The longest path in the Price model
Abstract
The Price model, the directed version of the BarabásiAlbert model, produces a growing directed acyclic graph. We look at variants of the model in which directed edges are added to the new vertex in one of two ways: using cumulative advantage (preferential attachment) choosing vertices in proportion to their degree, or with random attachment in which vertices are chosen uniformly at random. In such networks, the longest path is well defined and in some cases is known to be a better approximation to geodesics than the shortest path. We define a reverse greedy path and show both analytically and numerically that this scales with the logarithm of the size of the network with a coefficient given by the number of edges added using random attachment. This is a lower bound on the length of the longest path to any given vertex and we show numerically that the longest path also scales with the logarithm of the size of the network but with a larger coefficient that has some weak dependence on the parameters of the model.
 Publication:

Scientific Reports
 Pub Date:
 June 2020
 DOI:
 10.1038/s41598020674218
 arXiv:
 arXiv:1903.03667
 Bibcode:
 2020NatSR..1010503E
 Keywords:

 Physics  Physics and Society
 EPrint:
 Postpeerreview, precopyedit version of article to be published in Scientific Reports