Graph Clustering Via a Discrete Uncoupling Process
A discrete uncoupling process for finite spaces is introduced, called the Markov Cluster Process or the MCL process. The process is the engine for the graph clustering algorithm called the MCL algorithm. The MCL process takes a stochastic matrix as input, and then alternates expansion and inflation,...
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| Published in: | SIAM journal on matrix analysis and applications Vol. 30; no. 1; pp. 121 - 141 |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Philadelphia, PA
Society for Industrial and Applied Mathematics
01.01.2008
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| Subjects: | |
| ISSN: | 0895-4798, 1095-7162 |
| Online Access: | Get full text |
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| Abstract | A discrete uncoupling process for finite spaces is introduced, called the Markov Cluster Process or the MCL process. The process is the engine for the graph clustering algorithm called the MCL algorithm. The MCL process takes a stochastic matrix as input, and then alternates expansion and inflation, each step defining a stochastic matrix in terms of the previous one. Expansion corresponds with taking the $k$th power of a stochastic matrix, where $k\in\N$. Inflation corresponds with a parametrized operator $\Gamma_r$, $r\geq 0$, that maps the set of (column) stochastic matrices onto itself. The image $\Gamma_r M$ is obtained by raising each entry in $M$ to the $r$th power and rescaling each column to have sum 1 again. In practice the process converges very fast towards a limit that is invariant under both matrix multiplication and inflation, with quadratic convergence around the limit points. The heuristic behind the process is its expected behavior for (Markov) graphs possessing cluster structure. The process is typically applied to the matrix of random walks on a given graph $G$, and the connected components of (the graph associated with) the process limit generically allow a clustering interpretation of $G$. The limit is in general extremely sparse and iterands are sparse in a weighted sense, implying that the MCL algorithm is very fast and highly scalable. Several mathematical properties of the MCL process are established. Most notably, the process (and algorithm) iterands posses structural properties generalizing the mapping from process limits onto clusterings. The inflation operator $\Gamma_r$ maps the class of matrices that are diagonally similar to a symmetric matrix onto itself. The phrase diagonally positive semi-definite (dpsd) is used for matrices that are diagonally similar to a positive semi-definite matrix. For $r\in\N$ and for $M$ a stochastic dpsd matrix, the image $\Gamma_r M$ is again dpsd. Determinantal inequalities satisfied by a dpsd matrix $M$ imply a natural ordering among the diagonal elements of $M$, generalizing the mapping of process limits onto clusterings. The spectrum of $\Gamma_{\infty} M$ is of the form $\{0^{n-k}, 1^k\}$, where $k$ is the number of endclasses of the ordering associated with $M$, and $n$ is the dimension of $M$. This attests to the uncoupling effect of the inflation operator. |
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| AbstractList | A discrete uncoupling process for finite spaces is introduced, called the Markov Cluster Process or the MCL process. The process is the engine for the graph clustering algorithm called the MCL algorithm. The MCL process takes a stochastic matrix as input, and then alternates expansion and inflation, each step defining a stochastic matrix in terms of the previous one. Expansion corresponds with taking the $k$th power of a stochastic matrix, where $k\in\N$. Inflation corresponds with a parametrized operator $\Gamma_r$, $r\geq 0$, that maps the set of (column) stochastic matrices onto itself. The image $\Gamma_r M$ is obtained by raising each entry in $M$ to the $r$th power and rescaling each column to have sum 1 again. In practice the process converges very fast towards a limit that is invariant under both matrix multiplication and inflation, with quadratic convergence around the limit points. The heuristic behind the process is its expected behavior for (Markov) graphs possessing cluster structure. The process is typically applied to the matrix of random walks on a given graph $G$, and the connected components of (the graph associated with) the process limit generically allow a clustering interpretation of $G$. The limit is in general extremely sparse and iterands are sparse in a weighted sense, implying that the MCL algorithm is very fast and highly scalable. Several mathematical properties of the MCL process are established. Most notably, the process (and algorithm) iterands posses structural properties generalizing the mapping from process limits onto clusterings. The inflation operator $\Gamma_r$ maps the class of matrices that are diagonally similar to a symmetric matrix onto itself. The phrase diagonally positive semi-definite (dpsd) is used for matrices that are diagonally similar to a positive semi-definite matrix. For $r\in\N$ and for $M$ a stochastic dpsd matrix, the image $\Gamma_r M$ is again dpsd. Determinantal inequalities satisfied by a dpsd matrix $M$ imply a natural ordering among the diagonal elements of $M$, generalizing the mapping of process limits onto clusterings. The spectrum of $\Gamma_{\infty} M$ is of the form $\{0^{n-k}, 1^k\}$, where $k$ is the number of endclasses of the ordering associated with $M$, and $n$ is the dimension of $M$. This attests to the uncoupling effect of the inflation operator. |
| Author | Van Dongen, Stijn |
| Author_xml | – sequence: 1 givenname: Stijn surname: Van Dongen fullname: Van Dongen, Stijn |
| BackLink | http://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=21174063$$DView record in Pascal Francis |
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| ContentType | Journal Article |
| Copyright | 2015 INIST-CNRS [Copyright] © 2008 Society for Industrial and Applied Mathematics |
| Copyright_xml | – notice: 2015 INIST-CNRS – notice: [Copyright] © 2008 Society for Industrial and Applied Mathematics |
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| Keywords | Markov process Invariant Multiplication Random matrix Markov matrix Limit point Similarity Cluster analysis (statistics) Heuristics 05C90 Connected graph Invariance principle Mapping Image Stochastic process Input Column Ordering circulant matrices Behavior Fast algorithm Mathematical expansion Expansion Markov graph Stochastic matrix 05C85 Cluster Columns diagonal similarity Interpretation graph clustering Numerical analysis Symmetric matrix positive semi-definite matrices stochastic uncoupling Linear algebra Positive definite matrix M matrix 68R10 Power Inequality |
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| PublicationTitle | SIAM journal on matrix analysis and applications |
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| References | Cvetković D. (R9) 1995; 9 R61 R41 R40 R21 R23 R47 R24 R27 R26 R48 R29 R28 R1 R6 R7 R8 al. D. A. Lee et (R39) 2004; 32 R30 R52 R54 R31 R53 R33 R55 R16 R38 R15 R37 R18 R17 R19 al. L. D. Stein et (R60) 2003; 1 |
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| SubjectTerms | Algebra Algorithms Applied mathematics Bioinformatics Clustering Exact sciences and technology Graphs Heuristic Image retrieval Kinases Linear and multilinear algebra, matrix theory Markov processes Mathematics Multivariate analysis Numerical analysis Numerical analysis. Scientific computation Numerical linear algebra Peer to peer computing Probability and statistics Probability theory and stochastic processes Proteins Sciences and techniques of general use Statistics |
| Title | Graph Clustering Via a Discrete Uncoupling Process |
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