BOLLOB AS, B. (1985). Random graphs. Academic Press. Home/Search Document Not in Database Summary Related Articles Check |
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Network Topology Generators: Degree-Based vs. Structural - Hongsuda Tangmunarunkit.. (2002) (3 citations) (Correct)
....they merely attempt to create network topologies that embody the fundamental characteristics of real networks. The first network topology generator to become widely used in protocol simulations was developed by Waxman [47] This generator is a variant of the classical Erdos Renyi random graph [6]; its link creation probabilities are biased by Euclidean distance between the link endpoints. A later line of research, noting that real network topologies have a non random structure, emphasized the fundamental role of hierarchy. The following from [50] reflects this observation: the primary ....
....well [42] In Section 4, we describe the impact of policy on our conclusions. 3.1.2 Generators We consider three classes of network generators in this paper. The first category, random graph generators, is represented by the Waxman [47] generator. The classical Erdos Renyi random graph model [6] assigns a uniform probability for creating a link between any pair of nodes. The Waxman generator extends the classical model by randomly assigning nodes to locations on a plane and making the link creation probability a function of the Euclidean distance between the nodes. ....
BOLLOB AS, B. Random Graphs. Academic Press, Inc., Orlando, Florida, 1985.
The Chromatic Numbers of Random Hypergraphs - Krivelevich, Sudakov (1998) (1 citation) (Correct)
....log d=k and k 0 = n which can be split into exactly m independent sets, each of size k 0 . Then the event Y 0 implies that X mk 0 n(log d) On the other hand, to bound from below the probability that Y is positive we can use the following inequality (see, e.g. [5], p. 3) Y k 1 ; k m 1 P m 1 j=1 k j =k 0 n (i 1)k 0 km 1 n (i 1)k 0 r ) 3 9 where a l = k 1 ; k m j=1 k j =l Let k i 1 ; k i t be those from k 1 ; km which are greater than n( Since j=1 k j = l k 0 ....
B. Bollobas, Random graphs, Academic Press, London, 1985. 20
List Coloring of Random and Pseudo-Random Graphs - Alon, Krivelevich, Sudakov (1 citation) (Correct)
....the probability that the random graph G(n; p) lies in this family. In particular, we say that A holds almost surely (or a.s. for short) if the probability that G(n; p) satis es A tends to 1 as n tends to in nity. There are numerous papers dealing with random graphs, and the book of Bollob as [8] is an excellent extensive account of the known results in the subject proved before its publication in 1985. Answering an old question of Erd os and R enyi, Bollob as [9] proved that the chromatic number of the random graph G(n; 1=2) is (1 o(1) n= 2 log 2 n) almost surely. His result, together ....
B. Bollobas, Random Graphs, Academic Press, London, 1985.
The Diameter of Random Sparse Graphs - Fan Chung Linyuan (5 citations) (Correct)
....4 values. We also proved that the diameter of G(n; p) is almost surely equal to the diameter of its giant component if np 3:6. 1 Introduction Let G(n; p) denote a random graph on n vertices in which a pair of vertices appears as an edge of G(n; p) with probability p. The reader is referred to [8] for de nition and notation in random graphs. We will here brie y describe the history of work on the diameter of the random graph G(n; p) Klee and Larman [13] proved that for a xed integer d, G(n; p) has diameter d with probability approaching 1 as n goes to in nity if (pn) d 1 =n 0 and ....
....by Burtin [10, 11] Bollob as [9] showed that the diameter of a random graph G(n; p) is almost surely concentrated on at most four values if pn log n 1. Furthermore, it was pointed out that the diameter of a random graph is almost surely concentrated on at most two values if log n 1 (see [8] exercise 2, chapter 10) In the other direction, Luczak [15] examined the diameter of the random graph for the case of np 1. Luczak determined the limit distribution of the diameter of the random graph if (1 np)n 1. The diameter of G(n; p) almost surely either is equal to the diameter of ....
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B. Bollobas, Random Graphs Academic Press, (1985).
The Largest Eigenvalue of Sparse Random Graphs - Krivelevich, Sudakov (2003) (1 citation) (Correct)
....address: bsudakov math.princeton.edu. Research supported in part by NSF grants DMS 0106589, CCR 9987845 and by the State of New Jersey. the probability that G(n; p) has A tends to one as the number of vertices n tends to in nity. Necessary background information on random graphs may be found in [4], 8] It is important to observe that the adjacency matrix of the random graph G(n; p) can be viewed as a random symmetric matrix, whose diagonal entries are zeroes and whose entries above the diagonal are i.i.d. random variables, each taking value 1 with probability p and value 0 with ....
....connected components of G are of size at most (1 o(1) p , where o(1) 0 when p 1. iv) If p log n=n, then almost surely every vertex of G is contained in at most one cycle of length 4. Proof. Parts (i) and (ii) are well known and can be found, e.g. in the monograph of Bollob as [4]. To show (iii) it is enough to bound from above the expectation of the number Y of labeled trees on t = 1 1= log log n) p 2 vertices, contained in G(n; p) as subgraphs. Obviously this expectation is equal to EY = t (t=e) n enp t 2 p From Lemma 2.1 ....
B. Bollobas, Random graphs, Academic Press, New York, 1985. 11
Permutation Pseudographs and Contiguity - Catherine Greenhill Department (Correct)
....rming several conjectures of Janson [6, 5] First we introduce some necessary notation. Denote by G n;d the uniform probability space of d regular graphs on n vertices, where dn is even. As is usual in this area, we approach random d regular graphs via the standard pairing model (see Bollob as [1]) Consider dn labelled points, with d points in each of n buckets, and take a random perfect matching of the points. We call this uniform probability space P n;d . Letting the buckets be vertices and each pair represent an edge (joining the buckets containing the two endpoints of the pair) we ....
Bollobas, B. (1985) Random graphs, Academic Press. 28
Connectedness of Graphs Generated by a Random d-Process - Rucinski, Wormald (Correct)
....3, will be crucial for the present proof too. In the evolution of the ordinary random graph (which is the same as a d process except that there is no upper bound on vertex degree) connectedness occurs, with high probability, at the same moment as the last isolated vertex disappears (see [2] or [1]) We are unable to verify if this phenomenon holds also for d processes. Besides I i , another important random variable of the process is U i the number of unsaturated vertices in G i ; that is, vertices with current degree less than d. These are the vertices which are still in the game , ....
B. Bollobas, Random graphs, Academic Press, London, 1985.
The Choice Number of Dense Random Graphs - Krivelevich (Correct)
....2. 1) This can be done, for example, by bounding the total number of triangles in G(n; p) estimating the number of edges in each such U and then applying known lower bounds for the independence number of graphs with given number of vertices, average degree and number of triangles (see, e.g. [5], Ch. 12, Lemma 15) Of course, due to Proposition 2.1 we may assume that, say, p(n) n 1=5 . We omit further details of the proof. Acknowledgment. The author would like to thank Noga Alon for helpful discussions. ....
B. Bollobas, Random graphs, Academic Press, New York, 1985.
The Largest Eigenvalue of Sparse Random Graphs - Krivelevich, Sudakov (2003) (1 citation) Self-citation (As) (Correct)
....properties of random graphs are of interest. We say that a graph property holds almost surely, or a.s. for brevity, in G(n, p) if the probability that G(n, p) has tends to one as the number of vertices n tends to infinity. Necessary background information on random graphs may be found in [4] and [8] It is important to observe that the adjacency matrix of the random graph G(n, p) can be viewed as a random symmetric matrix whose diagonal entries are zeroes and whose entries above the diagonal are i.i.d. random variables, each taking value 1 with probability p and value 0 with ....
....connected components of G are of size at most (1 o(1) # p , where o(1) 0 when # p ##. iv) If p 6 log n n, then almost surely every vertex of G is contained in at most one cycle of length 6 4. Proof. Parts (i) and (ii) are well known and can be found, e.g. in the monograph by Bollob as [4]. To prove (iii) it is su#cient to bound from above the expectation of the number Y of labelled trees on t = 1 1 log log n)# p 2 vertices, contained in G(n, p)as subgraphs. Obviously this expectation is equal to EY = t (t e) n enp t 2 # p . From ....
Bollob as, B. (1985) Random Graphs, Academic Press, New York.
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