Geometric Extremal Graphical Models

• Geometric extremal graphical models are unified frameworks that exploit graph structures and gauge functions to represent and analyze extreme tail dependencies in high-dimensional random vectors.
• They factorize complex dependence structures via decomposable graphs, enabling modular inference and propagation of extremal dependence coefficients along network paths.
• Applications span environmental statistics, finance, and network risk, offering scalable and interpretable solutions for high-dimensional extreme event analysis.

Geometric extremal graphical models are a unified framework for modeling the extremal (tail) dependence structure of high-dimensional random vectors using the geometric structure of graphs and gauge functions. Central to this theory is the connection between large deviation geometry, conditional independence in multivariate extremes, and sparse statistical modeling. These models allow tractable representation and inference for complex extremal dependencies and conditional independence relations in large systems, unifying developments in max-stable, multivariate regular variation, and conditional extreme value theory. Applications extend from environmental statistics (e.g., river networks, meteorology) to finance and network risk modeling.

1. Mathematical Foundations: Gauge Functions and Limit Sets

The geometric approach begins with high-dimensional random vectors $\bm X=(X_1,\dots,X_d)$ with light-tailed marginal distributions (e.g., standard exponential or Laplace). Under suitable scaling—such as $r_n\sim\log n$—the point cloud $\{\bm X_1/r_n, \dotsc, \bm X_n/r_n\}$ concentrates in probability on a deterministic compact set

$G = \{\bm x \in \mathcal S^d : g(\bm x) \leq 1\}$

where $\mathcal S$ is $[0,\infty)$ for exponential or $\mathbb{R}$ for Laplace margins, and $g$ is a continuous, 1-homogeneous gauge function. The scaling property

$$-\frac{1}{t}\log f(t\bm x_t) \longrightarrow g(\bm x)\qquad (t\to\infty,\; \bm x_t\to\bm x)$$

directly links the tail structure of the joint density to the gauge $g$. Lower-dimensional marginal gauges are constructed as $g_J(\bm x_J) = \min_{x_i: i \notin J} g(\bm x)$. The shape of $G$ and analytic form of $g$ encode all aspects of extremal dependence and possible directions of co-occurring large values in the system (Papastathopoulos et al., 1 Jan 2026).

2. Graphical Structure and Factorization: Block Graphs and Hammersley–Clifford Decomposition

Geometric extremal graphical models impose sparsity in the gauge $g$ reflecting an undirected graph $\mathcal{G} = (V, E)$. For decomposable (i.e., chordal) graphs with clique set $\mathcal{C}$ and separator set $\mathcal{D}$, the gauge factorizes as

$$g(\bm x) = \sum_{C \in \mathcal{C}} g_C(\bm x_C) - \sum_{D \in \mathcal{D}} g_D(\bm x_D)$$

where $g_C$ is the clique marginal gauge. For block graphs (all separators are singletons), this simplifies to

$$g(\bm x) = \sum_{C \in \mathcal{C}} g_C(\bm x_C) - \sum_{j \in \mathcal{D}} |x_j|$$

This factorization mirrors the extremal Hammersley–Clifford theorem for density models, allowing modular specification and scalable inference (Engelke et al., 2018, Papastathopoulos et al., 1 Jan 2026).

Conditional independence in the tails is defined (for multivariate Pareto limits) not through classical independence, but via absence of direct interaction in $g$ or vanishing off-diagonal entries in extremal precision matrices. For block graphs, this implies that specifying bivariate or low-dimensional clique gauges suffices for consistent global modeling, subject to no further compatibility constraints (Hentschel et al., 2022, Wan et al., 2023).

3. Extremal Dependence Coefficients and Propagation

The geometric framework provides rigorous results on the behavior of extremal dependence measures on graphs, especially block graphs:

• Tail-dependence coefficient $\chi_{ij}$ quantifies the probability both $X_i$ and $X_j$ are simultaneously large:

$$\chi_{ij} = \lim_{u \to 1^-} \Pr[F_i(X_i) > u \mid F_j(X_j) > u] \in [0,1]$$

• Conditional extreme-value coefficients—location $\alpha_{j|i}$ and scale exponent $\beta_{j|i}$—describe the shape of $(X_j \mid X_i \to \infty)$. For exponential margins,

$$\alpha_{j|i} = \max \{\alpha \geq 0 : g_{\{i,j\}}(1,\alpha) = 1\}$$

The propagation theorem asserts that, on a block graph, $\alpha_{j|i}$ factors along the shortest path: $\alpha_{j|i} = \prod_{k=1}^m \alpha_{v_k|v_{k-1}}$, tracing from $i$ to $j$ (Papastathopoulos et al., 1 Jan 2026).

This propagation reflects that the influence of an extreme event at one node decays along network paths, directly paralleling graphical Markov properties in Gaussian models.

Additionally, the geometric perspective distinguishes between "conditional extremes" (directional, one variable large) and "joint extremes" (simultaneous extremes in a subset $A$), the latter characterized by testing if the vector $\bm z^A$ (with 1s for $A$ and sufficiently small values elsewhere) lies on $\partial G$ (Papastathopoulos et al., 1 Jan 2026).

4. Parametric and Algorithmic Instantiations

Several parametric families have been adapted to the geometric extremal graphical paradigm:

• Hüsler–Reiss models: Defined by a variogram matrix $\Gamma$, with likelihood and Markov properties governed by an associated extremal precision matrix $\Theta$, whose zeros encode graphical sparsity. For trees, $\Gamma_{ij}$ equals the sum of edgewise variograms along the path from $i$ to $j$. Block-graph structure is recovered via explicit metric gluing (Hentschel et al., 2022, Engelke et al., 2021, Wan et al., 2023).
• Max-linear models: On directed acyclic graphs or trees of transitive tournaments, these models admit explicit max-factor representations, stable tail dependence functions, and Markov-like conditional tail factorizations mirroring product-of-increments on trails (Segers et al., 2022).
• Partial Tail-Correlation Coefficient (PTCC): An analog of the Gaussian partial correlation, calculated as the normalization of the appropriate residual block in the estimated angular measure/covariance of the spectral measure, with zeros corresponding to partial tail-uncorrelatedness (Gong et al., 2022).

Algorithmic development includes:

• Extreme graphical lasso (EGlearn) and Laplacian-spectral-constraint methods for graph structure estimation from high-dimensional extremes (Hentschel et al., 2022, Wan et al., 2023, Gong et al., 2022).
• Minimum spanning tree or forward selection in the space of extremal variograms, facilitating scalable learning and model selection (Engelke et al., 2018).

5. Conditional Extreme Value Models and Statistical Inference

The conditional multivariate extreme value model (CMEVM) framework extends geometric graphical modeling to situations with both asymptotic dependence (AD) and asymptotic independence (AI). In this setup, marginal transformation to Laplace or exponential scale allows for centering and scaling so that the residual vector converges to a parametric distribution on the graphical model, supporting both fully-connected and sparse topologies. The inference strategy leverages profile likelihood and graphical lasso on transformed residuals to provide robust estimation and selection procedures in dimensions up to hundreds (Farrell et al., 2024).

For Hüsler–Reiss graphical models, inference proceeds via (i) thresholding and normalization to transform data to Pareto or exponential scale, (ii) estimating edgewise variograms or precision elements, (iii) global completion or penalized likelihood optimization under sparsity constraints, and (iv) combinatorial graph selection via pseudo-MBIC or majority-voting algorithms (Hentschel et al., 2022, Engelke et al., 2021).

6. Applications and Interpretable Network Structures

Empirical studies demonstrate that geometric extremal graphical models naturally recover meaningful network structures in real-world extremes:

• River network extremes: Block-graph or tree models capture both river-flow topology and meteorological proximity effects for flood risk along basin-wide measurement stations (Engelke et al., 2018, Engelke et al., 2021, Hentschel et al., 2022, Farrell et al., 2024, Gong et al., 2022).
• Financial risk and market contagion: In the context of exchange rates and stock returns, learned extreme graphs partition edges in line with trading blocs, market shocks, or geopolitical segmentation (Hentschel et al., 2022, Gong et al., 2022).
• Aviation delay networks: Sparse geometric extremal networks coincide with storm regions, flight hub structure, or regulatory perturbations (Hentschel et al., 2022).

A consistent finding is that learned graphical models, through the geometric framework, balance parsimony and predictive accuracy, outperforming dense models in high-dimensional regimes and offering interpretable summaries of extremal dependence.

7. Connections, Extensions, and Outlook

Geometric extremal graphical models provide a statistical apparatus paralleling Gaussian graphical models, but tailored to max-stable, Pareto, or other heavy/light-tailed regimes. Their direct interpretability via gauge factorization allows modular construction, propagation of dependence coefficients, and straightforward model selection.

Key extensions under active investigation include:

• Incorporation of mixture-of-max, sum-linear, or hybrid recursions for heavy-tailed or mixed marginal settings (Segers et al., 2022).
• Generalization from block graphs to arbitrary decomposable and non-decomposable graphs using matrix completions and penalized likelihood methods (Hentschel et al., 2022).
• Combination with conditional extreme value theory to jointly model both joint and conditional tail behavior in non-symmetric, asymptotically independent regimes (Farrell et al., 2024, Papastathopoulos et al., 1 Jan 2026).

The geometric factorization framework and propagation theorems for coefficients such as $\alpha_{j|i}$ and $\beta_{j|i}$ are central to achieving scalable, interpretable, and theoretically grounded inference in high-dimensional extremes (Papastathopoulos et al., 1 Jan 2026).