Erdős–Rényi Comparison Graph

• Erdős–Rényi Comparison Graph is a framework that contrasts classical, modified, and Bohman–Frieze models to analyze structural transitions and scaling laws in random networks.
• The methodology uses differential equations and branching process approximations to pinpoint critical thresholds and linear growth of the largest component.
• The framework benchmarks network heterogeneity and guides extensions to geometric, inhomogeneous, and empirical graph analyses by comparing deviations from mean-field behavior.

The Erdős–Rényi Comparison Graph refers to a class of models and analytical frameworks that leverage the well-understood homogeneous Erdős–Rényi (ER) random graph as a baseline or null for comparing structural properties—such as phase transitions, scaling of components, and network heterogeneity—in both standard and generalized random graph processes. The comparison is central for understanding universality, benchmarking invariants, and quantifying deviations in more complex or structured networks such as geometric, power-law, or modified ER models.

1. Classical and Modified Erdős–Rényi Models

The Erdős–Rényi model $G(n, p)$ consists of $n$ vertices, with each edge present independently with probability $p$. In the random edge-adding process $G(n, m)$, $m$ edges are chosen uniformly at random; its continuous-time version $G(n, t)$ lets edges appear independently at a uniform rate $1/n$, so that $G(n, t)$ contains each edge present by time $t$.

A modified ER process can start from an arbitrary initial (possibly non-empty) graph $F$ with components $C_i(F)$. Edges among remaining non-edges are then added at random via:

• $G(n, m; F)$: exactly $m$ new edges,
• $G^*(n, m; F)$: $m$ i.i.d. random edges with replacement,
• $G(n, t; F)$: each missing edge appears with rate $1/n$.

The Bohman–Frieze process (BF process) further modifies ER: candidate edges preferentially connect isolated vertices, shifting critical parameters but preserving global universality characteristics.

2. Critical Thresholds: Comparison and Universality

In $G(n, m)$, the emergence of a giant component is abrupt at $m = n/2 + o(n)$. Equivalently, the continuous-time threshold is $t_c = 1$.

For the modified ER from an initial graph $F$, the threshold depends on the second moment ("susceptibility") of component sizes:

$$s_2(F) = \frac{1}{n} \sum_i |C_i(F)|^2 \qquad t_c = \frac{1}{s_2(F)},\quad m_c = \frac{n}{2\,s_2(F)}$$

In the BF process, the critical time is $t_c \approx 1.1763$; this point is marked by divergence of susceptibility $s_2(BF(t))$.

Although these critical points differ, they organize the transition to a giant component in structurally analogous ways across these models.

3. Scaling Laws Near the Phase Transition

For $m=m_c + \varepsilon n$ or $t=t_c + \varepsilon$ ($0 < \varepsilon \ll 1$), the largest component $L_1$ grows linearly in each of the classical, modified, and BF processes:

• Classical ER:

$$L_1(G(n, m)) = (2\varepsilon + o(\varepsilon)) n \qquad \varepsilon \to 0^+$$

• Modified ER (from $F$):

$$L_1(G(n, t; F)) = \theta(\varepsilon) n, \qquad \theta(\varepsilon) \sim \frac{2 s_2(F)^3}{s_3(F)} \varepsilon$$

where $s_3(F) = \frac{1}{n} \sum_i |C_i(F)|^3$.

• Bohman–Frieze Process:

$$L_1(BF(t)) = (y + o(1)) \varepsilon n, \qquad y = \frac{2(1-x_1(t_c))}{B}$$

where $B$ arises in the asymptotic blow-up of $s_3(t) \sim B (t_c - t)^{-3}$, with $x_1(t)$ the limiting isolated vertex fraction.

All three models possess a scaling window of width $\Delta m = O(n^{2/3})$ and, within this window, a linear giant-component growth with model-specific slope constants.

4. Analytical Techniques: Differential Equations and Branching Processes

Analyses of phase transitions and scaling leverage the “collapse” of each initial-component into a supervertex, reducing the model to a rank-1 inhomogeneous random graph (see the Bollobás–Janson–Riordan kernel model), and the use of integral operators with kernels of the form $\kappa(x, y) = t x y$. The phase transition corresponds to the operator norm just exceeding unity.

The barely supercritical regime is governed by a mixed-Poisson Galton–Watson branching process approximation, culminating in a survival probability equation:

$p = 1 - \mathbb{E}[e^{-t Z p}]$

where $Z$ denotes the size of a randomly chosen pre-critical component.

For the BF process, a differential equation method tracks the evolution of quantities such as isolated vertices $x_1(t)$ and moments $s_k(t)$, with explosion rates described by ODEs $\dot{s}_k = F_k(x_1, s_2, \ldots, s_k)$. Deviations from criticality are captured by moment expansions and component survival equations.

5. Universality Class and Critical Behavior

Despite the diverse mechanisms for edge addition, all these graph processes—classical ER, modified ER, and the BF process—fall into the same critical universality class with respect to the emergence of the giant component. Characteristics include:

• Width of the critical window $\Delta m = O(n^{2/3})$ (or $\Delta t = O(n^{-1/3})$)
• Scaling law $L_1 \sim \kappa \varepsilon n$ with the same linear exponent
• The survival-probability branching process equation is universal

Slope constants:

Model	Slope $\kappa$	Critical Point $t_c$
ER	$2$	$1$
Modified ER	$\frac{2 s_2^3}{s_3}$	$\frac{1}{s_2}$
Bohman–Frieze	$y = \frac{2(1-x_1(t_c))}{B}$	$\approx 1.1763$

This universality is frequently interpreted in the language of mean-field models in statistical physics, with critical scaling and exponents identical across this family, even as the model-dependent constants vary.

6. Broader Impact: Benchmarks and Extensions

ER comparison graphs serve as null benchmarks for analyzing observed network heterogeneity and critical phenomena in empirical networks, power-law graphs, and geometric or inhomogeneous models. For instance, the Rényi index—a measure of degree heterogeneity—vanishes in ER graphs, but asymptotes to one in power-law networks, thus enabling quantitative comparisons of real networks’ heterogeneity to a randomized baseline (Yuan, 2023).

Extensions to geometric or intersection models (e.g., $G(n, r, p) = G(n, r) \cap G(n, p)$), as well as spatial graphs with decaying connection probabilities, further adopt ER threshold logic and scaling forms, often with analogous critical exponents and component scaling, thereby reaffirming the ER model as a central comparison object in random graph theory (Janson et al., 2010, Goriachkin et al., 2023, Bennett et al., 7 Nov 2024).

7. Methodological Significance and Ongoing Research

The Erdős–Rényi comparison framework unifies a diverse set of graph models and phase transition phenomena, providing both rigorous analytic leverage and essential applied benchmarks. Ongoing research continues to extend these universality insights to inhomogeneous, geometric, and multi-layered networks, as well as to establish precise coupling techniques and branching-process analogues for new models exhibiting near-mean-field phase behavior (Janson et al., 2010, Bennett et al., 7 Nov 2024, Goriachkin et al., 2023).