Mapping directed networks

Jonathan Crofts, Ernesto Estrada, Desmond Higham, Alan Taylor

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

We develop and test a new mapping that can be applied to directed unweighted networks. Although not a “matrix function” in the classical matrix theory sense, this mapping converts an unsymmetric matrix with entries of zero or one into a symmetric real-valued matrix of the same dimension that generally has both positive and negative entries. The mapping is designed to reveal approximate directed bipartite communities within a complex directed network; each such community is formed by two set of nodes S1 and S2 such that the connections involving these nodes are predominantly from a node in S1 and to a node in S2. The new mapping is motivated via the concept of alternating walks that successively respect and then violate the orientations of the links. Considering the combinatorics of these walks leads us to a matrix that can be neatly expressed via the singular value decomposition of the original adjacency matrix and hyperbolic functions. We argue that this new matrix mapping has advantages
over other, exponential-based measures. Its performance is illustrated on synthetic data, and we then show that it is able to reveal meaningful directed bipartite substructure in a network from neuroscience.
LanguageEnglish
Pages337-350
Number of pages14
JournalETNA - Electronic Transactions on Numerical Analysis
Volume37
Publication statusPublished - 2010

Fingerprint

Directed Network
Matrix Function
Vertex of a graph
Walk
Hyperbolic function
Neuroscience
Matrix Theory
Adjacency Matrix
Violate
Substructure
Synthetic Data
Singular value decomposition
Combinatorics
Complex Networks
Convert
Zero

Keywords

  • bipartivity
  • clustering
  • communities
  • exponential
  • neuroscience,
  • stickiness

Cite this

Crofts, J., Estrada, E., Higham, D., & Taylor, A. (2010). Mapping directed networks. ETNA - Electronic Transactions on Numerical Analysis, 37, 337-350.
Crofts, Jonathan ; Estrada, Ernesto ; Higham, Desmond ; Taylor, Alan. / Mapping directed networks. In: ETNA - Electronic Transactions on Numerical Analysis. 2010 ; Vol. 37. pp. 337-350.
@article{fb0c8dc134ef4891a47f790735a0493d,
title = "Mapping directed networks",
abstract = "We develop and test a new mapping that can be applied to directed unweighted networks. Although not a “matrix function” in the classical matrix theory sense, this mapping converts an unsymmetric matrix with entries of zero or one into a symmetric real-valued matrix of the same dimension that generally has both positive and negative entries. The mapping is designed to reveal approximate directed bipartite communities within a complex directed network; each such community is formed by two set of nodes S1 and S2 such that the connections involving these nodes are predominantly from a node in S1 and to a node in S2. The new mapping is motivated via the concept of alternating walks that successively respect and then violate the orientations of the links. Considering the combinatorics of these walks leads us to a matrix that can be neatly expressed via the singular value decomposition of the original adjacency matrix and hyperbolic functions. We argue that this new matrix mapping has advantages over other, exponential-based measures. Its performance is illustrated on synthetic data, and we then show that it is able to reveal meaningful directed bipartite substructure in a network from neuroscience.",
keywords = "bipartivity, clustering, communities, exponential, neuroscience,, stickiness",
author = "Jonathan Crofts and Ernesto Estrada and Desmond Higham and Alan Taylor",
year = "2010",
language = "English",
volume = "37",
pages = "337--350",
journal = "ETNA - Electronic Transactions on Numerical Analysis",
issn = "1068-9613",
publisher = "Kent State University",

}

Crofts, J, Estrada, E, Higham, D & Taylor, A 2010, 'Mapping directed networks' ETNA - Electronic Transactions on Numerical Analysis, vol. 37, pp. 337-350.

Mapping directed networks. / Crofts, Jonathan; Estrada, Ernesto; Higham, Desmond; Taylor, Alan.

In: ETNA - Electronic Transactions on Numerical Analysis, Vol. 37, 2010, p. 337-350.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Mapping directed networks

AU - Crofts, Jonathan

AU - Estrada, Ernesto

AU - Higham, Desmond

AU - Taylor, Alan

PY - 2010

Y1 - 2010

N2 - We develop and test a new mapping that can be applied to directed unweighted networks. Although not a “matrix function” in the classical matrix theory sense, this mapping converts an unsymmetric matrix with entries of zero or one into a symmetric real-valued matrix of the same dimension that generally has both positive and negative entries. The mapping is designed to reveal approximate directed bipartite communities within a complex directed network; each such community is formed by two set of nodes S1 and S2 such that the connections involving these nodes are predominantly from a node in S1 and to a node in S2. The new mapping is motivated via the concept of alternating walks that successively respect and then violate the orientations of the links. Considering the combinatorics of these walks leads us to a matrix that can be neatly expressed via the singular value decomposition of the original adjacency matrix and hyperbolic functions. We argue that this new matrix mapping has advantages over other, exponential-based measures. Its performance is illustrated on synthetic data, and we then show that it is able to reveal meaningful directed bipartite substructure in a network from neuroscience.

AB - We develop and test a new mapping that can be applied to directed unweighted networks. Although not a “matrix function” in the classical matrix theory sense, this mapping converts an unsymmetric matrix with entries of zero or one into a symmetric real-valued matrix of the same dimension that generally has both positive and negative entries. The mapping is designed to reveal approximate directed bipartite communities within a complex directed network; each such community is formed by two set of nodes S1 and S2 such that the connections involving these nodes are predominantly from a node in S1 and to a node in S2. The new mapping is motivated via the concept of alternating walks that successively respect and then violate the orientations of the links. Considering the combinatorics of these walks leads us to a matrix that can be neatly expressed via the singular value decomposition of the original adjacency matrix and hyperbolic functions. We argue that this new matrix mapping has advantages over other, exponential-based measures. Its performance is illustrated on synthetic data, and we then show that it is able to reveal meaningful directed bipartite substructure in a network from neuroscience.

KW - bipartivity

KW - clustering

KW - communities

KW - exponential

KW - neuroscience,

KW - stickiness

M3 - Article

VL - 37

SP - 337

EP - 350

JO - ETNA - Electronic Transactions on Numerical Analysis

T2 - ETNA - Electronic Transactions on Numerical Analysis

JF - ETNA - Electronic Transactions on Numerical Analysis

SN - 1068-9613

ER -

Crofts J, Estrada E, Higham D, Taylor A. Mapping directed networks. ETNA - Electronic Transactions on Numerical Analysis. 2010;37:337-350.