Condensation and optimal targeting of repeated forcing in heterogeneous networks

Many networks receive repeated external forcing through a subset of entry nodes, for example infections imported from another population, information seeded by a campaign, or invasive species arriving through ports. We study how concentrating such forcing on important entry nodes, with seeding probability ${\pi }_i\propto w_i^{\gamma }$, changes the size of the resulting cascades in a stochastic SIR model on bipartite networks. The response to one seeding event is linear in the degree of the seeded node, so the relative gain from targeting does not depend on transmissibility. The targeting distribution is a Gibbs measure. For importance tails $p(w)\sim w^{-a}$ and responses growing as $w^{\eta }$, the targeted response diverges at ${\gamma }_1=a-1-\eta$ and the forcing condenses on a few nodes at ${\gamma }_2=a-1$. Under repeated forcing, the burden-maximizing exponent ${\gamma }^{\ast }$ is strictly positive, and in large populations depletion of nodes already reached bounds it by the condensation point, ${\gamma }^{\ast }\le {\gamma }_2$. The optimum decays as ${\gamma }^{\ast }\simeq K\mathrm{/}x$ for intense forcing, where $x$ is the expected number of seeding events per entry node, and approaches the condensation point as ${\gamma }_2-{\gamma }^{\ast }\simeq (a-1)\mathrm{/}\ln (1\mathrm{/}x)$ for sparse forcing. A cut-off of the importance distribution rounds the transitions in populations smaller than a crossover size $n_c$ and removes them in larger ones. Networks calibrated to survey data on sexual partnerships have between $0.08n_c$ and $0.8n_c$ entry nodes, which places them in the finite-size regime, where the asymptotic results hold only qualitatively.