The scaling of the minimum sum of edge lengths in uniformly random trees

Abstract

The minimum linear arrangement problem on a network consists of finding the minimum sum of edge lengths that can be achieved when the vertices are arranged linearly. Although there are algorithms to solve this problem on trees in polynomial time, they have remained theoretical and have not been implemented in practical contexts to our knowledge. Here we use one of those algorithms to investigate the growth of this sum as a function of the size of the tree in uniformly random trees. We show that this sum is bounded above by its value in a star tree. We also show that the mean edge length grows logarithmically in optimal linear arrangements, in stark contrast to the linear growth that is expected on optimal arrangements of star trees or on random linear arrangements.

connections with theoretical physics have been expanded; in press in Journal of Statistical Mechanics (JSTAT)

References in corpus (3)

• The entropy of randomized network ensembles
• The entropic origin of disassortativity in complex networks
• Non-crossing dependencies: least effort, not grammar

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