Fundamental limits to identifying node and tie memory in temporal networks: marginal artefacts and spreading dynamics

Temporal-network models place the memory seen in contact data either on nodes, through self-excitation (branching ratio n_node), or on ties, through reinforcement (kappa). The two choices reproduce the same coarse statistics but predict different epidemics. We ask whether contact data can tell them apart and find two limits. The first is attribution. When the initiator of every event is recorded, the likelihood factorises into a timing block containing n_node and a selection block containing kappa, so the two are estimated independently. When it is not, as in proximity data, a missing-information term couples them, with a size set by the entropy of the initiator posterior. The second issue is interpretation. Even where the mechanisms separate, the fitted n_node might not accurately measure memory and it may be set by the distribution of waiting times between events. Shuffling each node's inter-event order, which keeps that distribution, typically changes the estimated n_node by less than 10%; this is a bias that longer records and a flexible scale-free kernel do not remove. Whether node memory is present must therefore be decided by model-free statistics, which find only weak order memory in the proximity records, far less than the fitted n_node around 0.9 would imply. Tie reinforcement remains recoverable, although a latent initiator inflates it by 28%-45%. The two limits lead to two recommendations: record who initiates each contact, and test a fitted node memory against shuffled and model-free benchmarks before interpreting it. On a high-school record, simulating from the fitted node memory under-predicts outbreak sizes by up to a factor of 2, whereas keeping tie reinforcement alone is closer at low transmissibility but overshoots at high.