text: **Conclusions**
  Our approach, Twigstats, transfers the power advantage of haplotype- 
  based approaches to a fully temporal framework, which is applica- 
  ble to*f*-statistics and enables previously unavailable unbiased and 
  time-stratified analyses of admixture. We demonstrated that Twigstats 
  enables fine-scale quantitative modelling of ancestry proportions, 
  revealing wide-ranging ancestry changes that affect northern and 
  central Europe during the Iron, Roman and Viking ages. We reveal evi- 
  dence of the southward and/or eastward expansion of individuals who 
  probably spoke Germanic languages and who had Scandinavian-related 
  ancestry in the first half of the first millennium ce. We note that 
  ‘Scandinavian-related’ in this context relates to the ancient genomes 
  available, and so it is entirely possible that these processes were driven, 
  for example, from regions in northern-central Europe. This could be 
  consistent with the attraction of the greater wealth, which tended to 
  build up among Rome’s immediate neighbours and may have played 
  a major role in vectors of migration internal to communities in Europe 
  who lived beyond the Roman frontier52. Later, patterns of gene flow 
  seem to have turned northwards, with the spread of Iron Age Central 
  Europe-related ancestry into Scandinavia. Overall, our approach can 
  be used for the reconstruction of new high-resolution genetic histories 
  around the world. 