## A Stochastic Uncoupling Process for Graphs (2000)

Venue: | NATIONAL RESEARCH INSTITUTE FOR MATHEMATICS AND COMPUTER SCIENCE IN THE |

Citations: | 2 - 1 self |

### BibTeX

@TECHREPORT{Dongen00astochastic,

author = {Stijn Van Dongen},

title = {A Stochastic Uncoupling Process for Graphs},

institution = {NATIONAL RESEARCH INSTITUTE FOR MATHEMATICS AND COMPUTER SCIENCE IN THE},

year = {2000}

}

### OpenURL

### Abstract

A discrete stochastic uncoupling process for finite spaces is introduced, called the Markov Cluster Process. The process takes a stochastic matrix as input, and then alternates flow expansion and flow inflation, each step defining a stochastic matrix in terms of the previous one. Flow expansion corresponds with taking the k th power of a stochastic matrix, where k 2 IN . Flow inflation corresponds with a parametrized operator \Gamma r , r 0, which 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 which is idempotent under both matrix multiplication and inflation, with quadratic convergence around the limit points. The limit is in general extremely sparse and the number of components of its associated graph may be larger than the number associated with the input matrix. This uncoupli...

### Citations

4822 |
Matrix Analysis
- Horn, Johnson
- 1990
(Show Context)
Citation Context ...ements are easy to verify. For extensive discussion of the majorization relationship between diagonal entries and eigenvalues of hermitian matrices, as well as results on interlacing inequalities see =-=[21]-=-. Statement b) follows from the fact that the compound operator distributes over matrix multiplication, and the fact that the compound of a positive (semi--) definite matrix is again positive (semi--)... |

743 |
Olkin Inequalities: Theory of Majorization and its Applications
- Marshall, I
- 1979
(Show Context)
Citation Context ...ful, since results from the theory of majorization of vectors do not carry over to matrices in such a straightforward way (i.e. the columns of one matrix majorizing the columns of another matrix). In =-=[25]-=- this issue is discussed at length. However, the statement clearly shows the inflationary or `decontracting' effect of \Gamma r , r ? 1, as opposed to the contracting effect of multiplication of nonne... |

512 |
Partitioning Sparse Matrices with Eigenvectors of Graphs
- Pothen, Simon, et al.
- 1990
(Show Context)
Citation Context ...evelop the framework in which the interplay of \Gamma r and Exp s can be studied. Hadamard--Schur theory was discussed in Section 3. Perron--Frobenius theory, graph partitioning by eigenvectors (e.g. =-=[32, 33]-=-), and work regarding the second largest eigenvalue of a graph (e.g. [1, 6]), form a natural source of inspiration. The theory of Perron complementation and stochastic complementation as introduced by... |

325 |
Nonnegative Matrices and Markov Chains
- Seneta
- 1981
(Show Context)
Citation Context ...action ratiosand the cross-ratio number OE by A = sup x;y d(Ax; Ay) d(x; y) OEA = min i;j;k;l A ik A jl A jk A il These are related to each other via A = 1 \Gamma p OEA 1 + p OEA (6.1) For proofs see =-=[3, 17, 34]-=-. The quantitysis used to measure the deviation of large products of nonnegative matrices from the set of rank 1 matrices (see e.g. [4, 5, 17, 34]). There is a straightforward connection between \Gamm... |

301 |
Circulant matrices
- DAVIS
- 1979
(Show Context)
Citation Context ... default parameters is unstable around the cyclic limit Fn . For the study of flip--flop equilibrium states the many results on circulant matrices are likely to be valuable, for example the monograph =-=[7]-=-, and the work on group majorization in the setting of circulant matrices in [16]. It may also be fruitful to investigate the relationship with Hilbert's distance and the contraction ratio for positiv... |

259 | Random walks on graphs: A survey
- Lovász
- 1993
(Show Context)
Citation Context ... the equilibrium distribution of M . In the theory of Markov chains, a stochastic diagonally symmetrizable matrix is called time reversible or said to satisfy the detailed balance condition (See e.g. =-=[24, 36]-=-). A slightly more general definition and different terminology was chosen here. The main reason is that the term `time reversible' is coupled tightly with the idea of studying a stochastic chain via ... |

225 | Graph Clustering by Flow Simulation - Dongen - 2000 |

179 |
Algorithms for Random Generation and Counting: A Markov Chain Approach
- Sinclair
- 1993
(Show Context)
Citation Context ... the equilibrium distribution of M . In the theory of Markov chains, a stochastic diagonally symmetrizable matrix is called time reversible or said to satisfy the detailed balance condition (See e.g. =-=[24, 36]-=-). A slightly more general definition and different terminology was chosen here. The main reason is that the term `time reversible' is coupled tightly with the idea of studying a stochastic chain via ... |

141 | Mathematical classification and clustering - Mirkin - 1996 |

75 | Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems
- Meyer
- 1989
(Show Context)
Citation Context ... form a natural source of inspiration. The theory of Perron complementation and stochastic complementation as introduced by Meyer may offer conceptual support in its focus on uncoupling Markov chains =-=[26, 27]-=-. There are also papers which address the topic of matrix structure when the subdominant eigenvalue is close to the dominant eigenvalue [18, 31]. The literature on the subject of diagonal similarity d... |

67 |
1, isoperimetric inequalities for graphs, and superconcentrators
- Alon, Milman
- 1985
(Show Context)
Citation Context ...ied. Hadamard--Schur theory was discussed in Section 3. Perron--Frobenius theory, graph partitioning by eigenvectors (e.g. [32, 33]), and work regarding the second largest eigenvalue of a graph (e.g. =-=[1, 6]-=-), form a natural source of inspiration. The theory of Perron complementation and stochastic complementation as introduced by Meyer may offer conceptual support in its focus on uncoupling Markov chain... |

50 | A cluster algorithm for graphs - Dongen - 2000 |

31 | Performance criteria for graph clustering and Markov cluster experiments
- Dongen, 2000b
(Show Context)
Citation Context ...m walk. Note: This report describes mathematical aspects of the MCL process. The process was introduced in [9] as a means for finding cluster structure in graphs. Cluster experiments are described in =-=[11]-=-. The work was carried out under project INS--3.2, Concept Building from Key--Phrases in Scientific Documents and Bottom Up Classification Methods in Mathematics. 1. Introduction The subject of this r... |

30 |
Nonnegative matrices
- Berman, Plemmons
- 1994
(Show Context)
Citation Context ...g from nonnegative idempotent matrices to overlapping clusterings in Definition 5. Its proof is given in [9] and can also be derived from the decomposition of nonnegative idempotent matrices given in =-=[2]-=-, page 65. Theorem 1 Let M be a nonnegative column allowable idempotent matrix of dimension n, let G be its associated graph. For s; t, nodes in G, write s ! t if there is an arc in G from s to t. By ... |

26 |
Special matrices and their applications in numerical mathematics
- Fiedler
- 1986
(Show Context)
Citation Context ...follows from the fact that the compound operator distributes over matrix multiplication, and the fact that the compound of a positive (semi--) definite matrix is again positive (semi--) definite. See =-=[14]-=- for an overview of results on compounds of matrices. c) follows from the fact that each term contributing to the principal minor in A is a product Q i Ak i k i+1 where each k i occurs once as a row i... |

15 |
Hilbert’s metric and positive contraction mappings in a Banach
- Bushell
- 1973
(Show Context)
Citation Context ...sitive vectors x and y both of dimension n is defined as d(x; y) = ln (max i x i y i )(max j y j x j ) = max i;j ln ` x i y j x j y i ' It can be defined in the more general setting of a Banach space =-=[4]-=-. Hilbert's metric is a genuine metric distance on the unit sphere in IR n , with respect to any vector norm (see [4]). For a positive matrix A define the contraction ratiosand the cross-ratio number ... |

13 |
On some methods for entropy maximization and matrix scaling", Linear Algebra and its Applications
- Elfving
- 1980
(Show Context)
Citation Context ...alue is close to the dominant eigenvalue [18, 31]. The literature on the subject of diagonal similarity does not seem to be of immediate further use, as it is often focussed on scaling problems (e.g. =-=[12, 20]-=-). Regarding flip--flop states, several interesting questions are open: i) For the MCL process with both parameter rows constant equal to 2, are there orbits of length greater than 2 in the class of d... |

12 |
Invariant subspaces for tightly clustered eigenvalues of tridiagonals
- Parlett
- 1996
(Show Context)
Citation Context ...eptual support in its focus on uncoupling Markov chains [26, 27]. There are also papers which address the topic of matrix structure when the subdominant eigenvalue is close to the dominant eigenvalue =-=[18, 31]-=-. The literature on the subject of diagonal similarity does not seem to be of immediate further use, as it is often focussed on scaling problems (e.g. [12, 20]). Regarding flip--flop states, several i... |

11 | On the structure of stochastic matrices with a subdominant eigenvalue near 1
- Hartfiel, Meyer
- 1998
(Show Context)
Citation Context ...eptual support in its focus on uncoupling Markov chains [26, 27]. There are also papers which address the topic of matrix structure when the subdominant eigenvalue is close to the dominant eigenvalue =-=[18, 31]-=-. The literature on the subject of diagonal similarity does not seem to be of immediate further use, as it is often focussed on scaling problems (e.g. [12, 20]). Regarding flip--flop states, several i... |

10 | Uncoupling the Perron eigenvector problem
- Meyer
- 1989
(Show Context)
Citation Context ... form a natural source of inspiration. The theory of Perron complementation and stochastic complementation as introduced by Meyer may offer conceptual support in its focus on uncoupling Markov chains =-=[26, 27]-=-. There are also papers which address the topic of matrix structure when the subdominant eigenvalue is close to the dominant eigenvalue [18, 31]. The literature on the subject of diagonal similarity d... |

9 |
On fractional Hadamard powers of positive definite matrices
- Fitzgerald, Horn
- 1977
(Show Context)
Citation Context ...ritten as the sum of weighted idempotents U i =si (S)u i u i . Statement i) now follows from setting E i = d t \Gamma1 U i d t . Statement ii) is adapted from a similar theorem by FitzGerald and Horn =-=[15]-=- for hermitian matrices. The proof of ii) follows from their argument for the hermitian case, given in the lemma below. 2 Lemma 5 [15] Let B be positive definite of dimension n. If Bnn ? 0 write bn fo... |

6 |
On products of non-negative matrices
- Hajnal
- 1976
(Show Context)
Citation Context ...mp k (A). It has dimension \Gamma n k \Delta . Its pq entry is equal to det A[up juq ], where u i is the i th k--tuple of distinct indices in the given ordering. Following terminology used in [5] and =-=[17]-=-, a nonnegative matrix is called column allowable if all its columns have at least one nonzero entry. With each square nonnegative matrix A of dimension n is a weighted graph G associated, defined on ... |

6 | Tree--tree matrices and other combinatorial problems from taxonomy - Hazewinkel - 1995 |

4 | Nonnegative Matrices. Wiley Interscience Series - Minc - 1988 |

3 |
Contractive inhomogeneous products of non-negative matrices
- Cohen
- 1979
(Show Context)
Citation Context ...itten Comp k (A). It has dimension \Gamma n k \Delta . Its pq entry is equal to det A[up juq ], where u i is the i th k--tuple of distinct indices in the given ordering. Following terminology used in =-=[5]-=- and [17], a nonnegative matrix is called column allowable if all its columns have at least one nonzero entry. With each square nonnegative matrix A of dimension n is a weighted graph G associated, de... |

3 |
Simić: The second largest eigenvalue of a graph–A survey
- Cvetković, S
- 1995
(Show Context)
Citation Context ...ied. Hadamard--Schur theory was discussed in Section 3. Perron--Frobenius theory, graph partitioning by eigenvectors (e.g. [32, 33]), and work regarding the second largest eigenvalue of a graph (e.g. =-=[1, 6]-=-), form a natural source of inspiration. The theory of Perron complementation and stochastic complementation as introduced by Meyer may offer conceptual support in its focus on uncoupling Markov chain... |

2 |
Structure of a matrix according to its second eigenvalue
- Powers
- 1987
(Show Context)
Citation Context ...evelop the framework in which the interplay of \Gamma r and Exp s can be studied. Hadamard--Schur theory was discussed in Section 3. Perron--Frobenius theory, graph partitioning by eigenvectors (e.g. =-=[32, 33]-=-), and work regarding the second largest eigenvalue of a graph (e.g. [1, 6]), form a natural source of inspiration. The theory of Perron complementation and stochastic complementation as introduced by... |

1 |
Lattice Theory. Number 25 in AMS Colloquium publications
- Birkhoff
- 1967
(Show Context)
Citation Context ...action ratiosand the cross-ratio number OE by A = sup x;y d(Ax; Ay) d(x; y) OEA = min i;j;k;l A ik A jl A jk A il These are related to each other via A = 1 \Gamma p OEA 1 + p OEA (6.1) For proofs see =-=[3, 17, 34]-=-. The quantitysis used to measure the deviation of large products of nonnegative matrices from the set of rank 1 matrices (see e.g. [4, 5, 17, 34]). There is a straightforward connection between \Gamm... |

1 |
Cyclic majorization and smoothing operators
- Giovagnoli, Wynn
- 1996
(Show Context)
Citation Context ...ip--flop equilibrium states the many results on circulant matrices are likely to be valuable, for example the monograph [7], and the work on group majorization in the setting of circulant matrices in =-=[16]-=-. It may also be fruitful to investigate the relationship with Hilbert's distance and the contraction ratio for positive matrices, introduced in Section 6. The MCL process converges quadratically in t... |

1 |
et al. Minimization of norms and the spectral radius of a sum of nonnegative matrices under diagonal equivalence
- Hershkowitz
- 1996
(Show Context)
Citation Context ...alue is close to the dominant eigenvalue [18, 31]. The literature on the subject of diagonal similarity does not seem to be of immediate further use, as it is often focussed on scaling problems (e.g. =-=[12, 20]-=-). Regarding flip--flop states, several interesting questions are open: i) For the MCL process with both parameter rows constant equal to 2, are there orbits of length greater than 2 in the class of d... |

1 | editors. Current trends in matrix theory - Uhlig, Grone - 1986 |