Collective Noise Filtering in Complex Networks

TL;DR

The Network Wiener Filter is introduced, a principled method for collective edge-level noise filtering that leverages both network topology and noise characteristics, to reduce error in the observed edge weights and to infer missing edge weights.

Abstract

Complex networks are powerful representations of complex systems across scales and domains, and the field is experiencing unprecedented growth in data availability. However, real-world network data often suffer from noise, biases, and missing data in the edge weights, which undermine the reliability of downstream network analyses. Standard noise filtering approaches, whether treating individual edges one-by-one or assuming a uniform global noise level, are suboptimal, because in reality both signal and noise can be heterogeneous and correlated across multiple edges. As a solution, we introduce the Network Wiener Filter, a principled method for collective edge-level noise filtering that leverages both network topology and noise characteristics, to reduce error in the observed edge weights and to infer missing edge weights. We demonstrate the broad practical efficacy of the Network Wiener Filter in two distinct settings, the genetic interaction network of the yeast S. cerevisiae and the Enron Corpus email network, noting compelling evidence of successful noise suppression in both applications. With the Network Wiener Filter, we advocate for a shift toward error-aware network science, one that embraces data imperfection as an inherent feature and learns to navigate it effectively.

Figures (4)

• Figure 1: Illustration of Wiener filtering for noise reduction across data types, and why networks are different. Starting from (A) a time series, (B) an image, and (C) a bipartite toy network, we add independent and identically distributed Gaussian white noise to obtain noisy observations. We then apply Wiener filtering to estimate the underlying signal from noisy observations using the known noise statistics (more details in SI Text [placeholder] and SI Figure [placeholder]). In time series and images, the classical Wiener filter exploits data-independent temporal or spatial proximity to capture signal correlations, with the intuition that nearby samples tend to be similar. In contrast, network data is inherently high-dimensional and requires a data-dependent adaptive treatment. The NetWF bridges this gap by using a network-informed edge-similarity measure that captures correlations globally across edges, as discussed in the main text. In the toy network, our NetWF reveals the original structure of two communities.
• Figure 2: Construction of our global edge-similarity measure, a key ingredient in the NetWF. We propose that edges are similar if their endpoints are similar. The similarity between two nodes is in turn quantified by correlating their connection profiles with all other nodes in the network. (A) In a directed network, for edges $\overrightarrow{AB}$ and $\overrightarrow{CD}$, we separately consider the two sources ($A$ vs. $C$) using their outgoing connection patterns and the two targets ($B$ vs. $D$) using their incoming connection patterns (Eq. \ref{['eqn:directed_edge_sim']}). (B) In an undirected network, the similarity between edges $\overline{AB}$ and $\overline{CD}$ is computed via considering the two configurations of endpoint matching, yielding a label-invariant similarity measure (Eq. \ref{['eqn:undirected_edge_sim']}).
• Figure 3: Evidence that the NetWF reduces experimental noise in the yeast S. cerevisiae genetic interaction (GI) network. (A) Validation of negative GI values against three biological benchmarks: protein--protein interaction (PPI), protein co-complex membership, and Gene Ontology (GO) co-annotation. Relative to the raw data and the optimal shrinker (OS) baseline, NetWF improves both area under the precision--recall curve (AUPRC) and fold enrichment across all benchmarks. (B) A 10-fold cross validation (CV) test: in each fold, a random subset of GI entries is masked and predicted from the remaining data. The NetWF and the OS achieve significantly lower average mean square error (MSE) than naïve mean imputation (MI), where the missing entries are imputed using the mean of available GI values in each gene. Error bars indicate standard errors across folds. (C) Example of inferring a novel GI within the exocyst complex (SEC15--SEC6) by the NetWF. Dashed edges highlight a few example profile similarities within the complex that facilitate information propagation, collectively shifting the GI value from zero to a strongly negative estimate. (D) Profile similarity networks (PSNs) showcasing similarities between gene pairs (Eq. \ref{['eqn:undirected_node_sim']}), before and after the NetWF is applied to the raw GI network. Edge width scales with similarity, and similarities below $0.2$ are omitted for clarity.
• Figure 4: Evidence that the NetWF reduces sampling noise in the Enron Corpus email network. (A) Full-year email exchange frequency network of 2001 between employees as nodes. Directed edges point from sender to recipient, with edge weight equal to email frequency (defined as the number of emails per month). Node size reflects the sum of incoming and outgoing weights. For clarity, edges with weight $<0.2$ are not shown. (B) Raw October 2001 snapshot, treated as a noisy calendar-month sample of the full-year network, exhibiting pronounced fluctuations in edge weights. (C) October 2001 after applying the NetWF, with the extreme weights reduced without compromising the global structure. (D) Relative to the raw October network and the OS baseline, the NetWF largely reduces the October MSE with respect to the full-year network. Repeating for all twelve calendar-month snapshots, the NetWF also attains the lowest and most stable average MSE. Error bars indicate standard errors across calendar months.