Quantum preferential attachment

TL;DR

This work proposes a model that uniformly randomly connects to any node within the proximity of the target, including, but not restricted to, the target itself, leading to two distinct classes of complex network architectures.

Abstract

The quantum internet is a rapidly developing technological reality, yet, it remains unclear what kind of quantum network structures might emerge. Since indirect quantum communication is already feasible and preserves absolute security of the communication channel, a new node joining the quantum network does not need to connect directly to its desired target. Instead, in our proposed quantum preferential attachment model, it uniformly randomly connects to any node within the proximity of the target, including, but not restricted to, the target itself. This local flexibility is found to qualitatively change the global network behavior, leading to two distinct classes of complex network architectures, both of which are small-world, but neither of which is scale-free. Our numerical findings are supported by rigorous analytic results, in a framework that incorporates quantum and classical variants of preferential attachment in a unified phase diagram. Besides quantum networks, we expect that our results will have broad implications for classical scenarios where there is flexibility in establishing new connections.

Figures (4)

• Figure 1: (a) The QPA mechanism. At each iteration, a new node selects an initial target via (nonlinear) preferential attachment. This target induces a Local-Area Network (LAN), displayed with graph distance $R=1$, within which the new node attaches to any member uniformly at random. (b) Entanglement swapping. Gray edges in the network represent optical fibers that enable entanglement distributions, and entangled qubits are illustrated as stars of matching color. To establish quantum communication between the new node and the initial target through another (relay) node, one performs a Bell measurement at the relay to exchange the green and cyan entanglements for the yellow entanglement.
• Figure 2: Phase diagram of the QPA and the CR models. Incoming nodes first identify an initial target based on its degree $d_i$, proportionally to $d_I^\alpha$. A subsequent redirection happens with probability $r_i=d_i/(d_i+1)$ in the QPA model, or with $r\in[0,1]$ in the CR model. Networks are visualized at $N=200$ nodes, where larger node size reflects higher degree and darker node shade reflects earlier entry into the network. Networks boxed in the same color indicate model equivalence, either exactly or in the asymptotic limit of $N\to\infty$.
• Figure 3: Simulation results. Panels show (a) the (effective) maximum-degree exponent, (b) the fraction of leaves, and (c) the network diameter (the longest distance between any two nodes in the network) for QPA and CR networks at $10^5$ nodes. In (a), the blue dashed curves mark the theoretical predictions for $\alpha \geq 1, r=1$ in the infinite-size limit, highlighting the jumps at $\alpha = 1$, based on Theorem \ref{['thm:layered_hierarchy']}. Across all observables, the QPA curves meet the $r = 1/2$ curves as $\alpha \to -\infty$, and would overlap with the $r = 1$ curves for $\alpha > 1$ in the absence of finite-size effects. The horizontal axis is presented using $\tanh(\alpha - 1)$. Each data point reports the mean and standard deviation over 50 simulations. For finite-size scalings of these results, see Fig. [SM figure placeholder].
• Figure 4: Degree distributions of the sublinear QPA model for $\alpha \leq 1$. Numerically, we observe a broad but not scale-free degree distribution that appears universal across all $\alpha \leq 1$. In particular, Theorem \ref{['thm:master_equation']} gives the exact recursive degree distribution for $\alpha = -\infty$, which agrees with the numerical results. A Weibull distribution with two parameters fits both the numerical results and the theory very well. Networks are grown to $10^5$ nodes, and results are averaged over 50 simulations.

Theorems & Definitions (31)

• Theorem A.1
• proof
• Theorem A.2
• proof
• Lemma 1
• proof : Proof of lemma \ref{['lemma:high_differences']}
• Lemma 2
• proof : Proof of lemma \ref{['lemma:low_prob_king_change_for_high_diff']}
• Theorem A.3
• proof