Joint work with Dániel Keliger · Preprint
Mean-field equations replace a high-dimensional stochastic system by a deterministic description of its average behaviour. On the complete graph, Kurtz’s classical theory for density-dependent Markov chains gives approximation errors of order $N^{-1/2}$. On a random network the randomness of the graph itself contributes a second, distinct source of error.
We study the N-Intertwined Mean-Field Approximation (NIMFA) for a broad class of Markov processes — including SIS- and SIR-type epidemic dynamics — on inhomogeneous random graphs of stochastic-block-model type, in the regime where the expected average degree $d$ grows polynomially in $N$.
The error depends on the initial condition
Network density alone does not determine the accuracy. For generic initial conditions the worst-case error is of order
$$ O\!\left(d^{-1/2}\right), $$and this rate is attained: for the catalyst process on an Erdős–Rényi graph with suitably chosen random initial conditions the error is exactly of this order, so the bound cannot be improved in general.
When the initial condition is fairly homogeneous — quantitatively, when the within-block variance of the initial state is $O(1/d)$ — averaging suppresses the leading graph-induced fluctuations and the error improves to
$$ O\!\left(\frac{1}{d}+\frac{1}{\sqrt{N}}\right). $$Interpretation
One way to read the results: in the homogeneous regime the error splits into a finite-population term of order $N^{-1/2}$, matching the classical complete-graph rate, and a network term of order $1/d$ coming from the randomness of the graph. When the initial condition is heterogeneous across nodes, this second effect dominates and degrades the overall rate to $d^{-1/2}$.
Two systems with the same graph distribution and the same dynamics can therefore exhibit different mean-field accuracy purely because they start from differently organised states. A localised or graph-correlated initial infection retains network randomness at first order, whereas a sufficiently homogeneous one averages much of it away.
Paper
Pierfrancesco Dionigi and Dániel Keliger, The effects of initial conditions on the accuracy of mean-field approximations of Markov processes on large random graphs, arXiv:2506.12872.