Table 2 Comparison of proposed Closed-Form (CF) approach with dGHD algorithm. We compared the proposed Closed-Form approach with dGHD, Louvain, Infomap and Spinglass techniques w.r.t. various evaluation metrics for random geometric (RG) and power law (PL) networks

d=0.15 (RG)

CF

0.935 ± 0.051

0.849 ± 0.037

0.846 ± 0.102

0.969 ± 0.011

0.983 ± 0.004

0.828 ± 0.068

0.078

d=0.15 (RG)

dGHD

0.926 ± 0.018

0.793 ± 0.021

0.878 ± 0.036

0.965 ± 0.005

0.974 ± 0.003

0.813 ± 0.026

1.0

d=0.15 (RG)

Louvain

0.980 ± 0.016

0.767 ± 0.052

1.0 ± 0.0

0.965 ± 0.028

0.960 ± 0.031

0.841 ± 0.113

0.012

d=0.15 (RG)

Infomap

0.843 ± 0.012

0.262 ± 0.015

1.0 ± 0.0

0.718 ± 0.022

0.685 ± 0.024

0.304 ± 0.024

0.018

d=0.15 (RG)

Spinglass

0.832 ± 0.011

0.249 ± 0.012

1.0 ± 0.0

0.699 ± 0.018

0.665 ± 0.021

0.285 ± 0.020

0.85

d=0.15, d ^′=0.3

CF

0.927 ± 0.048

0.839 ± 0.031

0.862 ± 0.098

0.969 ± 0.008

0.982 ± 0.005

0.825 ± 0.054

0.081

d=0.15, d ^′=0.3

dGHD

0.922 ± 0.022

0.806 ± 0.027

0.868 ± 0.045

0.966 ± 0.006

0.977 ± 0.004

0.816 ± 0.032

1.0

d=0.15, d ^′=0.3

Louvain

0.978 ± 0.018

0.887 ± 0.137

0.974 ± 0.042

0.982 ± 0.018

0.982 ± 0.023

0.916 ± 0.083

0.013

d=0.15, d ^′=0.3

Infomap

0.849 ± 0.008

0.269 ± 0.009

1.0 ± 0.0

0.728 ± 0.015

0.698 ± 0.016

0.316 ± 0.016

0.020

d=0.15, d ^′=0.3

Spinglass

0.859 ± 0.009

0.284 ± 0.013

1.0 ± 0.0

0.747 ± 0.016

0.719 ± 0.017

0.339 ± 0.019

0.92

d=0.3 (RG)

CF

0.877 ± 0.067

0.714 ± 0.075

0.789 ± 0.135

0.947 ± 0.016

0.975 ± 0.011

0.716 ± 0.099

0.083

d=0.3 (RG)

dGHD

0.724 ± 0.029

0.645 ± 0.049

0.577 ± 0.059

0.921 ± 0.007

0.971 ± 0.006

0.504 ± 0.051

1.0

d=0.3 (RG)

Louvain

0.866 ± 0.019

0.406 ± 0.061

1.0 ± 0.0

0.850 ± 0.034

0.833 ± 0.038

0.505 ± 0.072

0.013

d=0.3 (RG)

Infomap

0.677 ± 0.011

0.147 ± 0.004

1.0 ± 0.0

0.419 ± 0.019

0.354 ± 0.022

0.100 ± 0.008

0.021

d=0.3 (RG)

Spinglass

0.678 ± 0.011

0.148 ± 0.004

1.0 ± 0.0

0.420 ± 0.018

0.355 ± 0.021

0.100 ± 0.008

0.90

d=0.3, d ^′=0.5

CF

0.979 ± 0.005

0.771 ± 0.061

0.930 ± 0.082

0.965 ± 0.012

0.969 ± 0.011

0.821 ± 0.062

0.09

d=0.3, d ^′=0.5

dGHD

0.848 ± 0.071

0.700 ± 0.038

0.731 ± 0.148

0.941 ± 0.010

0.964 ± 0.009

0.672 ± 0.078

1.0

d=0.3, d ^′=0.5

Louvain

0.932 ± 0.029

0.478 ± 0.118

1.0 ± 0.0

0.879 ± 0.054

0.866 ± 0.059

0.582 ± 0.128

0.014

d=0.3, d ^′=0.5

Infomap

0.674 ± 0.010

0.145 ± 0.004

1.0 ± 0.0

0.413 ± 0.018

0.348 ± 0.020

0.097 ± 0.008

0.023

d=0.3, d ^′=0.5

Spinglass

0.711 ± 0.007

0.162 ± 0.003

1.0 ± 0.0

0.481 ± 0.013

0.423 ± 0.014

0.128 ± 0.006

0.94

α=2 (PL)

CF

0.797 ± 0.046

0.307 ± 0.307

0.792 ± 0.099

0.801 ± 0.018

0.349 ± 0.051

0.802 ± 0.022

0.09

α=2 (PL)

dGHD

0.797 ± 0.013

0.294 ± 0.009

0.809 ± 0.027

0.787 ± 0.008

0.333 ± 0.015

0.784 ± 0.009

1.0

α=2 (PL)

Louvain

0.780 ± 0.014

0.212 ± 0.010

1.0 ± 0.0

0.703 ± 0.018

0.272 ± 0.016

0.690 ± 0.011

0.015

α=2 (PL)

Infomap

0.665 ± 0.013

0.141 ± 0.004

1.0 ± 0.0

0.603 ± 0.018

0.162 ± 0.012

0.484 ± 0.019

0.026

α=2 (PL)

Spinglass

0.687 ± 0.014

0.153 ± 0.006

1.0 ± 0.0

0.645 ± 0.021

0.194 ± 0.011

0.527 ± 0.016

0.90

α=3 (PL)

CF

0.825 ± 0.019

0.345 ± 0.015

0.825 ± 0.035

0.826 ± 0.007

0.402 ± 0.024

0.826 ± 0.004

0.085

α=3 (PL)

dGHD

0.808 ± 0.027

0.327 ± 0.018

0.799 ± 0.050

0.816 ± 0.008

0.375 ± 0.031

0.817 ± 0.004

1.0

α=3 (PL)

Louvain

0.774 ± 0.015

0.233 ± 0.011

1.0 ± 0.0

0.736 ± 0.019

0.301 ± 0.009

0.732 ± 0.019

0.015

α=3 (PL)

Infomap

0.670 ± 0.014

0.168 ± 0.005

1.0 ± 0.0

0.635 ± 0.017

0.210 ± 0.014

0.532 ± 0.014

0.027

α=3 (PL)

Spinglass

0.694 ± 0.013

0.179 ± 0.007

1.0 ± 0.0

0.670 ± 0.023

0.232 ± 0.012

0.571 ± 0.017

0.94