Googling the brain: discovering hierarchical and asymmetric network structures, with applications in neuroscience

Crofts, J.J. ORCID: 0000-0001-7751-9984 and Higham, D.J., 2011. Googling the brain: discovering hierarchical and asymmetric network structures, with applications in neuroscience. Internet Mathematics, 7 (4), pp. 233-254. ISSN 1542-7951


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Hierarchical organisation is a common feature of many directed networks arising in nature and technology. For example, a well-defined message-passing framework based on managerial status typically exists in a business organisation. However, in many real-world networks such patterns of hierarchy are unlikely to be quite so transparent. Due to the nature in which empirical data is collated the nodes will often be ordered so as to obscure any underlying structure. In addition, the possibility of even a small number of links violating any overall “chain of command” makes the determination of such structures extremely challenging. Here we address the issue of how to reorder a directed network in order to reveal this type of hierarchy. In doing so we also look at the task of quantifying the level of hierarchy, given a particular node ordering. We look at a variety of approaches. Using ideas from the graph Laplacian literature, we show that a relevant discrete optimization problem leads to a natural hierarchical node ranking. We also show that this ranking arises via a maximum likelihood problem associated with a new range-dependent hierarchical random graph model. This random graph insight allows us to compute a likelihood ratio that quantifies the overall tendency for a given network to be hierarchical. We also develop a generalization of this node ordering algorithm based on the combinatorics of directed walks. In passing, we note that Google’s PageRank algorithm tackles a closely related problem, and may also be motivated from a combinatoric, walk-counting viewpoint. We illustrate the performance of the resulting algorithms on synthetic network data, and on a real-world network from neuroscience where results may be validated biologically.

Item Type: Journal article
Publication Title: Internet Mathematics
Creators: Crofts, J.J. and Higham, D.J.
Publisher: Taylor & Francis
Date: 2011
Volume: 7
Number: 4
ISSN: 1542-7951
Rights: This is an Author's Original Manuscript of an article whose final and definitive form, the Version of Record, has been published in Internet Mathematics, 2011, copyright Taylor & Francis, available online at:
Divisions: Schools > School of Science and Technology
Record created by: EPrints Services
Date Added: 09 Oct 2015 10:53
Last Modified: 09 Jun 2017 13:43

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