Chung–Lu Graphs Overview

• Chung–Lu graphs are inhomogeneous random graphs defined by vertex weights that establish expected degrees and independent edge probabilities.
• They generalize Erdős–Rényi models and capture scale-free phenomena by controlling degree distributions, phase transitions, and giant component emergence.
• Extensions incorporate geometry, reciprocity, and temporal dynamics, making Chung–Lu graphs a versatile null model for community detection and spectral studies.

Chung–Lu graphs are inhomogeneous random graphs specified by vertex weights or expected degrees, with edges inserted independently and probabilities proportional to products of endpoint weights. In the undirected setting, a standard formulation uses a weight vector $\mathbf w=(w_1,\dots,w_n)$ and total weight $W=\sum_{i=1}^n w_i$, with edge probability either $p_{ij}=w_iw_j/W$ in the admissible regime or $p_{ij}=\min\{1,w_iw_j/W\}$ in clipped formulations. In the directed setting, two weight sequences $(w_x)_{x\in[n]}$ and $(w'_y)_{y\in[n]}$ with equal total mass $w$ give $p_{xy}=(w_xw'_y/w)\wedge 1$. The model is an inhomogeneous generalization of Erdős–Rényi, a canonical expected-degree model, and a widely used null model for community detection, diffusion, routing, and spectral studies (Fasino et al., 2019, Bianchi et al., 2024).

1. Definition and basic formulations

The classical undirected Chung–Lu model generates a random graph on $n$ vertices from a nonnegative expected-degree vector $\mathbf w$. Each edge is placed independently with probability proportional to the product of the endpoint weights. When self-loops are allowed for mathematical convenience, the formula

$W=\sum_{i=1}^n w_i$0

gives the exact expected-degree identity

$W=\sum_{i=1}^n w_i$1

Several later works instead use the clipped form

$W=\sum_{i=1}^n w_i$2

which keeps the model well defined beyond the strict admissible regime but changes some exact identities. The directed version replaces one weight sequence by two, one for out-neighborhood formation and one for in-neighborhood formation, and places oriented edges independently (Fasino et al., 2019, Dionigi et al., 2022, Bianchi et al., 2024).

Variant	Edge probability	Notable point
Undirected admissible form	$W=\sum_{i=1}^n w_i$3	$W=\sum_{i=1}^n w_i$4
Undirected clipped form	$W=\sum_{i=1}^n w_i$5	used when probabilities need truncation
Directed Chung–Lu digraph	$W=\sum_{i=1}^n w_i$6	expected in- and out-structure separated

This factorized edge law makes Chung–Lu a weighted Erdős–Rényi model in which heterogeneity is encoded entirely through vertex weights. If all weights are equal, the model reduces to a homogeneous Erdős–Rényi graph or digraph. In the directed weakly sparse setting studied for random walks, equal weights recover a directed Erdős–Rényi model with

$W=\sum_{i=1}^n w_i$7

for a suitable constant $W=\sum_{i=1}^n w_i$8 (Bianchi et al., 2024).

2. Expected degrees, admissibility, and null-model status

A central feature of Chung–Lu graphs is that the parameter vector is interpreted as an expected-degree sequence. In the admissible undirected model, the standard condition

$W=\sum_{i=1}^n w_i$9

ensures that all edge probabilities lie in $p_{ij}=w_iw_j/W$0. Under this condition, the model preserves the exact identity $p_{ij}=w_iw_j/W$1, and the factorized edge probabilities do not impose degree-degree correlations between adjacent nodes. One paper explicitly characterizes Chung–Lu as the only model in that setting where edge probabilities factor as a product of separate functions of the endpoint weights and therefore do not introduce degree correlations between the degrees of two nodes joined by an edge (Fasino et al., 2019).

This admissibility requirement is not merely technical. When it fails and one replaces the formula by an ad hoc truncation such as

$p_{ij}=w_iw_j/W$2

the identity $p_{ij}=w_iw_j/W$3 is lost, and with it the controllability of expected degrees and the clean absence of degree correlations. That role as an analytically transparent, degree-driven baseline is why Chung–Lu is repeatedly used as a null model for community detection, modularity-based methods, spectral comparison, and algorithmic benchmarking. The same perspective underlies comparisons with Stochastic Kronecker Graphs, where the associated Chung–Lu model is obtained simply by matching the expected degree sequence implied by the Kronecker construction (Fasino et al., 2019, Pinar et al., 2011).

In directed settings, the analogous construction uses expected in-degrees and out-degrees. A basic form assigns a probability

$p_{ij}=w_iw_j/W$4

where $p_{ij}=w_iw_j/W$5 and $p_{ij}=w_iw_j/W$6 are expected out- and in-degrees and $p_{ij}=w_iw_j/W$7. This preserves the Chung–Lu principle while separating source and destination roles; later extensions add reciprocity or community structure without abandoning the factorized expected-degree logic (Burstein, 2017, Durak et al., 2012).

3. Scale-free parameterizations, giant components, and distances

A major use of Chung–Lu graphs is the generation and analysis of scale-free networks. One explicit power-law parameterization takes

$p_{ij}=w_iw_j/W$8

which yields an expected degree distribution with exponent $p_{ij}=w_iw_j/W$9. In the simplest case $p_{ij}=\min\{1,w_iw_j/W\}$0, admissibility can be enforced by explicit bounds on $p_{ij}=\min\{1,w_iw_j/W\}$1, and for $p_{ij}=\min\{1,w_iw_j/W\}$2 one can choose $p_{ij}=\min\{1,w_iw_j/W\}$3 and $p_{ij}=\min\{1,w_iw_j/W\}$4 so that the graph has a prescribed average expected degree $p_{ij}=\min\{1,w_iw_j/W\}$5, prescribed largest expected degree $p_{ij}=\min\{1,w_iw_j/W\}$6, and asymptotic average degree satisfying

$p_{ij}=\min\{1,w_iw_j/W\}$7

A convenient choice highlighted in the literature is

$p_{ij}=\min\{1,w_iw_j/W\}$8

which guarantees admissibility for sufficiently large $p_{ij}=\min\{1,w_iw_j/W\}$9 in the reported experiments (Fasino et al., 2019).

Connectivity phenomena in Chung–Lu graphs are often described through giant-component criteria. In one standard theorem, if $(w_x)_{x\in[n]}$0 and

$(w_x)_{x\in[n]}$1

then almost surely $(w_x)_{x\in[n]}$2 has a unique giant component, while if

$(w_x)_{x\in[n]}$3

for some $(w_x)_{x\in[n]}$4, then almost surely there is no giant component. A more explicit phase-transition analysis places Chung–Lu inside the Bollobás–Janson–Riordan inhomogeneous random graph framework with rank-1 kernel

$(w_x)_{x\in[n]}$5

The operator threshold is governed by $(w_x)_{x\in[n]}$6, and in the rank-1 case the critical parameter is

$(w_x)_{x\in[n]}$7

Near criticality, the giant component fraction $(w_x)_{x\in[n]}$8 has different asymptotics depending on the exponent: for $(w_x)_{x\in[n]}$9,

$(w'_y)_{y\in[n]}$0

whereas for $(w'_y)_{y\in[n]}$1,

$(w'_y)_{y\in[n]}$2

In the subcritical regime with $(w'_y)_{y\in[n]}$3 and $(w'_y)_{y\in[n]}$4, the largest component is at most of order

$(w'_y)_{y\in[n]}$5

with high probability (Ding et al., 2023, Fasino et al., 2019).

Distance asymptotics supply another canonical signature. For Chung–Lu graphs with power-law weights of exponent $(w'_y)_{y\in[n]}$6, and especially in the regime $(w'_y)_{y\in[n]}$7, the average distance satisfies

$(w'_y)_{y\in[n]}$8

A broad geometric extension preserving Chung–Lu-like marginal edge probabilities has the same leading asymptotic average distance, up to a factor $(w'_y)_{y\in[n]}$9. That class includes Chung–Lu graphs, hyperbolic random graphs, and geometric inhomogeneous random graphs, and the same work also proves a giant component of linear size, polylogarithmic size for all other components, and polylogarithmic diameter (Bringmann et al., 2016).

4. Directed Chung–Lu graphs and random walks

Directed Chung–Lu graphs are built from two positive weight sequences $w$0 and $w$1 on $w$2, with equal total mass $w$3, and independent oriented edges

$w$4

This is a soft-constraint model: the in- and out-degrees are not fixed exactly, only their expectations are prescribed. In the weakly sparse regime studied for mixing, the assumptions

$w$5

and

$w$6

imply $w$7, expected in- and out-degrees of order $w$8, and

$w$9

If there exists $p_{xy}=(w_xw'_y/w)\wedge 1$0 such that

$p_{xy}=(w_xw'_y/w)\wedge 1$1

then comparison with the Erdős–Rényi digraph and monotonicity of strong connectivity imply strong connectivity with high probability, hence irreducibility of the simple random walk and a unique stationary distribution $p_{xy}=(w_xw'_y/w)\wedge 1$2 (Bianchi et al., 2024).

For a realized digraph $p_{xy}=(w_xw'_y/w)\wedge 1$3, the transition matrix of the simple random walk is

$p_{xy}=(w_xw'_y/w)\wedge 1$4

and distance to stationarity is measured in total variation. The central mixing scale is the entropic time

$p_{xy}=(w_xw'_y/w)\wedge 1$5

where

$p_{xy}=(w_xw'_y/w)\wedge 1$6

Under the stated assumptions,

$p_{xy}=(w_xw'_y/w)\wedge 1$7

The main theorem proves a cutoff phenomenon at $p_{xy}=(w_xw'_y/w)\wedge 1$8: for fixed $p_{xy}=(w_xw'_y/w)\wedge 1$9, total variation is asymptotically near $n$0 before $n$1 and near $n$2 after $n$3, uniformly over the starting vertex. Moreover, in a critical window of size

$n$4

the cutoff profile converges to the Gaussian tail $n$5 under a mild non-degeneracy condition on the variance $n$6 (Bianchi et al., 2024).

Directed Chung–Lu graphs also admit spectral-radius asymptotics. For expected in-degree sequence $n$7, expected out-degree sequence $n$8, and $n$9, the adjacency spectral radius concentrates around

$\mathbf w$0

A community-structured extension, the Partitioned Chung–Lu model, replaces this scalar predictor by the spectral radius $\mathbf w$1 of a block-level matrix built from blockwise degree products. The analysis is path-counting based, using bounds on $\mathbf w$2 and $\mathbf w$3 to control $\mathbf w$4 (Burstein, 2017).

5. Spectral theory, normalized Laplacians, and PageRank

For undirected Chung–Lu graphs with prescribed degree sequence $\mathbf w$5, the adjacency matrix has rank-one expectation

$\mathbf w$6

where $\mathbf w$7. Consequently,

$\mathbf w$8

Under the assumptions

$\mathbf w$9

with $W=\sum_{i=1}^n w_i$00 and $W=\sum_{i=1}^n w_i$01, the principal eigenvalue and eigenvector obey Gaussian asymptotics. Specifically,

$W=\sum_{i=1}^n w_i$02

and with

$W=\sum_{i=1}^n w_i$03

one has

$W=\sum_{i=1}^n w_i$04

The principal eigenvector $W=\sum_{i=1}^n w_i$05 is asymptotically aligned with the normalized degree vector $W=\sum_{i=1}^n w_i$06, and for fixed coordinates there is also a pointwise central limit theorem under a stronger sparsity assumption (Dionigi et al., 2022).

The normalized Laplacian exhibits a separate global law. For a Chung–Lu graph $W=\sum_{i=1}^n w_i$07 with positive weights $W=\sum_{i=1}^n w_i$08, average weight

$W=\sum_{i=1}^n w_i$09

and minimum weight $W=\sum_{i=1}^n w_i$10, the normalized Laplacian

$W=\sum_{i=1}^n w_i$11

is defined using the Moore–Penrose pseudoinverse convention when zero degrees occur. Under

$W=\sum_{i=1}^n w_i$12

the empirical spectral distribution of the rescaled normalized adjacency,

$W=\sum_{i=1}^n w_i$13

converges weakly in probability to the semicircle law on $W=\sum_{i=1}^n w_i$14. This improves earlier work that established the semicircle law only for the proxy matrix $W=\sum_{i=1}^n w_i$15, where $W=\sum_{i=1}^n w_i$16 is the diagonal matrix of expected degrees (Chen et al., 23 Dec 2025).

PageRank on Chung–Lu graphs becomes asymptotically degree-driven under expansion-type assumptions. For personalized PageRank

$W=\sum_{i=1}^n w_i$17

on an undirected graph, if

$W=\sum_{i=1}^n w_i$18

then

$W=\sum_{i=1}^n w_i$19

with high probability. Under the stronger conditions $W=\sum_{i=1}^n w_i$20 for all $W=\sum_{i=1}^n w_i$21 and $W=\sum_{i=1}^n w_i$22 for some $W=\sum_{i=1}^n w_i$23, the same approximation holds coordinatewise: $W=\sum_{i=1}^n w_i$24 In this regime, PageRank is asymptotically a mixture of the restart distribution and the degree distribution (Avrachenkov et al., 2017, Avrachenkov et al., 2015).

6. Extensions, comparative models, and applications

Chung–Lu graphs are used both as a generator and as a baseline to which more structured models are compared. For large scale-free network generation, explicit admissible power-law constructions make it possible to control the largest expected degree and the average expected degree independently while preserving a power-law expected degree profile and, under suitable conditions, a giant component. Because the standard practical generators can underproduce low-degree vertices, especially degree-1 vertices, one later correction models the output degree sequence by a Poisson-based linear system $W=\sum_{i=1}^n w_i$25, truncates to a finite matrix $W=\sum_{i=1}^n w_i$26, and computes an adjusted input from

$W=\sum_{i=1}^n w_i$27

That “shifted Chung–Lu” construction is reported to reduce output error substantially for feasible sparse power-law distributions (Fasino et al., 2019, Brissette et al., 2021).

Several important variants modify the baseline while preserving the degree-oriented logic. The Transitive Chung–Lu model mixes ordinary Chung–Lu edge generation with a friend-of-a-friend mechanism controlled by a single parameter $W=\sum_{i=1}^n w_i$28, while preserving the expected degree distribution of the input graph. For directed networks with reciprocity, the Fast Reciprocal Directed model extends the Chung–Lu philosophy by separately generating reciprocal edges and one-way edges so as to match in-degree, out-degree, and reciprocal-degree distributions; the algorithm requires $W=\sum_{i=1}^n w_i$29 random numbers for a graph with $W=\sum_{i=1}^n w_i$30 edges (III et al., 2012, Durak et al., 2012).

Other extensions introduce geometry, temporal dynamics, or graph alignment. A geometric Chung–Lu model augments the expected-degree term by a distance-dependent factor,

$W=\sum_{i=1}^n w_i$31

and uses boundary corrections to fit the Drosophila Medulla connectome. A dynamic Chung–Lu model lets each edge alternate between present and absent states while preserving the stationary marginal

$W=\sum_{i=1}^n w_i$32

and estimates the underlying parameters from repeated snapshots of the total number of edges via method-of-moments equations and a delta-method central limit theorem. For correlated Chung–Lu graphs, the $W=\sum_{i=1}^n w_i$33-core estimator achieves exact recovery of the latent permutation when

$W=\sum_{i=1}^n w_i$34

under the stated sparsity condition on the maximum weight (Agarwala et al., 2021, Hazra et al., 17 Feb 2025, Rácz et al., 2023).

Chung–Lu also functions as a comparative control model in the analysis of other random graph families. One study shows that the probability matrix of Stochastic Kronecker Graphs is often nearly identical to that of an associated Chung–Lu model, and proves exact equality when

$W=\sum_{i=1}^n w_i$35

Another study compares Barabási–Albert and Chung–Lu graphs after transferring the empirical BA degree profile into Chung–Lu parameters: the bulk spectral distributions become similar above a threshold near $W=\sum_{i=1}^n w_i$36, but the extreme eigenvalues and principal eigenvector remain distinguishable. These comparisons reinforce the role of Chung–Lu as a flexible, analytically tractable control model rather than a universal substitute for models with clustering, attachment dependence, or strong mesoscopic structure (Pinar et al., 2011, Glos, 2018).

A recurring limitation is that the simplest Chung–Lu construction deliberately suppresses effects not encoded by the degree sequence. It does not by itself model clustering, reciprocity, latent geometry, or temporal persistence. In heavy-tailed regimes this simplification also shapes rare-event behavior: for Chung–Lu graphs with regularly varying weights, an excess of order $W=\sum_{i=1}^n w_i$37 edges is asymptotically dominated by the appearance of a small number of hubs, and the edge-count large deviations are governed by a stepwise rate function

$W=\sum_{i=1}^n w_i$38

That result emphasizes the same structural fact that underlies much of the model’s theory: heterogeneity enters through vertex weights, and extreme graph events are often controlled by extreme weights (Stegehuis et al., 2022).