Node Ordering for Rescalable Network Summarization (or, the Apparent Magic of Word Frequency and Age of Acquisition in the Lexicon)

David Liben-Nowell, Violet Brown, Xi Chen, Maryam Hedayati, Camden Sikes, Julia Strand, Tegan Wilson

Abstract:

How can we “scale down” an n-node network G to a smaller network
G′, with k ≪ n nodes, so that G′ (approximately) maintains the important struc-
tural properties of G? There is a voluminous literature on many versions of this
problem if k is given in advance, but one’s tolerance for approximation (and the
resulting value of k) will vary. Here, then, we formulate a “rescalable” version of
this approximation task for complex networks. Specifically, we propose a node
ordering version of graph summarization: permute the nodes of G so that the sub-
graph induced by the first k nodes is a good size-k approximation of G, averaged
over the full range of possible sizes k. We consider as a case study the phono-
logical network of English words, and discover two natural word orders (word
frequency and age of acquisition) that do a surprisingly good job of rescalably
summarizing the lexicon