scholarly journals Ramsey properties of randomly perturbed graphs: cliques and cycles

2020 ◽  
Vol 29 (6) ◽  
pp. 830-867 ◽  
Author(s):  
Shagnik Das ◽  
Andrew Treglown
Keyword(s):  
Random Graph ◽  
High Probability ◽  
Graph Model ◽  
Significant Progress ◽  
Random Choice ◽  
Ramsey Problem ◽  
Random Graph Model ◽  
Dense Graph ◽  
Precise Solution ◽  
Analogous Question

AbstractGiven graphs H1, H2, a graph G is (H1, H2) -Ramsey if, for every colouring of the edges of G with red and blue, there is a red copy of H1 or a blue copy of H2. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs. This is a random graph model introduced by Bohman, Frieze and Martin [8] in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali [30] in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (K3, Kt) -Ramsey (for t ≽ 3). They also raised the question of generalizing this result to pairs of graphs other than (K3, Kt). We make significant progress on this question, giving a precise solution in the case when H1 = Ks and H2 = Kt where s, t ≽ 5. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the (K3, Kt) -Ramsey question. Moreover, we give bounds for the corresponding (K4, Kt) -Ramsey question; together with a construction of Powierski [37] this resolves the (K4, K4) -Ramsey problem.We also give a precise solution to the analogous question in the case when both H1 = Cs and H2 = Ct are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalization of the Krivelevich, Sudakov and Tetali [30] result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (Cs, Kt) -Ramsey (for odd s ≽ 5 and t ≽ 4).To prove our results we combine a mixture of approaches, employing the container method, the regularity method as well as dependent random choice, and apply robust extensions of recent asymmetric random Ramsey results.

scholarly journals Finding Hidden Cliques in Linear Time with High Probability

2013 ◽  
Vol 23 (1) ◽  
pp. 29-49 ◽  
Author(s):  
YAEL DEKEL ◽  
ORI GUREL-GUREVICH ◽  
YUVAL PERES
Keyword(s):  
Random Graph ◽  
Failure Probability ◽  
High Probability ◽  
Linear Time ◽  
Success Probability ◽  
Graph Model ◽  
General Setting ◽  
Random Graph Model ◽  
Spectral Techniques ◽  
Random Subset

We are given a graph G with n vertices, where a random subset of k vertices has been made into a clique, and the remaining edges are chosen independently with probability $\frac12$. This random graph model is denoted $G(n,\frac12,k)$. The hidden clique problem is to design an algorithm that finds the k-clique in polynomial time with high probability. An algorithm due to Alon, Krivelevich and Sudakov [3] uses spectral techniques to find the hidden clique with high probability when $k = c \sqrt{n}$ for a sufficiently large constant c > 0. Recently, an algorithm that solves the same problem was proposed by Feige and Ron [12]. It has the advantages of being simpler and more intuitive, and of an improved running time of O(n2). However, the analysis in [12] gives a success probability of only 2/3. In this paper we present a new algorithm for finding hidden cliques that both runs in time O(n2) (that is, linear in the size of the input) and has a failure probability that tends to 0 as n tends to ∞. We develop this algorithm in the more general setting where the clique is replaced by a dense random graph.

scholarly journals Local Resilience and Hamiltonicity Maker–Breaker Games in Random Regular Graphs

2010 ◽  
Vol 20 (2) ◽  
pp. 173-211 ◽  
Author(s):  
SONNY BEN-SHIMON ◽  
MICHAEL KRIVELEVICH ◽  
BENNY SUDAKOV
Keyword(s):  
Random Graph ◽  
Regular Graph ◽  
High Probability ◽  
Perfect Matching ◽  
Winning Strategy ◽  
Graph Model ◽  
Vertex Connectivity ◽  
Random Graph Model ◽  
Graph Properties ◽  
Local Resilience

For an increasing monotone graph propertythelocal resilienceof a graphGwith respect tois the minimalrfor which there exists a subgraphH⊆Gwith all degrees at mostr, such that the removal of the edges ofHfromGcreates a graph that does not possess. This notion, which was implicitly studied for somead hocproperties, was recently treated in a more systematic way in a paper by Sudakov and Vu. Most research conducted with respect to this distance notion focused on the binomial random graph model(n, p) and some families of pseudo-random graphs with respect to several graph properties, such as containing a perfect matching and being Hamiltonian, to name a few. In this paper we continue to explore the local resilience notion, but turn our attention to random and pseudo-randomregulargraphs of constant degree. We investigate the local resilience of the typical randomd-regular graph with respect to edge and vertex connectivity, containing a perfect matching, and being Hamiltonian. In particular, we prove that for every positive ϵ and large enough values ofd, with high probability, the local resilience of the randomd-regular graph,n, d, with respect to being Hamiltonian, is at least (1−ϵ)d/6. We also prove that for the binomial random graph model(n, p), for every positive ϵ > 0 and large enough values ofK, ifp>$\frac{K\ln n}{n}$then, with high probability, the local resilience of(n, p) with respect to being Hamiltonian is at least (1−ϵ)np/6. Finally, we apply similar techniques to positional games, and prove that ifdis large enough then, with high probability, a typical randomd-regular graphGis such that, in the unbiased Maker–Breaker game played on the edges ofG, Maker has a winning strategy to create a Hamilton cycle.

Random graphs

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Large Scale ◽  
Random Network ◽  
Graph Model ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Random Graph Model ◽  
Definition Of ◽  
Average Size

An introduction to the mathematics of the Poisson random graph, the simplest model of a random network. The chapter starts with a definition of the model, followed by derivations of basic properties like the mean degree, degree distribution, and clustering coefficient. This is followed with a detailed derivation of the large-scale structural properties of random graphs, including the position of the phase transition at which a giant component appears, the size of the giant component, the average size of the small components, and the expected diameter of the network. The chapter ends with a discussion of some of the shortcomings of the random graph model.

The Impact of Edge Correlations in Random Networks

2021 ◽  
Vol 30 (4) ◽  
pp. 525-537
Author(s):  
András Faragó ◽  
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Real Life ◽  
Graph Model ◽  
Random Networks ◽  
Random Graph Model ◽  
Special Cases ◽  
Graph Parameters ◽  
The Impact ◽  
Good Number

Random graphs are frequently used models of real-life random networks. The classical Erdös–Rényi random graph model is very well explored and has numerous nontrivial properties. In particular, a good number of important graph parameters that are hard to compute in the deterministic case often become much easier in random graphs. However, a fundamental restriction in the Erdös–Rényi random graph is that the edges are required to be probabilistically independent. This is a severe restriction, which does not hold in most real-life networks. We consider more general random graphs in which the edges may be dependent. Specifically, two models are analyzed. The first one is called a p-robust random graph. It is defined by the requirement that each edge exist with probability at least p, no matter how we condition on the presence/absence of other edges. It is significantly more general than assuming independent edges existing with probability p, as exemplified via several special cases. The second model considers the case when the edges are positively correlated, which means that the edge probability is at least p for each edge, no matter how we condition on the presence of other edges (but absence is not considered). We prove some interesting, nontrivial properties about both models.

scholarly journals A theoretical and empirical comparison of the temporal exponential random graph model and the stochastic actor-oriented model

2019 ◽  
Vol 7 (1) ◽  
pp. 20-51 ◽  
Author(s):  
Philip Leifeld ◽  
Skyler J. Cranmer
Keyword(s):  
Network Analysis ◽  
Random Graph ◽  
Graph Model ◽  
Real Data ◽  
The Other ◽  
Exponential Random Graph Model ◽  
Empirical Comparison ◽  
Random Graph Model ◽  
Exponential Random Graph ◽  
Longitudinal Network Analysis

AbstractThe temporal exponential random graph model (TERGM) and the stochastic actor-oriented model (SAOM, e.g., SIENA) are popular models for longitudinal network analysis. We compare these models theoretically, via simulation, and through a real-data example in order to assess their relative strengths and weaknesses. Though we do not aim to make a general claim about either being superior to the other across all specifications, we highlight several theoretical differences the analyst might consider and find that with some specifications, the two models behave very similarly, while each model out-predicts the other one the more the specific assumptions of the respective model are met.

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