Susceptibility Analysis in Complex Systems

• Susceptibility analysis is a method that quantifies how systems respond to external perturbations by measuring changes through derivatives or covariances.
• It employs analytical and numerical approaches, such as 1/N expansion, FRG, and topological data analysis, to reveal critical transitions and system robustness.
• Applications range from optimizing material properties and predicting epidemic spread to interpreting neural network behavior, highlighting its broad impact.

Susceptibility analysis refers to a broad class of quantitative and computational methods used to assess the sensitivity or responsiveness of a physical, biological, social, or artificial system to external perturbations. The notion of susceptibility originated in statistical physics as a measure of the response—such as magnetization—to external fields, but has since found widespread application in condensed matter theory, dynamical systems, network science, epidemiology, social dynamics, machine learning, and neurobiology. Susceptibility is often formalized as a derivative or covariance between an observable and a perturbative parameter, and is central for interpreting phase transitions, system robustness, risk profiles, and the structure of complex models.

1. Principles and Definitions

Susceptibility (commonly denoted as χ) generally quantifies the linear response of an order parameter or system observable φ in the presence of an infinitesimal external perturbation h. Mathematically, this is written as

$$\chi = \frac{\partial\langle\phi\rangle}{\partial h} \bigg|_{h=0}$$

where ⟨φ⟩ is the expectation value of φ under the (possibly perturbed) system state. The external field h might correspond to a magnetic field in physics, an infectious pressure in epidemiology, or a feature perturbation in machine learning. Susceptibility can also be framed in terms of the covariance between observables and the generator of the perturbation, as in

$\chi = -\text{Cov}[\phi, \Delta L]$

where ΔL is the change in log-likelihood or potential due to the applied perturbation (Baker et al., 25 Apr 2025).

In statistical mechanics, susceptibility often exhibits singular or divergent behavior near a phase transition, marking critical points of qualitative change in the system.

2. Analytical and Numerical Approaches

A wide array of theoretical and computational frameworks have been developed to analyze susceptibility in complex systems:

• 1/N Expansion and Saddle-Point Analysis: Extensively used in quantum field theory to analyze the O(N) model, where susceptibility is found from the saddle-point equations for auxiliary fields (Branchina et al., 2013). As the external source J → 0 in the broken phase, the longitudinal susceptibility exhibits power-law or logarithmic scaling with J:

$$\chi_L^{-1}(J) \sim J^{\epsilon/2} \;\;\; (2 < d < 4), \qquad \chi_L^{-1}(J) \sim [\ln J]^{-1} \;\; (d=4)$$

where ε = 4 - d.

• Functional Renormalization Group (FRG): Used to investigate susceptibility flow under changes of coarse-graining scale, allowing nonperturbative treatment of strong fluctuations (Branchina et al., 2013). In FRG, the scaling of susceptibility with control parameters and the convexity of the effective potential are accessible.
• Signal Detection Theory (SDT): In the context of human or algorithmic error analysis (e.g., phishing susceptibility), SDT provides a rigorous decomposition of truth sensitivity (discrimination d′) and bias (criterion c) in decision-making (Unchit et al., 2020, Nahon et al., 31 May 2024).
• Computational Models (Machine Learning, Networks): In social influence and misinformation analysis, susceptibility is modeled as a latent user attribute inferred via neural networks, regression on exposure-response statistics, or response to simulated perturbations in data distributions (Liu et al., 2023, Baker et al., 25 Apr 2025, Luceri et al., 17 Jun 2024).
• Topological Data Analysis (TDA): For complex time series such as brain signals, susceptibility to pathological events (e.g., seizures) is probed by quantifying the persistence and entropy of topological features (connected components, cycles) reconstructed from delay-embedded LFP recordings (Lucas et al., 2 Dec 2024).
• Physical Modeling and Phase Decomposition: In material science, analyzing differential susceptibility curves (the derivative of magnetization with respect to field, χ(H) = dM/dH) using Lorentzian peak fits allows robust decomposition of multiphase materials (Perevertov, 1 Jul 2024).

3. Case Studies and Applications

Domain	Observable / Perturbation	Methodology
O(N) Quantum Field Theory	Longitudinal 2-pt function, source J	1/N, FRG, Callan–Symanzik
Magnetic Materials	Differential M(H) loop	Lorentzian decomp., Preisach model
Infectious Disease Spread	Outbreak probabilities, local risk	Spatial GP, dimension reduction
Online Social Influence	Adoption probability by exposure	Network regression, GFP analysis
Machine Learning Interpretability	NN response to data distribution shift	Linear response, SGLD sampling
Brain Dynamics	Topological structure of LFP time series	TDA (persistent homology)
Biospeckle Contrast (Microbiology)	ROI image contrast under treatment	Time-series PCA, clustering

Physical Examples: In the O(N) theory, the scaling of susceptibility with external source and its vanishing rate encode universal features of spontaneous symmetry breaking (Branchina et al., 2013).

Material Analysis: Lorentzian modeling of DS curves enables phase attribution and robust baseline correction, reducing noise compared to classical Preisach approaches (Perevertov, 1 Jul 2024).

Epidemiology: Spatial susceptibility mapping using Gaussian processes enables the identification of high-risk regions for targeted intervention, even in the presence of spatially correlated unobserved confounders (Li et al., 2021).

Social Networks: In social systems, the Susceptibility Paradox indicates that individuals prone to influence are often connected, creating clusters of high susceptibility and reinforcing diffusion dynamics. Predictive models show that a user's susceptibility can be inferred primarily from that of their peers (Luceri et al., 17 Jun 2024).

Neural Networks: Linear response analysis reveals that susceptibilities, efficiently estimated via local SGLD, uncover the modular organization of LLMs and yield high-resolution attribution to individual input tokens (Baker et al., 25 Apr 2025).

4. Scaling Laws and Critical Behavior

Power-law scaling and critical exponents frequently emerge in susceptibility analysis near phase transitions or bifurcation points. For the O(N) theory in $2

$$\chi_L^{-1}(J) \propto J^{\epsilon/2}, \;\; \epsilon = 4-d$$

while for $d=4$ the behavior is logarithmic. These results are robust across analytic (1/N, FRG, Callan–Symanzik) and numerical methods (Branchina et al., 2013).

In statistical inference and field theory, this scaling is intimately related to universality classes and the underlying geometry/topology of the model's state space.

5. Interpretation, Attribution, and Response Matrices

Susceptibility analysis often yields high-fidelity, interpretable attributions:

• Per-token/decomposition: In neural networks, susceptibilities can be factorized into signed, per-sample or per-token attributions (Baker et al., 25 Apr 2025).
• Response matrices and modularity: Measuring responses to a range of perturbations across observables enables matrix factorization (e.g., SVD), revealing low-rank structure and functional modules such as "multigram circuits" or induction heads in transformers (Baker et al., 25 Apr 2025).
• Peak fitting in physical systems: Lorentzian deconvolution produces direct phase-specific measures (position, width, amplitude) for multiphase material analysis (Perevertov, 1 Jul 2024).

6. Broader Implications and Comparative Insights

Susceptibility analysis serves as a unifying quantitative concept for:

• Detecting and characterizing phase transitions, including reentrant behavior in strongly correlated or constrained systems (Neto et al., 2016).
• Identifying latent risk factors and guiding resource allocation in epidemiology, disaster preparedness, and infrastructure (Li et al., 2021, Alvioli et al., 2021, Heyns et al., 2020).
• Designing targeted interventions against social risks, misinformation propagation, or system vulnerabilities, guided by quantified homophily and susceptibility mapping (Luceri et al., 17 Jun 2024, Liu et al., 2023).
• Interpreting and debugging deep learning models via provable local response theory (Baker et al., 25 Apr 2025).

Advantages of Susceptibility Analysis:

• Robustness to noise, especially when using topological measures or variance-based decompositions.
• High interpretability, with clear mathematical and statistical definitions.
• Compatibility with both analytic and numerical methods, including simulation, probabilistic modeling, and deep learning frameworks.

Limitations and Caveats:

• Some methods (e.g., Lorentzian modeling) are tailored to specific system types and may not generalize to all material classes (Perevertov, 1 Jul 2024).
• For strongly nonlinear systems, linear susceptibility may inadequately capture higher-order effects.
• Interpretation depends critically on the choice of observable and perturbation.

7. Future Directions

• Hybrid approaches: Integrating topological, statistical, and machine learning techniques for complex susceptibility landscapes (e.g., in brain–machine interfaces or resilient infrastructure).
• Beyond linear response: Development of nonlinear susceptibility measures to capture higher-order effects, especially in out-of-equilibrium systems or adversarial contexts (e.g., adversarial examples in neural networks).
• Spatial-temporal and multi-modal susceptibility mapping: Combining spatiotemporal data, network structure, and domain knowledge to produce layered risk and response maps across domains.

Susceptibility analysis remains a cornerstone methodology for disentangling sensitivity and robustness across domains, enabling deep insights from physical sciences to data-driven applications.