Table 1 Correlation matrix of imperial density and the predictor variables tested in this paper (lower left: Pearson correlation coefficients, upper right: Spearman rank correlations).

	Total imperial density	Duration of agriculture (best)	Duration of agriculture (min)	Duration of agriculture (max)	Agricultural productivity	Distance from Steppe (max)	Distance from Steppe (min)	Distance from first empires	Elevation
Total imperial density		0.57	0.61	0.54	0.01	0.71	0.72	0.57	0.22
Duration of agriculture (best)	0.64		0.99	0.99	−0.08	0.43	0.43	0.54	0.13
Duration of agriculture (min)	0.68	0.99		0.98	−0.08	0.48	0.47	0.55	0.15
Duration of agriculture (max)	0.61	0.99	0.97		−0.06	0.39	0.39	0.52	0.12
Agricultural productivity	0.01	−0.07	−0.06	−0.06		−0.11	−0.07	−0.15	−0.12
Distance from steppe (max)	0.63	0.46	0.51	0.42	−0.09		0.96	0.72	0.14
Distance from steppe (min)	0.62	0.47	0.51	0.43	−0.07	0.97		0.64	0.13
Distance from first empires	0.59	0.51	0.52	0.48	−0.12	0.77	0.67		0.20
Elevation	0.21	0.16	0.18	0.14	−0.13	0.13	0.13	0.16

Different measures of duration of agriculture and distance for the Steppe are given to capture the uncertainty in these variables. Imperial density is generally greater closer to the steppe and with increasing duration of agriculture. There is not a strong relationship between imperial density and potential productivity. There is a strong relationship between imperial density and the control variable of distance from first empires, but the relationship with elevation (terrain ruggedness) is fairly weak and in the opposite direction to that predicted. There are also substantial-positive relationship between the measures of duration of agriculture, the measures of distance from the steppe, and the distance from first empires.

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