### Abstract

Although there are well-known heuristics for the blockmodeling of one-mode, unsigned, deterministic networks using structural equivalence, the potential benefits of exact algorithms that generate globally optimal solutions are many. In this paper we extend the applicability of one such method - integer programming - to exploratory blockmodeling. Specifically, leveraging the work of Brusco and Steinley (2009), we use the isomorphic properties of the image matrix to develop a minimal, representative set of image matrices with P positions. Not only does this drastically reduce the total number of image matrices the researcher must fit, but it also simultaneously solves all blockmodels with less than P positions. We demonstrate and prove the latter using the structural equivalence of positions, and we subsequently develop a minimal set of image matrices for four or fewer positions. These developments are illustrated using Fine's well-known Sharpstone Auto Little League Team network (1987), and we use the results to discover new structural features. In our account, competing, globally optimal alternatives are embraced as equally compelling, coexisting representations of a complex culture.

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

Pages (from-to) | 93-106 |

Number of pages | 14 |

Journal | Social Networks |

Volume | 47 |

DOIs | |

State | Published - Oct 1 2016 |

### Fingerprint

### Keywords

- Blockmodeling
- Integer programming
- Isomorphism
- One-mode network
- Social position
- Structural equivalence

### ASJC Scopus subject areas

- Sociology and Political Science
- Social Sciences(all)
- Anthropology
- Psychology(all)

### Cite this

**Exploratory blockmodeling for one-mode, unsigned, deterministic networks using integer programming and structural equivalence.** / Dabkowski, Matthew; Fan, Neng; Breiger, Ronald L.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Exploratory blockmodeling for one-mode, unsigned, deterministic networks using integer programming and structural equivalence

AU - Dabkowski, Matthew

AU - Fan, Neng

AU - Breiger, Ronald L

PY - 2016/10/1

Y1 - 2016/10/1

N2 - Although there are well-known heuristics for the blockmodeling of one-mode, unsigned, deterministic networks using structural equivalence, the potential benefits of exact algorithms that generate globally optimal solutions are many. In this paper we extend the applicability of one such method - integer programming - to exploratory blockmodeling. Specifically, leveraging the work of Brusco and Steinley (2009), we use the isomorphic properties of the image matrix to develop a minimal, representative set of image matrices with P positions. Not only does this drastically reduce the total number of image matrices the researcher must fit, but it also simultaneously solves all blockmodels with less than P positions. We demonstrate and prove the latter using the structural equivalence of positions, and we subsequently develop a minimal set of image matrices for four or fewer positions. These developments are illustrated using Fine's well-known Sharpstone Auto Little League Team network (1987), and we use the results to discover new structural features. In our account, competing, globally optimal alternatives are embraced as equally compelling, coexisting representations of a complex culture.

AB - Although there are well-known heuristics for the blockmodeling of one-mode, unsigned, deterministic networks using structural equivalence, the potential benefits of exact algorithms that generate globally optimal solutions are many. In this paper we extend the applicability of one such method - integer programming - to exploratory blockmodeling. Specifically, leveraging the work of Brusco and Steinley (2009), we use the isomorphic properties of the image matrix to develop a minimal, representative set of image matrices with P positions. Not only does this drastically reduce the total number of image matrices the researcher must fit, but it also simultaneously solves all blockmodels with less than P positions. We demonstrate and prove the latter using the structural equivalence of positions, and we subsequently develop a minimal set of image matrices for four or fewer positions. These developments are illustrated using Fine's well-known Sharpstone Auto Little League Team network (1987), and we use the results to discover new structural features. In our account, competing, globally optimal alternatives are embraced as equally compelling, coexisting representations of a complex culture.

KW - Blockmodeling

KW - Integer programming

KW - Isomorphism

KW - One-mode network

KW - Social position

KW - Structural equivalence

UR - http://www.scopus.com/inward/record.url?scp=84973333470&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84973333470&partnerID=8YFLogxK

U2 - 10.1016/j.socnet.2016.05.005

DO - 10.1016/j.socnet.2016.05.005

M3 - Article

AN - SCOPUS:84973333470

VL - 47

SP - 93

EP - 106

JO - Social Networks

JF - Social Networks

SN - 0378-8733

ER -