Integrative Learning of Dynamically Evolving Multiplex Graphs and Nodal Attributes Using Neural Network Gaussian Processes with an Application to Dynamic Terrorism Graphs

Abstract

Exploring the dynamic co-evolution of multiplex graphs and nodal attributes is a compelling question in criminal and terrorism networks. This article is motivated by the study of dynamically evolving interactions among prominent terrorist organizations, considering various organizational attributes like size, ideology, leadership, and operational capacity. Statistically principled integration of multiplex graphs with nodal attributes is significantly challenging due to the need to leverage shared information within and across layers, account for uncertainty in predicting unobserved links, and capture temporal evolution of node attributes. These difficulties increase when layers are partially observed, as in terrorism networks where connections are deliberately hidden to obscure key relationships. To address these challenges, we present a principled methodological framework to integrate the multiplex graph layers and nodal attributes. The approach employs time-varying stochastic latent factor models, leveraging shared latent factors to capture graph structure and its co-evolution with node attributes. Latent factors are modeled using Gaussian processes with an infinitely wide deep neural network-based covariance function, termed neural network Gaussian processes (NN-GP). The NN-GP framework on latent factors exploits the predictive power of Bayesian deep neural network architecture while propagating uncertainty for reliability. Simulation studies highlight superior performance of the proposed approach in achieving inferential objectives. The approach, termed as dynamic joint learner, enables predictive inference (with uncertainty) of diverse unobserved dynamic relationships among prominent terrorist organizations and their organization-specific attributes, as well as clustering behavior in terms of friend-and-foe relationships, which could be informative in counter-terrorism research.

Figures (5)

• Figure 1: Diagrammatic representation of a two-layer multiplex graph with nodal attributes over $T$ time points between terrorist organizations (nodes). The two layers are represented by "Alliance" and "Rivalry" between organizations. A black edge between nodes denotes a known (observed) link between them. A white edge with a "?" denotes an unknown (unobserved) link between the corresponding nodes.
• Figure 2: Diagram of the dependencies among observations (${\boldsymbol A} ^{(o)}_{l}(t)$), $x_{j,k}(t)$), latent variables ($\mu(t)$, ${\boldsymbol \zeta} _{j}(t)$, ${\boldsymbol \xi} _{j,l}(t)$, ${\boldsymbol \alpha} _{k, l}(t)$, $\eta_{k}(t)$), and parameters ($\sigma^{2}_{k}$, ${\boldsymbol \beta} _{\mu}$, ${\boldsymbol \beta} _{\zeta}$, ${\boldsymbol \beta} _{\xi}$, ${\boldsymbol \beta} _{\alpha}$, ${\boldsymbol \beta} _{\eta}$).
• Figure 3: First row shows results for edge prediction and point prediction of nodal attributes for the proposed DJL, along with DML, JLAFAC(first) and JLAFAC(second) at in-sample, missing and out-of-sample scenarios. The second row shows coverage and length of 95% prediction intervals (PIs) for nodal attributes for all competitors at in-sample, missing and out-of-sample scenarios.
• Figure 4: Visual represention of the dynamic relationships between between Hizballah (Hi), Hamas (Ha), and the Palestinian Islamic Jihad (PIJ) over time. The edges in 2010 are dotted and have a check mark to represent that they were unobserved and correctly predicted by DJL as being present.
• Figure 5: Plots over time of the first two principal components of the latent factors, ${\boldsymbol \zeta} _{j}(t)$, for each organization after performing Procrustes transformations.