Parallel and I/O-Efficient Algorithms for Non-Linear Preferential Attachment

Preferential attachment lies at the heart of many network models aiming to replicate features of real world networks. To simulate the attachment process, conduct statistical tests, or obtain input data for benchmarks, efficient algorithms are required that are capable of generating large graphs according to these models.

Existing graph generators are optimized for the most simple model, where new nodes that arrive in the network are connected to earlier nodes with a probability P(h) ∝ d that depends linearly on the degree d of the earlier node h. Yet, some networks are better explained by a more general attachment probability P(h) ∝ f (d) for some function f : ℕ → ℝ. Here, the polynomial case f(d) = d^α where α ∈ ℝ _>0 is of particular interest.

In this paper, we present efficient algorithms that generate graphs according to the more general models. We first design a simple yet optimal sequential algorithm for the polynomial model. We then parallelize the algorithm by identifying batches of independent samples and obtain a near-optimal speedup when adding many nodes. In addition, we present an I/O-efficient algorithm that can even be used for the fully general model. To showcase the efficiency and scalability of our algorithms, we conduct an experimental study and compare their performance to existing solutions.

Funding This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under grant ME 2088/5-1 (FOR 2975 — Algorithms, Dynamics, and Information Flow in Networks).

Supplemental material The internal memory algorithms are maintained at https://github.com/massive-graphs/nonlinear-preferential-attachment. The implementation of the dynamic weighted sampling data structure of [28] is independently maintained as the Rust crate https://crates.io/crates/dynamic-weighted-index. The external memory algorithms are available at https://github.com/massive-graphs/extmem-nlpa. An archive containing the frozen source code of all experiments and most of the raw data collected can be found on https://zenodo.org/record/7318118.

^* Link to full version: https://arxiv.org/abs/2211.06884