Results 1  10
of
23
A Critical Point For Random Graphs With A Given Degree Sequence
, 2000
"... Given a sequence of nonnegative real numbers 0 ; 1 ; : : : which sum to 1, we consider random graphs having approximately i n vertices of degree i. Essentially, we show that if P i(i \Gamma 2) i ? 0 then such graphs almost surely have a giant component, while if P i(i \Gamma 2) i ! 0 the ..."
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Cited by 289 (7 self)
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Given a sequence of nonnegative real numbers 0 ; 1 ; : : : which sum to 1, we consider random graphs having approximately i n vertices of degree i. Essentially, we show that if P i(i \Gamma 2) i ? 0 then such graphs almost surely have a giant component, while if P i(i \Gamma 2) i ! 0 then almost surely all components in such graphs are small. We can apply these results to G n;p ; G n;M , and other wellknown models of random graphs. There are also applications related to the chromatic number of sparse random graphs.
Sybilguard: Defending against sybil attacks via social networks
 In ACM SIGCOMM ’06
, 2006
"... Peertopeer and other decentralized, distributed systems are known to be particularly vulnerable to sybil attacks. In a sybil attack, a malicious user obtains multiple fake identities and pretends to be multiple, distinct nodes in the system. By controlling a large fraction of the nodes in the syst ..."
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Cited by 208 (6 self)
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Peertopeer and other decentralized, distributed systems are known to be particularly vulnerable to sybil attacks. In a sybil attack, a malicious user obtains multiple fake identities and pretends to be multiple, distinct nodes in the system. By controlling a large fraction of the nodes in the system, the malicious user is able to “out vote” the honest users in collaborative tasks such as Byzantine failure defenses. This paper presents SybilGuard, anovelprotocolfor limiting the corruptive influences of sybil attacks. Our protocol is based on the “social network ” among user identities, where an edge between two identities indicates a humanestablished trust relationship. Malicious users can create many identities but few trust relationships. Thus, there is a disproportionatelysmall “cut ” in the graph between the sybil nodes and the honest nodes. SybilGuard exploits this property to bound the number of identities a malicious user can create. We show the effectiveness of SybilGuard both analytically and experimentally.
The degree sequence of a scalefree random graph process. Random Structures and Algorithms
, 2001
"... ABSTRACT: Recently, Barabási and Albert [2] suggested modeling complex realworld networks such as the worldwide web as follows:consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number of earlier vertices, selected with probabilities proport ..."
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Cited by 156 (2 self)
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ABSTRACT: Recently, Barabási and Albert [2] suggested modeling complex realworld networks such as the worldwide web as follows:consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number of earlier vertices, selected with probabilities proportional to their degrees. In [2] and, with Jeong, in [3], Barabási and Albert suggested that after many steps the proportion P�d � of vertices with degree d should obey a power law P�d � α d −γ. They obtained γ = 2�9 ± 0�1 by experiment and gave a simple heuristic argument suggesting that γ = 3. Here we obtain P�d � asymptotically for all d ≤ n 1/15, where n is the number of vertices, proving as a consequence that γ = 3.
Simulated annealing for graph bisection
 in Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science
, 1993
"... We resolve in the affirmative a question of Boppana and Bui: whether simulated annealing can, with high probability and in polynomial time, find the optimal bisection of a random graph in Gnpr when p r = O(n*’) for A 5 2. (The random graph model Gnpr specifies a “planted ” bisection of density r, ..."
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Cited by 34 (1 self)
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We resolve in the affirmative a question of Boppana and Bui: whether simulated annealing can, with high probability and in polynomial time, find the optimal bisection of a random graph in Gnpr when p r = O(n*’) for A 5 2. (The random graph model Gnpr specifies a “planted ” bisection of density r, separating two n/2vertex subsets of slightly higher density p.) We show that simulated “annealing ” at an appropriate fixed temperature (i.e., the Metropolis algorithm) finds the unique smallest bisection in O(n2+‘) steps with very high probability, provided A> 1116. (By using a slightly modified neighborhood structure, the number of steps can be reduced to O(n’+‘).) We leave open the question of whether annealing is effective for A in the range 312 < A 5 1116, whose lower limit represents the threshold at which the planted bisection becomes lost amongst other random small bisections. It also remains open whether hillclimbing (i.e., annealing at temperature 0) solves the same problem. 1
Cores in random hypergraphs and boolean formulas
, 2003
"... We describe a technique for determining the thresholds for the appearance of cores in random structures. We use it to determine (i) the threshold for the appearance of a kcore in a random runiform hypergraph for all r; k * 2; r + k? 4, and (ii) the threshold for the pure literal rule to find a sa ..."
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Cited by 26 (3 self)
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We describe a technique for determining the thresholds for the appearance of cores in random structures. We use it to determine (i) the threshold for the appearance of a kcore in a random runiform hypergraph for all r; k * 2; r + k? 4, and (ii) the threshold for the pure literal rule to find a satisfying assignment for a random instance of rSAT, r * 3.
Optimal Construction of EdgeDisjoint Paths in Random Regular Graphs
, 2000
"... Given a graph G = (V, E) and a set of n pairs of vertices in V, we are interested in finding for each pair (ai, bi), a path connecting ai to bi, such that the set of n paths so found is edgedisjoint. (For arbitrary graphs the problem is Af7>complete, although it is in T' if n is fixed.) We pre ..."
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Cited by 21 (1 self)
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Given a graph G = (V, E) and a set of n pairs of vertices in V, we are interested in finding for each pair (ai, bi), a path connecting ai to bi, such that the set of n paths so found is edgedisjoint. (For arbitrary graphs the problem is Af7>complete, although it is in T' if n is fixed.) We present a polynomial time randomized algorithm for finding the optimal number of edge disjoint paths (up to constant factors) in the random regular graph Gn,r, for r sufficiently large. (The graph is chosen first, then an adversary chooses the pairs of endpoints.) 1
EdgeDisjoint Paths in Expander Graphs
, 2000
"... Given a graph G = (V, E) and a set of n pairs of vertices in V, we are interested in finding for each pair (ai, bi), a path connecting ai to bi, such that the set of n paths so found is edgedisjoint. (For arbitrary graphs the problem is AfPcomplete, although it is in 7 > if n is fixed.) We pre ..."
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Cited by 19 (0 self)
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Given a graph G = (V, E) and a set of n pairs of vertices in V, we are interested in finding for each pair (ai, bi), a path connecting ai to bi, such that the set of n paths so found is edgedisjoint. (For arbitrary graphs the problem is AfPcomplete, although it is in 7 > if n is fixed.) We present a polynomial time randomized algorithm for finding edge disjoint paths in an rregular expander graph G. We show that if G has sufficiently strong expansion properties and r is sufficiently large then all sets of n = f(n/log n) pairs of vertices can be joined. This is within a constant factor of best possible.
Concentration
, 1998
"... Upper bounds on probabilities of large deviations for sums of bounded independent random variables may be extended to handle functions which depend in a limited way on a number of independent random variables. This ‘method of bounded differences’ has over the last dozen or so years had a great impac ..."
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Cited by 17 (2 self)
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Upper bounds on probabilities of large deviations for sums of bounded independent random variables may be extended to handle functions which depend in a limited way on a number of independent random variables. This ‘method of bounded differences’ has over the last dozen or so years had a great impact in probabilistic methods in discrete mathematics and in the mathematics of operational research and theoretical computer science. Recently Talagrand introduced an exciting new method for bounding probabilities of large deviations, which often proves superior to the bounded differences approach. In this paper we
Linear Extensions Of A Random Partial Order
 Ann. Appl. Probab
, 1994
"... . We study asymptotics of the number of linear extensions of the random Gn;p partial order, where p is fixed and n !1. In particular, it is shown that the distribution is asymptotically lognormal. 1. Introduction and results One of the standard models for a random partial order is the G n;p order ..."
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Cited by 8 (1 self)
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. We study asymptotics of the number of linear extensions of the random Gn;p partial order, where p is fixed and n !1. In particular, it is shown that the distribution is asymptotically lognormal. 1. Introduction and results One of the standard models for a random partial order is the G n;p order, defined as follows. Let ! denote the natural order on the vertex set [n] = f1; : : : ; ng. A graph G on [n] induces a partial order on that vertex set, viz. the transitive closure of the relation fi ! j and there is an edge ijg. I.e., if each edge ij in G, with i ! j, is directed from i to j, i OE j iff there exists a directed path from i to j in G. The random G n;p order is obtained by applying this procedure to the random graph G n;p . Here, as throughout, we let G n;p denote a random graph in G(n; p), i.e., a graph on [n] such that each possible edge appears with probability p, independently of all other edges. We assume throughout that 0 ! p ! 1 and set q = 1 \Gamma p. This G n;p o...
On concentration of probability
 Combinatorics, Probability and Computing
"... Abstract. We give a survey of several methods to obtain sharp concentration results, typically with exponentially small error probabilities, for random variables occuring in combinatorial probability. ..."
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Cited by 8 (0 self)
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Abstract. We give a survey of several methods to obtain sharp concentration results, typically with exponentially small error probabilities, for random variables occuring in combinatorial probability.