Geometric Inhomogeneous Random Graphs

• GIRGs are random graph models where vertices occupy a geometric space and edge connection probabilities depend on power-law weights and distances, ensuring scale-free degree distributions.
• They preserve key features such as high clustering coefficients, short average distances, and a balance between local connectivity and hub-driven interactions.
• GIRGs support efficient algorithms for generation, routing, and percolation analyses, with multiple formulations that reveal phase transitions and scalability in network structures.

Geometric Inhomogeneous Random Graphs (GIRGs) are random graph models for scale-free networks that combine geometry, inhomogeneity, and randomness: vertices live in a geometric space, nearby vertices connect more easily, heavier vertices act as hubs, and edges are formed independently with probabilities depending on weights and distances (Chiu et al., 1 Jun 2026). They were introduced as a technically simpler generalization of hyperbolic random graphs, while preserving heavy-tailed degrees, short paths, and high clustering (Bringmann et al., 2015). The literature contains several equivalent fixed-$n$, Poissonized, threshold, soft, metric, and non-metric formulations, but the recurring theme is the same: GIRGs interpolate between geometry-driven locality and hub-driven connectivity.

1. Probabilistic definition and model variants

A standard GIRG formulation places vertices in a $d$-dimensional torus of volume $n$, assigns i.i.d. power-law weights $w_v\ge 1$ with exponent $\tau\in(2,3)$, and inserts each edge independently with probability

$$p_{uv}\coloneqq \Theta\!\left(\min\left\{\left(\frac{w_u w_v}{\|x_u-x_v\|^d}\right)^\alpha,\,1\right\}\right),$$

where $\alpha>1$ is a geometric decay parameter and $\|\cdot\|$ is the torus distance; the paper notes that any norm gives the same asymptotics (Chiu et al., 1 Jun 2026). In a fixed-size torus formulation with $V=\{1,\dots,n\}$, i.i.d. Pareto weights of exponent $\tau\in(2,3)$, and $d$0, the edge law is written explicitly as

$d$1

with $d$2 and $d$3 (Michielan et al., 2021). Across papers, the power-law exponent is variously denoted by $d$4 or $d$5, and the geometric decay parameter by $d$6 or $d$7; these are notational variants of the same general design principle.

The standard metric choice is the toroidal $d$8 distance, but later work generalized GIRGs to Boolean Distance Functions (BDFs), defined recursively by coordinatewise minima and maxima. This class includes the max-norm and the minimum-component distance (MCD), and permits hierarchical feature geometries in which similarity in some coordinates may suffice for closeness, rather than requiring similarity in all coordinates (Kaufmann et al., 2024). This broadening is important because many structural properties of GIRGs persist across distance functions, whereas separator structure and expansion can change sharply with the geometry.

A closely related threshold, or zero-temperature, version replaces soft probabilities by a deterministic rule. In one such formulation on a torus of volume $d$9, vertices form a Poisson point process of intensity $n$0, each vertex receives an independent power-law weight with exponent $n$1, and an edge $n$2 is present iff

$n$3

This threshold GIRG is the zero-temperature analogue of the general model and is particularly useful for diameter and connectivity arguments (Benjert et al., 14 Oct 2025).

2. Local limits, degree law, giant component, and distances

A fundamental feature of GIRGs is that geometry changes local structure without destroying the scale-free degree law. Averaging over positions yields

$n$4

and for a vertex of weight $n$5, the degree satisfies

$n$6

so weights encode expected degrees exactly in the intended Chung–Lu-like sense (Chiu et al., 1 Jun 2026). In a more general local-limit framework, after blowing up positions by $n$7, finite GIRGs converge locally in probability to an infinite Poisson-GIRG on $n$8 with i.i.d. weights and limiting connection kernel. The degree of the root in this local limit is mixed Poisson, and the degree sequence of a uniformly chosen finite vertex is uniformly integrable (Hofstad et al., 2021).

These local limits imply convergence of clustering observables. In the general spatial inhomogeneous framework covering GIRGs, the local clustering coefficient, the clustering function conditioned on degree $n$9, and—under an additional integrability condition—the global clustering coefficient converge to their infinite-volume counterparts (Hofstad et al., 2021). For the original GIRG model, the mean clustering coefficient is $w_v\ge 1$0 with high probability, in sharp contrast to non-geometric inhomogeneous random graphs such as Chung–Lu, where clustering vanishes polynomially in $w_v\ge 1$1 (Bringmann et al., 2015).

On global connectivity, GIRGs in the infinite-variance regime have a unique giant component of linear size, and all other components are polylogarithmic (Chiu et al., 1 Jun 2026). In the threshold model with Pareto exponent $w_v\ge 1$2, the existence of a giant component can be strengthened to a stretched-exponential probability statement: the graph contains a connected component of size $w_v\ge 1$3 with probability

$w_v\ge 1$4

and the same argument yields analogous “partial giant” results in large induced spatial subgraphs (Bläsius et al., 2023).

Distance behavior is bifurcated between typical and extremal scales. For $w_v\ge 1$5, average distance in the giant component is

$w_v\ge 1$6

so GIRGs are ultra-small worlds in the usual sense (Chiu et al., 1 Jun 2026). By contrast, for threshold GIRGs the diameter is $w_v\ge 1$7: the paper proves an $w_v\ge 1$8 upper bound for $w_v\ge 1$9, an $\tau\in(2,3)$0 lower bound for all $\tau\in(2,3)$1, and hence logarithmic diameter in the scale-free regime (Benjert et al., 14 Oct 2025). This coexistence of $\tau\in(2,3)$2 average distance and $\tau\in(2,3)$3 diameter is a defining feature of GIRG geometry.

3. Clique structure, clustering mechanisms, and motif phase transitions

Clique counts in GIRGs exhibit a precise phase transition between geometry-dominated and hub-dominated behavior. For a fixed $\tau\in(2,3)$4, the number of $\tau\in(2,3)$5-cliques $\tau\in(2,3)$6 in a Pareto-weight GIRG has two regimes governed by

$\tau\in(2,3)$7

If $\tau\in(2,3)$8, then

$\tau\in(2,3)$9

and the dominant cliques are non-geometric: their vertices have weights $$p_{uv}\coloneqq \Theta\!\left(\min\left\{\left(\frac{w_u w_v}{\|x_u-x_v\|^d}\right)^\alpha,\,1\right\}\right),$$0 and lie at constant macroscopic distances. If $$p_{uv}\coloneqq \Theta\!\left(\min\left\{\left(\frac{w_u w_v}{\|x_u-x_v\|^d}\right)^\alpha,\,1\right\}\right),$$1, then

$$p_{uv}\coloneqq \Theta\!\left(\min\left\{\left(\frac{w_u w_v}{\|x_u-x_v\|^d}\right)^\alpha,\,1\right\}\right),$$2

and the dominant cliques are geometric: their vertices have low weights and concentrate at distances $$p_{uv}\coloneqq \Theta\!\left(\min\left\{\left(\frac{w_u w_v}{\|x_u-x_v\|^d}\right)^\alpha,\,1\right\}\right),$$3 (Michielan et al., 2021). A particularly sharp corollary is that for $$p_{uv}\coloneqq \Theta\!\left(\min\left\{\left(\frac{w_u w_v}{\|x_u-x_v\|^d}\right)^\alpha,\,1\right\}\right),$$4, all finite-size cliques are asymptotically non-geometric: the clique count exponent matches that of non-geometric scale-free models, so geometry is asymptotically irrelevant at the level of fixed-size clique counts (Michielan et al., 2021).

The same paper identifies the optimal clique type by maximizing an explicit exponent

$$p_{uv}\coloneqq \Theta\!\left(\min\left\{\left(\frac{w_u w_v}{\|x_u-x_v\|^d}\right)^\alpha,\,1\right\}\right),$$5

which encodes the trade-off between rare high weights and rare geometric proximity. The resulting localized clique counts are self-averaging for the dominant type, so the phase transition is not merely first-moment behavior but describes typical realizations (Michielan et al., 2021).

Maximal cliques behave differently from fixed-size clique counts. In several GIRG variants, the number of maximal cliques is super-polynomial because the power-law core contains induced co-matchings. For instance, in a threshold $$p_{uv}\coloneqq \Theta\!\left(\min\left\{\left(\frac{w_u w_v}{\|x_u-x_v\|^d}\right)^\alpha,\,1\right\}\right),$$6-dimensional GIRG on the torus with $$p_{uv}\coloneqq \Theta\!\left(\min\left\{\left(\frac{w_u w_v}{\|x_u-x_v\|^d}\right)^\alpha,\,1\right\}\right),$$7, the number $$p_{uv}\coloneqq \Theta\!\left(\min\left\{\left(\frac{w_u w_v}{\|x_u-x_v\|^d}\right)^\alpha,\,1\right\}\right),$$8 of maximal cliques satisfies

$$p_{uv}\coloneqq \Theta\!\left(\min\left\{\left(\frac{w_u w_v}{\|x_u-x_v\|^d}\right)^\alpha,\,1\right\}\right),$$9

for any constants $\alpha>1$0 and $\alpha>1$1 (Bläsius et al., 2023). The same paper explains why this does not contradict empirical observations of few maximal cliques in realistic network sizes: linear lower-order terms, such as maximal $\alpha>1$2-cliques induced by degree-$\alpha>1$3 vertices, dominate until astronomically large $\alpha>1$4, so the super-polynomial asymptotics are effectively invisible in practical ranges (Bläsius et al., 2023).

4. Routing, weighted distances, percolation, and rare-event geometry

GIRGs are not only structurally small-world; they are also algorithmically navigable. In the regime $\alpha>1$5 and $\alpha>1$6, purely geometric routing—using only neighbor positions and the target position, but no weight information—succeeds with probability $\alpha>1$7 and, conditioned on success, finds a path of length

$\alpha>1$8

asymptotically almost surely (Chiu et al., 1 Jun 2026). This matches the asymptotic guarantee for weight-aware greedy routing and the average graph distance, and the proof identifies a two-phase trajectory: first an ascent to the heavy core, then efficient navigation within and out of that core toward the target (Chiu et al., 1 Jun 2026).

When i.i.d. edge-lengths $\alpha>1$9 are added, GIRGs admit a sharp explosion criterion. In the infinite-volume EGIRG, define the explosion time $\|\cdot\|$0 as the limit of the weighted radius needed to reach the $\|\cdot\|$1-th vertex as $\|\cdot\|$2. If $\|\cdot\|$3, then

$\|\cdot\|$4

If the weight law is power-law with $\|\cdot\|$5, then EGIRG is explosive if and only if

$\|\cdot\|$6

which is exactly the age-dependent branching-process explosion criterion in the infinite-variance regime (Ansari et al., 2018). Under the same assumptions, typical weighted distances within the giant or infinite component converge in distribution, resolving the finite-size question that motivated the paper (Ansari et al., 2018).

Bootstrap percolation on GIRGs reveals another geometry-sensitive phase transition. Starting from a source region $\|\cdot\|$7 of expected size $\|\cdot\|$8, with each vertex in $\|\cdot\|$9 infected independently with probability $V=\{1,\dots,n\}$0, the critical localized infection rate is

$V=\{1,\dots,n\}$1

If $V=\{1,\dots,n\}$2, then $V=\{1,\dots,n\}$3 w.h.p.; if $V=\{1,\dots,n\}$4, then $V=\{1,\dots,n\}$5 w.h.p.; and in the critical window both extinction and macroscopic outbreak occur with positive probability (Koch et al., 2016). In the supercritical regime, the number of rounds needed to infect a linear fraction is $V=\{1,\dots,n\}$6, and the infection time of a given vertex is governed jointly by its weight and its distance from the source (Koch et al., 2016).

A continuum GIRG analysis of annulus crossings refines the subcritical geometry of rare long-range connections. In the quantitatively subcritical phase $V=\{1,\dots,n\}$7, $V=\{1,\dots,n\}$8, $V=\{1,\dots,n\}$9, annulus crossing probabilities are asymptotically equivalent to one-edge or two-edge crossing probabilities depending on the radii and the balance between $\tau\in(2,3)$0 and $\tau\in(2,3)$1; the paper also derives subcritical one-arm exponents for a typical point (Jacob et al., 27 Sep 2025). This suggests that, on large scales below percolation, GIRG connectivity is often controlled by very short exceptional paths rather than long combinatorial chains.

5. Geometry beyond metrics: separators, robustness, and expansion

The introduction of Boolean Distance Functions turns GIRGs into a family of hierarchical feature-space models. A BDF is built recursively from coordinatewise torus distances using minima and maxima, and its small-ball volume satisfies

$\tau\in(2,3)$2

where the depth $\tau\in(2,3)$3 is defined recursively from the min/max structure (Kaufmann et al., 2024). This framework captures both standard max-norm GIRGs and non-metric models such as MCD, while preserving power-law degrees, the giant component, and small-world behavior.

Within this enlarged class, separator structure is completely classified. If the BDF is Single-Coordinate Outer-Max (SCOM), meaning

$\tau\in(2,3)$4

for some coordinate $\tau\in(2,3)$5, then with high probability there exists a set of $\tau\in(2,3)$6 edges whose deletion splits the graph into two connected components of size $\tau\in(2,3)$7 (Kaufmann et al., 2024). If the BDF is non-SCOM, then the opposite holds: any deletion that separates two linear-size components must remove $\tau\in(2,3)$8 edges (Kaufmann et al., 2024). Thus sublinear cuts are precisely the SCOM case; outside that class, BDF-GIRGs are robust in the same sense as MCD-GIRGs.

The paper also proves that every BDF satisfies a stochastic triangle inequality with $\tau\in(2,3)$9, and therefore every BDF-GIRG has clustering coefficient $d$00 with high probability (Kaufmann et al., 2024). This is notable because separator structure changes radically across BDFs, whereas clustering survives throughout the class.

For MCD-GIRGs, expansion becomes the dominant phenomenon. In dimension $d$01, with $d$02 and $d$03, the subgraph induced by vertices of weight at least $d$04, with $d$05, is an expander with high probability; more precisely, every set $d$06 of size at most a fixed fraction of the core has expansion factor at least

$d$07

for suitable constants $d$08 and $d$09 (Kaufmann et al., 24 Jun 2025). This sharply contrasts with metric-based GIRGs, where small separators exist at all scales. A plausible implication is that “similar in some dimensions” geometry produces a network core that is algorithmically closer to an expander, whereas “similar in all dimensions” geometry preserves geometric bottlenecks.

Empirical expressivity studies reinforce this distinction. GIRGs with metric and non-metric distances, torus or cube topology, and both power-law and degree-replicating weights were evaluated against 104 Facebook networks. The results provide evidence that GIRGs are more realistic candidates with respect to closeness centrality, betweenness centrality, local clustering coefficient, and graph effective diameter, while also showing that outer-min, non-metric distances better capture partial similarity effects in some settings; at the same time, GIRGs face difficulties reproducing higher variance and more extreme values of graph statistics observed in real networks (Dayan et al., 2024).

6. Hyperbolic connections, efficient generation, and algorithmic role

Hyperbolic random graphs are embedded in the GIRG framework. Under the mapping

$d$10

a hyperbolic random graph with radial coordinate $d$11 and angle $d$12 becomes a $d$13 GIRG, with $d$14 and $d$15; in particular, threshold hyperbolic graphs correspond to GIRGs with $d$16 and $d$17 (Bringmann et al., 2015). This is why many HRG results can be reinterpreted as statements about one-dimensional GIRGs.

One reason GIRGs became central is algorithmic tractability. A sampling algorithm for GIRGs can generate a random graph from the model in expected linear time $d$18, improving over the best-known sampling algorithm for hyperbolic random graphs by a factor $d$19 in the comparison stated by the paper (Bringmann et al., 2015). The same work proves that standard GIRGs have clustering coefficient $d$20, admit small separators in the metric model, and can be compressed using an expected linear number of bits, with constant-time degree and neighbor access in the compressed representation (Bringmann et al., 2015).

Subsequent work turned this into an efficient practical generator. A later implementation gives an expected $d$21 generator for GIRGs and HRGs, supports non-zero temperature and higher-dimensional geometries, and includes an efficient procedure for determining the non-trivial dependency between expected average degree and the input parameters, so that the desired expected average degree can be given directly as input (Bläsius et al., 2019). The implementation also shows that a straight-forward HRG-as-GIRG inclusion is not exact in practice, although the discrepancy is negligible for most use cases (Bläsius et al., 2019).

These algorithmic developments explain why GIRGs now function as a workhorse model. They are analytically rich enough to support local-limit theory, motif asymptotics, routing theorems, percolation analyses, separator classifications, and diameter bounds, yet structured enough to permit exact large-scale generation and empirical fitting. In that sense, GIRGs occupy a distinctive position among network models: they retain the scale-free and clustered features associated with real networks while exposing a geometry that can be tuned, generalized, or replaced to isolate the mechanisms behind connectivity, motifs, and algorithmic performance.