Network Diffusion Models — SIR, SIS, and Independent Cascade

Use SIR when recovery confers lasting immunity — as in many infectious diseases or one-time technology adoptions — so that a node cannot return to susceptible after recovering. Use SIS when the condition is recurring or immunity is temporary, such as seasonal illness or rumour re-exposure, where nodes cycle between susceptible and infected indefinitely. Choose Independent Cascade or Linear Threshold when the focus is on social influence and a node can become active at most once; these models are better suited to marketing, information spread, or behaviour adoption problems where the goal is to maximise the final number of activated nodes from a chosen seed set.

There is no universal rule, but in practice 100–1000 repetitions are commonly used in the literature, with more repetitions needed as network size increases or as outcome variance is high. Report the number of repetitions and the variance of the key metrics (peak infection, final activation fraction) so readers can assess stability.

If you have time-stamped case data and a known network, τ can be estimated by fitting the compartmental ODE system to observed curves using least-squares or maximum-likelihood methods, or by Bayesian inference. For social influence settings, edge-level activation probabilities are sometimes set proportional to observed interaction frequencies. When no data are available, τ is typically varied in a sensitivity analysis across a plausible range informed by the literature.

Yes. Edge weights can encode heterogeneous transmission probabilities — a strongly tied pair has a higher τ than a weakly tied pair. Directed edges model asymmetric influence, where node A can infect B but B cannot directly infect A. Both extensions are straightforward to implement in full network simulation, though mean-field approximations become less accurate and often need to be replaced by degree-based heterogeneous mean-field or pair-approximation methods.