Table 1.

Permutation-based regression results showing the effects of density ($D$) and centralization ($C_D$) on productivity

Collaboration type	Model	$D$	$D^{\mathit{\text{2}}}$	$C_D$	$D$:$C_D$	$D^2$:$C_D$	Adjusted $R^2$
Local to local	1.1	0.047		−0.024			−0.109
	1.2	−0.017	0.068	−0.026			−0.177
	1.3	0.078		−0.040	−0.045		−0.176
	1.4	−0.090	0.258	−0.086	−0.159		−0.243
	1.5	1.994	−5.095	0.796	−2.536	6.282	−0.198
Local to regional	2.1	−0.045		−0.073+			0.101*
	2.2	−0.432**	0.405*	−0.095*			0.188**
	2.3	−0.010*		−0.084*	0.058*		0.184*
	2.4	−0.313	0.249	−0.092*	0.033		0.184*
	2.5	−0.300	0.233	−0.096*	−0.011	0.040	0.165*
Regional to regional	3.1	−0.031*		−0.106***			0.757***
	3.2	0.316	−0.351	−0.102***			0.766***
	3.3	−0.284*		−0.104***	−0.035*		0.774***
	3.4	0.032	−0.061	−0.104***	−0.031		0.769***
	3.5	0.205	−0.272	−0.074***	−0.823*	0.803*	0.790***
All combined	4.1	−0.048*					0.116*
	4.2	0.625***	−0.676***				0.300***

Local to local adjusted R² values can be interpreted as R² = 0; they are negative because of compensating for multiple variables in models with low explanatory power. All coefficients are standardized to help interpret interaction terms (Materials and Methods). The local level outlier D = 1 was removed from analyses (details are in ^¶). C_D is excluded when analyzing combined edges because it is undefined at the aggregate level (Materials and Methods). Therefore, models 4.1 and 4.2 only consider D and D²; models 4.3–4.5 are ignored because interactions are not possible. Colons represent interactions between variables. Significance values are ⁺<0.1, *<0.05, **<0.01, ***<0.001.