Where \(V_{i}^{t-1}\ \)is the set of premises that are active in the period \(t-1\), and \({\ V}_{i}^{t}\) is the set of premises that moved at least one animal at time \(t\). For both analysis (loyalty and node-loyalty), we considered a span time \(t\) of one year.
The yearly \({\ \varphi\ }_{i}^{t-1,\ t}\)and\({\ \theta}_{i}^{t-1,\ t}\)are expressed in a value between zero and one, where a \(\theta_{i}^{t-1,\ t}\) of one indicates that 100% of links with same origin and destination were preserved (100% loyalty between premises); and a value of one in\({\ \varphi\ }_{i}^{t-1,\ t}\) means that all premises were active in two consecutive periods, but not is necessarily trading with the same commercial partner.