Drawing Clusterings of Bipartite Graphs PDF

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Drawing Clusterings of Bipartite Graphs

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Drawing Clusterings of Bipartite Graphs

Thibault Marette

1andStefan Neumann2 1Ecole Normale Supérieure de Lyon, France2KTH Royal Institute of Technology, Stockholm, SwedenAbstract

We study the drawing of a givenoverlapping

clusterings of bipartite graphs. Our approach does not require a (dis-)similarity function be- tween the clusters that shall be visualized. We present:

An algorithm for drawing overlapping clusters

A pre-processing routine that decreases the

computational complexity

A novel objective function to assess the

drawing qualitySeriation Problem

Goal: Reorder the matrix to find "dense" blocks

Given a quality metric, find row and column

permutations to optimize the visualization [2]

Pattern-agnostic:no prior kno wledgeo verthe

structure of the data

Cannot visualize agivenclusteringargmaxπq(π;M) or argminπq(π;M)Drawing Overlapping Clusterings

In a bipartite graphG= (R?C,E), aclusteris de-

fined by(R??R,C??C). Clustering algorithms find dense subgraphs{(R?,C?)}ki=1.

Tiles are colored according to which clusters the

corresponding rows and columns belong to

Projection over the rows and columns

Complex clusterings appear in Data Mining [3, 4]simple clustering (disjoint clusters)complex clustering (overlapping clusters)Problem Summary

Input:

The bipartite graphG

A complex clustering, i.e., clusters{(R?i,C?i)}ki=1•Output:

A permutation of the rows and of the columns that

visualizes the given clustering{(R?i,C?i)}ki=1

To get the permutations, we maximize an objective

function that measures the drawing quality: argmax

πR,πCk

X i=0S(clusteri,(πR,πC)), where

S(cluster) =X

rect?R(cluster)area(rect) 2.

It is founded on two properties:

The score of two clusters should be evaluated

independently:S( )=S( ) + S( )•

A cluster is well drawn when it appears in the

drawing as a large consecutive rectangleS( )>S( )>S( )Algorithm Overview

Initial biadjacency matrixpre-processing

orderingpost-processing

Blocked-matrix ordered matrixfinal matrix•

Pre-processing: Reduce the computational

complexity by creatingblocksof rows and columns

Ordering: Provide an orderingof the blocksto

draw clusters as good as possible.

There are two components in this step:

Objective function: Assert the quality of the ordering

Optimization algorithm: Find the best permutation

possible to maximize the objective function

Post-processing: Augment the ordering of the

previous phase to get a better global picturePre-processing

Goal:Utilize the clustering information to reduce

the computational complexity of the problem.

Formblocksof columns (and rows) that belong

exactly to the same set of clustersConsequence:We only need to reorder the blocks! This reorders the rows and columns implicitly.Ordering

Goal:Compute an ordering of the blocks to draw

the clusters as good as possible.

We maximize the objective function detailed on

the left columnPost-processing

Goal:Reordergroups ofblocks such that the ob-

jective function does not decrease, but we increase the global quality of the picture.> -→Partition the columns so that any pair of columns that have a common cluster are in the same setResults Use the visualization to compare clusteringsPCV algorithm Basso algorithm Assess the local imperfections of a clusteringReferences [1] CorinnaV ehlow,F abianB eck,and D .W eiskopf.Visualizing group structures in graphs: A survey.Computer Graphics

Forum, 36, 2017.

[2] PanpanXu, Nan Cao, Huamin Qu, and J. Stask o.In ter- active visual co-cluster analysis of bipartite graphs.Paci- ficVis, pages 32-39, 2016. [3] PauliMiettinen and Stefan Neumann. Recen tdev elopments in boolean matrix factorization. InIJCAI, pages 4922-4928, 2020.
[4] S.C.Madei raand A.L. Oliv eira.Biclustering algorithms for biological data analysis: a survey.IEEE/ACM Trans- actions on Computational Biology and Bioinformatics,

1(1):24-45, 2004.

Contact Information:thibault.marette@ens-lyon.fr

Supported by the ERC Advanced Grant REBOUND (834862) and the EC H2020 RIA project SoBigData++ (871042).quotesdbs_dbs12.pdfusesText_18

[PDF] Drawing Clusterings of Bipartite Graphs

Drawing Clusterings of Bipartite Graphs

Thibault Marette

1andStefan Neumann2 1Ecole Normale Supérieure de Lyon, France2KTH Royal Institute of Technology, Stockholm, SwedenAbstract

We study the drawing of a givenoverlapping

An algorithm for drawing overlapping clusters

A pre-processing routine that decreases the

A novel objective function to assess the

Goal: Reorder the matrix to find "dense" blocks

Given a quality metric, find row and column

Pattern-agnostic:no prior kno wledgeo verthe

In a bipartite graphG= (R?C,E), aclusteris de-

Tiles are colored according to which clusters the

Projection over the rows and columns

Input:

The bipartite graphG

A permutation of the rows and of the columns that

To get the permutations, we maximize an objective

πR,πCk

S(cluster) =X

It is founded on two properties:

The score of two clusters should be evaluated

A cluster is well drawn when it appears in the

Initial biadjacency matrixpre-processing

Blocked-matrix ordered matrixfinal matrix•

Pre-processing: Reduce the computational

Ordering: Provide an orderingof the blocksto

There are two components in this step:

Optimization algorithm: Find the best permutation

Post-processing: Augment the ordering of the

Goal:Utilize the clustering information to reduce

Formblocksof columns (and rows) that belong

Goal:Compute an ordering of the blocks to draw

We maximize the objective function detailed on

Goal:Reordergroups ofblocks such that the ob-

Forum, 36, 2017.

1(1):24-45, 2004.

Contact Information:thibault.marette@ens-lyon.fr