We study a family of Crump–Mode–Jagers branching processes in a random environment that explode, i.e. that grow infinitely large in finite time with positive probability. Building on recent work of Iyer and the author (‘On the structure of genealogical trees associated with explosive Crump–Mode–Jagers branching processes’, arXiv:2311.14664, 2023), we weaken certain assumptions required to prove that the branching process, at the time of explosion, contains a (unique) individual with infinite offspring. We then apply these results to super-linear preferential attachment models. In particular, we fill gaps in some of the cases analysed in Appendix A of the work of Iyer and the author and study a large range of previously unattainable cases.

We study the number of triangles $T_n$ in the sparse $\beta$-model on n vertices, a random graph model that captures degree heterogeneity in real-world networks. Using the norms of the heterogeneity parameter vector, we first determine the asymptotic mean and variance of $T_n$. Next, by applying the Malliavin–Stein method, we derive a non-asymptotic upper bound on the Kolmogorov distance between the normalized $T_n$ and the standard normal distribution. Under an additional assumption on degree heterogeneity, we further prove the asymptotic normality for $T_n$ as $n\to\infty$.

Let $X_1,\ldots, X_n$ be independent integers distributed uniformly on [M], $M\ge 2$. A partition S of [n] into $\nu$ non-empty subsets $S_1,\ldots, S_{\nu}$ is called perfect if all $\nu$ values $\sum_{j\in S_{\alpha}}X_j$ are equal. For a perfect partition to exist, $\sum_j X_j$ has to be divisible by $\nu$. In 2001, for $\nu=2$, Christian Borgs, Jennifer Chayes, and the author proved that, conditioned on $\sum_j X_j$ being even, with high probability a perfect partition exists if $\kappa\;:\!=\; \lim {{n}/{\log M}}>{{1}/{\log 2}}$, and that with high probability no perfect partition exists if $\kappa<{{1}/{\log 2}}$. Responding to a question by George Varghese, we prove that for $\nu\ge 3$ with high probability no perfect partition exists if $\kappa<{{2}/{\log \nu}}$, which is twice as large as the naive threshold $1/\log 3$ for $\nu=3$. We identify the range of $\kappa$ where the expected number of perfect partitions is exponentially high. We show that for $\kappa> {{2(\nu-1)}/{\log[(1-2\nu^{-2})^{-1}]}}$ the total number of perfect partitions is exponentially high with probability $\gtrsim (1+\nu^2)^{-1}$, i.e. below $1/\nu$, the limiting probability that $\sum_j X_j$ is divisible by $\nu$.

Confirming a conjecture of Erdős on the chromatic number of Kneser hypergraphs, Alon, Frankl and Lovász proved that in any $q$-colouring of the edges of the complete $r$-uniform hypergraph, there exists a monochromatic matching of size $\lfloor \frac {n+q-1}{r+q-1}\rfloor$. In this paper, we prove a transference version of this theorem. More precisely, for fixed $q$ and $r$, we show that with high probability, a monochromatic matching of approximately the same size exists in any $q$-colouring of a random hypergraph, already when the average degree is a sufficiently large constant. In fact, our main new result is a defect version of the Alon–Frankl–Lovász theorem for almost complete hypergraphs. From this, the transference version is obtained via a variant of the weak hypergraph regularity lemma. The proof of the defect version uses tools from extremal set theory developed in the study of the Erdős matching conjecture.

The purpose of this paper is to analyze the degree index and the clustering index in dense random graphs. The degree index in our setup is a certain measure of degree irregularity whose basic properties are well studied in the literature, and the corresponding theoretical analysis in a random graph setup turns out to be tractable. On the other hand, the clustering index, based on a similar reasoning, is first introduced in this paper. Computing exact expressions for the expected clustering index turns out to be more challenging even in the case of Erdős–Rényi graphs, and our results are on obtaining relevant upper bounds. These are also complemented with observations based on Monte Carlo simulations. In addition to the Erdős–Rényi case, we also present a simulation-based analysis for random regular graphs, the Barabási–Albert model, and the Watts–Strogatz model.

Given a sequence of graphs $G_n$ and a fixed graph H, denote by $T(H, G_n)$ the number of monochromatic copies of the graph H in a uniformly random c-coloring of the vertices of $G_n$. In this paper we study the joint distribution of a finite collection of monochromatic graph counts in networks with multiple layers (multiplex networks). Specifically, given a finite collection of graphs $H_1, H_2, \ldots, H_d$ we derive the joint distribution of $(T(H_1, G_n^{(1)}), T(H_2, G_n^{(2)}), \ldots, T(H_d, G_n^{(d)}))$, where $\mathbf{G}_n = (G_n^{(1)}, G_n^{(2)}, \ldots, G_n^{(d)})$ is a collection of dense graphs on the same vertex set converging in the multiplex cut-metric. The limiting distribution is the sum of two independent components: a multivariate Gaussian and a sum of independent bivariate stochastic integrals. This extends previous results on the marginal convergence of monochromatic subgraphs in a sequence of graphs to the joint convergence of a finite collection of monochromatic subgraphs in a sequence of multiplex networks. Several applications and examples are discussed.

We identify the size of the largest connected component in a subcritical inhomogeneous random graph with a kernel of preferential attachment type. The component is polynomial in the graph size with an explicitly given exponent, which is strictly larger than the exponent for the largest degree in the graph. This is in stark contrast to the behaviour of inhomogeneous random graphs with a kernel of rank one. Our proof uses local approximation by branching random walks going well beyond the weak local limit and novel results on subcritical killed branching random walks.

In this work, we develop a unified framework for establishing sharp threshold results for various Ramsey properties. To achieve this, we view such properties as noncolourability of auxiliary hypergraphs. Our main technical result gives sufficient conditions on a sequence of such hypergraphs that guarantee that this noncolourability property has a sharp threshold in subhypergraphs induced by random subsets of the vertices.

Furthermore, we verify these conditions in several cases of interest. In the classical setting of Ramsey theory for graphs, we show that the property of being Ramsey for a graph H in r colours has a sharp threshold in $G_{n,p}$, for all $r \geqslant 2$ and all H in a class of graphs that includes all cliques and cycles. In the arithmetic setting, we establish sharpness of thresholds for the properties corresponding to van der Waerden’s theorem and Schur’s theorem, also in any number of colours.

Distributed ledgers, including blockchain and other decentralized databases, are designed to store information online where all trusted network members can update the data with transparency. The dynamics of a ledger’s development can be mathematically represented by a directed acyclic graph (DAG). In this paper, we study a DAG model that considers batch arrivals and random delay of attachment. We analyze the asymptotic behavior of this model by letting the arrival rate go to infinity and the inter-arrival time go to zero. We establish that the number of leaves in the DAG, as well as various random variables characterizing the vertices in the DAG, can be approximated by its fluid limit, represented as the solution to a set of delayed partial differential equations. Furthermore, we establish the stable state of this fluid limit and validate our findings through simulations.

In this paper we study degree-penalized contact processes on Galton-Watson (GW) trees and the configuration model. The model we consider is a modification of the usual contact process on a graph. In particular, each vertex can be either infected or healthy. When infected, each vertex heals at rate one. Also, when infected, a vertex u with degree $d_u$ infects its neighboring vertex v with degree $d_v$ with rate $\lambda / f(d_u, d_v)$ for some positive function f. In the case $f(d_u, d_v)=\max (d_u, d_v)^\mu $ for some $\mu \ge 0$, the infection is slowed down to and from high-degree vertices. This is in line with arguments used in social network science: people with many contacts do not have the time to infect their neighbors at the same rate as people with fewer contacts.

We show that new phase transitions occur in terms of the parameter $\mu $ (at $1/2$) and the degree distribution D of the GW tree.

• • When $\mu \ge 1$, the process goes extinct for all distributions D for all sufficiently small $\lambda>0$;

• • When $\mu \in [1/2, 1)$, and the tail of D weakly follows a power law with tail-exponent less than $1-\mu $, the process survives globally but not locally for all $\lambda $ small enough;

• • When $\mu \in [1/2, 1)$, and $\mathbb {E}[D^{1-\mu }]<\infty $, the process goes extinct almost surely, for all $\lambda $ small enough;

• • When $\mu <1/2$, and D is heavier than stretched exponential with stretch-exponent $1-2\mu $, the process survives (locally) with positive probability for all $\lambda>0$.

We also study the product case, where $f(d_u,d_v)=(d_u d_v)^\mu $. In that case, the situation for $\mu < 1/2$ is the same as the one described above, but $\mu \ge 1/2$ always leads to a subcritical contact process for small enough $\lambda>0$ on all graphs. Furthermore, for finite random graphs with prescribed degree sequences, we establish the corresponding phase transitions in terms of the length of survival.

We study the bilateral preference graphs $\mathit{LK}(n, k)$ of La and Kabkab, obtained as follows. Put independent and uniform [0, 1] weights on the edges of the complete graph $K_n$. Then, each edge (i, j) is included in $\mathit{LK}(n,k)$ if it is bilaterally preferred, in the sense that it is among the k edges of lowest weight incident to vertex i, and among the k edges of lowest weight incident to vertex j. We show that $k = \log(n)$ is the connectivity threshold, solving a conjecture of La and Kabkab, and obtaining finer results about the window. We also investigate the asymptotic behavior of the average degree of vertices in $\mathit{LK}(n, k)$ as $n\rightarrow\infty$.

It is well known that almost all graphs are canonizable by a simple combinatorial routine known as colour refinement, also referred to as the 1-dimensional Weisfeiler–Leman algorithm. With high probability, this method assigns a unique label to each vertex of a random input graph and, hence, it is applicable only to asymmetric graphs. The strength of combinatorial refinement techniques becomes a subtle issue if the input graphs are highly symmetric. We prove that the combination of colour refinement and vertex individualization yields a canonical labelling for almost all circulant digraphs (i.e., Cayley digraphs of a cyclic group). This result provides first evidence of good average-case performance of combinatorial refinement within the class of vertex-transitive graphs. Remarkably, we do not even need the full power of the colour refinement algorithm. We show that the canonical label of a vertex $v$ can be obtained just by counting walks of each length from $v$ to an individualized vertex. Our analysis also implies that almost all circulant graphs are compact in the sense of Tinhofer, that is, their polytops of fractional automorphisms are integral. Finally, we show that a canonical Cayley representation can be constructed for almost all circulant graphs by the more powerful 2-dimensional Weisfeiler–Leman algorithm.

Let $X_H$ be the number of copies of a fixed graph H in G(n,p). In 2016, Gilmer and Kopparty conjectured that a local central limit theorem should hold for $X_H$ as long as H is connected, $p\gg n^{-1/m(H)}$ and $n^2(1-p)\gg 1$, where m(H) denotes the m-density of H. Recently, Sah and Sawhney showed that the Gilmer–Kopparty conjecture holds for constant p. In this paper, we show that the Gilmer–Kopparty conjecture holds for triangle counts in the sparse range. More precisely, if $p \in (4n^{-1/2}, 1/2)$, then

\[ {} {} {}\sup_{x\in \mathcal{L}}\left| \dfrac{1}{\sqrt{2\pi}}e^{-x^2/2}-\sigma\cdot \mathbb{P}(X^* = x)\right|=n^{-1/2+o(1)}p^{1/2}, {} {}\]

$\sigma^2 = \mathbb{V}\text{ar}(X_{K_3})$

$X^{*}=(X_{K_3}-\mathbb{E}(X_{K_3}))/\sigma$

$\mathcal{L}$

$X^*$

$n^{-1}\ll p \lt c$

$c\in (0,1)$

$\ell_1$

$m_2$

We consider the problem of detecting whether a power-law inhomogeneous random graph contains a geometric community, and we frame this as a hypothesis-testing problem. More precisely, we assume that we are given a sample from an unknown distribution on the space of graphs on n vertices. Under the null hypothesis, the sample originates from the inhomogeneous random graph with a heavy-tailed degree sequence. Under the alternative hypothesis, $k=o(n)$ vertices are given spatial locations and connect following the geometric inhomogeneous random graph connection rule. The remaining $n-k$ vertices follow the inhomogeneous random graph connection rule. We propose a simple and efficient test based on counting normalized triangles to differentiate between the two hypotheses. We prove that our test correctly detects the presence of the community with high probability as $n\to\infty$, and identifies large-degree vertices of the community with high probability.

The generalised random graph contains n vertices with positive i.i.d. weights. The probability of adding an edge between two vertices is increasing in their weights. We require the weight distribution to have finite second moments, and study the point process $\mathcal{C}_n$ on $\{3,4,\ldots\}$, which counts how many cycles of the respective length are present in the graph. We establish convergence of $\mathcal{C}_n$ to a Poisson point process. Under the stronger assumption of the weights having finite fourth moments we provide the following results. When $\mathcal{C}_n$ is evaluated on a bounded set A, we provide a rate of convergence. If the graph is additionally subcritical, we extend this to unbounded sets A at the cost of a slower rate of convergence. From this we deduce the limiting distribution of the length of the shortest and longest cycles when the graph is subcritical, including rates of convergence. All mentioned results also apply to the Chung–Lu model and the Norros–Reittu model.

We describe the asymptotic behaviour of large degrees in random hyperbolic graphs for all values of the curvature parameter $\alpha$. We prove that, with high probability, the node degrees satisfy the following ordering property: the ranking of the nodes by decreasing degree coincides with the ranking of the nodes by increasing distance to the centre, at least up to any constant rank. In the sparse regime $\alpha>\tfrac{1}{2}$, the rank at which these two rankings cease to coincide is $n^{1/(1+8\alpha)+o(1)}$. We also provide a quantitative description of the large degrees by proving the convergence in distribution of the normalised degree process towards a Poisson point process. In particular, this establishes the convergence in distribution of the normalised maximum degree of the graph. A transition occurs at $\alpha = \tfrac{1}{2}$, which corresponds to the connectivity threshold of the model. For $\alpha < \tfrac{1}{2}$, the maximum degree is of order $n - O(n^{\alpha + 1/2})$, whereas for $\alpha \geq \tfrac{1}{2}$, the maximum degree is of order $n^{1/(2\alpha)}$. In the $\alpha < \tfrac{1}{2}$ and $\alpha > \tfrac{1}{2}$ cases, the limit distribution of the maximum degree belongs to the class of extreme value distributions (Weibull for $\alpha < \tfrac{1}{2}$ and Fréchet for $\alpha > \tfrac{1}{2}$). This refines previous estimates on the maximum degree for $\alpha > \tfrac{1}{2}$ and extends the study of large degrees to the dense regime $\alpha \leq \tfrac{1}{2}$.

A seminal result of Komlós, Sárközy, and Szemerédi states that any $n$-vertex graph $G$ with minimum degree at least $(1/2+\alpha )n$ contains every $n$-vertex tree $T$ of bounded degree. Recently, Pham, Sah, Sawhney, and Simkin extended this result to show that such graphs $G$ in fact support an optimally spread distribution on copies of a given $T$, which implies, using the recent breakthroughs on the Kahn-Kalai conjecture, the robustness result that $T$ is a subgraph of sparse random subgraphs of $G$ as well. Pham, Sah, Sawhney, and Simkin construct their optimally spread distribution by following closely the original proof of the Komlós-Sárközy-Szemerédi theorem which uses the blow-up lemma and the Szemerédi regularity lemma. We give an alternative, regularity-free construction that instead uses the Komlós-Sárközy-Szemerédi theorem (which has a regularity-free proof due to Kathapurkar and Montgomery) as a black box. Our proof is based on the simple and general insight that, if $G$ has linear minimum degree, almost all constant-sized subgraphs of $G$ inherit the same minimum degree condition that $G$ has.

In this paper we consider a dynamic Erdős–Rényi graph in which edges, according to an alternating renewal process, change from present to absent and vice versa. The objective is to estimate the on- and off-time distributions while only observing the aggregate number of edges. This inverse problem is dealt with, in a parametric context, by setting up an estimator based on the method of moments. We provide conditions under which the estimator is asymptotically normal, and we point out how the corresponding covariance matrix can be identified. We also demonstrate how to adapt the estimation procedure if alternative subgraph counts are observed, such as the number of wedges or triangles.

Let $K^r_n$ be the complete $r$-uniform hypergraph on $n$ vertices, that is, the hypergraph whose vertex set is $[n] \, :\! = \{1,2,\ldots ,n\}$ and whose edge set is $\binom {[n]}{r}$. We form $G^r(n,p)$ by retaining each edge of $K^r_n$ independently with probability $p$. An $r$-uniform hypergraph $H\subseteq G$ is $F$-saturated if $H$ does not contain any copy of $F$, but any missing edge of $H$ in $G$ creates a copy of $F$. Furthermore, we say that $H$ is weakly $F$-saturated in $G$ if $H$ does not contain any copy of $F$, but the missing edges of $H$ in $G$ can be added back one-by-one, in some order, such that every edge creates a new copy of $F$. The smallest number of edges in an $F$-saturated hypergraph in $G$ is denoted by ${\textit {sat}}(G,F)$, and in a weakly $F$-saturated hypergraph in $G$ by $\mathop {\mbox{$w$-${sat}$}}\! (G,F)$. In 2017, Korándi and Sudakov initiated the study of saturation in random graphs, showing that for constant $p$, with high probability ${\textit {sat}}(G(n,p),K_s)=(1+o(1))n\log _{\frac {1}{1-p}}n$, and $\mathop {\mbox{$w$-${sat}$}}\! (G(n,p),K_s)=\mathop {\mbox{$w$-${sat}$}}\! (K_n,K_s)$. Generalising their results, in this paper, we solve the saturation problem for random hypergraphs $G^r(n,p)$ for cliques $K_s^r$, for every $2\le r \lt s$ and constant $p$.

We derive large- and moderate-deviation results in random networks given as planar directed navigations on homogeneous Poisson point processes. In this non-Markovian routing scheme, starting from the origin, at each consecutive step a Poisson point is joined by an edge to its nearest Poisson point to the right within a cone. We establish precise exponential rates of decay for the probability that the vertical displacement of the random path is unexpectedly large. The proofs rest on controlling the dependencies of the individual steps and the randomness in the horizontal displacement as well as renewal-process arguments.