Erdős Graphs: Extremal & Random Models

• Erdős graphs are a class of graph models that blend deterministic extremal constructions with probabilistic and dynamic variants, including Erdős–Rényi and Turán-type examples.
• They exhibit key phenomena such as the Erdős–Hajnal property, nonhamiltonian extremal bounds, and precise spectral edge localization based on local graph structures.
• These models inform practical applications in graph alignment, network de-anonymization, and evolving networks, unifying combinatorial, probabilistic, and dynamic approaches.

Erdős graphs are a class of objects within extremal graph theory and random graph theory that encompass diverse phenomena connected to the foundational work of Paul Erdős. The term typically refers to several structurally or probabilistically defined families and extremal examples within classical problems of combinatorics and graph theory. Key contexts include stable graphs relevant to the Erdős–Hajnal conjecture, extremal nonhamiltonian graphs in relation to Turán-type problems, and random graphs such as Erdős–Rényi models at critical thresholds.

1. The Erdős–Hajnal Property and Stable Graphs

A central object of paper is the Erdős–Hajnal property, which conjectures the existence of large homogeneous sets (either cliques or independent sets) in graphs belonging to hereditary families. Specifically, for any finite class $\mathcal{F}$ of graphs, there exists $\varepsilon > 0$ such that every $G \in \mathcal{F}$ of order $n$ contains a clique or independent set of size at least $n^\varepsilon$:

$$\forall \mathcal{F}\;\exists \varepsilon>0\;\forall G\in\mathcal{F},\; |V(G)|=n\;\; \Bigl(\omega(G)\ge n^{\varepsilon}\;\vee\;\alpha(G)\ge n^{\varepsilon}\Bigr)$$

where $\omega(G)$ and $\alpha(G)$ denote the clique and independent-set numbers, respectively (Chernikov et al., 2015).

The absence of the $m$-order property—a configuration isomorphic to the half-graph of height $m$—characterizes so-called stable graphs. Chernikov and Starchenko demonstrated that, for every $m\geq 1$, there exists $\delta(m) > 0$ such that any $n$-vertex graph omitting the $m$-order property must contain a clique or independent set of size at least $n^{\delta(m)}$. Their proof employs model-theoretic techniques and an explicit dimension map $\delta(A)=\mathrm{st}(\log|A|/\log|V|)$ on pseudo-finite ultraproducts, circumventing traditional regularity lemmata and leveraging local stability/rank arguments.

The implication is a polynomial-size Erdős–Hajnal phenomenon for hereditary subclasses excluding the half-graph and its complements, extending also to stable hypergraphs and other settings where model-theoretic tameness (stability, NIP, distal) is present (Chernikov et al., 2015).

2. Extremal Erdős Graphs in Turán-Type and Hamiltonicity Problems

Another classical context is the extremal construction of nonhamiltonian graphs maximizing the number of edges subject to minimum degree constraints. Define, for integers $1 \le d \le \lfloor (n-1)/2 \rfloor$,

$h(n,d) = \binom{n-d}{2} + d^2$

The graph $H_{n,d}$ is constructed as a clique of order $n-d$ with $d$ additional vertices, each connected to the same $d$ vertices of the clique. Erdős proved that, for $n \geq 6d$, every nonhamiltonian $n$-vertex graph of minimum degree at least $d$ has at most $h(n,d)$ edges. This bound is sharp and achieved by $H_{n,d}$ (Füredi et al., 2017).

Subsequent work established stability results: if the edge count exceeds $h(n,d+1)$, the graph is forced to be a subgraph of an extremal configuration ($H_{n,d}$ or a related glued-clique construction). Critically, $H_{n,d}$ also maximizes the count of copies of any fixed subgraph $F$, for sufficiently large $n$ relative to $d$ and $|F|$. Further generalizations provide tight extremal and stability results for clique counts $N_k(G)$ and a full classification for almost extremal nonhamiltonian graphs, with low-degree sporadic configurations required in some cases.

Graph Class	Extremal Construction	Key Bound $h(n,d)$
Nonhamiltonian	$H_{n,d}$	$h(n,d) = \binom{n-d}{2} + d^2$

These results encapsulate how certain deterministic Erdős graphs—$H_{n,d}$ in particular—act as extremal objects for a range of combinatorial parameters (Füredi et al., 2017).

3. Erdős–Rényi Graphs and Spectral Structure

The probabilistic model popularized by Erdős and Rényi, $G(n,p)$, is a paradigmatic example of a random graph studied extensively for threshold phenomena and spectral properties. Recent research analyzes the spectral edge of sparse Erdős–Rényi graphs $G(N, d/N)$, with $d$ in slowly growing or nearly constant regimes.

In the range $\log^{-1/9} N \leq d \leq \log^{1/40} N$, the largest eigenvalues (“edge eigenvalues”) of the adjacency matrix are determined by local neighborhoods around vertices of (near) maximal degree. For the $K = \exp(\log^{1/8} N)$ largest eigenvalues, there exist vertices $x$ such that

$$\lambda = \sqrt{\,\alpha_x + \frac{\beta_x}{\alpha_x} + \frac{d^2+d}{\alpha_x} + O((d^{3/2}+1)u^{-11/6})}$$

where $\alpha_x$ is the degree, and $\beta_x$ the number of distance-two neighbors of $x$. The corresponding eigenvectors are exponentially localized around these high-degree vertices. Moreover, after appropriate centering and scaling, the extremal eigenvalues converge to a Poisson point process, a phenomenon previously conjectured to occur only for larger $d$ but now verified asymptotically down to near-logarithmic $d$ (Hiesmayr et al., 2023).

These results elucidate how, in Erdős–Rényi graphs, rare local structure entirely governs the spectral edge, and the top eigenvectors are tightly localized—a stark contrast with delocalized eigenvectors of denser or regular random graphs.

4. Correlated Erdős–Rényi Graphs and Exact Alignment

In the context of network alignment, two correlated Erdős–Rényi graphs $(G_a, G_b)$ are generated on the same vertex set, with edges present independently according to a joint distribution parameterized by $(p_{11}, p_{10}, p_{01}, p_{00})$ to encode correlation. After random label permutation of one graph, the recovery of the hidden permutation—“graph alignment”—exhibits a sharp information-theoretic threshold.

For positive correlation ($p_{11}p_{00} > p_{10}p_{01}$), the alignment recovery is possible if and only if $p_{11}n > (1+o(1))\log n$, analogous to the connectivity threshold for $G(n,p)$. The analysis involves bounding the probability that an incorrect permutation achieves smaller or equal edge discrepancy, leading to union bounds over the symmetric group. In the sparse regime, exact recovery becomes information-theoretically possible at this logarithmic edge-density threshold (Cullina et al., 2017).

This model clarifies fundamental limits of de-anonymization and alignment in large sparse random networks and underscores the combinatorial sharpness typical of Erdős-type graph models.

5. Dynamic Erdős–Rényi Graphs

Dynamic analogues of the Erdős–Rényi model generalize static edge probabilities to time-dependent, environment-driven, or stochastic regime-switching settings. Two models are notable:

1. Regime-Switching Model: Each edge undergoes birth and death transitions at rates determined by an external Markov process $X(t)$; conditional on $X(t)=i$, edge activation and deletion occur at rates $\lambda_i$, $\mu_i$.
2. Periodic Resampling Model: At each time epoch, transition probabilities for edges are resampled (possibly from a continuous distribution).

Quantitative analysis yields explicit formulae for the stationary mean, variance, and factorial moments of the number of edges $Y(t)$—via generating functions and coupled ODE systems—in both models. Functionally, in the fast-environment limit, the edge count approaches a time-inhomogeneous binomial law, and its diffusion scaling limit is Ornstein–Uhlenbeck (Mandjes et al., 2017).

Sample-path large deviations principles are established, with variational rate functionals capturing both local jump dynamics (via Cramér transforms) and environmental randomness. Such dynamic Erdős graphs supply a versatile framework for time-evolving networks in settings such as chemical reaction networks, adaptive communication networks, and road or social systems subject to regime changes.

6. Broader Implications and Theoretical Relationships

Research on Erdős graphs continues to bridge combinatorial extremal theory, probabilistic methods, and logic/model theory. Results on stable graphs and the Erdős–Hajnal property exemplify applications of stability theory and pseudo-finite analysis, linking homogeneous set existence to the tameness of definable structures (Chernikov et al., 2015). Turán-type extremal constructions such as $H_{n,d}$ provide deterministic counterpoints to the typical case behavior in random graphs, while the sharp algebraic and probabilistic thresholds in random and dynamic Erdős graphs reveal the fine interplay between deterministic structure and stochasticity.

Theoretical advances in these areas influence directions in computational complexity (e.g., graph alignment under correlation), network science (robustness of dynamically evolving systems), and mathematical logic (model-theoretic classification of combinatorial phenomena). The continuing unification of extremal, probabilistic, spectral, and structural perspectives underpins the centrality of Erdős graphs across contemporary graph theory.