Rank-1 Inhomogeneous Random Graph

• Rank-1 inhomogeneous random graphs are models where vertex heterogeneity is encoded by single weight coordinates, with edge probabilities proportional to weight products.
• They include formulations such as Chung–Lu, Norros–Reittu, and the generalized random graph, illustrating distinct phase transitions and scaling limits.
• The analysis uses exploration processes, continuum limits, and spectral methods to reveal Gaussian fluctuation behavior and distinct universality classes.

A rank-1 inhomogeneous random graph is a random graph model in which vertex heterogeneity is encoded by a single weight coordinate, so that the connection mechanism factors through products of vertex weights. In the finite-vertex formulations, vertices $i$ carry positive weights $w_i$, and the edge probability between $i$ and $j$ depends on $w_i w_j$; in the graphon formulation, the kernel is of product form $r(x,y)=v(x)v(y)$. The classical Erdős–Rényi model is recovered by taking constant weights, while nonconstant weights generate heterogeneous expected degrees and heavy-tailed degree laws within an edge-independent framework (Amini et al., 2014, Chakrabarty et al., 2020, Jr, 2 Jan 2025).

1. Model class and standard formulations

The defining feature of the rank-1 kernel structure is that the kernel can be written as the outer product of a single vector of weights with itself, so there is only one “dimension” of heterogeneity. In the Chung–Lu formulation on $[n]$, the edge probability is

$$p_{ij}=\min\left\{\frac{w_i w_j}{W_{[n]}},1\right\},\qquad W_{[n]}=\sum_{k=1}^n w_k,$$

and the expected degree of vertex $i$ is approximated by its weight $w_i$ (Amini et al., 2014). In the Norros–Reittu, or Poissonian, formulation,

$w_i$0

while the generalized random graph uses

$w_i$1

In dense inhomogeneous Erdős–Rényi models, the rank-1 graphon is written as $w_i$2 (Hofstad et al., 2014, Giardinà et al., 14 Feb 2025, Chakrabarty et al., 2020).

Formulation	Edge rule	Source
Chung–Lu	$w_i$3	(Amini et al., 2014)
Norros–Reittu / Poissonian	$w_i$4	(Hofstad et al., 2014)
Generalized random graph	$w_i$5	(Giardinà et al., 14 Feb 2025)
Rank-1 graphon	$w_i$6	(Chakrabarty et al., 2020)

Several asymptotic theories are shared across these realizations. Under the asymptotic equivalence quoted for critical rank-one models, the same continuum scaling results hold for the Chung–Lu model and the Britton-Deijfen-Löf model as for the Norros–Reittu model, provided the stated moment and regularity assumptions are satisfied (Bhamidi et al., 2014). This does not mean that all finite-$w_i$7 formulations are identical, but it does mean that a large part of the asymptotic component geometry is formulation-invariant in the regimes covered by those results.

2. Phase transition and component-size regimes

For the supercritical Poissonian model with edge probability

$w_i$8

a uniformly chosen vertex has empirical weight $w_i$9, and if $i$0 with $i$1, then the critical threshold is

$i$2

The giant component exists only if $i$3 (Jr, 2 Jan 2025). In the Norros–Reittu scaling, the corresponding criterion is expressed through

$i$4

with subcritical behavior when $i$5, a giant component when $i$6, and criticality at $i$7 (Hofstad et al., 2014, Bhamidi et al., 2014).

At criticality with finite-moment weights, the largest components exhibit the same macroscopic scaling exponents as in the Erdős–Rényi critical window. In the rank-one Norros–Reittu model, one perturbs the weights by

$i$8

and the ordered components, viewed as measured metric spaces with graph distances scaled by $i$9 and masses by $j$0, converge in the Gromov–Hausdorff–Prokhorov topology to rescaled versions of the critical Erdős–Rényi limit objects. In this regime, component sizes are of order $j$1 and graph distances are of order $j$2 (Bhamidi et al., 2014).

Heavy-tailed critical regimes form a distinct class. When the limiting degree law has finite variance but infinite third moment, with power-law exponent $j$3, the ordered critical cluster sizes satisfy

$j$4

and the limit is described through excursion lengths of a thinned Lévy process. The tails of the largest-cluster scaling limit admit precise asymptotics with an exponential rate $j$5, a polynomial prefactor $j$6, and explicit correction terms (Hofstad et al., 2014). A plausible implication is that the product-kernel formalism supports more than one critical universality class, depending on which moments of the weight law remain finite.

3. Supercritical fluctuations and subcritical contrast

Above the phase transition, the size and the total weight of the giant component admit a process-level central limit theorem. If

$j$7

and $j$8, $j$9 are the deterministic approximations defined from the empirical weight sequence, then for all $w_i w_j$0,

$w_i w_j$1

converges in distribution, as a process on $w_i w_j$2, to a centered continuous Gaussian process in $w_i w_j$3 with an explicit covariance structure derived from jointly centered Gaussian processes $w_i w_j$4 (Jr, 2 Jan 2025). The same result recovers the classical Erdős–Rényi case when $w_i w_j$5.

In the barely supercritical regime, exponential concentration estimates quantify how closely the rank-1 model follows Erdős–Rényi behavior. When the weights have a finite fourth moment and the mean degree is slightly larger than $w_i w_j$6, the largest component has size of order $w_i w_j$7, total weight of order $w_i w_j$8, and surplus of order $w_i w_j$9, with bad-event probabilities bounded by $r(x,y)=v(x)v(y)$0. The constants depend on the moments of the weight law, with

$r(x,y)=v(x)v(y)$1

and the bounds are uniform in the barely supercritical parameter $r(x,y)=v(x)v(y)$2 (Safsafi, 2020).

Subcritical behavior exposes a limitation of any blanket “universality” claim for heavy-tailed inhomogeneous random graphs. For preferential-attachment-type kernels, the largest subcritical component has exponent

$r(x,y)=v(x)v(y)$3

and $r(x,y)=v(x)v(y)$4, so the largest component is polynomially larger than the maximum degree. In stark contrast, for the rank-1 kernel the size of the largest component scales as the maximum degree itself, with exponent $r(x,y)=v(x)v(y)$5 (Mörters et al., 7 Mar 2025). This directly identifies the product-kernel case as a distinct universality class rather than a generic proxy for all heavy-tailed inhomogeneous graphs.

4. Exploration processes, continuum limits, and metric structure

A central methodological feature of rank-1 models is that their component structure can be encoded by one-dimensional exploration processes. In the supercritical fluctuation theory, breadth-first walks driven by exponential random variables generalize the Aldous–Limic–Sellke representations, and weighted empirical-process CLTs supply the Gaussian limits (Jr, 2 Jan 2025). In the critical metric-space theory, connected components are constructed through tilted $r(x,y)=v(x)v(y)$6-trees, and tail bounds for $r(x,y)=v(x)v(y)$7-tree heights provide the tightness needed for convergence in Gromov–Hausdorff–Prokhorov topology (Bhamidi et al., 2014).

The continuum formulation goes further by embedding discrete multiplicative random graphs into Galton–Watson forests and then into Lévy processes. Continuous multiplicative graphs are constructed from the excursions of a Lévy-type encoding process; the associated height process defines a real-tree metric, and surplus edges appear as Poissonian “pinching” identifications along excursion intervals. These continuum objects are expected to be the universal limits for graph families related to the multiplicative coalescent, including Erdős–Rényi, rank-one inhomogeneous random graphs of various types, and the configuration model (Broutin et al., 2018). This suggests that the rank-1 condition is not merely a degree-modeling device, but also a setting in which the limiting metric geometry can be made explicit.

The same critical geometry underlies other optimization problems. For the minimum spanning tree on the complete graph with inhomogeneous random edge capacities determined by vertex weights, the expected diameter and typical distances are of order $r(x,y)=v(x)v(y)$8 under finite-moment conditions, and the proof is based on detailed study of rank-1 critical inhomogeneous random graphs together with couplings to Galton–Watson trees (Safsafi, 2020).

5. Spectral, extremal, and subgraph observables

In dense inhomogeneous Erdős–Rényi graphs with reference graphon $r(x,y)=v(x)v(y)$9, the maximal eigenvalue $[n]$0 satisfies a large deviation principle with speed $[n]$1 and rate function

$[n]$2

When $[n]$3 has rank $[n]$4, $[n]$5, the typical operator norm is $[n]$6, the rate function is quadratic near $[n]$7,

$[n]$8

and the minimizing perturbations are balanced global perturbations which are themselves rank $[n]$9 near the minimum (Chakrabarty et al., 2020). The later graphon LDP work weakens the regularity assumptions to $$p_{ij}=\min\left\{\frac{w_i w_j}{W_{[n]}},1\right\},\qquad W_{[n]}=\sum_{k=1}^n w_k,$$0 and identifies an explicit good rate function $$p_{ij}=\min\left\{\frac{w_i w_j}{W_{[n]}},1\right\},\qquad W_{[n]}=\sum_{k=1}^n w_k,$$1, making the rank-1 variational analysis more tractable (Markering, 2020).

Sparse spectral asymptotics show a different mechanism. For an inhomogeneous Erdős–Rényi graph with rank-1 kernel

$$p_{ij}=\min\left\{\frac{w_i w_j}{W_{[n]}},1\right\},\qquad W_{[n]}=\sum_{k=1}^n w_k,$$2

the largest eigenvalue of the adjacency matrix satisfies

$$p_{ij}=\min\left\{\frac{w_i w_j}{W_{[n]}},1\right\},\qquad W_{[n]}=\sum_{k=1}^n w_k,$$3

is separated from the spectral bulk, and after centering and scaling by $$p_{ij}=\min\left\{\frac{w_i w_j}{W_{[n]}},1\right\},\qquad W_{[n]}=\sum_{k=1}^n w_k,$$4 converges to a Gaussian law when the largest eigenvalue of the kernel has multiplicity $$p_{ij}=\min\left\{\frac{w_i w_j}{W_{[n]}},1\right\},\qquad W_{[n]}=\sum_{k=1}^n w_k,$$5. The associated eigenvector aligns asymptotically with the discretized eigenfunction $$p_{ij}=\min\left\{\frac{w_i w_j}{W_{[n]}},1\right\},\qquad W_{[n]}=\sum_{k=1}^n w_k,$$6 (Chakrabarty et al., 2019).

Extremal combinatorial observables are also sharply controlled in this class. The clique number in rank-1 random graphs is concentrated on at most two consecutive integers, its order is primarily determined by the overall edge density, and for sparse enough graphs it is always bounded, so the effect of inhomogeneity vanishes (Bogerd et al., 2018). At the level of induced subgraphs, power-law rank-1 inhomogeneous random graphs differ detectably from uniform random graphs with the same approximate degree sequence: certain induced subgraphs, notably starting at size $$p_{ij}=\min\left\{\frac{w_i w_j}{W_{[n]}},1\right\},\qquad W_{[n]}=\sum_{k=1}^n w_k,$$7, appear polynomially more often in the uniform model, and this supports a linear-time randomized algorithm for distinguishing the two models (Stegehuis, 2021).

6. Dynamics, percolation, and annealed spin systems

Bootstrap percolation on rank-1 kernels admits a law of large numbers governed by a fixed-point equation involving the size-biased weight law. With activation threshold $$p_{ij}=\min\left\{\frac{w_i w_j}{W_{[n]}},1\right\},\qquad W_{[n]}=\sum_{k=1}^n w_k,$$8 and initial infection probability $$p_{ij}=\min\left\{\frac{w_i w_j}{W_{[n]}},1\right\},\qquad W_{[n]}=\sum_{k=1}^n w_k,$$9, if $i$0 is the smallest positive solution of

$i$1

then the final infected fraction converges in probability to

$i$2

For power-law weights with exponent in $i$3, there is a critical function $i$4: if $i$5, the process does not evolve at all with high probability, whereas if $i$6, then with high probability the final infected set is linear (Amini et al., 2014).

The contact process exhibits long metastable plateaux on supercritical rank-one graphs with sufficiently large mean degree. Under the hypotheses stated as (H1)–(H3), for any $i$7 there exist $i$8 such that

$i$9

where $w_i$0 is the extinction time started from full occupancy, and

$w_i$1

The mechanism relies on a linear number of disjoint star subgraphs that sustain infection and transfer it between one another (Can, 2016).

Annealed spin systems provide another direction in which the rank-1 structure remains analytically explicit. For the annealed ferromagnetic $w_i$2-state Potts model on sparse rank-1 random graphs with edge probability $w_i$3, the thermodynamic limit of the pressure per particle exists under general conditions. In the infinite-variance weight case, the critical temperature equals infinity. For finite-variance weights, a general condition yields a first-order phase transition for all $w_i$4, and in the Pareto case the transition is first order when $w_i$5, but for $w_i$6 it is second order on $w_i$7 and first order for $w_i$8 (Giardinà et al., 14 Feb 2025). For the Ising case, the annealed generalized random graph maps to a rank-1 inhomogeneous Curie–Weiss model with couplings

$w_i$9

and with $w_i$00 replaced by $w_i$01 in the annealed graph formulation. The critical exponents are classical when the fourth moment of the weight distribution is finite, but for power-law weights with $w_i$02 they depend sensitively on $w_i$03, and the total spin at criticality satisfies a non-classical limit theorem (Dommers et al., 2015).