Product rule wins a competitive game

Beveridge, Andrew; Bohman, Tom; Frieze, Alan; Pikhurko, Oleg

doi:10.1090/s0002-9939-07-08853-3

Cited by 17 publications

(22 citation statements)

References 9 publications

(13 reference statements)

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“…In nature, its formation generally takes place in specialized, self-assembled compartments, such as vesicles or layered macromolecular structures, where domains of acidic proteins induce oriented nucleation (5,6). Avoiding the complexity and dynamics of the biological mineralization systems, template-directed CaCO 3 mineralization has been studied in vitro through the use of two-dimensional (2D) molecular assemblies as model systems (7).…”

Section: Wwwsciencemagorg Science Vol 323 13 March 2009mentioning

confidence: 99%

Explosive Percolation in Random Networks

Achlioptas

D’Souza

Spencer

2009

Science

611

873

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. 41. We thank N. C. Harata for help with high-frequency imaging and data analysis, R. J. Reimer and members of the Tsien lab for comments, J. W. Mulholland and J. J. Perrino for help with imaging, and X. Gao and M. Bruchez for consultation on quantum dots. Supported by grants from the Grass Foundation (Q.Z.), the National Institute A large system is said to undergo a phase transition when one or more of its properties change abruptly after a slight change in a controlling variable. Besides water turning into ice or steam, other prototypical phase transitions are the spontaneous emergence of magnetization and superconductivity in metals, the epidemic spread of disease, and the dramatic change in connectivity of networks and lattices known as percolation. Perhaps the most fundamental characteristic of a phase transition is its order, i.e., whether the macroscopic quantity it affects changes continuously or discontinuously at the transition. Continuous (smooth) transitions are called second-order and include many magnetization phenomena, whereas discontinuous (abrupt) transitions are called first-order, a familiar example being the discontinuous drop in entropy when liquid water turns into solid ice at 0°C.We consider percolation phase transitions in models of random network formation. In the classic Erdös-Rényi (ER) model (1), we start with n isolated vertices (points) and add edges (connections) one by one, each edge formed by picking two vertices uniformly at random and connecting them (Fig. 1A). At any given moment, the (connected) component of a vertex v is the set of vertices that can be reached from v by traversing edges. Components merge under ER as if attracted by gravitation. This is because every time an edge is added, the probability two given components will be merged is proportional to the number of possible edges between them which, in turn, is equal to the product of their respective sizes (number of vertices).One of the most studied phenomena in probability theory is the percolation transition of ER random networks, also known as the emergence of a giant component. When rn edges have been added, if r < ½, the largest component remains miniscule, its number of vertices C scaling as log n; in contrast, if r > ½, there is a component of size linear in n. Specifically, C ≈ (4r − 2)n for r slightly greater than ½ and, thus, the fraction of vertices in the largest component undergoes a continuous phase transition at r = ½ (Fig. 1C). Such continuity has been considered a basic characteristic of percolation transitions, occurring in models ranging from classic percolation in the two-dimensional grid to random graph models of social networks (2).Here, we show that percolation transitions in random networks can be discontinuous. We demonstrate this result for models similar to ER, thus also establishing that altering a networkformation process slightly can affect it dramatically, changing the order of its percolation transition. Concretely, we consider models that, like ER, start with n isolated vertices an...

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Section: Wwwsciencemagorg Science Vol 323 13 March 2009mentioning

confidence: 99%

Explosive Percolation in Random Networks

Achlioptas

D’Souza

Spencer

2009

Science

611

873

View full text Add to dashboard Cite

show abstract

“…, and so n 5 p 6 , roughly the expected number of copies of K 2,3 in the random graph, is a positive power of n. So, since K 2,3 is balanced, Theorem 2.6 bounds the number of copies of K 2,3 …”

Section: B1 Balanced Graphsmentioning

confidence: 98%

“…Note that n −1/2 p n −1/2 log n. Our result follows from the following three claims: Proof of (i). In the random graph, the expected codegree is roughly np 2 2, but since we do not know how far p exceeds n −1/2 , we need a slightly more careful argument. Let S be the sum of the codegrees {u,v} d(u, v) over all unordered pairs {u, v}, and let us decompose S = S 1 + S 2 + S 3 , where S 1 is the contribution from summands with d(u, v) ∈ {0, 1}, S 2 is the contribution from summands with 2 ≤ d(u, v) ≤ 4np 2 , and S 3 is the remainder.…”

Section: Theorem the Threshold For Avoiding K 4 In The Achlioptas Prmentioning

confidence: 99%

“…)n(np) 2 . On the other hand, Lemma 8.1 shows that whp, G m has the property that all codegrees are at most np 2 log n, and for every integer 4 ≤ k ≤ log n, the number of pairs with codegree at least knp 2 is at most n 2 /k 3 .…”

mentioning

confidence: 99%

“…On the other hand, Lemma 8.1 shows that whp, G m has the property that all codegrees are at most np 2 log n, and for every integer 4 ≤ k ≤ log n, the number of pairs with codegree at least knp 2 is at most n 2 /k 3 . Conditioning on this, we may then bound S 3 , the sum of codegrees which exceed 4np 2 , by: Also, S 1 , the sum of codegrees which are in {0, 1}, is trivially at most n 2 n 3 p 2 since we assumed p n −1/2 . So, S 2 , the sum of codegrees between 2 and 4np 2 , is at least…”

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Avoiding small subgraphs in Achlioptas processes

Krivelevich

Loh

Sudakov

2008

Random Struct Algorithms

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ABSTRACT:For a fixed integer r, consider the following random process. At each round, one is presented with r random edges from the edge set of the complete graph on n vertices, and is asked to choose one of them. The selected edges are collected into a graph, which thus grows at the rate of one edge per round. This is a natural generalization of what is known in the literature as an Achlioptas process (the original version has r = 2), which has been studied by many researchers, mainly in the context of delaying or accelerating the appearance of the giant component.In this article, we investigate the small subgraph problem for Achlioptas processes. That is, given a fixed graph H, we study whether there is an online algorithm that substantially delays or accelerates a typical appearance of H, compared to its threshold of appearance in the random graph G(n, M). It is easy to see that one cannot accelerate the appearance of any fixed graph by more than the constant factor r, so we concentrate on the task of avoiding H. We determine thresholds for the avoidance of all cycles C t , cliques K t , and complete bipartite graphs K t,t , in every Achlioptas process with parameter r ≥ 2.

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Ramsey games with giants

Bohman¹,

Frieze²,

Krivelevich³

et al. 2010

Random Struct Algorithms

Self Cite

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ABSTRACT:The classical result in the theory of random graphs, proved by Erdős and Rényi in 1960, concerns the threshold for the appearance of the giant component in the random graph process. We consider a variant of this problem, with a Ramsey flavor. Now, each random edge that arrives in a sequence of rounds must be colored with one of r colors. The goal can be either to create a giant component in every color class, or alternatively, to avoid it in every color. One can analyze the offline or online setting for this problem. In this paper, we consider all these variants and provide nontrivial upper and lower bounds; in certain cases (like online avoidance) the obtained bounds are asymptotically tight.

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Product rule wins a competitive game

Cited by 17 publications

References 9 publications

Explosive Percolation in Random Networks

Explosive Percolation in Random Networks

Avoiding small subgraphs in Achlioptas processes

Ramsey games with giants

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