Sensitivity of wardrop equilibria: revisited

For single-commodity networks, the increase of the price of anarchy is bounded by a factor of $$(1+\varepsilon )^p$$ from above, when the travel demand is increased by a factor of $$1+\varepsilon $$ and the latency functions are polynomials of degree at most p. We show that the same upper bound holds for multi-commodity networks and provide a lower bound as well.