Geo-Foci Model: Cross-Domain Focal Organization

• The Geo-Foci Model is a family of frameworks where a defined focus object, such as a subduction zone or shared urban place, organizes system behavior.
• It establishes spatial or structural relations—like distances, elliptical regions, or semantic labels—to interpret observable patterns across earthquake science, social networks, and optics.
• Applications span modeling earthquake intensity, social flux estimation, geopolitical centrality, and optical imaging, providing actionable insights and model validation.

The Geo-Foci Model is not a single canonical formalism in the supplied literature. Rather, the expression is used, reconstructed, or naturally interpreted as a family of models in which a spatial, geometric, semantic, or institutional focus governs organization, interaction, inference, or design. In different domains, the focus may be a geological structure that modulates earthquake intensity, the paired foci of an ellipse that regulate social flux, shared urban places that act as extra-network foci, geographic entities central to a news story, a capital that anchors military power, or literal optical or parabola foci that constrain physical behavior. This suggests a shared modeling logic: observable patterns are explained by how entities are positioned relative to one or more privileged focal structures (Choiruddin et al., 2021, Herrera-Yagüe et al., 2013, Brown et al., 2013, Ariyarathne et al., 28 Feb 2026, Kuperman, 2010, Jaud, 2022).

1. Conceptual scope and recurrent structure

Across the supplied papers, Geo-Foci formulations share three recurring elements. First, they define a focus object: for example, nearest subduction zones and volcanoes in earthquake occurrence modeling, the two locations forming an ellipse in social-flux estimation, or a fixed capital in geopolitical competition (Choiruddin et al., 2021, Herrera-Yagüe et al., 2013, Kuperman, 2010). Second, they specify a spatial or structural relation between observations and that focus object, such as nearest distance, membership in an ellipse, border distance from a capital, or cross-correlation with a stress kernel (Lomax, 2024). Third, they use that relation to build an interpretable target quantity: intensity, flux, military power, slip potential, or focal placement.

The supplied literature also shows that “focus” has multiple technical meanings. In social-network theory, it follows Scott Feld’s focused organization theory, where homes, workplaces, restaurants, bars, shops, parks, or transport hubs act as extra-network foci for tie formation (Brown et al., 2013). In journalism, geo-foci are the counties, cities, states, or countries central to an article’s subject matter (Ariyarathne et al., 28 Feb 2026). In optics and billiards, the term is literal: prescribed optical foci at several wavelengths, or the loci of parabola foci in gravitational motion (Doskolovich et al., 2018, Jaud, 2022). A common misconception is therefore to treat Geo-Foci as one standardized model class; the supplied papers instead support a broader view in which it is a cross-domain focal-design principle rather than a unique methodology.

2. Geological-focus formulations in earthquake science

In "Quantifying effect of geological factor on distribution of earthquake occurrences by inhomogeneous Cox processes" (Choiruddin et al., 2021), the focal objects are geological structures. Earthquake epicenters in Sulawesi and Maluku are modeled as a spatial point process whose inhomogeneous intensity depends on distances to the nearest subduction zone, volcano, and fault, all measured in units of 100 km. The concrete Poisson comparison model is

$$\lambda(u;)=\exp(\beta_0 + \sum_{i=1}^3 \beta_i z_i(u)).$$

Within this formulation, negative coefficients mean that earthquake risk increases as distance to the geological feature decreases. The reported substantive result is that subduction zone and volcano distances are significant, with fitted values

$$\hat \theta_{z_1}=-0.363,\quad \exp(\hat \theta)=0.696,\quad 1/\exp(\hat \theta)=1.437$$

and

$$\hat \theta_{z_2}=-0.276,\quad \exp(\hat \theta)=0.759,\quad 1/\exp(\hat \theta)=1.318,$$

while fault distance is not significant at the 0.05 level. The clustering component is handled through inhomogeneous Cox-process structure, with comparisons among Thomas, Cauchy, variance-Gamma, and log-Gaussian Cox processes; model selection uses AIC and envelope tests, and the paper concludes that the Cauchy and variance-Gamma cluster models fit well the major earthquake distribution in Sulawesi and Maluku (Choiruddin et al., 2021).

A second seismological use appears in "Mapping finite-fault slip with spatial correlation between seismicity and point-source Coulomb failure stress change" (Lomax, 2024). Here the focal object is not a geological covariate but a masked point-source $\Delta$CFS kernel. The method maps finite-fault slip directly from aftershocks and seismicity by correlating the 3D distribution of relocated seismicity with the stress field expected from a source mechanism and a chosen receiver-fault geometry. The governing stress quantity is

$$\Delta \mathrm{CFS} = \Delta \tau + \mu' \Delta \sigma_n,$$

and the seismicity-stress finite-fault field is obtained by 3D cross-correlation,

$$F(i,j,k) = \sum_{l,m,n} \mathrm{ACFS}(1+i+l,\; 1+j+m,\; 1+k+n)\, S(l,m,n).$$

The output is explicitly a relative slip potential field rather than absolute slip or moment. Synthetic tests recover the location and lateral extent of imposed slip patches, and benchmark applications show agreement with independent estimates for the 2004 Mw 6.0 Parkfield and 2021 Mw 6.0 Antelope Valley earthquakes. For the 2018 Mw 7.1 Anchorage earthquake, the method favors rupture on the gently east-dipping plane, using seismicity to resolve fault-plane ambiguity (Lomax, 2024).

Taken together, these earthquake papers show two distinct Geo-Foci logics. One treats geological structures as covariates that modulate first-order intensity; the other treats stress kernels as focal templates against which seismicity is back-projected. In both cases, the explanatory variable is explicitly spatial and physically interpretable.

3. Elliptic and place-focused models of social organization

In "The elliptic model for social fluxes" (Herrera-Yagüe et al., 2013), the focus is geometrically bilateral. The model estimates the number of social ties between locations $i$ and $j$ by making it inversely proportional to the population inside the smallest ellipse with foci at $i$ and $j$ that contains the two circles of radius $$\hat \theta_{z_1}=-0.363,\quad \exp(\hat \theta)=0.696,\quad 1/\exp(\hat \theta)=1.437$$0 centered at the two locations. If $$\hat \theta_{z_1}=-0.363,\quad \exp(\hat \theta)=0.696,\quad 1/\exp(\hat \theta)=1.437$$1 denotes the population inside that ellipse, the model is

$$\hat \theta_{z_1}=-0.363,\quad \exp(\hat \theta)=0.696,\quad 1/\exp(\hat \theta)=1.437$$2

Because the ellipse with foci $$\hat \theta_{z_1}=-0.363,\quad \exp(\hat \theta)=0.696,\quad 1/\exp(\hat \theta)=1.437$$3 and $$\hat \theta_{z_1}=-0.363,\quad \exp(\hat \theta)=0.696,\quad 1/\exp(\hat \theta)=1.437$$4 is the same as the ellipse with foci $$\hat \theta_{z_1}=-0.363,\quad \exp(\hat \theta)=0.696,\quad 1/\exp(\hat \theta)=1.437$$5 and $$\hat \theta_{z_1}=-0.363,\quad \exp(\hat \theta)=0.696,\quad 1/\exp(\hat \theta)=1.437$$6, the model is symmetric:

$$\hat \theta_{z_1}=-0.363,\quad \exp(\hat \theta)=0.696,\quad 1/\exp(\hat \theta)=1.437$$7

That symmetry is the key difference from the radiation model, whose one-sided circle geometry leads to $$\hat \theta_{z_1}=-0.363,\quad \exp(\hat \theta)=0.696,\quad 1/\exp(\hat \theta)=1.437$$8. Evaluation on anonymous call detail records from France, Portugal, and Spain, comprising over 7 billion calls and about 25 million users, shows that the elliptic model outperforms both the radiation model and the bilateral symmetrized radiation model for communication fluxes. For intracity prediction, the paper introduces a correction by replacing $$\hat \theta_{z_1}=-0.363,\quad \exp(\hat \theta)=0.696,\quad 1/\exp(\hat \theta)=1.437$$9 with $$\hat \theta_{z_2}=-0.276,\quad \exp(\hat \theta)=0.759,\quad 1/\exp(\hat \theta)=1.318,$$0, with best performance near

$$\hat \theta_{z_2}=-0.276,\quad \exp(\hat \theta)=0.759,\quad 1/\exp(\hat \theta)=1.318,$$1

This use of Geo-Foci is exact in the geometric sense: the shared interaction region is defined by two literal foci (Herrera-Yagüe et al., 2013).

In "A place-focused model for social networks in cities" (Brown et al., 2013), the focal objects are not geometric foci but places. The paper draws directly on Scott Feld’s focused organization theory and argues that city social networks are organized around shared spaces such as homes, workplaces, restaurants, bars, shops, parks, or transport hubs. Using Foursquare posts on Twitter over 10 months (Nov 2010–Sep 2011), estimated to cover 20–25% of all Foursquare check-ins during that period, the authors construct city social networks in Atlanta, Boston, Chicago, Minneapolis, and Seattle. Empirical findings include approximately power-law degree distributions with average exponent

$$\hat \theta_{z_2}=-0.276,\quad \exp(\hat \theta)=0.759,\quad 1/\exp(\hat \theta)=1.318,$$2

high clustering, short path lengths, and strong community structure. Most social triangles share at least one common place, with reported proportions of 0.90 in Atlanta, 0.81 in Boston, 0.80 in Chicago, 0.71 in Minneapolis, and 0.84 in Seattle.

The generative model assigns users to venues using place popularity and geographic distance, then creates social ties with category-dependent probabilities. For place $$\hat \theta_{z_2}=-0.276,\quad \exp(\hat \theta)=0.759,\quad 1/\exp(\hat \theta)=1.318,$$3, the tie probability is

$$\hat \theta_{z_2}=-0.276,\quad \exp(\hat \theta)=0.759,\quad 1/\exp(\hat \theta)=1.318,$$4

with

$$\hat \theta_{z_2}=-0.276,\quad \exp(\hat \theta)=0.759,\quad 1/\exp(\hat \theta)=1.318,$$5

An intra-place triadic closure step adds links with probability 0.15 among co-visitors who already share a friend. The full model reproduces clustering $$\hat \theta_{z_2}=-0.276,\quad \exp(\hat \theta)=0.759,\quad 1/\exp(\hat \theta)=1.318,$$6, average path length $$\hat \theta_{z_2}=-0.276,\quad \exp(\hat \theta)=0.759,\quad 1/\exp(\hat \theta)=1.318,$$7, and modularity $$\hat \theta_{z_2}=-0.276,\quad \exp(\hat \theta)=0.759,\quad 1/\exp(\hat \theta)=1.318,$$8, close to the empirical city networks. The broader significance is that Geo-Foci here refers to focal organization around shared places, not to an ellipse or a latent field (Brown et al., 2013).

4. Semantic, geophysical, and perceptual focus in contemporary AI

In "Identifying the Geographic Foci of US Local News" (Ariyarathne et al., 28 Feb 2026), Geo-Foci becomes a semantic-labeling task. The proposed NLGF (News Lab Geo-Focus) model first classifies a news article into one of five geo-focus levels,

$$\hat \theta_{z_2}=-0.276,\quad \exp(\hat \theta)=0.759,\quad 1/\exp(\hat \theta)=1.318,$$9

and then identifies the actual place names central to the story. The dataset is a balanced dataset of 1,250 US local news articles, evenly split across the five geo-focus levels, with strong reliability: Cohen’s $\Delta$0, Krippendorff’s $\Delta$1 for geo-focus level, and Krippendorff’s $\Delta$2 for geo-foci. After spaCy NER and LLM-based toponym disambiguation, each toponym receives an Initial Geo-Focus Level (IGL), and the model computes 15 spatial-semantic features. The classifier is XGBoost, tuned with stratified 5-fold cross-validation, with best hyperparameters learning rate 0.2, tree depth 6, number of estimators 25, and subsample ratio 0.9. Geo-focus scoring then uses

$\Delta$3

NLGF achieves Precision = 0.89, Recall = 0.89, $\Delta$4 for geo-focus level classification, and Precision = 0.86, Recall = 0.89, $\Delta$5 for geo-foci identification. A notable finding is that GPT-4o outperforms all eight evaluated geographic entity disambiguation methods, with GPE F1: 0.948, LOC F1: 0.813, and FAC F1: 0.964 on the gold-standard set (Ariyarathne et al., 28 Feb 2026).

A different use of the term appears in "Foundation Models for Geophysics: Review and Perspective" (Liu et al., 2024). There, GeoFMs are a family of large, pretrained, generalizable models for exploration geophysics rather than a single architecture. The paper organizes them into a hierarchy consisting of Data basis, Task-specific models, Modality-specific models, Multimodal models, and Geophysical agent and copilot. The development workflow comprises Data preparation, Pretraining, Multimodal alignment, and Task-specific adaptation, with emphasis on self-supervised learning, contrastive pretraining, parameter-efficient fine-tuning (PEFT), instruction tuning, and alignment tuning. This is a broader, systems-level use of focal modeling: not a single focus geometry, but a hierarchy in which multiple geophysical modalities are aligned around shared latent representations (Liu et al., 2024).

In "GeoFocus: Blending Efficient Global-to-Local Perception for Multimodal Geometry Problem-Solving" (Deng et al., 9 Feb 2026), the focal distinction is perceptual rather than geographic. The framework combines a VertexLang Topology Percepter for global figure structure with a Critical Local Perceptor for theorem-relevant local relations. The local module uses thirteen theory-based perception templates, boosting critical local feature coverage by 61% compared to previous methods. The global module encodes topology through VertexLang, which uses a circle radius dictionary, a vertex coordinate dictionary,

$\Delta$6

and a connectivity dictionary. Relative to code-based encodings, VertexLang reduces global perception training time by 20%. On Geo3K, GeoQA, and FormalGeo7K, GeoFocus achieves a 4.7% accuracy improvement over leading specialized models, with GeoFocus-7B reporting 55.3 / 71.9 / 63.5 on the three benchmarks. Here “focus” denotes the joint necessity of global topology perception and critical local structure in geometry problem solving (Deng et al., 9 Feb 2026).

5. Capital-centered focality in geopolitical division

In "A model for the emergence of geopolitical division" (Kuperman, 2010), the focus is the capital as a center of power. The world is represented as a square lattice; each country is a connected cluster of sites with area $\Delta$7, border set $\Delta$8, and fixed capital $\Delta$9. The simplest military-power formulation is

$$\Delta \mathrm{CFS} = \Delta \tau + \mu' \Delta \sigma_n,$$0

where perimeter $$\Delta \mathrm{CFS} = \Delta \tau + \mu' \Delta \sigma_n,$$1 penalizes overextended states. The central innovation is the border-distance penalty

$$\Delta \mathrm{CFS} = \Delta \tau + \mu' \Delta \sigma_n,$$2

with a coastline-weighted extension using $$\Delta \mathrm{CFS} = \Delta \tau + \mu' \Delta \sigma_n,$$3, and the resulting power law

$$\Delta \mathrm{CFS} = \Delta \tau + \mu' \Delta \sigma_n,$$4

A more local conflict version further uses

$$\Delta \mathrm{CFS} = \Delta \tau + \mu' \Delta \sigma_n,$$5

Competitive dynamics operate through pairwise contests between neighboring countries. If adjacent states have powers $$\Delta \mathrm{CFS} = \Delta \tau + \mu' \Delta \sigma_n,$$6, the stronger wins with probability

$$\Delta \mathrm{CFS} = \Delta \tau + \mu' \Delta \sigma_n,$$7

and the weaker with probability $$\Delta \mathrm{CFS} = \Delta \tau + \mu' \Delta \sigma_n,$$8. On a $$\Delta \mathrm{CFS} = \Delta \tau + \mu' \Delta \sigma_n,$$9 lattice starting from 400 countries, each initially a $$F(i,j,k) = \sum_{l,m,n} \mathrm{ACFS}(1+i+l,\; 1+j+m,\; 1+k+n)\, S(l,m,n).$$0 square with the capital at the barycenter, the model generates metastable partitions rather than inevitable universal empire when military power depends on geometry. Analytical results show, for the capital-distance formulation, that

$$F(i,j,k) = \sum_{l,m,n} \mathrm{ACFS}(1+i+l,\; 1+j+m,\; 1+k+n)\, S(l,m,n).$$1

and

$$F(i,j,k) = \sum_{l,m,n} \mathrm{ACFS}(1+i+l,\; 1+j+m,\; 1+k+n)\, S(l,m,n).$$2

For $$F(i,j,k) = \sum_{l,m,n} \mathrm{ACFS}(1+i+l,\; 1+j+m,\; 1+k+n)\, S(l,m,n).$$3, this becomes

$$F(i,j,k) = \sum_{l,m,n} \mathrm{ACFS}(1+i+l,\; 1+j+m,\; 1+k+n)\, S(l,m,n).$$4

The paper interprets these results as an evolutionary explanation for why many capitals in Eurasia are centrally located and far from coasts or borders. In Geo-Foci terms, the capital is the focal point from which territorial coherence and defensive efficiency are measured (Kuperman, 2010).

6. Literal focal engineering in optics and dynamics

In optics, Geo-Foci becomes a design problem for prescribed focal positions. "Multifocal diffractive lens generating several fixed foci at different design wavelengths" (Doskolovich et al., 2018) proposes a spectral multifocal zone plate (SMZP) that, when combined with a refractive lens of focal length $$F(i,j,k) = \sum_{l,m,n} \mathrm{ACFS}(1+i+l,\; 1+j+m,\; 1+k+n)\, S(l,m,n).$$5, produces focal positions

$$F(i,j,k) = \sum_{l,m,n} \mathrm{ACFS}(1+i+l,\; 1+j+m,\; 1+k+n)\, S(l,m,n).$$6

The central contribution is multiwavelength optimization so that these foci remain fixed for several design wavelengths. The reported trifocal example uses $$F(i,j,k) = \sum_{l,m,n} \mathrm{ACFS}(1+i+l,\; 1+j+m,\; 1+k+n)\, S(l,m,n).$$7 nm, $$F(i,j,k) = \sum_{l,m,n} \mathrm{ACFS}(1+i+l,\; 1+j+m,\; 1+k+n)\, S(l,m,n).$$8 nm, $$F(i,j,k) = \sum_{l,m,n} \mathrm{ACFS}(1+i+l,\; 1+j+m,\; 1+k+n)\, S(l,m,n).$$9 nm, with $i$0 mm, $i$1 mm, $i$2, $i$3, and $i$4 mm, yielding

$i$5

Fabrication by direct laser writing and experiments with an Ekspla NT242 tunable laser confirm three sharp foci at the design wavelengths (Doskolovich et al., 2018).

"Three-dimensional array foci of generalized Fibonacci photon sieves" (Zhang et al., 2015) extends literal focus engineering to aperiodic diffractive optics. A generalized Fibonacci photon sieve (GFiPS) is defined through

$i$6

and zone radii obey

$i$7

The parameters $i$8 and the optical path difference scaling factor $i$9 determine the number and spacing of the resulting foci. The paper reports, for $j$0, two focal lengths $j$1 and $j$2, and for $j$3, $j$4 and $j$5. With phase modulation, the same framework generates three-dimensional array foci, including a $j$6 array and a $j$7 array with focal planes at 2.414 cm, 3.164 cm, and 3.914 cm (Zhang et al., 2015).

In "Gravitational billiards bouncing inside general domains -- foci curves and confined domains" (Jaud, 2022), focal geometry is again literal, but dynamical rather than optical. A point particle moves under uniform gravity and reflects inside a mirror $j$8. Each flight segment is a parabola, with motion

$j$9

and focus

$i$0

The paper’s main contribution is the derivation of foci curves, explicit envelope curves on which all consecutive flight-parabola foci lie for fixed energy. From the foci curve, the authors derive envelope curves for the family of flight parabolas, and together with the mirror these define a confined domain for all possible particle trajectories in the non-periodic orbit case. For a parabolic mirror, the foci curve becomes a circle; for a straight-line mirror, a straight line; and for a hyperbolic mirror, a more intricate curve with vertical asymptotes $i$1 (Jaud, 2022).

Across these optical and dynamical examples, Geo-Foci refers not to covariates or semantic labels but to the explicit control or characterization of physical focal structure. The underlying theme nevertheless remains the same: system behavior is organized by a privileged focal geometry, and the model becomes interpretable because that geometry is specified directly.