Transient and slim versus recurrent and fat: Random walks and the trees they grow

Abstract

Network growth models that embody principles such as preferential attachment and local attachment rules have received much attention over the last decade. Among various approaches, random walks have been leveraged to capture such principles. In this paper we consider the No Restart Random Walk (NRRW) model where a walker builds its graph (tree) while moving around. In particular, the walker takes s steps (a parameter) on the current graph. A new node with degree one is added to the graph and connected to the node currently occupied by the walker. The walker then resumes, taking another s steps, and the process repeats. We analyze this process from the perspective of the walker and the network, showing a fundamental dichotomy between transience and recurrence for the walker as well as power law and exponential degree distribution for the network. More precisely, we prove the following results: s = 1: the random walk is transient and the degree of every node is bounded from above by a geometric distribution. s even: the random walk is recurrent and the degree of non-leaf nodes is bounded from below by a power law distribution with exponent decreasing in s. We also provide a lower bound for the fraction of leaves in the graph, and for s = 2 our bound implies that the fraction of leaves goes to one as the graph size goes to infinity. NRRW exhibits an interesting mutual dependency between graph building and random walking that is fundamentally influenced by the parity of s. Understanding this kind of coupled dynamics is an important step towards modeling more realistic network growth processes.

On a random walk that grows its own tree

“…In essence the proof consists in establishing a coupling between the random walk in BGRW and a biased random walk on Z. While this can be readily achieved for p sufficiently large (e.g., for p = 1 see [13], but also for p > 2/3, more generally), establishing such a coupling for all positive p turns out to be surprisingly laborious. We do that by resorting to a grass-roots argument which we believe closely mimic the actual behaviour of the walker in BGRW.…”

“…A more formal definition is given in Section 2. This model is a variant of the Non-Restart Random Walk (NRRW) proposed and studied in [2,13]. There, the initial graph is a vertex with a loop edge, and new leafs are created every s steps for a fixed s > 0.…”