Table 3 Transformation of an equation into another by adding or multiplying by constants (Group IDs correspond to clusters in Fig. 2)

1

\( {D}_{Chord}=\sqrt{2\left(1-\frac{a}{\sqrt{\left(a+b\right)\left(a+c\right)}}\right)} \) (Eq 30)

\( =\frac{1}{\sqrt{2}}2\sqrt{\left(1-\frac{a}{\sqrt{\left(a+b\right)\left(a+c\right)}}\right)}=\frac{1}{\sqrt{2}}{D}_{Hellinger} \) (Eq.29)

2

\( {D}_{Mean- Manhattan}=\frac{b+c}{a+b+c+d} \) (Eq.20)

\( =\frac{1}{M}\left(b+c\right)=\frac{1}{M}{D}_{Hamming} \) (Eq.15)

\( {D}_{Vari}=\frac{b+c}{4\left(a+b+c+d\right)} \) (Eq.23)

\( =\frac{1}{4M}\left(b+c\right)=\frac{1}{4M}{D}_{Hamming} \) (Eq.15)

3

\( {S}_{Russell\&Rao}=\frac{a}{a+b+c+d} \) (Eq.14)

\( =\frac{1}{M}a=\frac{1}{M}{S}_{Intersection} \) (Eq.12)

4

\( {S}_{Baroni- Urbani\& Buser-2}=\frac{\sqrt{ad}+a-\left(b+c\right)}{\sqrt{ad}+a+b+c} \) (Eq.72)

\( =2\frac{\sqrt{ad}+a}{\sqrt{ad}+a+b+c}-1=\left[2 \times {S}_{Baroni- Urbani\& Buser-1}\right] \) ^.(Eq.71)

5

\( {S}_{Kulczynski-2}=\frac{\frac{a}{2}\left(2a+b+c\right)}{\left(a+b\right)\left(a+c\right)} \) (Eq.41)

\( =\frac{1}{2}\left(\frac{a}{a+b}+\frac{a}{a+c}\right)=\frac{1}{2}{S}_{Johnson} \) (Eq.43)

\( {S}_{Driver\& Kroeber}=\frac{a}{2}\left(\frac{1}{a+b}+\frac{1}{a+c}\right) \) (Eq.42)

\( =\frac{1}{2}\left(\frac{a}{a+b}+\frac{a}{a+c}\right)=\frac{1}{2}{S}_{Johnson} \) (Eq.43)

\( {S}_{Johnson}=\frac{a}{a+b}+\frac{a}{a+c} \) (Eq.43)

\( =1+\left(\frac{a^2-bc}{\left(a+b\right)\left(a+c\right)}\right)=1+{S}_{McConnaughey} \) (Eq.39)

6

\( {S}_{Dice-1/ Czekanowski}=\frac{2a}{2a+b+c} \) (Eq.3)

\( =2\frac{a}{2a+b+c}=2 \times {S}_{Dice-2} \) (Eq.2)

7

S _Innerproduct  = a + d (Eq.13)

\( =M\frac{a+d}{a+b+c+d}=M \times {S}_{Sokal\& Michener} \) (Eq.7)

\( {S}_{Hamann}=\frac{\left(a+d\right)-\left(b+c\right)}{a+b+c+d} \) (Eq.67)

\( =2\left(\frac{a+d}{a+b+c+d}\right)-1=\left[2 \times {S}_{Sokal\& Michener}\right]-1 \) (Eq.7)