Dynamic Heterogeneity in Complex Systems

• Dynamic heterogeneity refers to transient, spatially correlated fluctuations in local particle mobility in disordered matter, defining regions of fast and slow dynamics.
• It is quantified using methods like four-point correlation functions, non-Gaussian metrics, and overlap analysis, which reveal insights into structural relaxation and fragility.
• This concept has broad applications in glass physics, jamming, active matter, and complex systems, providing practical tools to predict material responses under diverse conditions.

Dynamic heterogeneity denotes the transient, spatially correlated fluctuation of local dynamical behavior in a disordered system: over a time window comparable to structural relaxation, some regions or particles are highly mobile while others remain nearly immobile, and these fast and slow domains evolve in time rather than forming a permanent pattern. In glass-forming liquids and amorphous materials, this phenomenon has become a central statistical-mechanical signature of slow relaxation, because it links microscopic intermittency, cooperative motion, nonexponential relaxation, and growing dynamic length scales; related usages now also appear in jamming, active matter, driven colloids, granular deformation, and several non-glassy heterogeneous dynamical systems (Berthier et al., 2010, Berthier, 2011).

1. Definition and conceptual basis

In the glass-physics sense, dynamic heterogeneity is not merely a broad spectrum of relaxation times. It is the existence of temporary spatial fluctuations in local mobility, with mobile particles tending to cluster near other mobile particles and immobile regions likewise appearing in correlated patches. The central physical point is that amorphous materials are structurally disordered yet dynamically collective: the slowing down of relaxation is not spatially uniform, and the relevant correlations are dynamical rather than conventional static density fluctuations (Berthier, 2011).

A standard formalization introduces a local mobility variable $c_i(t,0)$, for example

$$c_i(t,0)=e^{-|\mathbf r_i(t)-\mathbf r_i(0)|^2/d^2},$$

and the mobility field

$$c(\mathbf r;t,0)=\sum_{i=1}^N c_i(t,0)\,\delta[\mathbf r-\mathbf r_i(0)].$$

Dynamic heterogeneity is then encoded in higher-order correlations of this field, especially the four-point function

$$G_4(r;t)=\langle c(\mathbf r;t,0)c(\mathbf 0;t,0)\rangle-\langle c(\mathbf r;t,0)\rangle^2,$$

its Fourier transform $S_4(q;t)$, and the associated susceptibility

$$\chi_4(t)=\int d^d\mathbf r\, G_4(r;t) =N\left[\langle C(t,0)^2\rangle-\langle C(t,0)\rangle^2\right],$$

with $C(t,0)=\int d^d\mathbf r\, c(\mathbf r;t,0)$. If a dominant dynamic length scale exists, the large-distance decay is often represented in Ornstein–Zernike-like form,

$G_4(r;t)\approx \frac{A(t)}{r^{p}}e^{-r/\xi_4(t)},$

so that $\chi_4(t)$ estimates the size or volume of a correlated rearranging region (Berthier et al., 2010).

This formulation also clarifies a recurrent misconception. A broad relaxation spectrum or a nonexponential average correlator does not by itself specify dynamic heterogeneity; the stronger statement is that the underlying particle motions or local mobilities are spatially correlated. That distinction is explicit in polymer simulations of Johari–Goldstein relaxation, where a broad distribution of relaxation times is noted not to imply non-Gaussian displacement statistics automatically (Puosi et al., 2021).

2. Statistical observables and inference strategies

In practice, DH is inferred through several complementary observables rather than a single universal scalar. Structural relaxation in supercooled liquids is commonly defined from the self-intermediate scattering function

$$F_s(k,t)=\frac{1}{N}\left\langle \sum_{j=1}^{N} \exp\left[i\vec{k}\cdot(\vec r_j(t)-\vec r_j(0))\right]\right\rangle, \qquad F_s(k,\tau)=e^{-1},$$

while overlap-based definitions use

$$c_i(t,0)=e^{-|\mathbf r_i(t)-\mathbf r_i(0)|^2/d^2},$$0

A moment-based DH measure is the non-Gaussian parameter

$$c_i(t,0)=e^{-|\mathbf r_i(t)-\mathbf r_i(0)|^2/d^2},$$1

or, in one-dimensional form,

$$c_i(t,0)=e^{-|\mathbf r_i(t)-\mathbf r_i(0)|^2/d^2},$$2

Its peak is widely used as a characteristic DH time or amplitude, but four-point quantities remain the more direct probes of correlated mobility (Wang et al., 2017).

A standard overlap-based four-point construction defines

$$c_i(t,0)=e^{-|\mathbf r_i(t)-\mathbf r_i(0)|^2/d^2},$$3

and a spatial four-point function

$$c_i(t,0)=e^{-|\mathbf r_i(t)-\mathbf r_i(0)|^2/d^2},$$4

In the floating-sphere Faraday-wave system, the peak of $$c_i(t,0)=e^{-|\mathbf r_i(t)-\mathbf r_i(0)|^2/d^2},$$5 defines a heterogeneity time scale $$c_i(t,0)=e^{-|\mathbf r_i(t)-\mathbf r_i(0)|^2/d^2},$$6, while a heterogeneity length $$c_i(t,0)=e^{-|\mathbf r_i(t)-\mathbf r_i(0)|^2/d^2},$$7 is extracted by fitting $$c_i(t,0)=e^{-|\mathbf r_i(t)-\mathbf r_i(0)|^2/d^2},$$8 to

$$c_i(t,0)=e^{-|\mathbf r_i(t)-\mathbf r_i(0)|^2/d^2},$$9

with $$c(\mathbf r;t,0)=\sum_{i=1}^N c_i(t,0)\,\delta[\mathbf r-\mathbf r_i(0)].$$0 (Sanlı et al., 2013).

Recent work has emphasized that moment-based non-Gaussianity is not exhaustive. An information-theoretic alternative defines the non-Gaussian information

$$c(\mathbf r;t,0)=\sum_{i=1}^N c_i(t,0)\,\delta[\mathbf r-\mathbf r_i(0)].$$1

equivalently the Kullback–Leibler divergence between the observed self-van Hove function and the Gaussian reference with the same first two moments. The corresponding DH time,

$$c(\mathbf r;t,0)=\sum_{i=1}^N c_i(t,0)\,\delta[\mathbf r-\mathbf r_i(0)].$$2

can differ substantially from the conventional

$$c(\mathbf r;t,0)=\sum_{i=1}^N c_i(t,0)\,\delta[\mathbf r-\mathbf r_i(0)].$$3

especially as intermittency strengthens near the glass transition (Vaibhav et al., 2023).

Characterization is not restricted to particle mobility fields. In quenched Ising dynamics, the dynamical cluster size heterogeneity

$$c(\mathbf r;t,0)=\sum_{i=1}^N c_i(t,0)\,\delta[\mathbf r-\mathbf r_i(0)].$$4

detects short-time percolative and long-time coarsening regimes, showing that DH-like diagnostics can also be framed geometrically when the relevant fluctuating objects are domains rather than mobile particles (Azevedo-Lopes et al., 2019).

3. Structural relaxation, fragility, and scaling in supercooled liquids

A central question in glass physics is whether there is a simple quantitative link between structural relaxation and DH across different glass formers. Molecular-dynamics simulations of six model glass-forming liquids show that, within each system, $$c(\mathbf r;t,0)=\sum_{i=1}^N c_i(t,0)\,\delta[\mathbf r-\mathbf r_i(0)].$$5 increases as the structural relaxation time $$c(\mathbf r;t,0)=\sum_{i=1}^N c_i(t,0)\,\delta[\mathbf r-\mathbf r_i(0)].$$6 increases, but more importantly that at fixed $$c(\mathbf r;t,0)=\sum_{i=1}^N c_i(t,0)\,\delta[\mathbf r-\mathbf r_i(0)].$$7, systems with larger kinetic fragility $$c(\mathbf r;t,0)=\sum_{i=1}^N c_i(t,0)\,\delta[\mathbf r-\mathbf r_i(0)].$$8 have larger $$c(\mathbf r;t,0)=\sum_{i=1}^N c_i(t,0)\,\delta[\mathbf r-\mathbf r_i(0)].$$9. This establishes a positive correlation between fragility and DH in those model liquids and motivates the notion of comparing different systems at identical dynamic heterogeneity, meaning equal $$G_4(r;t)=\langle c(\mathbf r;t,0)c(\mathbf 0;t,0)\rangle-\langle c(\mathbf r;t,0)\rangle^2,$$0 (Wang et al., 2017).

Under this iso-$$G_4(r;t)=\langle c(\mathbf r;t,0)c(\mathbf 0;t,0)\rangle-\langle c(\mathbf r;t,0)\rangle^2,$$1 condition, the relaxation time required to reach the same DH defines a characteristic system-dependent scale $$G_4(r;t)=\langle c(\mathbf r;t,0)c(\mathbf 0;t,0)\rangle-\langle c(\mathbf r;t,0)\rangle^2,$$2, leading to the collapse

$$G_4(r;t)=\langle c(\mathbf r;t,0)c(\mathbf 0;t,0)\rangle-\langle c(\mathbf r;t,0)\rangle^2,$$3

The hidden time scale is then tied to kinetic fragility by

$$G_4(r;t)=\langle c(\mathbf r;t,0)c(\mathbf 0;t,0)\rangle-\langle c(\mathbf r;t,0)\rangle^2,$$4

with $$G_4(r;t)=\langle c(\mathbf r;t,0)c(\mathbf 0;t,0)\rangle-\langle c(\mathbf r;t,0)\rangle^2,$$5 for the chosen reference $$G_4(r;t)=\langle c(\mathbf r;t,0)c(\mathbf 0;t,0)\rangle-\langle c(\mathbf r;t,0)\rangle^2,$$6. The same study also introduces a characteristic temperature $$G_4(r;t)=\langle c(\mathbf r;t,0)c(\mathbf 0;t,0)\rangle-\langle c(\mathbf r;t,0)\rangle^2,$$7 from identical DH in temperature space,

$$G_4(r;t)=\langle c(\mathbf r;t,0)c(\mathbf 0;t,0)\rangle-\langle c(\mathbf r;t,0)\rangle^2,$$8

and combines the two scalings into

$$G_4(r;t)=\langle c(\mathbf r;t,0)c(\mathbf 0;t,0)\rangle-\langle c(\mathbf r;t,0)\rangle^2,$$9

For the master curve in reduced units, the Vogel-Fulcher-Tammann form

$S_4(q;t)$0

fits the full collapse better than mode-coupling, ECG, Avramov-Milchev, or MYEGA forms, which fit only parts of the curve (Wang et al., 2017).

The extraction of a DH timescale itself is not unique. In the Kob–Andersen mixture, the information-theoretic $S_4(q;t)$1 grows faster than the moment-based $S_4(q;t)$2, with

$S_4(q;t)$3

so that $S_4(q;t)$4. At lower temperature, $S_4(q;t)$5 continues to grow even after $S_4(q;t)$6 has peaked, implying that low-order moments can underestimate both the extent and duration of DH (Vaibhav et al., 2023).

A distinct proposal for the origin of DH growth interprets supercooled-liquid heterogeneity as the finite-temperature manifestation of zero-temperature avalanche criticality. In the three-dimensional Kob–Andersen liquid, the dynamical correlation length and peak four-point susceptibility were found to scale as

$S_4(q;t)$7

$S_4(q;t)$8

with

$S_4(q;t)$9

The same work reports a finite-size scaling collapse of $$\chi_4(t)=\int d^d\mathbf r\, G_4(r;t) =N\left[\langle C(t,0)^2\rangle-\langle C(t,0)\rangle^2\right],$$0 using independently determined exponents, and explicitly states that the avalanche-criticality picture explains the temperature and system-size dependence of DH above $$\chi_4(t)=\int d^d\mathbf r\, G_4(r;t) =N\left[\langle C(t,0)^2\rangle-\langle C(t,0)\rangle^2\right],$$1, while the observed saturation below $$\chi_4(t)=\int d^d\mathbf r\, G_4(r;t) =N\left[\langle C(t,0)^2\rangle-\langle C(t,0)\rangle^2\right],$$2 remains an open problem (Oyama et al., 4 Apr 2026).

4. Secondary relaxation, interfaces, and materials-specific mechanisms

Dynamic heterogeneity is not confined to the primary $$\chi_4(t)=\int d^d\mathbf r\, G_4(r;t) =N\left[\langle C(t,0)^2\rangle-\langle C(t,0)\rangle^2\right],$$3-relaxation window. In a model polymer exhibiting Johari–Goldstein relaxation, the non-Gaussian parameter develops two peaks that grow on cooling, with maxima located near the JG and structural relaxation times. The first maximum is associated with localized secondary mobility, the second with large-scale structural rearrangement. The temporal evolution is explicitly nonmonotonic: fast local bond reorientation during the JG process partially averages out previously visible heterogeneity before DH grows again at the $$\chi_4(t)=\int d^d\mathbf r\, G_4(r;t) =N\left[\langle C(t,0)^2\rangle-\langle C(t,0)\rangle^2\right],$$4 scale. A subset-memory analysis further shows that the monomers mobile during the JG window are only weakly predictive of those mobile during structural relaxation (Puosi et al., 2021).

A closely related but more specific correspondence appears in simulations of the metallic glass-forming alloy Al$$\chi_4(t)=\int d^d\mathbf r\, G_4(r;t) =N\left[\langle C(t,0)^2\rangle-\langle C(t,0)\rangle^2\right],$$5Sm$$\chi_4(t)=\int d^d\mathbf r\, G_4(r;t) =N\left[\langle C(t,0)^2\rangle-\langle C(t,0)\rangle^2\right],$$6. There, the lifetime of mobile particle clusters $$\chi_4(t)=\int d^d\mathbf r\, G_4(r;t) =N\left[\langle C(t,0)^2\rangle-\langle C(t,0)\rangle^2\right],$$7 nearly coincides with the Johari–Goldstein relaxation time $$\chi_4(t)=\int d^d\mathbf r\, G_4(r;t) =N\left[\langle C(t,0)^2\rangle-\langle C(t,0)\rangle^2\right],$$8, while the previously established correspondence $$\chi_4(t)=\int d^d\mathbf r\, G_4(r;t) =N\left[\langle C(t,0)^2\rangle-\langle C(t,0)\rangle^2\right],$$9 links structural relaxation to the lifetime of immobile clusters. The same study reports

$C(t,0)=\int d^d\mathbf r\, c(\mathbf r;t,0)$0

that the hopping time associated with diffusion coincides with $C(t,0)=\int d^d\mathbf r\, c(\mathbf r;t,0)$1 to within numerical uncertainty, and that the fragile-to-strong transition does not greatly alter the geometrical nature of DH in this material. Mobile clusters and string-like exchange motions thus provide a concrete heterogeneous-dynamics interpretation of JG relaxation in this alloy (Zhang et al., 2021).

Interfacial dynamics can also tune DH in ways that differ from the common glass-forming trend. In athermal polymer nanocomposite films built from polymer-grafted nanoparticles in a chemically identical matrix, the control parameter is the grafted-to-matrix molecular-weight ratio

$C(t,0)=\int d^d\mathbf r\, c(\mathbf r;t,0)$2

The penetration depth

$C(t,0)=\int d^d\mathbf r\, c(\mathbf r;t,0)$3

is reported to be proportional to $C(t,0)=\int d^d\mathbf r\, c(\mathbf r;t,0)$4, so increasing $C(t,0)=\int d^d\mathbf r\, c(\mathbf r;t,0)$5 increases entropic compatibility and matrix penetration into the grafted layer. The study finds that fragility $C(t,0)=\int d^d\mathbf r\, c(\mathbf r;t,0)$6 increases with $C(t,0)=\int d^d\mathbf r\, c(\mathbf r;t,0)$7 while DH decreases with $C(t,0)=\int d^d\mathbf r\, c(\mathbf r;t,0)$8, quantified through the experimentally estimated Kohlrausch-Watts-Williams parameter and the simulated peak non-Gaussian parameter. It explicitly describes this as an anti-correlation between fragility and DH, contrary to most earlier observations on glasses and polymer nanocomposites (Begam et al., 2018).

5. Nonequilibrium, active, jammed, and sheared manifestations

Many nonequilibrium systems exhibit DH without being equilibrium glass formers. A dense monolayer of macroscopic spheres floating on chaotic capillary Faraday waves shows caging, subdiffusion, and later diffusion once the large-scale convective mean flow is subtracted. Standard four-point methods then yield power-law growth of both the heterogeneity time and length scales on approaching jamming,

$C(t,0)=\int d^d\mathbf r\, c(\mathbf r;t,0)$9

$G_4(r;t)\approx \frac{A(t)}{r^{p}}e^{-r/\xi_4(t)},$0

with $G_4(r;t)\approx \frac{A(t)}{r^{p}}e^{-r/\xi_4(t)},$1, and

$G_4(r;t)\approx \frac{A(t)}{r^{p}}e^{-r/\xi_4(t)},$2

These exponents are reported to be consistent with dense colloidal suspensions and sheared microgels, and the conclusions are robust to whether convective subtraction is performed (Sanlı et al., 2013).

In a constantly driven oppositely charged binary colloidal suspension, DH is strongest not in the homogeneous state or the fully developed lane state but in the intermediate pre-lane regime. The transverse overlap relaxation broadens, the four-point susceptibility loses a sharp single peak at intermediate field, and tagged fast/slow particle substructures relax at distinct rates. At larger field, the same susceptibility develops two peaks, indicating separation of fast and slow processes in the lane-forming state (Dutta, 2017).

Active glasses provide a stronger challenge to equilibrium intuition. Simulations and active mode-coupling theory show that activity can make two states have nearly identical structural relaxation times $G_4(r;t)\approx \frac{A(t)}{r^{p}}e^{-r/\xi_4(t)},$3 and similar $G_4(r;t)\approx \frac{A(t)}{r^{p}}e^{-r/\xi_4(t)},$4 while exhibiting very different four-point susceptibility peaks $G_4(r;t)\approx \frac{A(t)}{r^{p}}e^{-r/\xi_4(t)},$5 and DH lengths $G_4(r;t)\approx \frac{A(t)}{r^{p}}e^{-r/\xi_4(t)},$6. The same work estimates $G_4(r;t)\approx \frac{A(t)}{r^{p}}e^{-r/\xi_4(t)},$7 using four independent procedures—finite-size block scaling of $G_4(r;t)\approx \frac{A(t)}{r^{p}}e^{-r/\xi_4(t)},$8, block analysis of the self van Hove function, spatial displacement-displacement correlation, and Ornstein–Zernike fits of the four-point structure factor—and reports dramatic growth of $G_4(r;t)\approx \frac{A(t)}{r^{p}}e^{-r/\xi_4(t)},$9 under activity. For comparable ranges of $\chi_4(t)$0, $\chi_4(t)$1 in the most active system grows by about a factor of $\chi_4(t)$2, versus only $\chi_4(t)$3–$\chi_4(t)$4 in the passive system, directly demonstrating decoupling between average relaxation and heterogeneity (Paul et al., 2021).

In two-dimensional active glasses, interpretation of $\chi_4(t)$5 requires an additional caveat. Run-and-tumble simulations find a new short-time peak in $\chi_4(t)$6, associated with long-wavelength phonon-like fluctuations amplified by activity. The effect is strongest in $\chi_4(t)$7, present in $\chi_4(t)$8, and absent in $\chi_4(t)$9 for the studied system sizes, and is interpreted as a dynamical signature of violation of the equilibrium Mermin-Wagner-Hohenberg scenario. After subtracting cage motion through cage-relative observables, activity suppresses DH in the non-MRCO glass former but enhances it in the MRCO-containing one, even while the cage-relative relaxation time decreases monotonically. This makes clear that increasing activity does not have a single generic effect on DH; local ordering and long-wavelength modes both matter (Dey et al., 8 Jan 2026).

Sheared granular matter displays a similarly nontrivial regime dependence. In three-dimensional triaxial compression with in-situ X-ray tomography, DH is quantified from $$F_s(k,t)=\frac{1}{N}\left\langle \sum_{j=1}^{N} \exp\left[i\vec{k}\cdot(\vec r_j(t)-\vec r_j(0))\right]\right\rangle, \qquad F_s(k,\tau)=e^{-1},$$0-based four-point spatial correlations,

$$F_s(k,t)=\frac{1}{N}\left\langle \sum_{j=1}^{N} \exp\left[i\vec{k}\cdot(\vec r_j(t)-\vec r_j(0))\right]\right\rangle, \qquad F_s(k,\tau)=e^{-1},$$1

fit by

$$F_s(k,t)=\frac{1}{N}\left\langle \sum_{j=1}^{N} \exp\left[i\vec{k}\cdot(\vec r_j(t)-\vec r_j(0))\right]\right\rangle, \qquad F_s(k,\tau)=e^{-1},$$2

and from a facilitation ratio

$$F_s(k,t)=\frac{1}{N}\left\langle \sum_{j=1}^{N} \exp\left[i\vec{k}\cdot(\vec r_j(t)-\vec r_j(0))\right]\right\rangle, \qquad F_s(k,\tau)=e^{-1},$$3

Under large strain increments of $$F_s(k,t)=\frac{1}{N}\left\langle \sum_{j=1}^{N} \exp\left[i\vec{k}\cdot(\vec r_j(t)-\vec r_j(0))\right]\right\rangle, \qquad F_s(k,\tau)=e^{-1},$$4 axial strain, both DH and dynamic facilitation are strongest in the transition regime between the initially elastic and critical-state regimes; the transition-regime correlation length is reported to be approximately $$F_s(k,t)=\frac{1}{N}\left\langle \sum_{j=1}^{N} \exp\left[i\vec{k}\cdot(\vec r_j(t)-\vec r_j(0))\right]\right\rangle, \qquad F_s(k,\tau)=e^{-1},$$5. Once a persistent shear band forms, DH and facilitation weaken inside the band and are stronger outside it. Under small increments of $$F_s(k,t)=\frac{1}{N}\left\langle \sum_{j=1}^{N} \exp\left[i\vec{k}\cdot(\vec r_j(t)-\vec r_j(0))\right]\right\rangle, \qquad F_s(k,\tau)=e^{-1},$$6, both measures remain suppressed across all regimes, implying that correlated rearrangements require a minimum deformation magnitude to organize collectively (Lee et al., 25 Sep 2025).

6. Broader extensions of the term

Outside glass physics, “dynamic heterogeneity” is increasingly used for time-dependent heterogeneity in intrinsic parameters, coefficients, or interaction structure. In recurrent neural populations with heterogeneous intrinsic timescales, heterogeneous dynamical mean-field theory does not reduce to a single effective neuron equation; instead it yields a family of neuron-specific mean-field equations and a common stability condition

$$F_s(k,t)=\frac{1}{N}\left\langle \sum_{j=1}^{N} \exp\left[i\vec{k}\cdot(\vec r_j(t)-\vec r_j(0))\right]\right\rangle, \qquad F_s(k,\tau)=e^{-1},$$7

In that framework, graded-persistent-activity neurons with very long timescales lower $$F_s(k,t)=\frac{1}{N}\left\langle \sum_{j=1}^{N} \exp\left[i\vec{k}\cdot(\vec r_j(t)-\vec r_j(0))\right]\right\rangle, \qquad F_s(k,\tau)=e^{-1},$$8 and expand the dynamical regime, whereas heterogeneous adaptation can raise $$F_s(k,t)=\frac{1}{N}\left\langle \sum_{j=1}^{N} \exp\left[i\vec{k}\cdot(\vec r_j(t)-\vec r_j(0))\right]\right\rangle, \qquad F_s(k,\tau)=e^{-1},$$9 and shrink it (Tomita et al., 2024).

In macroeconomics, dynamic heterogeneity refers to time-varying cross-sectional distributions that feed back into aggregate propagation. A semi-structural approximation around the representative-agent allocation shows that aggregate shocks alter the shares of constrained households and firms, dispersion in consumption shares, and dispersion in firms’ marginal revenue products, and that these changing distributions modify future aggregate responses. The paper explicitly treats those moments as state variables relevant for fiscal multipliers and shock transmission (Tryphonides, 2020).

In panel econometrics, a dynamic heterogeneous distribution regression model assigns each unit a threshold-dependent coefficient function $$c_i(t,0)=e^{-|\mathbf r_i(t)-\mathbf r_i(0)|^2/d^2},$$00 in

$$c_i(t,0)=e^{-|\mathbf r_i(t)-\mathbf r_i(0)|^2/d^2},$$01

allowing predicted actual distributions, counterfactual distributions, stationary distributions, and quantile effects to vary across workers and across the outcome distribution. The application to PSID labor-income data reports strong heterogeneity in persistence and substantial variation in poverty-trap incidence (Fernandez-Val et al., 2022).

In network science, related terminology denotes evolving heterogeneity in node types and temporal tie behavior. DHNet addresses common-community detection in dynamic heterogeneous networks through a modularity-based framework that accommodates multiple node and edge types without requiring the number of communities to be known a priori (Zhang et al., 2022). A distinct two-way heterogeneity model for dynamic networks assigns each node one parameter for baseline tie propensity and another for tie retention, so that

$$c_i(t,0)=e^{-|\mathbf r_i(t)-\mathbf r_i(0)|^2/d^2},$$02

thereby separating static from dynamic heterogeneity (Jiang et al., 2023).

This suggests that the phrase now has a broader methodological life than its original condensed-matter meaning. Its canonical usage remains the spatio-temporal fluctuation of local mobility in disordered matter, quantified by four-point functions, overlap fluctuations, and related response measures; but in adjacent fields it has become a general descriptor for heterogeneity that evolves in time and changes the stability, propagation, or inference properties of the system under study.