## 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.