--- title: Rank-1 Inhomogeneous Random Graph url: https://www.emergentmind.com/topics/rank-1-inhomogeneous-random-graph type: topic --- # Rank-1 Inhomogeneous Random Graph 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 [1402.2815], [2008.08367], [2501.01354]. ## 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 \(w_i\) [1402.2815]. In the Norros–Reittu, or Poissonian, formulation, \[ p_{ij}=1-\exp\left(-\frac{w_i w_j}{\ell_n}\right),\qquad \ell_n=\sum_{i=1}^n w_i, \] while the generalized random graph uses \[ \mathbb{P}(I_{i,j}=1)=\frac{w_i w_j}{\ell_n+w_i w_j}. \] In dense inhomogeneous Erdős–Rényi models, the rank-1 graphon is written as \(r(x,y)=v(x)v(y)\) [1404.1727], [2502.10553], [2008.08367]. | Formulation | Edge rule | Source | |---|---|---| | Chung–Lu | \(\min\{w_i w_j/W_{[n]},1\}\) | [1402.2815] | | Norros–Reittu / Poissonian | \(1-\exp(-w_i w_j/\ell_n)\) | [1404.1727] | | Generalized random graph | \(w_i w_j/(\ell_n+w_i w_j)\) | [2502.10553] | | Rank-1 graphon | \(r(x,y)=v(x)v(y)\) | [2008.08367] | 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 [1404.4118]. This does not mean that all finite-\(n\) 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 \[ 1-\exp\left(-\lambda w_i w_j/n\right), \] a uniformly chosen vertex has empirical weight \(W_n\), and if \(W_n \xrightarrow{d} W\) with \(\mathbb{E}[W_n^2]\to \mathbb{E}[W^2]<\infty\), then the critical threshold is \[ \lambda_*=\frac{1}{\mathbb{E}[W^2]}. \] The giant component exists only if \(\lambda>\lambda_*\) [2501.01354]. In the Norros–Reittu scaling, the corresponding criterion is expressed through \[ \nu=\frac{\mathbb{E}[W^2]}{\mathbb{E}[W]}, \] with subcritical behavior when \(\nu<1\), a giant component when \(\nu>1\), and criticality at \(\nu=1\) [1404.1727], [1404.4118]. 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 \[ w_i\mapsto \left(1+\lambda n^{-1/3}\right)w_i, \] and the ordered components, viewed as measured metric spaces with graph distances scaled by \(n^{-1/3}\) and masses by \(n^{-2/3}\), 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 \(n^{2/3}\) and graph distances are of order \(n^{1/3}\) [1404.4118]. Heavy-tailed critical regimes form a distinct class. When the limiting degree law has finite variance but infinite third moment, with power-law exponent \(\tau\in(3,4)\), the ordered critical cluster sizes satisfy \[ \left(|\mathcal C_{(1)}|n^{-\rho},|\mathcal C_{(2)}|n^{-\rho},\ldots\right)\xrightarrow{d}(\gamma_1(\lambda),\gamma_2(\lambda),\ldots), \qquad \rho=\frac{\tau-2}{\tau-1}, \] 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 \(-Iu^{\tau-1}\), a polynomial prefactor \(u^{-(\tau-1)/2}\), and explicit correction terms [1404.1727]. 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 \[ L_n(\lambda)=\text{size of the largest component},\qquad V_n(\lambda)=\text{total weight of the largest component}, \] and \(\theta^{(n)}(\lambda)\), \(\varrho^{(n)}(\lambda)\) are the deterministic approximations defined from the empirical weight sequence, then for all \(\lambda>\lambda_*\), \[ \left( \frac{L_n(\lambda)-\varrho^{(n)}(\lambda)n}{\sqrt n}, \frac{V_n(\lambda)-\theta^{(n)}(\lambda)n}{\sqrt n} \right) \] converges in distribution, as a process on \((\lambda_*,\infty)\), to a centered continuous Gaussian process in \(\mathbb{R}^2\) with an explicit covariance structure derived from jointly centered Gaussian processes \(\Psi_0,\Psi_1\) [2501.01354]. The same result recovers the classical Erdős–Rényi case when \(w_i\equiv 1\). 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 \(1\), the largest component has size of order \(2fn^{2/3}/C\), total weight of order \(2fn/C\), and surplus of order \(f^2\), with bad-event probabilities bounded by \(A\exp(-Af)\). The constants depend on the moments of the weight law, with \[ C=\frac{\mathbb{E}[W^3]}{(\mathbb{E}[W^2])^2}, \] and the bounds are uniform in the barely supercritical parameter \(f\) [2001.00882]. 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 \[ \rho_-=\frac{1}{2}-\sqrt{\left(\frac12-\gamma\right)^2-\beta(1-2\gamma)}, \] and \(\rho_->\gamma\), 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 \(\gamma\) [2503.05469]. 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 [2501.01354]. In the critical metric-space theory, connected components are constructed through tilted \(\mathbf p\)-trees, and tail bounds for \(\mathbf p\)-tree heights provide the tightness needed for convergence in Gromov–Hausdorff–Prokhorov topology [1404.4118]. 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 [1804.05871]. 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 \(n^{1/3}\) 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 [2012.01664]. ## 5. Spectral, extremal, and subgraph observables In dense inhomogeneous Erdős–Rényi graphs with reference graphon \(r\), the maximal eigenvalue \(\lambda_N/N\) satisfies a large deviation principle with speed \(N^2\) and rate function \[ \psi_r(\beta)=\inf_{h\in\mathcal W:\,\|T_h\|=\beta} I_r(h), \qquad I_r(h)=\int_{[0,1]^2} R(h(x,y)\mid r(x,y))\,dx\,dy. \] When \(r\) has rank \(1\), \(r(x,y)=v(x)v(y)\), the typical operator norm is \(C_r=m_2\), the rate function is quadratic near \(C_r\), \[ \psi_r(\beta)=[1+o(1)]K_r(\beta-C_r)^2,\qquad K_r=\frac{m_2^2}{2(m_3-m_2)}, \] and the minimizing perturbations are balanced global perturbations which are themselves rank \(1\) near the minimum [2008.08367]. The later graphon LDP work weakens the regularity assumptions to \(\log r,\log(1-r)\in L^1([0,1]^2)\) and identifies an explicit good rate function \(J_r\), making the rank-1 variational analysis more tractable [2010.03504]. Sparse spectral asymptotics show a different mechanism. For an inhomogeneous Erdős–Rényi graph with rank-1 kernel \[ f(x,y)=\theta r(x)r(y), \] the largest eigenvalue of the adjacency matrix satisfies \[ \lambda_1(A_N)=N\varepsilon_N\theta(1+o_p(1)), \] is separated from the spectral bulk, and after centering and scaling by \(\varepsilon_N^{-1/2}\) converges to a Gaussian law when the largest eigenvalue of the kernel has multiplicity \(1\). The associated eigenvector aligns asymptotically with the discretized eigenfunction \(r\) [1911.08244]. 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 [1805.01688]. 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 \(6\), appear polynomially more often in the uniform model, and this supports a linear-time randomized algorithm for distinguishing the two models [2102.09315]. ## 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 \(r\ge 2\) and initial infection probability \(p\), if \(\hat y\) is the smallest positive solution of \[ f_r(y;W_F^*,p)=(1-p)\,\mathbb E[\psi_r(W_F^*y)]+p-y=0, \] then the final infected fraction converges in probability to \[ (1-p)\,\mathbb E[\psi_r(W_F\hat y)]+p. \] For power-law weights with exponent in \((2,3)\), there is a critical function \(a_c(n)=o(n)\): if \(a(n)\ll a_c(n)\), the process does not evolve at all with high probability, whereas if \(a(n)\gg a_c(n)\), then with high probability the final infected set is linear [1402.2815]. 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 \(\lambda>0\) there exist \(c,C>0\) such that \[ \mathbb P\big(\exp(cn)\tau(q)\) [2502.10553]. For the Ising case, the annealed generalized random graph maps to a rank-1 inhomogeneous Curie–Weiss model with couplings \[ J_{ij}(\beta)=\beta\frac{w_iw_j}{\sum_k w_k}, \] and with \(\beta\) replaced by \(\sinh(\beta)\) 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 \(\tau\in(3,5)\) they depend sensitively on \(\tau\), and the total spin at criticality satisfies a non-classical limit theorem [1509.07327]. Source: https://www.emergentmind.com/topics/rank-1-inhomogeneous-random-graph