Learning of networked spreading models from noisy and incomplete data

TL;DR

Addressing learning of spreading dynamics from incomplete and uncertain data, the paper introduces SLICER+ based on Dynamic Message Passing to jointly infer network structure and Independent Cascade parameters $\alpha_{ij}$. It integrates prior information and uncertainty models via a KL-based objective and a Lagrangian constraint framework, achieving linear per-iteration complexity $O(|E|T|S|)$. Empirical results on synthetic networks and real-world data demonstrate robustness to missing data, noisy timestamps, and partial observability, enabling accurate structure recovery and parameter estimation. This work advances practical diffusion modeling by providing a universal, scalable method that accommodates multiple data imperfections simultaneously.

Abstract

Recent years have seen a lot of progress in algorithms for learning parameters of spreading dynamics from both full and partial data. Some of the remaining challenges include model selection under the scenarios of unknown network structure, noisy data, missing observations in time, as well as an efficient incorporation of prior information to minimize the number of samples required for an accurate learning. Here, we introduce a universal learning method based on scalable dynamic message-passing technique that addresses these challenges often encountered in real data. The algorithm leverages available prior knowledge on the model and on the data, and reconstructs both network structure and parameters of a spreading model. We show that a linear computational complexity of the method with the key model parameters makes the algorithm scalable to large network instances.

Figures (10)

• Figure 1: Schematic representation of different scenarios of incomplete and uncertain data that we consider in learning of networked spreading models. We focus on the general setting where a fraction of the nodes never reports information (depicted as empty nodes). In addition, we treat cases where network structure is not known or known only partially; where the number of data samples is small, but prior information on the parameters is available; and where the observed node activation timestamps are missing or noisy. At the end of our experimental evaluation, we address learning of real-world network instances under a combination of these scenarios.
• Figure 2: Heat maps of average difference between inferred $\alpha_{ij}$ and real $\alpha^*_{ij}$ parameters in $\ell_1$ norm, as a function of the number of available cascades and the fraction of unobserved nodes. Each heat map represents a different network type, but all of them have the same color scale. Each point is averaged over 5 different networks and 5 different sets of parameters $\alpha^*_{ij}$ (sampled from a uniform distribution in the range $[0,1]$). All networks contain $N=100$ nodes and all but the tree have average degree $\langle k \rangle = 3$. Unobserved nodes were picked at random. All cascades had length $T=5$. Note that $\xi = 33\%$ corresponds to a random guess.
• Figure 3: (a) The structure of the lattice expanded by additional fake (inactive) edges. The structure learning task is to find the true edges, which were used to produce the observed cascades. (b) Average ROC curve surface, as a function of the number of available cascades for square lattice with additional fake (inactive) edges in the case where a fraction $\xi$ of nodes is unobserved. Each point is averaged over five different sets of parameters $\alpha^*_{ij}$ (sampled from a uniform distribution in the range $[0,1]$). Network contains $N=100$ nodes. Unobserved nodes were picked at random. All cascades had length $T=5$.
• Figure 4: Structure learning starting with a super-set of edges. Average ROC curve surface, as a function of the number of available cascades for different network types in the case where a fraction $\xi$ of nodes is unobserved and there is a known set of potential edges, where the number of fake edges is equal to the true ones. Each point is averaged over five different networks and five different sets of parameters $\alpha^*_{ij}$ (sampled from a uniform distribution in the range $[0,1]$). All networks contain $N=100$ nodes and all but the tree have average degree $\langle k \rangle = 3$. Unobserved nodes were picked at random. All cascades had length $T=5$.
• Figure 5: Learning of simple graphs with equal transmission probabilities on all edges. Average difference between inferred $\alpha_{ij}$ and real $\alpha^*_{ij}$ parameters in $\ell_1$ norm, as a function of the number of available cascades for different network types in the case where a fraction $\xi$ of nodes is unobserved. The dashed gray line denotes the benchmark case where one is using standard SLICER without imposing the constraint of equal parameters on all edges. The triangles corresponds to the case where SLICER includes the knowledge about equal parameters. Each point is averaged over 5 different networks with parameters $\forall_{(i, j) \in E} \, \alpha^*_{ij} = 0.5$. All networks contain $N=100$ nodes and all but the tree have average degree equal to $\langle k \rangle = 3$. Unobserved nodes were picked at random. All cascades had length equal to $T=5$.