Observer-Based Source Localization in Tree Infection Networks via Laplace Transforms

TL;DR

The paper tackles locating the infection source in SI dynamics on trees when only a subset of nodes are observed. It develops a Laplace-transform based framework to characterize the distribution of observed infection times and shows identifiability of the source under this representation. Two estimators are proposed: a Laplace-inversion style hat estimator that minimizes the sup norm between empirical and model transforms, and a variance-reducing check estimator based on conditional transforms; both are evaluated on synthetic trees and a river network across various edge-delay models. The work also highlights fundamental limitations when extending to graphs with cycles, where multiple competing paths produce complex mixtures that challenge traditional spanning-tree reductions. Overall, the approach yields scale-invariant, model-flexible source localization with practical performance on trees and highlights key open problems for general networks.

Abstract

We address the problem of localizing the source of infection in an undirected, tree-structured network under a susceptible-infected outbreak model. The infection propagates with independent random time increments (i.e., edge-delays) between neighboring nodes, while only the infection times of a subset of nodes can be observed. We show that a reduced set of observers may be sufficient, in the statistical sense, to localize the source and characterize its identifiability via the joint Laplace transform of the observers' infection times. Using the explicit form of these transforms in terms of the edge-delay probability distributions, we propose scale-invariant least-squares estimators of the source. We evaluate their performance on synthetic trees and on a river network, demonstrating accurate localization under diverse edge-delay models. To conclude, we highlight overlooked technical challenges for observer-based source localization on networks with cycles, where standard spanning-tree reductions may be ill-posed.

Figures (9)

• Figure 1: Diagram of an infection tree with observer nodes labeled 1 through 9. It contains four equivalence classes with nodes colored blue, white, yellow, and green. The boundaries of these classes are $\{7,8,9\}$, $\{2,3,5,4\}$, $\{2\}$, and $\{1,2\}$, respectively. The white, yellow, and green classes form a star arrangement (centered at node 2). These three classes are feasible only when observer 2 is the first to become infected; in which case, observers 1 through 5 are sufficient to estimate the source. However, if observer 3 is the first to be infected, only the white class remains feasible, and observers 2 through 5 are sufficient to estimate the source.
• Figure 2: Toy diagram illustrating an infection tree where all the observer nodes, labeled $0$ through $(n+1)$, are leaves except for node $0$, which is the center of the star arrangement of equivalence classes when $\tau_0<\tau_i$ for $i=1,\ldots,(n+1)$.
• Figure 3: Example of an infection tree with observers labeled 1 to 3 (colored gray), non-observer nodes labeled $u$, $v$, and $w$, and edges set $\{a,b,c,d,e\}$.
• Figure 4: Diagram of a path infection network with a single observer (top row) and confusion matrices for source localization based on the $\hat{s}$ estimator when using i.i.d. PosNormal (middle row), Uniform (bottom left), Exponential (bottom center), and AbsCauchy (bottom right) edge delay distributions. Each of these was run with 1,000 samples for each possible true source. The darker the shading along the diagonals and the lighter the shading off them, the better the source localization performance.
• Figure 5: Left: Average edge-distance (i.e., number of edges) between $\hat{s}$ and $s$ in infection trees with only 2 observers, as the size of randomly generated trees increases. Each tree size had 1,000 samples. Right: Average edge-distance in randomly generated trees with 100 nodes, as the number of observers increases. Each number of observers had 1,000 samples. In all the plots, the shaded bands represent $\pm$ one standard deviation from the mean.

Theorems & Definitions (18)

• Remark 1
• Lemma 2.1
• proof
• Theorem 2.2
• Theorem 2.3
• Theorem 3.1
• proof
• Corollary 4.1
• Theorem 4.2
• Corollary 4.3