Random Vertex Research Articles

Degree evolution in a general growing network

We consider the preferential attachment model introduced by Deijfen and Lindholm (2009) in which, at every discrete time step: (i) either we add a vertex and connect it to an older vertex; or (ii) we add an edge between two random vertices; or (iii) we delete one edge. We show that, when the deletion probability equals 1/3, the expected degree of any given vertex grows logarithmically, thus correcting a statement made in Lindholm and Vallier (2011). Moreover we show that, when the deletion probability is strictly less than 1/3, then the function which scales the expected degree of a given vertex, identified in Lindholm and Vallier (2011), also guarantees almost sure convergence for the degree process of a given vertex.

Local weak limit of preferential attachment random trees with additive fitness

AbstractWe consider linear preferential attachment trees with additive fitness, where fitness is the random initial vertex attractiveness. We show that when the fitnesses are independent and identically distributed and have positive bounded support, the local weak limit can be constructed using a sequence of mixed Poisson point processes. We also provide a rate of convergence for the total variation distance between the r-neighbourhoods of a uniformly chosen vertex in the preferential attachment tree and the root vertex of the local weak limit. The proof uses a Pólya urn representation of the model, for which we give new estimates for the beta and product beta variables in its construction. As applications, we obtain limiting results and convergence rates for the degrees of the uniformly chosen vertex and its ancestors, where the latter are the vertices that are on the path between the uniformly chosen vertex and the initial vertex.

Power of $k$ Choices in the Semi-Random Graph Process

The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed monotone graph property, the objective of the player is to force the graph to satisfy this property with high probability in as few rounds as possible.
 In this paper, we introduce a natural generalization of this game in which $k$ random vertices $u_1, \ldots, u_k$ are presented to the player in each round. She needs to select one of the presented vertices and connect to any vertex she wants. We focus on the following three monotone properties: minimum degree at least $\ell$, the existence of a perfect matching, and the existence of a Hamiltonian cycle.

Prominent examples of flip processes

AbstractFlip processes, introduced in [Garbe, Hladký, Šileikis, Skerman: From flip processes to dynamical systems on graphons], are a class of random graph processes defined using a rule which is just a function from all labelled graphs of a fixed order into itself. The process starts with an arbitrary given ‐vertex graph . In each step, the graph is obtained by sampling random vertices of and replacing the induced graph by .Using the formalism of dynamical systems on graphons associated to each such flip process from ibid. we study several specific flip processes, including the triangle removal flip process and its generalizations, ‘extremist flip processes’ (in which is either a clique or an independent set, depending on whether has less or more than half of all potential edges), and ‘ignorant flip processes’ in which the output does not depend on .

Graph isomorphism: physical resources, optimization models, and algebraic characterizations

In the (G, H)-isomorphism game, a verifier interacts with two non-communicating players (called provers), by privately sending each of them a random vertex from either G or H. The goal of the players is to convince the verifier that the graphs G and H are isomorphic. In recent work along with Atserias et al. (J Comb Theory Ser B 136:89–328, 2019) we showed that a verifier can be convinced that two non-isomorphic graphs are isomorphic, if the provers are allowed to share quantum resources. In this paper we model classical and quantum graph isomorphism by linear constraints over certain complicated convex cones, which we then relax to a pair of tractable convex models (semidefinite programs). Our main result is a complete algebraic characterization of the corresponding equivalence relations on graphs in terms of appropriate matrix algebras. Our techniques are an interesting mix of algebra, combinatorics, optimization, and quantum information.

A fourth‐moment phenomenon for asymptotic normality of monochromatic subgraphs

Given a graph sequence and a simple connected subgraph , we denote by the number of monochromatic copies of in a uniformly random vertex coloring of with colors. We prove a central limit theorem for (we denote the appropriately centered and rescaled statistic as ) with explicit error rates. The error rates arise from graph counts of collections formed by joining copies of which we callgood joins. Good joins are closely related to the fourth moment of , which allows us to show afourth moment phenomenonfor the central limit theorem. For , we show that converges in distribution to whenever its fourth moment converges to 3. We show the convergence of the fourth moment is necessary to obtain a normal limit when .

Local weak convergence for sparse networks of interacting processes

We study the limiting behavior of interacting particle systems indexed by large sparse graphs, which evolve either according to a discrete time Markov chain or a diffusion, in which particles interact directly only with their nearest neighbors in the graph. To encode sparsity we work in the framework of local weak convergence of marked (random) graphs. We show that the joint law of the particle system varies continuously with respect to local weak convergence of the underlying graph marked with the initial conditions. In addition, we show that the global empirical measure converges to a nonrandom limit for a large class of graph sequences including sparse Erdős–Rényi graphs and configuration models, whereas the empirical measure of the connected component of a uniformly random vertex converges to a random limit. Along the way, we develop some related results on the time-propagation of ergodicity and empirical field convergence, as well as some general results on local weak convergence of Gibbs measures in the uniqueness regime which appear to be new. The results obtained here are also useful for obtaining autonomous descriptions of marginal dynamics of interacting diffusions and Markov chains on sparse graphs. While limits of interacting particle systems on dense graphs have been extensively studied, there are relatively few works that have studied the sparse regime in generality.

On the generation of metric TSP instances with a large integrality gap by branch-and-cut

This paper introduces a computational method for generating metric Travelling Salesman Problem (TSP) instances having a large integrality gap. The method is based on the solution of an integer programming problem, called IH-OPT, that takes as input a fractional solution of the Subtour Elimination Problem (SEP) on a TSP instance and computes a TSP instance having an integrality gap larger than or equal to the integrality gap of the first instance. The decision variables of IH-OPT are the entries of the TSP cost matrix, and the constraints are defined by the intersection of the metric cone with an exponential number of inequalities, one for each possible TSP tour. Given the very large number of constraints, we have implemented a branch-and-cut algorithm for solving IH-OPT. Then, by sampling cost vectors over the metric polytope and by solving the corresponding SEP, we can generate random fractional vertices of the SEP polytope. If we solve the IH-OPT problem for every sampled vertex using our branch-and-cut algorithm, we can select the generated TSP instance (i.e., cost vector), yielding the longest runtime for Concorde, the state-of-the-art TSP solver. Our computational results show that our method is very effective in producing challenging instances. As a by-product, we release the Hard-TSPLIB, a library of 41 small metric TSP instances which have a large integrality gap and are challenging in terms of runtime for Concorde.

Anticoncentration in Ramsey graphs and a proof of the Erdős–McKay conjecture

Abstract An n-vertex graph is called C-Ramsey if it has no clique or independent set of size $C\log _2 n$ (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a C-Ramsey graph. This brings together two ongoing lines of research: the study of ‘random-like’ properties of Ramsey graphs and the study of small-ball probability for low-degree polynomials of independent random variables. The proof proceeds via an ‘additive structure’ dichotomy on the degree sequence and involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics and low-rank approximation. In particular, a key ingredient is a new sharpened version of the quadratic Carbery–Wright theorem on small-ball probability for polynomials of Gaussians, which we believe is of independent interest. One of the consequences of our result is the resolution of an old conjecture of Erdős and McKay, for which Erdős reiterated in several of his open problem collections and for which he offered one of his notorious monetary prizes.

Random plane increasing trees: Asymptotic enumeration of vertices by distance from leaves

AbstractWe prove that for any fixed , the probability that a random vertex of a random increasing plane tree is of rank , that is, the probability that a random vertex is at distance from the leaves, converges to a constant as the size of the tree goes to infinity. We prove that , so that the tail of the limiting rank distribution is super‐exponentially narrow. We prove that the latter property holds uniformly for all finite as well. More generally, we prove that the ranks of a finite uniformly random set of vertices are asymptotically independent, each with distribution . We compute the exact value of for , demonstrating that the limiting expected fraction of vertices with rank is … We show that with probability the highest rank of a vertex in the tree is sandwiched between and , and that this rank is asymptotic to with probability .

Effects of general non-magnetic quenched disorder on a spin-density-wave quantum critical metallic system in two spatial dimension

We investigate the effects of general non-magnetic quenched disorder on the two-dimensional spin-density-wave (SDW) quantum critical metallic system using a renormalization group (RG) method. We consider (i) all possible scattering channels by a random charge potential for a fermion field and (ii) a random mass term for a SDW boson order parameter as effects of the non-magnetic quenched disorder. From the one-loop analysis, we find a weakly disordered non-Fermi liquid metallic fixed point (random mass disordered non-Fermi liquid fixed point) when only the random boson mass vertex is considered. However, in the general case where all disorder vertices are considered, it turns out that there is no stable fixed point and the low-energy RG flows are governed by the large random charge potential vertices especially channels in the ‘Direct’ category. Focusing on the physical meanings of the low-energy RG flows, we provide a detailed explanation of the one-loop results. Beyond the one-loop level, we first discuss partial two-loop corrections to the random charge potential vertices. Furthermore, we examine the possibility of different low-energy RG flows compared to that of the one-loop results by considering the two-loop corrections to the random boson mass vertex and, discuss low-energy properties in relation to the random singlet phase. Related to physical properties, we calculate anomalous dimensions of the four superconducting channels in the one-loop level.

Assessing Graph Robustness through Modified Zagreb Index

Graph robustness or network robustness is the ability that a graph or a network preserves its connectivity or other properties after the loss of vertices and edges, which has been a central problem in the research of complex networks. In this paper, we introduce the Modified Zagreb index and Modified Zagreb index centrality as novel measures to study graph robustness. We theoretically find some relationships between these novel measures and some other graph measures. Then, we use Modified Zagreb index centrality to analyze the robustness of BA scale-free networks, ER random graphs and WS small world networks under deliberate or random vertex attacks. We also study the correlations between this new measure and some other existed measures. Finally, we use Modified Zagreb index centrality to study the robustness of two real world networks. All these results demonstrate the efficiency of Modified Zagreb index centrality for assessing the graph robustness.

Improving Network Representation Learning via Dynamic Random Walk, Self-Attention and Vertex Attributes-Driven Laplacian Space Optimization.

Network data analysis is a crucial method for mining complicated object interactions. In recent years, random walk and neural-language-model-based network representation learning (NRL) approaches have been widely used for network data analysis. However, these NRL approaches suffer from the following deficiencies: firstly, because the random walk procedure is based on symmetric node similarity and fixed probability distribution, the sampled vertices’ sequences may lose local community structure information; secondly, because the feature extraction capacity of the shallow neural language model is limited, they can only extract the local structural features of networks; and thirdly, these approaches require specially designed mechanisms for different downstream tasks to integrate vertex attributes of various types. We conducted an in-depth investigation to address the aforementioned issues and propose a novel general NRL framework called dynamic structure and vertex attribute fusion network embedding, which firstly defines an asymmetric similarity and h-hop dynamic random walk strategy to guide the random walk process to preserve the network’s local community structure in walked vertex sequences. Next, we train a self-attention-based sequence prediction model on the walked vertex sequences to simultaneously learn the vertices’ local and global structural features. Finally, we introduce an attributes-driven Laplacian space optimization to converge the process of structural feature extraction and attribute feature extraction. The proposed approach is exhaustively evaluated by means of node visualization and classification on multiple benchmark datasets, and achieves superior results compared to baseline approaches.

On measuring network robustness for weighted networks

Network robustness measures how well network structure is strong and healthy when it is under attack, such as vertices joining and leaving. It has been widely used in many applications, such as information diffusion, disease transmission, and network security. However, existing metrics, including node connectivity, edge connectivity, and graph expansion, can be suboptimal for measuring network robustness since they are inefficient to be computed and cannot directly apply to the weighted networks or disconnected networks. In this paper, we define the \({\mathcal {R}}\)-energy as a new robustness measurement for weighted networks based on the method of spectral analysis. \({\mathcal {R}}\)-energy can cope with disconnected networks and is efficient to compute with a time complexity of \(O(|V|+|E|)\), where V and E are sets of vertices and edges in the network, respectively. Our experiments illustrate the rationality and efficiency of computing \({\mathcal {R}}\)-energy: (1) Removal of high degree vertices reduces network robustness more than that of random or small degree vertices; (2) it takes as little as 120 s to compute for a network with about 6M vertices and 33M edges. We can further detect events occurring in a dynamic Twitter network with about 130K users and discover interesting weekly tweeting trends by tracking changes to \({\mathcal {R}}\)-energy.

On scaling limits of random trees and maps with a prescribed degree sequence

We study a configuration model on bipartite planar maps in which, given $n$ even integers, one samples a planar map with $n$ faces uniformly at random with these face degrees. We prove that when suitably rescaled, such maps always admit nontrivial subsequential limits as $n \to \infty$ in the Gromov-Hausdorff-Prokhorov topology. Further, we show that they converge in distribution towards the celebrated Brownian sphere, and more generally a Brownian disk for maps with a boundary, if and only if there is no inner face with a macroscopic degree, or, if the perimeter is too big, the maps degenerate and converge to the Brownian tree. By first sampling the degrees at random with an appropriate distribution, this model recovers that of size-conditioned Boltzmann maps associated with critical weights in the domain of attraction of a stable law with index $\alpha\in [1,2]$. The Brownian tree and disks then appear respectively in the case $\alpha=1$ and $\alpha=2$, whereas in the case $\alpha \in (1,2)$ our results partially recover previous known ones. Our proofs rely on known bijections with labelled plane trees, which are similarly sampled uniformly at random given $n$ outdegrees. Along the way, we obtain some results on the geometry of such trees, such as a convergence to the Brownian tree but only in the weaker sense of subtrees spanned by random vertices, which are of independent interest.

Formula omitted]-variable first-order logic of preferential attachment random graphs

We study logical limit laws for preferential attachment random graphs. In this random graph model, vertices and edges are introduced recursively: at time 1, we start with vertices 0,1 and m edges between them. At step n+1 the vertex n+1 is introduced together with m edges joining the new vertex with m vertices chosen from 1,…,n independently with probabilities proportional to their degrees plus a positive parameter δ. We prove that this random graph obeys the convergence law for first-order sentences with at most m−2 variables.

γ-variable first-order logic of uniform attachment random graphs

We study logical limit laws for uniform attachment random graphs. In this random graph model, vertices and edges are introduced recursively: at time n+1, the vertex n+1 is introduced together with m edges joining the new vertex with m different vertices chosen uniformly at random from 1,…,n. We prove that this random graph obeys convergence law for first-order sentences with at most m−2 variables.

Depth of vertices with high degree in random recursive trees

Let $T_n$ be a random recursive tree with $n$ nodes. List vertices of $T_n$ in decreasing order of degree as $v^1,\ldots,v^n$, and write $d^i$ and $h^i$ for the degree of $v^i$ and the distance of $v^i$ from the root, respectively. We prove that, as $n \to \infty$ along suitable subsequences, \[ \bigg(d^i - \lfloor \log_2 n \rfloor, \frac{h^i - \mu\ln n}{\sqrt{\sigma^2\ln n}}\bigg) \to ((P_i,i \ge 1),(N_i,i \ge 1))\, , \] where $\mu=1-(\log_2 e)/2$, $\sigma^2=1-(\log_2 e)/4$, $(P_i,i \ge 1)$ is a Poisson point process on $\mathbb{Z}$ and $(N_i,i \ge 1)$ is a vector of independent standard Gaussians. We additionally establish joint normality for the depths of uniformly random vertices in $T_n$, which extends results for the case of a single random vertex. The joint limit holds even if the random vertices are conditioned to have large degree, provided the normalizing constants are adjusted accordingly; however, both the mean and variance of the conditinal depths remain of orden $\ln n$. Our results are based on a simple relationship between random recursive trees and Kingman's $n$-coalescent; a utility that seems to have been largely overlooked.

MDA + RRT: A general approach for resolving the problem of angle constraint for hyper-redundant manipulator

In this paper, a cable-driven hyper-redundant manipulator with 17 degrees of freedom is presented for narrow and complex environments firstly. Based on the mechanical structure, the MDA + RRT algorithms is proposed for path planning considering the maximum deflection angle of joint. First, the transformation relationship between the path parameters and manipulator parameters is obtained through the theorem of sine of triangles and two extreme points, which can convert limitation of the deflection angle of the joint to the limitation of the deflection angle of the path. Then, three corresponding improved algorithms are proposed: MDA-RRT, MDA-RRT*, MDA-QRRT* on the basis of the traditional RRT, RRT* and Q-RRT* algorithms. Finally, six algorithms are applied to plan the path of avoiding three different obstacles respectively by simulation. The results demonstrate that, three improved algorithms can guarantee that the planned path satisfies the constraint of deflection angle of joint without increasing the computational amount compared with the three traditional algorithms, only by limiting the selection range of the next random vertex. Among the six algorithms, the feasibility and optimization of the MDA-QRRT* algorithm are the best and the feasibility and optimization of the three improved algorithms are also better than the corresponding traditional algorithms.

Greedy Matching in Bipartite Random Graphs

This paper studies the performance of greedy matching algorithms on bipartite graphs [Formula: see text]. We focus primarily on three classical algorithms: [Formula: see text], which sequentially selects random edges from [Formula: see text]; [Formula: see text], which sequentially matches random vertices in [Formula: see text] to random neighbors; and [Formula: see text], which generates a random priority order over vertices in [Formula: see text] and then sequentially matches random vertices in [Formula: see text] to their highest-priority remaining neighbor. Prior work has focused on identifying the worst-case approximation ratio for each algorithm. This guarantee is highest for [Formula: see text] and lowest for [Formula: see text]. Our work instead studies the average performance of these algorithms when the edge set [Formula: see text] is random. Our first result compares [Formula: see text] and [Formula: see text] and shows that on average, [Formula: see text] produces more matches. This result holds for finite graphs (in contrast to previous asymptotic results) and also applies to “many to one” matching in which each vertex in [Formula: see text] can match with multiple vertices in [Formula: see text]. Our second result compares [Formula: see text] and [Formula: see text] and shows that the better worst-case guarantee of [Formula: see text] does not translate into better average performance. In “one to one” settings where each vertex in [Formula: see text] can match with only one vertex in [Formula: see text], the algorithms result in the same number of matches. When each vertex in [Formula: see text] can match with two vertices in [Formula: see text] produces more matches than [Formula: see text].