Link Prediction via Convex Nonnegative Matrix Factorization on Multiscale Blocks

Algorithm 2

Algorithm for 𝑀-CNMF.

input: 𝐴^𝑜//observed network

    𝐿, 𝐾//desired rank, scale parameter

output: 𝐴//predicting matrix

𝐴=(0)

find community structure 𝐶={𝐶₁,…,𝐶_𝑚}

for 𝐶_𝑖∈𝐶

 if size(𝐶_𝑖)>𝐾

  divide 𝐶_𝑖 into int(size(𝐶_𝑖)/𝐾+1) equal parts

  append each part to 𝐶

  delete 𝐶_𝑖

 end if

end for

for 𝐶_𝑖∈𝐶

 if size(𝐶_𝑖)<𝐾

  for 𝐶_𝑗∈𝐶, 𝑖≠𝑗

   if size(𝐶_𝑖∪𝐶_𝑗)<𝐾

    𝐶_𝑖=𝐶_𝑖∪𝐶_𝑗

    delete 𝐶_𝑗

   end if

  end for

 end if

end for

partition matrix into 𝐵={𝐵₁,…,𝐵_|𝐶|²} by 𝐶

for block 𝐵_𝑖∈𝐵

 do CNMF with rank min(𝑐𝑜𝑙(𝐵_𝑖),𝑟𝑜𝑤(𝐵_𝑖),𝐿)

 𝐴=𝐴+𝐵_𝑖𝑊𝐺//sum the corresponding entities

end for