On connectivity,conductance and bootstrap percolation for a random k‐out,age‐biased graph

A uniform attachment graph (with parameter k), denoted G_n,k in the paper, is a random graph on the vertex set [n], where each vertex v makes k selections from [v ? 1] uniformly and independently, and these selections determine the edge set. We study several aspects of this graph. Our motivation comes from two similarly constructed, well‐studied random graphs: k‐out graphs and preferential attachment graphs. In this paper, we find the asymptotic distribution of its minimum degree and connectivity, and study the expansion properties of G_n,k to show that the conductance of G_n,k is of order . We also study the bootstrap percolation on G_n,k, where r infected neighbors infect a vertex, and show that if the probability of initial infection of a vertex is negligible compared to then with high probability (whp) the disease will not spread to the whole vertex set, and if this probability exceeds by a sub‐logarithmical factor then the disease whp will spread to the whole vertex set.     

On percolation and ‐hardness

The edge‐percolation and vertex‐percolation random graph models start with an arbitrary graph G, and randomly delete edges or vertices of G with some fixed probability. We study the computational complexity of problems whose inputs are obtained by applying percolation to worst‐case instances. Specifically, we show that a number of classical ‐hard problems on graphs remain essentially as hard on percolated instances as they are in the worst‐case (assuming ). We also prove hardness results for other ‐hard problems such as Constraint Satisfaction Problems and Subset‐Sum, with suitable definitions of random deletions. Along the way, we establish that for any given graph G the independence number and the chromatic number are robust to percolation in the following sense. Given a graph G, let be the graph obtained by randomly deleting edges of G with some probability . We show that if is small, then remains small with probability at least 0.99. Similarly, we show that if is large, then remains large with probability at least 0.99. We believe these results are of independent interest.     

Asymptotics in percolation on high‐girth expanders

We consider supercritical bond percolation on a family of high‐girth ‐regular expanders. The previous study of Alon, Benjamini and Stacey established that its critical probability for the appearance of a linear‐sized (“giant”) component is . Our main result recovers the sharp asymptotics of the size and degree distribution of the vertices in the giant and its 2‐core at any . It was further shown in the previous study that the second largest component, at any , has size at most for some . We show that, unlike the situation in the classical Erd?s‐Rényi random graph, the second largest component in bond percolation on a regular expander, even with an arbitrarily large girth, can have size for arbitrarily close to 1. Moreover, as a by‐product of that construction, we answer negatively a question of Benjamini on the relation between the diameter of a component in percolation on expanders and the existence of a giant component. Finally, we establish other typical features of the giant component, for example, the existence of a linear path.     

Competition in growth and urns

We study survival among two competing types in two settings: a planar growth model related to two‐neighbor bootstrap percolation, and a system of urns with graph‐based interactions. In the planar growth model, uncolored sites are given a color at rate 0, 1 or , depending on whether they have zero, one, or at least two neighbors of that color. In the urn scheme, each vertex of a graph G has an associated urn containing some number of either blue or red balls (but not both). At each time step, a ball is chosen uniformly at random from all those currently present in the system, a ball of the same color is added to each neighboring urn, and balls in the same urn but of different colors annihilate on a one‐for‐one basis. We show that, for every connected graph G and every initial configuration, only one color survives almost surely. As a corollary, we deduce that in the two‐type growth model on , one of the colors only infects a finite number of sites with probability one. We also discuss generalizations to higher dimensions and multi‐type processes, and list a number of open problems and conjectures.     

Semi‐random graph process

We introduce and study a novel semi‐random multigraph process, described as follows. The process starts with an empty graph on n vertices. In every round of the process, one vertex v of the graph is picked uniformly at random and independently of all previous rounds. We then choose an additional vertex (according to a strategy of our choice) and connect it by an edge to v. For various natural monotone increasing graph properties , we prove tight upper and lower bounds on the minimum (extended over the set of all possible strategies) number of rounds required by the process to obtain, with high probability, a graph that satisfies . Along the way, we show that the process is general enough to approximate (using suitable strategies) several well‐studied random graph models.     

Slightly subcritical hypercube percolation

We study bond percolation on the hypercube {0,1}^m in the slightly subcritical regime where p = p_c(1 ? ε_m) and ε_m = o(1) but ε_m ? 2^?m/3 and study the clusters of largest volume and diameter. We establish that with high probability the largest component has cardinality , that the maximal diameter of all clusters is , and that the maximal mixing time of all clusters is . These results hold in different levels of generality, and in particular, some of the estimates hold for various classes of graphs such as high‐dimensional tori, expanders of high degree and girth, products of complete graphs, and infinite lattices in high dimensions.     

The time of graph bootstrap percolation

Graph bootstrap percolation, introduced by Bollobás in 1968, is a cellular automaton defined as follows. Given a “small” graph H and a “large” graph , in consecutive steps we obtain from G_t by adding to it all new edges e such that contains a new copy of H. We say that G percolates if for some , we have G_t = K_n. For H = K_r, the question about the size of the smallest percolating graphs was independently answered by Alon, Frankl and Kalai in the 1980's. Recently, Balogh, Bollobás and Morris considered graph bootstrap percolation for and studied the critical probability , for the event that the graph percolates with high probability. In this paper, using the same setup, we determine, up to a logarithmic factor, the critical probability for percolation by time t for all © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 143–168, 2017     

Sharp asymptotic for the chemical distance in long‐range percolation

We consider instances of long‐range percolation on and , where points at distance r get connected by an edge with probability proportional to r^?s, for s ∈ (d,2d), and study the asymptotic of the graph‐theoretical (a.k.a. chemical) distance D(x,y) between x and y in the limit as |x ? y|→∞. For the model on we show that, in probability as |x|→∞, the distance D(0,x) is squeezed between two positive multiples of , where for γ: = s/(2d). For the model on we show that D(0,xr) is, in probability as r→∞ for any nonzero , asymptotic to for φ a positive, continuous (deterministic) function obeying φ(r^γ) = φ(r) for all r > 1. The proof of the asymptotic scaling is based on a subadditive argument along a continuum of doubly‐exponential sequences of scales. The results strengthen considerably the conclusions obtained earlier by the first author. Still, significant open questions remain.     

Line percolation

We study a new geometric bootstrap percolation model, line percolation, on the d‐dimensional integer grid . In line percolation with infection parameter r, infection spreads from a subset of initially infected lattice points as follows: if there exists an axis‐parallel line L with r or more infected lattice points on it, then every lattice point of on L gets infected, and we repeat this until the infection can no longer spread. The elements of the set A are usually chosen independently, with some density p, and the main question is to determine , the density at which percolation (infection of the entire grid) becomes likely. In this paper, we determine up to a multiplicative factor of and up to a multiplicative constant as for every fixed . We also determine the size of the minimal percolating sets in all dimensions and for all values of the infection parameter.     

New methods to bound the critical probability in fractal percolation

We study the critical probability p_c(M) in two‐dimensional M‐adic fractal percolation. To find lower bounds, we compare fractal percolation with site percolation. Fundamentally new is the construction of a computable increasing sequence that converges to p_c(M). We prove that and . For the upper bounds, we introduce an iterative random process on a finite alphabet , which is easier to analyze than the original process. We show that and . © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 710–730, 2015     

Information percolation and cutoff for the random‐cluster model

We consider the random‐cluster model (RCM) on with parameters p∈(0,1) and q ≥ 1. This is a generalization of the standard bond percolation (with edges open independently with probability p) which is biased by a factor q raised to the number of connected components. We study the well‐known Fortuin‐Kasteleyn (FK)‐dynamics on this model where the update at an edge depends on the global geometry of the system unlike the Glauber heat‐bath dynamics for spin systems, and prove that for all small enough p (depending on the dimension) and any q>1, the FK‐dynamics exhibits the cutoff phenomenon at with a window size , where λ_∞ is the large n limit of the spectral gap of the process. Our proof extends the information percolation framework of Lubetzky and Sly to the RCM and also relies on the arguments of Blanca and Sinclair who proved a sharp mixing time bound for the planar version. A key aspect of our proof is the analysis of the effect of a sequence of dependent (across time) Bernoulli percolations extracted from the graphical construction of the dynamics, on how information propagates.     

A phase transition in the evolution of bootstrap percolation processes on preferential attachment graphs

The theme of this paper is the analysis of bootstrap percolation processes on random graphs generated by preferential attachment. This is a class of infection processes where vertices have two states: they are either infected or susceptible. At each round every susceptible vertex which has at least infected neighbours becomes infected and remains so forever. Assume that initially vertices are randomly infected, where is the total number of vertices of the graph. Suppose also that , where is the average degree. We determine a critical function such that when , complete infection occurs with high probability as , but when , then with high probability the process evolves only for a bounded number of rounds and the final set of infected vertices is asymptotically equal to .     

Seeded graph matching via large neighborhood statistics

We study a noisy graph isomorphism problem, where the goal is to perfectly recover the vertex correspondence between two edge‐correlated graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. We show that it is possible to achieve the information‐theoretic limit of graph sparsity in time polynomial in the number of vertices n. Moreover, we show the number of seeds needed for perfect recovery in polynomial‐time can be as low as in the sparse graph regime (with the average degree smaller than ) and in the dense graph regime, for a small positive constant . Unlike previous work on graph matching, which used small neighborhoods or small subgraphs with a logarithmic number of vertices in order to match vertices, our algorithms match vertices if their large neighborhoods have a significant overlap in the number of seeds.     

The size Ramsey number of short subdivisions of bounded degree graphs

For graphs G and F, write if any coloring of the edges of G with colors yields a monochromatic copy of the graph F. Suppose is obtained from a graph S with s vertices and maximum degree d by subdividing its edges h times (that is, by replacing the edges of S by paths of length h + 1). We prove that there exists a graph G with no more than edges for which holds, provided that . We also extend this result to the case in which Q is a graph with maximum degree d on q vertices with the property that every pair of vertices of degree greater than 2 are distance at least h + 1 apart. This complements work of Pak regarding the size Ramsey number of “long subdivisions” of bounded degree graphs.     

Powers of tight Hamilton cycles in randomly perturbed hypergraphs

For k ≥ 2 and r ≥ 1 such that k + r ≥ 4, we prove that, for any α > 0, there exists ε > 0 such that the union of an n‐vertex k‐graph with minimum codegree and a binomial random k‐graph with on the same vertex set contains the rth power of a tight Hamilton cycle with high probability. This result for r = 1 was first proved by McDowell and Mycroft.     

Finding cliques using few probes

Consider algorithms with unbounded computation time that probe the entries of the adjacency matrix of an n vertex graph, and need to output a clique. We show that if the input graph is drawn at random from (and hence is likely to have a clique of size roughly ), then for every δ<2 and constant ?, there is an α<2 (that may depend on δ and ?) such that no algorithm that makes n^δ probes in ? rounds is likely (over the choice of the random graph) to output a clique of size larger than .     

Random perfect graphs

We investigate the asymptotic structure of a random perfect graph P_n sampled uniformly from the set of perfect graphs on vertex set . Our approach is based on the result of Prömel and Steger that almost all perfect graphs are generalised split graphs, together with a method to generate such graphs almost uniformly. We show that the distribution of the maximum of the stability number and clique number is close to a concentrated distribution L(n) which plays an important role in our generation method. We also prove that the probability that P_n contains any given graph H as an induced subgraph is asymptotically 0 or or 1. Further we show that almost all perfect graphs are 2‐clique‐colorable, improving a result of Bacsó et al. from 2004; they are almost all Hamiltonian; they almost all have connectivity equal to their minimum degree; they are almost all in class one (edge‐colorable using Δ colors, where Δ is the maximum degree); and a sequence of independently and uniformly sampled perfect graphs of increasing size converges almost surely to the graphon .     

The critical probability for confetti percolation equals 1/2

In the confetti percolation model, or two‐coloured dead leaves model, radius one disks arrive on the plane according to a space‐time Poisson process. Each disk is coloured black with probability p and white with probability . In this paper we show that the critical probability for confetti percolation equals 1/2. That is, if p > 1/2 then a.s. there is an unbounded curve in the plane all of whose points are black; while if then a.s. all connected components of the set of black points are bounded. This answers a question of Benjamini and Schramm [1]. The proof builds on earlier work by Hirsch [7] and makes use of an adaptation of a sharp thresholds result of Bourgain. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 679–697, 2017