivity frameworks are required to achieve effective retrieval.
