Joint work with T. Zhao, B. Maga, G. Ódor, K. Soni, A. Salova, B. Hao, M. Abért and I. A. Kovács · Preprint
Classical preferential attachment assumes a new node connects directly to a selected target, with high-degree targets attracting more links. In a quantum network this assumption can be relaxed: indirect quantum communication is already feasible and preserves the security of the channel, so a node joining the network need not attach to its desired target directly. In the model it instead connects uniformly at random to any node within a short graph distance of that target — the target itself included.
In the most local setting, proximity radius $R=1$, meaning at most a single entanglement swap, the initial target is selected with degree bias $d_i^{\alpha}$ and the effective attachment weight of vertex $i$ becomes
$$ w_i^{Q}(\alpha) \;=\; \frac{d_i^{\alpha}}{d_i+1} \;+\; \sum_{j\sim i} \frac{d_j^{\alpha}}{d_j+1}. $$The first term is the case where $i$ is selected as target and the new node attaches to $i$ itself, the factor $1/(d_i+1)$ being the probability of not being redirected. The second collects the cases where $i$ is reached through a neighbouring target.
Main finding
This local freedom changes the global geometry of the growing network. The model produces two distinct classes of network architecture: a hierarchical superlinear phase, and a (sub)linear phase with a broad, Weibull-like degree distribution. Both are small-world, but neither is scale-free.
The numerical findings are supported by rigorous analysis within a unified phase diagram that also contains the classical variants of preferential attachment.
Although the mechanism is motivated by quantum communication, we expect the results to carry over to classical settings — biological and social — where there is comparable flexibility in establishing new connections.
Paper
T. Zhao, B. Maga, P. Dionigi, G. Ódor, K. Soni, A. Salova, B. Hao, M. Abért, I. A. Kovács, Quantum preferential attachment, arXiv:2512.22542.