Co-evolving dynamic networks

TL;DR

This paper introduces a versatile co-evolving network model driven by local exploration, analyzing its properties, phase transitions, and asymptotic behaviors, connecting to known models like preferential attachment and PageRank-based networks.

Contribution

It develops a general framework for co-evolving networks with local exploration, deriving local limits, degree and PageRank distributions, and identifying phase transitions and regimes.

Findings

Identifies 'fringe' and 'non-fringe' regimes with distinct behaviors.

Shows condensation phenomenon where root degree scales with network size.

Establishes phase transitions in height and PageRank distributions.

Abstract

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.