  We propose a general class of co-evolving tree network models driven by local
exploration where new vertices attach to the current network via randomly
sampling a vertex and then exploring the graph for a random number of steps in
the direction of the root, connecting to the terminal vertex. Specific choices
of the exploration step distribution lead to the well-studied affine
preferential attachment and uniform attachment models, as well as less well
understood dynamic network models with global attachment functionals such as
PageRank scores [Chebolu-Melsted (2008)]. We obtain local weak limits for such
networks and use them to derive asymptotics for the limiting empirical degree
and PageRank distribution. We also quantify asymptotics for the degree and
PageRank of fixed vertices, including the root, and the height of the network.
Two distinct regimes are seen to emerge, based on the expected exploration
distance of incoming vertices, which we call the `fringe' and `non-fringe'
regimes. These regimes are shown to exhibit different qualitative and
quantitative properties. In particular, networks in the non-fringe regime
undergo `condensation' where the root degree grows at the same rate as the
network size. Networks in the fringe regime do not exhibit condensation.
Non-trivial phase transition phenomena are displayed for the height and the
PageRank distribution, the latter connecting to the well known power-law
hypothesis. In the process, we develop a general set of techniques involving
local limits, infinite-dimensional urn models, related multitype branching
processes and corresponding Perron-Frobenius theory, branching random walks,
and in particular relating tail exponents of various functionals to the scaling
exponents of quasi-stationary distributions of associated random walks. These
techniques are expected to shed light on a variety of other co-evolving network
models.
