Rank-1 Inhomogeneous Random Graphs

• Rank-1 inhomogeneous random graphs are defined by vertex weights with edge probabilities proportional to the product of weights, serving as a flexible model for complex networks.
• They exhibit clearly defined phase transitions, scaling windows, and universality classes, influenced by the underlying weight distribution and heavy-tailed behaviors.
• Their applications span random processes and statistical mechanics, offering insights into phenomena like bootstrap percolation, contact processes, and spectral properties.

A rank-1 inhomogeneous random graph is a probabilistic graph model in which each vertex is endowed with a (typically nonnegative) weight, and edges between vertex pairs are formed independently with probability proportional to the product of their associated weights. This model generalizes classical Erdős–Rényi graphs and is central to the analysis of real-world complex networks exhibiting heavy-tailed degree distributions, heterogeneous local structures, and rich phase transition phenomena. The rank-1 structure is also foundational for understanding universality in critical random graphs, spectral properties, random processes on graphs, and statistical mechanics on random networks.

1. Model Definition and Fundamental Properties

A rank-1 inhomogeneous random graph (commonly referred to as the Norros–Reittu or Chung–Lu model) on a finite vertex set $V_n=\{1,\dots,n\}$ is defined by a weight vector $w=(w_1,\ldots,w_n)\in(0,\infty)^n$, and total weight $\ell_n=\sum_{i=1}^n w_i$.

The edge probability between distinct vertices $i\ne j$ is model-dependent but always of the form:

• Norros–Reittu: $p_{ij} = 1 - \exp(-w_i w_j/\ell_n)$,
• Chung–Lu: $p_{ij} = \min\{w_i w_j/\ell_n, 1\}$,
• Logistic: $p_{ij} = w_i w_j/(w_i w_j+\ell_n)$.

These models are equivalent in the sparse and critical regimes up to $o(1)$-errors on probability scales relevant for component structure and local limits (Bhamidi et al., 2014). The expected degree of vertex $i$ is $\mathbb{E}[\deg(i)]\approx w_i$, and if the empirical distribution of the $w=(w_1,\ldots,w_n)\in(0,\infty)^n$0 converges weakly to a law $w=(w_1,\ldots,w_n)\in(0,\infty)^n$1 with finite mean $w=(w_1,\ldots,w_n)\in(0,\infty)^n$2, the degree $w=(w_1,\ldots,w_n)\in(0,\infty)^n$3 is asymptotically Poisson-mixed,

$w=(w_1,\ldots,w_n)\in(0,\infty)^n$4

and a uniform random vertex has degree distribution $w=(w_1,\ldots,w_n)\in(0,\infty)^n$5 (Giardinà et al., 14 Feb 2025, Dommers et al., 2015). For power-law weights $w=(w_1,\ldots,w_n)\in(0,\infty)^n$6 with $w=(w_1,\ldots,w_n)\in(0,\infty)^n$7, the degree tail matches that of $w=(w_1,\ldots,w_n)\in(0,\infty)^n$8: $w=(w_1,\ldots,w_n)\in(0,\infty)^n$9 see (Stegehuis, 2021, Amini et al., 2014).

2. Criticality, Scaling Windows and Universality

Let $\ell_n=\sum_{i=1}^n w_i$0, and set $\ell_n=\sum_{i=1}^n w_i$1. The phase transition for the existence of a giant component is governed by: $\ell_n=\sum_{i=1}^n w_i$2

• Subcritical: $\ell_n=\sum_{i=1}^n w_i$3 — all components are sublinear in $\ell_n=\sum_{i=1}^n w_i$4.
• Supercritical: $\ell_n=\sum_{i=1}^n w_i$5 — a unique giant component of linear size emerges.
• Critical window: Adjust $\ell_n=\sum_{i=1}^n w_i$6; then largest clusters are of order $\ell_n=\sum_{i=1}^n w_i$7 and, upon rescaling graph distances by $\ell_n=\sum_{i=1}^n w_i$8, one observes convergence to continuum tree + shortcuts structures (Bhamidi et al., 2014).

This scaling limit is universal under mild moment and independence assumptions, encompassing the configuration model and general kernel-based inhomogeneous random graphs. When $\ell_n=\sum_{i=1}^n w_i$9 but $i\ne j$0, different scaling regimes and nonclassical universality classes arise (Hofstad et al., 2014).

3. Component Sizes, Structure, and Fluctuations

In the critical window, the vector of largest component sizes (scaled by $i\ne j$1) converges in distribution to a nontrivial sequence associated with excursions and Poissonian shortcuts on Brownian or Lévy processes:

• For standard finite third moment: scaling limits are continuum random trees plus a Poisson set of “shortcut” edges (Bhamidi et al., 2014, Broutin et al., 2018).
• For power-law weights $i\ne j$2, $i\ne j$3: component sizes scale as $i\ne j$4 with $i\ne j$5, and the scaling limit is expressed in terms of excursions of a “thinned Lévy process” (Hofstad et al., 2014).
• Above criticality, the (volume, size) of the largest component obeys process-level functional central limit theorems: fluctuations are Gaussian, with explicit covariance determined by the underlying weight law (Jr, 2 Jan 2025).

A breadth-first-walk or exploration process representation enables sharp exponential tail bounds on component sizes, total component weights, and surplus edges, both in the critical and barely-supercritical regimes, leveraging martingale decompositions and size-biased sampling concentration (Safsafi, 2020).

4. Local Limits, Functionals, and Subgraph Statistics

The local weak limit of sparse rank-1 IRGs is a delayed Galton–Watson tree: the root has offspring $i\ne j$6, descendants have offspring given by the size-biased law (Sturm et al., 2024). This underpins local central limit theorems for a wide class of graph functionals.

A central limit theorem applies to any (reasonable) functional $i\ne j$7 that satisfies a “good local approximation” (GLA) property, i.e., its fluctuations under local re-randomization can be controlled by nearby neighborhoods that (in the limit) look like trees (Sturm et al., 2024).

Counts of small subgraphs (motifs) in the power-law rank-1 IRG are determined via optimization over degree-weight bands, characterized by power-of-$i\ne j$8 scalings of the representative weights. Theorems identify the precise exponent in $i\ne j$9 for induced copies of any connected motif $p_{ij} = 1 - \exp(-w_i w_j/\ell_n)$0, revealing concentration onto sets of vertices whose empirical degrees are in small prescribed intervals. Notably, these results form the basis for randomized linear-time algorithms to distinguish IRGs from uniform random graphs with similar degree statistics, via the detection (or absence) of witness subgraphs whose frequency differs polynomially between models (Stegehuis, 2021).

5. Extreme Value and Spectral Theory

The maximal clique number in rank-1 IRGs is sharply concentrated on at most two consecutive integers, with the leading order set by the typical edge density, and inhomogeneity only manifesting in lower-order corrections or $p_{ij} = 1 - \exp(-w_i w_j/\ell_n)$1 factors. In sparse regimes, the dependence on individual weights vanishes (Bogerd et al., 2018).

Spectral properties are dominated by the rank-1 structure of the expected adjacency matrix. In dense regimes, the largest eigenvalue separates (as a “spike”) from the bulk, has Gaussian fluctuations, and the associated top eigenvector aligns closely with the population weight profile (Chakrabarty et al., 2019). Large deviation principles for the top eigenvalue are governed by explicit variational problems in graphon space, specialized to rank-1 by functional expansions around the reference kernel $p_{ij} = 1 - \exp(-w_i w_j/\ell_n)$2 (Chakrabarty et al., 2020).

6. Random Processes and Statistical Mechanics on Rank-1 IRGs

Random processes, such as bootstrap percolation, the contact process, and models from statistical mechanics (Ising and Potts), display phase transition behavior that depends sensitively on the moments of the weight distribution:

• Bootstrap percolation: For power-law $p_{ij} = 1 - \exp(-w_i w_j/\ell_n)$3, the critical initial infection fraction is drastically reduced due to “hubs,” with the critical scale $p_{ij} = 1 - \exp(-w_i w_j/\ell_n)$4 determined explicitly (Amini et al., 2014).
• Contact process: On supercritical IRGs, survival/extinction times are exponential in $p_{ij} = 1 - \exp(-w_i w_j/\ell_n)$5 for every $p_{ij} = 1 - \exp(-w_i w_j/\ell_n)$6, underpinned by the abundance of high-degree vertices that serve as persistent infection reservoirs (Can, 2016).
• Ising and Potts models (annealed): The critical temperature and scaling exponents diverge from classical mean-field values when the fourth moment diverges. Phase transitions may become first order for $p_{ij} = 1 - \exp(-w_i w_j/\ell_n)$7 Potts with weights having finite variance, but are second order in the infinite variance regime. The precise boundary in the Pareto case is set by a root $p_{ij} = 1 - \exp(-w_i w_j/\ell_n)$8 of an explicit equation (Giardinà et al., 14 Feb 2025, Dommers et al., 2015).

The critical exponents, limiting laws (including non-Gaussian limits at criticality), and universality classes are completely dictated by the weight tail, with finite fourth moment marking the threshold between classical and novel universality (Dommers et al., 2015).

7. Scaling Limits and Continuum Random Graph Objects

Rank-1 IRGs in the critical regime converge in the Gromov–Hausdorff–Prokhorov topology to continuum random graphs constructed as real trees (encoded via Brownian or Lévy processes) with a Poissonian surplus edge structure (“continuum trees plus shortcuts”) (Bhamidi et al., 2014, Broutin et al., 2018). The explicit continuum embedding employs growth processes, pruning, and Poisson pinching on height processes, with limit fractal dimensions determined by the parameters of the (possibly stable) Lévy process (Broutin et al., 2018). These results firmly establish “Lévy graphs” and their variants as the universal limits for wide classes of random graph models at criticality.

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References

(Stegehuis, 2021, Bhamidi et al., 2014, Hofstad et al., 2014, Sturm et al., 2024, Amini et al., 2014, Can, 2016, Dommers et al., 2015, Bogerd et al., 2018, Broutin et al., 2018, Safsafi, 2020, Giardinà et al., 14 Feb 2025, Chakrabarty et al., 2019, Chakrabarty et al., 2020, Jr, 2 Jan 2025).