The Longest Path in the Price Model
Talk given at The Conference on Complex Systems (CCS) Singapore 2019, 4/10/19
The Price Model (Price, 1965) is the directed network version of the Barabási-Albert model
(1999). Price was motivated by citation networks which encode the fundamental arrow of time
in the network; bibliographies can only refer to older documents. Edges are therefore directed
and there are no cycles in the network so a citation network is an example of a Directed Acyclic
Graph. Mathematically, one distinctive property of a Directed Acyclic Graph is that the longest
path between nodes is both well defined and often more meaningful than the shortest path. For
instance, most of our knowledge of the Price Model did not come directly from the original 1965
paper but via longer routes through more recent work.
In this work, we explore the scaling of the longest path length in Price model using analytical and
numerical methods (see arXiv:1903.03667). Nodes are added sequentially along with m new
edges from an existing node s to each new node t. The source s of each edge is chosen in one of
two ways: with probability (1-p) we choose s uniformly at random from the set of existing vertices,
otherwise s is chosen in proportion to its current out-degree (Price’s “cumulative advantage”).
We measure the longest path from the first node (s = 1) to each node t in the network. This is
bounded from below by l(t), the length of the reverse greedy path leading to t. This path ends
at t and, working backwards, each edge (u,v) in the reverse greedy path is chosen such that
the difference (v − u) is minimised. We have found that the reverse greedy path length in the
mean-field approximation is proportional to m (1-p) ln(t). We confirm this
leading order behaviour using numerical simulations. We
also show numerically that the ratio of longest to greedy path length is roughly constant but it
does have a weak dependence on the m and p parameters. One insight is that these long path
scales are dominated by edges added to nodes chosen randomly, on average m(1-p) edges per node,which are typically shorter than edges added with cumulative attachment.