[Paper Review] A theory of maximum likelihood for weighted infection graphs

Justin Khim, Po‐Ling Loh|arXiv (Cornell University)|Jun 13, 2018

Data-Driven Disease Surveillance30 references4 citations

TL;DR

This paper proposes a maximum likelihood estimation framework for inferring edge weights in weighted infection graphs where transmission probabilities depend on cumulative infections and edge covariates. It establishes consistency and asymptotic normality under a log-linear weight model, leveraging martingale convergence and Pólya urn theory, and provides algorithms for both ordered and unordered transmission data with theoretical guarantees, validated on synthetic and Ebola outbreak data.

ABSTRACT

We study the problem of parameter estimation based on infection data from an epidemic outbreak on a graph. We assume that successive infections occur via contagion; i.e., transmissions can only spread across existing directed edges in the graph. Our stochastic spreading model allows individual nodes to be infected more than once, and the probability of the transmission spreading across a particular edge is proportional to both the cumulative number of times the source nodes has been infected in previous stages of the epidemic and the weight parameter of the edge. We propose a maximum likelihood estimator for inferring the unknown edge weights when full information is available concerning the order and identity of successive edge transmissions. When the weights take a particular form as exponential functions of a linear combination of known edge covariates, we show that maximum likelihood estimation amounts to optimizing a convex function, and produces a solution that is both consistent and asymptotically normal. Our proofs are based on martingale convergence theorems and the theory of weighted Pólya urns. We also show how our theory may be generalized to settings where the weights are not exponential. Finally, we analyze the case where the available infection data comes in the form of an unordered set of edge transmissions. We propose two algorithms for weight parameter estimation in this setting and derive corresponding theoretical guarantees. Our methods are validated using both synthetic data and real-world data from the Ebola spread in West Africa.

Motivation & Objective

• To develop a statistical framework for estimating unknown edge weights in weighted infection graphs based on epidemic transmission data.
• To model transmission probabilities as proportional to both source node infection counts and edge weights, allowing repeated infections.
• To provide theoretical guarantees—consistency and asymptotic normality—for maximum likelihood estimation under a log-linear weight model.
• To extend the framework to settings with unordered transmission data using two proposed algorithms.
• To validate the method on real-world Ebola transmission data and synthetic networks, identifying key covariates influencing spread.

Proposed method

• Formulates a stochastic spreading model where transmission across an edge depends on the source node's cumulative infections and the edge's weight parameter.
• Uses maximum likelihood estimation (MLE) to infer edge weights when the full sequence of transmissions is observed.
• Assumes edge weights are exponential functions of a linear combination of known edge covariates, reducing MLE to convex optimization.
• Employs martingale convergence theorems and weighted Pólya urn processes to prove consistency and asymptotic normality of the MLE.
• Proposes two algorithms for the unordered transmission case: one based on iterative reweighting and another using a surrogate likelihood function.
• Derives theoretical convergence guarantees for both algorithms under mild regularity conditions.

Experimental results

Research questions

• RQ1 Which edge covariates most strongly influence the spread of an infection in a network?
• RQ2 Can maximum likelihood estimation consistently recover edge weights in a weighted infection graph with repeated infections?
• RQ3 How can statistical inference be performed when only an unordered set of transmissions is observed?
• RQ4 What theoretical properties—such as consistency and asymptotic normality—can be established for MLE in this non-i.i.d. epidemic setting?
• RQ5 How do edge weights based on covariates like population density, travel time, and borders affect disease spread, as seen in real-world Ebola data?

Key findings

• The MLE for edge weights is consistent and asymptotically normal when weights are modeled as exponential functions of edge covariates.
• The convexity of the log-likelihood under the log-linear weight model enables efficient optimization and theoretical analysis.
• In the Ebola data analysis, the distance between regions and the populations of source and destination regions were the most significant predictors, with t-statistics exceeding 1,000.
• International borders had a strong positive effect (coefficient = 3.027), while cross-border transmission between specific countries like Liberia and Sierra Leone showed large negative coefficients (e.g., -3.866 for Liberia to Sierra Leone).
• The model identified shared language and population density as significant covariates, with standardized coefficients of 0.845 and 0.312, respectively.
• The proposed algorithms for unordered data achieve theoretical convergence and perform well on both synthetic and real-world data, demonstrating robustness to data structure.