Given a sequence of non-negative 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 well-known models of random graphs. There are also applications related to the chromatic number of sparse random graphs.

ABSTRACT: Recently, Barabási and Albert [2] suggested modeling complex real-world 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.

Peer-to-peer 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 human-established trust relationship. Malicious users can create many identities but few trust relationships. Thus, there is a disproportionately-small “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.

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/2-vertex 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

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 edge-disjoint. (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

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 k-core in a random r-uniform 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 r-SAT, r * 3.

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

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 edge-disjoint. (For arbitrary graphs the problem is AfP-complete, although it is in 7 > if n is fixed.) We present a polynomial time randomized algorithm for finding edge disjoint paths in an r-regular 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.

. 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 log-normal. 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...

A model of commodity trading consists of n traders, each bringing to the market his own individual good, and each having his own preference for the goods on the market. The trade results in a so-called core allocation, that is, an exchange of goods which cannot be destabilized by a coalition of traders. Shapley and Scarf, who proposed Supported in part by NSF grant CCR-9024935 y Supported in part by NSF grant DMS-9002347 the model, proved the existence of such an exchange by means of an algorithm invented by Gale. The algorithm determines sequentially a cyclic decomposition of the set of traders into trading groups with equally priced goods that satisfies the stability requirement. In this paper the work of the algorithm is studied under an assumption that the traders' individual preferences are independent and uniform. It is shown that the decreasing sequence of the market sizes has the same distribution as a Markov chain f i g on integers in which the next state 0 is obtai...