Optimal Online Probe Allocation for Classical and Quantum Network Tomography

TL;DR

This work tackles the problem of efficiently recovering network parameters by sequentially selecting probing experiments in network tomography. It introduces Opal, an online probe allocation algorithm that combines online experimental design with maximum likelihood estimation to minimize regret relative to the optimal allocation, under Lipschitz continuity and finite-confidence-interval conditions. The authors provide a formal online framework applicable to both classical loss-tomography and quantum bit-flip tomography, derive a general regret bound, and validate the approach through analytical results and extensive simulations. In particular, Opal achieves near-optimal convergence rates in classical star networks and provable regret in quantum star networks, with substantial empirical improvements over baselines, including a 13.64% improvement in Roofnet for classical tomography. The methods offer a principled, scalable path to adaptive tomography in both traditional and quantum networking contexts.

Abstract

How to efficiently perform network tomography is a fundamental problem in network management and monitoring. A network tomography task usually consists of applying multiple probing experiments, e.g., across different paths or via different casts (e.g., unicast and multicast). We study how to optimize the network tomography process through online sequential decision-making. From the methodology perspective, we introduce an online probe allocation algorithm that sequentially performs network tomography based on the principles of optimal experimental design and the maximum likelihood estimation. We rigorously analyze the regret of the algorithm under the conditions that i) the optimal allocation is Lipschitz continuous in the parameters being estimated and ii) the parameter estimators satisfy a concentration property. From the application perspective, we present two case studies: a) the classical lossy packet-switched network and b) the quantum bit-flip network. We show that both cases fulfill the two theoretical conditions and provide their corresponding regrets when deploying our proposed online probe allocation algorithm. Besides case studies with theoretical guarantees, we also conduct simulations to compare our proposed algorithm with existing methods and demonstrate our algorithm's effectiveness in a broader range of scenarios. In an experiment on the Roofnet topology, our algorithm improves the estimation accuracy by 13.64% compared with the state-of-the-art baseline.

Figures (11)

• Figure 1: Paper structure and contribution summary
• Figure 2: A star network with $4$ nodes: Denote the central node as $4$, and the peripheral nodes are denoted as $1,2,3.$ The solid lines represent the loss characteristics of the network (modeled by Bernoulli random variables $X_1, X_2, X_3$), the dashed blue line represents a unicast probe (probe at node $1$, receive at node $2$, or vice versa), and the two dotted red lines represent a multicast probe (probe at node $1$, duplicate at intermediate node $4$, receive at nodes $2$ and $3$).
• Figure 3: Root independent multicast for quantum star network tomography. The root $\ell_{\text{root}} = 1$ prepares state $\ket{+}$, transmits it to the central node, and an $(L-1)$ qubit state is prepared from the application of CNOT gates. Each output is transmitted to an end node via the links representing the quantum channels $\mathcal{E}_{\ell}$, for $\ell = 2,\ldots, L$. $\sigma_{\text{Z}}$ basis measurements performed in the end nodes generate boolean random variables $\bm Y_{\ell_{\text{root}}} \in \{0,1\}^{L-1}$.
• Figure 4: A high-level illustration of the novel bounding technique used in Proof of Theorem \ref{['thm:convergence-rate']} (esp. \ref{['eq:phi_star_diff']}). The thick solid line is the chased allocation $\hat{\phi}_{m,t}^*$ varying over time, and the dashed black line is the confidence interval of the ground truth allocation $\phi_{m}^*$. The derivation of \ref{['eq:phi_star_diff']} comes from two simplifications of the chasing step of Line \ref{['line:chase']} of Algorithm \ref{['alg:chase']} in Figs. \ref{['fig:horizontal_line']} and \ref{['fig:zigzag_line']}. Fig. \ref{['fig:horizontal_line']}: if the chased estimated optimal allocation $\hat{\bm\phi}_t^*$ is fixed over time $t$, then the actual allocation $\bm\phi_t$ is close to the chased allocation up to a distance of $1/t$; Fig. \ref{['fig:zigzag_line']}: if the chased allocation $\hat{\bm\phi}_t^*$ is changing over time $t$ but its variation is bounded by an interval with width $2\epsilon$, then the empirical allocation is close to the final chased allocation up to $2\epsilon + 1/t$. These two simplifications lead to \ref{['eq:phi_star_diff']}, illustrated in Fig. \ref{['fig:shrinking_zigzag_line']}.
• Figure 5: Topology illustration for the three types of networks used in our experiments: (a) A $40$-node star network, where one central node is connected to all other nodes. (b) A random Erdős-Rényi (ER) network with $20$ nodes and approximately $35$ links, and (c) a Roofnet topology with $37$ nodes and $114$ links, based on real-world data aguayo2004link.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Lemma 3
• Lemma 4
• proof : Proof of Lemma \ref{['lma:MLE-estimator-fix-radius']}
• proof : Proof of Lemma \ref{['lma:inverse-matrix-fix-entry']}