Network evolution with self-reinforcement

Abstract

We study a new class of preferential attachment trees with \emph{self-reinforcement}. At each time, each vertex is assigned a weight equal to the cumulative sum over past times of an affine function of its degree. A new vertex attaches itself via a single edge to an already present vertex with a probability proportional to the current weight of that vertex. This ``integrated popularity'' rule builds long memory directly into the attachment mechanism, thereby destroying the Markov and partial-exchangeability features that underlie the classical analysis of affine preferential attachment models. More broadly, the model connects to applied-probability work on long-memory self-interacting processes (such as the elephant random walk), emphasizing how non-Markovian reinforcement reshapes asymptotic behaviour. Despite this loss of structure, we identify an explicit exponent $ϕ=ϕ(δ)$ governing both local and global growth: typical degrees at time $n$ scale as $n^{1/ϕ}$, and the empirical degree distribution converges to a power-law with a tail exponent $ϕ+1$. We further prove Benjamini--Schramm local convergence to an infinite random rooted tree characterized via an embedded continuous-time branching process. The limiting tree is a \texttt{sin}-tree, and is \emph{not} the Pólya-type limiting tree arising in the non-reinforced setting. Our results provide a tractable probabilistic description of a natural ``memoryful'' network-growth mechanism, and quantify precisely how reinforcement renormalizes the classical preferential-attachment exponents.