Diffusion-Network Alignment: An Efficient Algorithm and Explicit Probability Bounds

Abstract: This paper studies a variation of the classic network alignment problem, named diffusion-network alignment. The goal is to align the vertices of a rooted diffusion tree to the vertices of a network, where the diffusion tree could be from a communication trace or contact tracing, and the network could be an online or offline social network. Different from the classic network alignment where both networks are fully observed, this model captures the information asymmetry of two networks. To solve this problem, this paper presents an efficient algorithm based on tree correlation tests to extract alignment information from local neighborhoods. We analyze the performance of the algorithm in the sparse graph regime and show that with high probability, all matched pairs are correct. Furthermore, for each vertex on the diffusion tree, this paper establishes an explicit lower bound on the probability that the vertex is correctly matched. These lower bounds are depth-dependent and increase as vertices get closer to the root.

• The paper proposes an efficient two-pass algorithm for reconstructing node alignment between a diffusion tree and a fully observable network.
• It utilizes tree correlation testing and Poisson thinning to derive monotonic, depth-dependent matching probability bounds.
• The method guarantees no false matches with high probability, robustly addressing the information asymmetry in diffusion processes.

Diffusion-Network Alignment: Efficient Algorithms and Explicit Probability Bounds

Problem Formulation and Motivation

The paper "Diffusion-Network Alignment: An Efficient Algorithm and Explicit Probability Bounds" (2606.12879) introduces a variant of the classic network alignment problem termed diffusion-network alignment. In this setting, instead of observing two fully known correlated graphs as in traditional problems, one observes only a diffusion tree from the first network—a typical scenario in information diffusion or epidemic traces—while the second network remains fully observable. The underlying vertex correspondence between the networks, which encodes the latent identity alignment, must be inferred.

This formulation models settings where information about one network is incomplete, originating from a stochastic diffusion process governed by the Independent Cascade (IC) model, while a reference network is available in full. Classical methods such as subgraph counting are inadequate here, as they require access to both complete networks. The authors address the core algorithmic and probabilistic challenges this asymmetry produces, particularly in the regime of sparse random graphs.

Figure 1: An example of the IC model and its corresponding diffusion tree.

Model and Technical Framework

The mathematical model is instantiated using the correlated Erdős–Rényi (ER) graph pair paradigm with parameters $(n, \lambda, s)$. The diffusion process on the first graph, $G_1$, is driven by the IC model with constant edge-activation probability $q$, resulting in a rooted tree $T_{\mathrm{IC}}$. The task is to reconstruct the hidden permutation $\pi^*$ which aligns vertices of $T_{\mathrm{IC}}$ with those of a permuted copy of the second network $G_2$.

Critically, the diffusion tree, typically a small random fraction of the original network, preserves only local structural information correlated with $G_2$. This architecture necessitates inference strategies that go beyond classic ML-based graph alignment and explore statistical correlations in highly restricted subgraph structures.

Algorithmic Approach: Two-Pass Tree Correlation Propagation

The authors propose an efficient algorithm that generalizes the tree correlation testing paradigm to handle matching at all depths in the diffusion tree, not merely at the root. The algorithm proceeds in two passes:

1. Upward Pass: Starting from the leaves, local evidence for candidate alignments is aggregated upward using a set of four structural criteria based on subtree correlation likelihood ratios. This ensures that matching information can be certified even for non-root, non-leaf vertices.
2. Downward Pass: Matching information accumulated at higher levels (closer to the root) is recursively propagated downward, allowing for additional matches based on parent-child relationships and further correlation evidence.

Local correlation testing between diffusion subtrees and breadth-first search (BFS) trees in $G_2$ is implemented via a Poisson Galton–Watson approximation, greatly leveraging known likelihood-ratio characterizations and hypothesis-testing optimality in such branching structures.

Figure 2: Illustration of the four matching criteria.

Statistical Guarantees: Explicit Depth-Dependent Probability Bounds

The core contribution resides in the explicit, constant lower bounds on matching probability for individual vertices, parameterized by their depth in the diffusion tree. These bounds are constructed recursively using Poisson thinning arguments and are strictly increasing as the depth decreases (i.e., as vertices approach the root).

Specifically, let $p_d$ denote the minimal matching probability for a vertex at depth $G_1$0. The authors prove that $G_1$1, where $G_1$2 is the limit as $G_1$3, and $G_1$4 is the bound for the leaves. The guarantee is that, with high probability $G_1$5, all matches produced are either correct or empty—no false matches are output. For large depth, the per-vertex matching probability approaches $G_1$6 (the fixed-point value of the recursion), and the root matching probability is even higher, reflecting the information gain from propagation.

Figure 3: An example of the sequence $G_1$7 showing monotonic growth and convergence.

Theoretical Analysis and Proof Techniques

Analysis is carried out under the constant-degree (sparse) regime, drawing on both algorithmic and probabilistic tools:

• Random graph coupling: The distribution of local neighborhoods in $G_1$8 and $G_1$9 is shown to converge in total variation to correlated Poisson Galton–Watson trees, justifying the use of tree-based likelihood-ratio hypothesis testing.
• Error control: The high-probability guarantee of correctness is established via a careful path overlap and cycle analysis, ruling out spurious matches arising from non-tree-like intersections between subtrees and BFS neighborhoods. Harris' inequality and concentration-of-measure arguments are leveraged.
• Recursive bounds: For each depth, explicit recursive formulas relate the success probability at depth $q$0 to that at depth $q$1, accounting for both strong (parent-child) and weak (correlation test only) signals.
• Efficient computation: The algorithm is shown to run in time $q$2, determined mainly by the number of required tree likelihood-ratio computations.

Discussion and Implications

This work makes two strong claims:

1. Global correctness guarantee: With high probability, every matched pair is either correct or abstained—no false positives are output.
2. Vertex-wise depth-dependent probability bounds: The probability of correct matching can be made explicit, monotonic in depth, and is uniformly bounded away from zero for all depths above the leaves. These bounds are significantly sharper for vertices close to the root.

Remarkably, these guarantees are robust to the sparsity and stochastic incompleteness inherent in the diffusion process. The work reveals a computationally efficient (polynomial-time) regime that aligns with, but does not achieve, the known information-theoretic optima for partial recovery in classical network alignment. For example, in parameter regimes where the information-theoretic matching fraction is $q$3, the algorithm guarantees a root matching fraction around $q$4—indicating an explicit gap and open problem for future research.

Practical implications extend directly to scenarios of social network deanonymization via cascades, epidemic source tracing, and analysis of incomplete communication logs, providing robust, principled tools for cross-network inference when only propagating traces can be partially observed.

Figure 4: Intersection patterns of candidate paths showing the combinatorial cases analyzed in false positive exclusion.

Figure 5: Examples of overlap configurations for path pairs in candidate matches.

Future Directions

Building on this foundation, open research avenues include:

• Tightening the matching probability lower bounds towards the information-theoretic limit;
• Theoretical characterization of the improvement due to the downward pass, especially for leaf and near-leaf vertices;
• Generalization to richer graph models (e.g., attributions, communities, or time-varying networks);
• Relaxation to non-tree diffusion substructures (cycles/forests) and more general diffusion models;
• Data-driven evaluation of the practical performance and robustness in real networks with noise and model inconsistencies.

Conclusion

The paper rigorously formulates and addresses the diffusion-network alignment problem, providing the first practical, theoretically-analyzed algorithm for the setting of information-asymmetric graph matching. The results establish concrete, depth-dependent recovery guarantees and advance the theory of partially-observed network inference, with immediate implications for real-world applications in privacy, epidemiology, and digital forensics.

Figure 6: Notational illustration: subtree structure, canonical order, and parent/child relationships pivotal to upward and downward passes.

Figure 7: Visual representation of event structure and overlapping sets in the error analysis.

Figure 8: Conditioned graph construction used in the proof of high-probability correctness and Poisson limit approximation.