### Abstract

This paper is concerned with the problem of fuzzy aggregation of a network with nonnegative weights on its edges into a small number of clusters. Specifically, we want to optimally define a probability of affiliation of each of the n nodes of the network to each of m « n clusters or aggregates. In contrast to statistics-based approaches to this problem, we take a purely dynamical perspective by analyzing the discrete-time Markov chain associated with the network and mapping it onto a Markov chain describing transitions between the clusters. We show that every such aggregated Markov chain and affiliation function can be lifted again onto the full network to define the so-called lifted transition matrix between the nodes of the network. The optimal aggregated Markov chain and affiliation function can then be determined by minimizing some appropriately defined distance between the lifted transition matrix and the transition matrix of the original chain. In general, the resulting constrained nonlinear minimization problem comes out having continuous level sets of minimizers. We exploit this fact to devise an algorithm for identification of the optimal cluster number by choosing specific minimizers from the level sets. Numerical minimization is performed by some appropriately adapted version of restricted line search using projected gradient descent. The resulting algorithmic scheme is shown to perform well on several test examples.

Original language | English (US) |
---|---|

Pages (from-to) | 1535-1561 |

Number of pages | 27 |

Journal | Multiscale Modeling and Simulation |

Volume | 8 |

Issue number | 4 |

DOIs | |

State | Published - 2010 |

### Fingerprint

### Keywords

- Affiliation function
- Clustering
- Constrained nonlinear minimization problem
- Fuzzy aggregation
- Markov chain
- Network
- Numerical minimization
- Partitioning
- Random walk
- Restricted line search

### ASJC Scopus subject areas

- Modeling and Simulation
- Chemistry(all)
- Computer Science Applications
- Ecological Modeling
- Physics and Astronomy(all)

### Cite this

*Multiscale Modeling and Simulation*,

*8*(4), 1535-1561. https://doi.org/10.1137/090758519

**Optimal fuzzy aggregation of networks.** / Sarich, Marco; Schütte, Christof; Vanden Eijnden, Eric.

Research output: Contribution to journal › Article

*Multiscale Modeling and Simulation*, vol. 8, no. 4, pp. 1535-1561. https://doi.org/10.1137/090758519

}

TY - JOUR

T1 - Optimal fuzzy aggregation of networks

AU - Sarich, Marco

AU - Schütte, Christof

AU - Vanden Eijnden, Eric

PY - 2010

Y1 - 2010

N2 - This paper is concerned with the problem of fuzzy aggregation of a network with nonnegative weights on its edges into a small number of clusters. Specifically, we want to optimally define a probability of affiliation of each of the n nodes of the network to each of m « n clusters or aggregates. In contrast to statistics-based approaches to this problem, we take a purely dynamical perspective by analyzing the discrete-time Markov chain associated with the network and mapping it onto a Markov chain describing transitions between the clusters. We show that every such aggregated Markov chain and affiliation function can be lifted again onto the full network to define the so-called lifted transition matrix between the nodes of the network. The optimal aggregated Markov chain and affiliation function can then be determined by minimizing some appropriately defined distance between the lifted transition matrix and the transition matrix of the original chain. In general, the resulting constrained nonlinear minimization problem comes out having continuous level sets of minimizers. We exploit this fact to devise an algorithm for identification of the optimal cluster number by choosing specific minimizers from the level sets. Numerical minimization is performed by some appropriately adapted version of restricted line search using projected gradient descent. The resulting algorithmic scheme is shown to perform well on several test examples.

AB - This paper is concerned with the problem of fuzzy aggregation of a network with nonnegative weights on its edges into a small number of clusters. Specifically, we want to optimally define a probability of affiliation of each of the n nodes of the network to each of m « n clusters or aggregates. In contrast to statistics-based approaches to this problem, we take a purely dynamical perspective by analyzing the discrete-time Markov chain associated with the network and mapping it onto a Markov chain describing transitions between the clusters. We show that every such aggregated Markov chain and affiliation function can be lifted again onto the full network to define the so-called lifted transition matrix between the nodes of the network. The optimal aggregated Markov chain and affiliation function can then be determined by minimizing some appropriately defined distance between the lifted transition matrix and the transition matrix of the original chain. In general, the resulting constrained nonlinear minimization problem comes out having continuous level sets of minimizers. We exploit this fact to devise an algorithm for identification of the optimal cluster number by choosing specific minimizers from the level sets. Numerical minimization is performed by some appropriately adapted version of restricted line search using projected gradient descent. The resulting algorithmic scheme is shown to perform well on several test examples.

KW - Affiliation function

KW - Clustering

KW - Constrained nonlinear minimization problem

KW - Fuzzy aggregation

KW - Markov chain

KW - Network

KW - Numerical minimization

KW - Partitioning

KW - Random walk

KW - Restricted line search

UR - http://www.scopus.com/inward/record.url?scp=77956821564&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77956821564&partnerID=8YFLogxK

U2 - 10.1137/090758519

DO - 10.1137/090758519

M3 - Article

VL - 8

SP - 1535

EP - 1561

JO - Multiscale Modeling and Simulation

JF - Multiscale Modeling and Simulation

SN - 1540-3459

IS - 4

ER -