Log-Mobility Ratio Overview

• Log-mobility ratio is a quantitative metric that captures logarithmic scaling effects and memory-induced constraints in mobility across diverse systems.
• It is applied in stochastic models, epidemic spread, convex geometry, human trajectories, queueing theory, and privacy protocols to reveal underlying scaling laws and phase transitions.
• Its computation employs logarithmic transformations to quantify phenomena such as diffusion slowdown, contact restoration, geometric efficiency, and secure leakage, enabling actionable insights.

A log-mobility ratio is a quantitative and structural measure that emerges in multiple scientific domains to characterize growth rates, scaling relationships, and diffusion properties connected to mobility or compositional change, typically by exploiting logarithmic transformations or asymptotic behaviors. The concept spans stochastic models of mobility with memory, statistical epidemic modeling, geometric analysis in high-dimensional function spaces, human trajectory studies, queueing theory in networks, cryptographic privacy leakage evaluations, and compositional biomarker selection. Across these domains, the log-mobility ratio is used as either a direct observable, an asymptotic scaling law, an affine-invariant metric, or an interpretable derived quantity that captures the interplay between memory, randomness, dimensional structure, and statistical regularity in mobility.

1. Stochastic Mobility Models: Logarithmic Diffusion and Ratio

In memory-augmented random walk models (Choi et al., 2012), log-mobility ratios surface as markers of "logarithmic diffusion," wherein the mean square displacement (MSD) and number of distinct sites visited transition from power-law scaling to logarithmic time dependence. The model is parameterized by a memory coefficient $\alpha$ and an impulse coefficient $p$, producing the following mobility regimes:

• Ordinary diffusion: $\alpha = 0$, MSD scales as $\langle r^2(t) \rangle \sim t$.
• Subdiffusive regime: $0 < \alpha < \alpha_c$, MSD exhibits $\nu < \frac{1}{2}$ exponent.
• Trapping: $\alpha > \alpha_c$, long-range revisitation induces $\langle r^2(t) \rangle \rightarrow$ constant, $\langle S(t)\rangle \sim \log t$.

The log-mobility ratio in this context expresses the ultraslow growth—distinct from power-law diffusion—providing a quantitative signature of memory-induced kinetic constraints. This ratio can be interpreted as the logarithmic rate at which spatial exploration proceeds in the presence of memory trapping and impulsive escapes, analogous to Sinai diffusion.

2. Epidemic Spread and Logarithmic Mobility Scaling

The synchronous SIR model with dilution and mobility (Silva et al., 2014) demonstrates a functional interdependence between population density, individual movement probability, and epidemic critical thresholds. Empirically, the critical immunization rate $c_c$ obeys a logarithmic law with respect to density $\rho$:

• Without mobility: $c_c(\rho) = a + b\ln \rho$, illustrating that the loss of contact due to dilution results in a logarithmic reduction of epidemic threshold.

Introducing mobility (random hops of probability $p$), $c_c$ scales as a power law in $p$ for each $\rho$, $c_c(p) = \gamma(\rho) p^{\beta(\rho)}$. The nuanced competition between these logarithmic and power-law effects encapsulates the "log-mobility ratio," which quantifies how increases in mobility offset the dilution-driven decrease in epidemic threshold, effectively boosting contact restoration in the system.

3. Functional Geometry: Integral Ratios for Log-Concave Mobility

The concept generalizes in convex geometric analysis of log-concave functions (Alonso-Gutiérrez et al., 2015). Here, the "integral ratio" $\mathrm{I.rat}(f)$ is defined for a normalized log-concave function $f:\mathbb{R}^n\rightarrow\mathbb{R}$:

$$\mathrm{I.rat}(f) = \frac{\int_{\mathbb{R}^n} f(x)\, dx}{t_0\|f\|_\infty |\mathcal{E}_{t_0}|},$$

where $t_0$ is chosen to maximize $t |\mathcal{E}_t|$ (the scaled volume of John’s ellipsoid at level set $t$). This ratio is affine invariant and, for indicator functions of convex bodies, reduces to the classical volume ratio. The log-mobility ratio in this geometric context is essentially $\log \mathrm{I.rat}(f)$, quantifying how "spread out" or "mobile" $f$ is relative to its maximal ellipsoidal subset. The extension enables functional versions of reverse isoperimetric inequalities and stability statements, linking mobility concepts to entropy and geometric efficiency.

4. Human Trajectory Analysis: Log-Normal Distributions and Mobility Ratios

High-resolution digital trace analysis (Alessandretti et al., 2016) reveals that both displacement lengths ($\Delta r$) and waiting times ($\Delta t$) in human mobility obey log-normal statistics:

$$P(x) = \frac{1}{\sqrt{2\pi}\sigma x}\exp\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right),$$

with $\mu$ and $\sigma$ empirically fitted.

A derived log-mobility ratio, $$R = \frac{\ln \Delta r - \mu_r}{\ln \Delta t - \mu_t}$$, can serve as a normalized comparative metric, contrasting behavioral regularity in spatial and temporal domains. This supports quantification of the relative propensity for movement versus pausing, allows cross-context comparisons, and encompasses the emergence of routine-driven time-scale peaks overlaying the heavy-tailed envelope of mobility measures.

5. Queueing Theory: Logarithmic Performance in Mobile Networks

In processor-sharing queues for wireless cells with mobile and static users (Simatos et al., 2019), the impact of mobility is quantified through a transition in heavy-traffic scaling:

• Static regime ($\theta=0$): queue length grows as $1/(1-\rho)$.
• Mobile regime ($\theta>0$): queue length grows as $-\log(1-\rho)$.

The log-mobility improvement is formalized via a fixed-point equation balancing external arrivals and mobility-driven departures. The log-mobility ratio here characterizes the logarithmic reduction in congestion, underpinning fundamental system design principles for load balancing and resource allocation in dense mobile networks.

6. Privacy Leakage and Log-Ratio Mobility

In secure computation and differential privacy (Haitner et al., 2021), a protocol’s leakage is measured not by statistical distance, but by log-ratio closeness:

$$D_0, D_1 \text{ are } (\epsilon, \delta)-\text{log-ratio close if} \;\; \Pr_{x \sim D_0}[x \in A] \leq e^{\epsilon}\Pr_{x \sim D_1}[x \in A] + \delta,\;\; \forall A.$$

This multiplicative measure is termed a log-mobility ratio in the context of quantifying "changeability" of output distributions under input flips. The protocol can be amplified into an OT channel when accuracy $\alpha \in \Omega(\epsilon^2)$; otherwise, if $\alpha \in o(\epsilon^2)$, OT is not implied. This analytic approach delineates phase transitions for differential privacy and compositional security guarantees.

7. Statistical Modelling: Compositional Data and Log-Mobility Ratios

In high-dimensional biomarker selection (Ma et al., 2023), log ratios (particularly balances between groups of variables) are central to constructing interpretable, scale-invariant predictors. The supervised log ratio (SLR) method screens features using univariate regressions and identifies balances via clustering, then expresses biomarkers as

$$B(X; I_+, I_-) = \log\left(\frac{g(X_{I_+})}{g(X_{I_-})}\right),$$

where $g(\cdot)$ denotes geometric mean. Adaptations can be made for mobility-related applications, where the log-mobility ratio is constructed analogously to capture groupwise differences in movement or compositional change.

Table: Representative Formulations of Log-Mobility Ratio

Domain	Mathematical Form/Scaling	Contextual Usage
Mobility with Memory	$\langle S(t)\rangle \sim \log t$	Ultraslow spatial expansion
SIR Epidemic	$c_c(\rho) = a + b\ln \rho$	Dilution-mobility competing effects
Convex Geometry	$\log \mathrm{I.rat}(f)$	Functional affine invariance
Human Mobility Stats	$$R = \frac{\ln \Delta r - \mu_r}{\ln \Delta t - \mu_t}$$	Normalized spatial-temporal comparison
Queueing Systems	$-\log(1-\rho)$	Heavy traffic regime scaling improvement
Privacy Protocols	$(\epsilon, \delta)$-log-ratio closeness	Leakage quantification, amplification
Compositional Data	$B(X; I_+, I_-) = \log(g(X_{I_+})/g(X_{I_-}))$	Group-level compositional biomarker

Summary

The log-mobility ratio encapsulates logarithmic scaling effects, affine-invariant measures, or statistical ratios that quantify diffusion, growth, mixing, congestion, privacy leakage, or compositional change as a function of mobility and memory processes in the underlying system. Whether serving as a direct observable or as an analytic tool for phase transitions and stability analysis, the concept provides a unifying quantitative language for characterizing limited, ultraslow, or structure-dependent mobility in discrete, continuous, or functional settings.