Evidential Reconstruction of Network from Time Series

TL;DR

A framework that is based on the Dempster-Shafer evidence theory to infer network structures directly from time series is proposed and suggests that evidential reasoning offers a powerful and scalable approach for uncovering the structural organization of complex systems.

Abstract

Reconstructing the topology of complex networks from observational data remains a central challenge in network science. Here we propose a framework that is based on the Dempster-Shafer evidence theory to infer network structures directly from time series. By integrating multi-source information within an evidential reasoning scheme, the method captures underlying interaction patterns with high fidelity. Tests on three representative network models Barabasi-Albert Network, Erdos-Renyi Network, and Watts-Strogatz Network-show that the reconstruction accuracy is consistently high and remains robust against increases in network size and density. The application of the framework to real-world datasets from diverse domains further confirms its stability and general applicability. These results suggest that evidential reasoning offers a powerful and scalable approach for uncovering the structural organization of complex systems, especially when dealing with uncertain or incomplete multi-source data.

Figures (14)

• Figure 1: The process of evidential network reconstruction. The subfigure (a) shows the overall reconstruction procedure, in which Basic Probability Assignments (BPA) are constructed from time-series observations of an unknown network, followed by the fusion of multi-source time-series BPA, ultimately reconstructing the topology of the unknown network based on decision criteria. The subfigure (b) specifically demonstrates the extraction of two types of BPA—from both positive and negation perspectives-from a single time series, and their fusion into the BPA for that series. The subfigure (c) presents two decision methods: the Decision Rule Based on Minimum Robustness (DR-MR) and the Decision Rule Based on Maximum Similarity (DR-MS), with the former providing the upper bound of the decision range for the latter.
• Figure 2: The association relationships between network nodes from time series. Specifically: (a) shows that nodes are considered associated in a given time step if they meet specific conditions; (b) outlines the criteria for determining association between nodes; (c) describes the conditions for confirming non-association; (d) represents cases of uncertainty, which provide no valid information regarding node associations.
• Figure 3: The process of generating $\mathbf{M^{(A)}}$ and $\mathbf{M^{(N)}}$ from time-eries data, and further constructing BPA matrices. Specifically, the $\mathbf{M^{(A)}}$ is built based on the number of associations between node pairs, while the $\mathbf{M^{(N)}}$ is derived from the number of non-associations between node pairs. Using these two matrices together with the BPA generation function (equations. \ref{['eq1']}-\ref{['eq6']}), six BPA matrices are produced. Only two of them, $m^{(A)}\{T\}$ and $m^{(N)}\{T\}$, are shown in this figure.
• Figure 4: The generation of BPA from $\mathbf{M^{(N)}}$. It depicting how BPA is derived from a non-associated adjacency matrix. For a given node pair, if the cumulative association result is x, it corresponds to a belief assignment: $m^{(N)}\{T\}$ indicates that the degree of belief in the existence of an association between this node pair is 0.22, while $m^{(N)}\{T,F\}$ indicates that the degree of belief in being uncertain about its connection state is 0.78. Similarly, BPA from $\mathbf{M^{(A)}}$ can be generated in an analogous manner using this method.
• Figure 5: The changes in uncertainty during the fusion process of six BPA matrices. It demonstrates that the first fusion step (combining $m_A\{T\}$, $m_A\{F\}$, $m_A\{T,F\}$ with $m_N\{T\}$, $m_N\{F\}$, $m_N\{T,F\}$) reduces uncertainty. Subsequently, during the second fusion step (which integrates information from different time series), uncertainty may either increase or decrease—if contradictory time series are incorporated, uncertainty rises; otherwise, it continues to decrease.