[Paper Review] Loss Tomography from Tree Topologies to General Topologies

Weiping Zhu, Ke Deng|arXiv (Cornell University)|Apr 30, 2011

Statistical Methods and Inference24 references3 citations

TL;DR

This paper proposes a novel likelihood-based framework for loss rate estimation in general network topologies, extending existing tree-topology methods. It introduces minimal sufficient statistics, link-based and path-based likelihood equations, and a divide-and-conquer strategy that decomposes intersecting networks into independent trees, enabling maximum likelihood estimation with provable concavity and convergence guarantees.

ABSTRACT

Loss tomography has received considerable attention in recent years and a number of estimators based on maximum likelihood (ML) or Bayesian principles have been proposed. Almost all of the estimators are devoted to the tree topology despite the general topology is more common in practice. There has been few likelihood function devoted to the general topology, not to mention the estimator. To overcome this, two sets of sufficient statistics for the tree and general topologies, respectively, are proposed in this paper. Using the statistics, two likelihood functions, one for a topology, are proposed here and subsequently two likelihood equations for the general topology, one is link-based and the other is path-based, are obtained. In addition, a dependence between subtrees in terms of their estimates is identified for the general topology and a divide-and-conquer strategy is proposed to deal with the dependence, which divides a general network into two types of independent trees. Further, two algorithms, one for a type of the independent trees, are proposed to estimate the loss rates of each type.

Motivation & Objective

• Address the lack of maximum likelihood estimators for loss tomography in general topologies, which are more common than tree topologies in practice.
• Overcome limitations in existing estimators that ignore sufficient statistics and fail to model dependencies in intersecting network substructures.
• Develop analytical likelihood functions and estimators that account for probe correlations across multiple sources and intersected subtrees in general topologies.
• Establish a theoretical foundation for loss rate estimation in general topologies by proving concavity of the solution space, ensuring convergence of iterative algorithms to MLEs.
• Propose a divide-and-conquer strategy to decompose complex networks into independent tree-like components, enabling scalable and consistent estimation.

Proposed method

• Introduce a set of minimal sufficient statistics for both tree and general topologies, enabling complete information capture from end-to-end probe measurements.
• Derive two likelihood equations for general topologies: one link-based and one path-based, both generalizing existing tree-topology formulations.
• Identify and model dependence between estimates in intersected subnetworks, leading to a required estimation order for consistency.
• Propose a divide-and-conquer decomposition strategy that splits a general network into two types of independent trees: non-intersecting and intersecting subtrees.
• Develop two estimators—one for descendant trees (using existing tree-topology methods) and one for ancestor trees (a new estimator for intersected regions).
• Use fixed-point and iterative procedures (e.g., EM) to solve the likelihood equations, with theoretical guarantees of convergence to MLEs due to concave solution space.

Experimental results

Research questions

• RQ1 How can sufficient statistics be defined for general network topologies to ensure all relevant information is captured from end-to-end probe measurements?
• RQ2 What are the analytical forms of the likelihood functions and equations for loss rate estimation in general topologies, and how do they generalize from tree topologies?
• RQ3 How can dependencies between estimates in intersected network substructures be modeled and resolved to ensure consistent and accurate inference?
• RQ4 What decomposition strategy enables scalable and consistent estimation in general topologies by transforming them into independent tree components?
• RQ5 Under what conditions can iterative algorithms like EM or fixed-point methods converge to the maximum likelihood estimate in general topologies?

Key findings

• The proposed link-based and path-based likelihood equations for general topologies generalize existing tree-topology estimators and maintain similar structural forms.
• The solution space for both estimators is proven to be concave under a Bernoulli loss model, guaranteeing convergence of iterative algorithms like EM to the MLE.
• The divide-and-conquer strategy successfully decomposes a general network into two types of independent trees—descendant and ancestor trees—enabling modular estimation.
• Simulations show that relative error for links in shared subtrees drops to around 5% when background TCP traffic is removed, indicating strong model correspondence is critical for accuracy.
• Even with reduced TCP window size or half the number of flows, relative error remains similar, indicating that TCP loss processes are statistically consistent and independent of flow count or window size.
• The accuracy of estimation is highly sensitive to model correspondence between the estimator and background traffic; mismatch leads to persistent errors, even with increased probing intensity.