Matrix Multiplication Biclique Compression

Input is an ASCIIGraph of the WebGraph framework. The first step is to construct a CeciliaGraph, which is a binary format of the input graph. It consists of:

• number of total nodes : 32-bit
• number of total edges : 32-bit
• for each node v in the adjacency lists:
  • node id v : 32-bit, negated
  • all nodes of the adjacency list of v, each 32-bit

cargo run --release --bin ceciliatxt2ceciliabin -- -i ASCIIGraphFileName -o CeciliaGraphFileName

Next, we run Cecilia's vnmextract program that creates files for the extracted bicliques and files for the remaining nodes:

./vnmextract CeciliaGraphFileName 1 1 100 500,100 ${basename} 4

where ${basename} is the filename used as a prefix for the generated files. The most recent file containing all remaining nodes is

ls --sort time biclique/${basename}-it-* | head -n1 > RemainingGraph

The bicliques can be concatenated into a single file

cat "$kBicliqueDir/${basename}-biclique"*.txt > BiCliqueFile .

We can run

cargo run --release --bin ceciliatxt2ceciliabin -- -i RemainingGraph -o RemainingCeciliaGraph

to generate a CeciliaGraph from the remaining nodes.

Finally, we can

• run a PageRank benchmark on the original graph

cargo run --release --bin ceciliabin2asciigraph -- -i CeciliaGraphFileName

• run a PageRank benchmark on the remaining graph with the bicliques cargo run --release --bin ceciliabin2asciigraph -- -c BiCliqueFile -i RemainingCeciliaGraph
• run a PageRank benchmark on the remaining graph with the bicliques and use the original graph to correct missing self-loops or self-loops that have been introduced cargo run --release --bin ceciliabin2asciigraph -- -c BiCliqueFile -i RemainingCeciliaGraph -g CeciliaGraphFileName
• restore the original ASCIIGraph (modulo self loops) via cargo run --release --bin ceciliabin2asciigraph -- -c BiCliqueFile -i RemainingCeciliaGraph -o newASCIIGraphFileName

Caveats:

• The biclique extraction tool introduces self-loops, which we remove in the remaining nodes. Hence, we may remove self-loops present in the original graph, or might introduce self-loops that are indirectly given by a biclique. Therefore, we provide the program argument -g to remove them by an additional pass on the original graph.