Showing papers by "Abraham D. Flaxman published in 2007"

A geometric preferential attachment model of networks II

Abstract: A detailed understanding of expansion in complex networks can greatly aid in the design and analysis of algorithms for a variety of important network tasks, including routing messages, ranking nodes, and compressing graphs. This has motivated several recent investigations of expansion properties in real-world graphs and also in random models of real-world graphs, like the preferential attachment graph. The results point to a gap between real-world observations and theoretical models. Some real-world graphs are expanders and others are not, but a graph generated by the preferential attachment model is an expander whp. We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with power-law degree distribution where the expansion property depends on a tunable parameter of the model. The vertices of Gn are n sequentially generated points x1, x2, ..., xn chosen uniformly at random from the unit sphere in R3. After generating xt, we randomly connect it to m points from those points in x1, x2, ..., xt-1....

Maximum Matchings in Regular Graphs of High Girth

Abstract: Let $G=(V,E)$ be any $d$-regular graph with girth $g$ on $n$ vertices, for $d \geq 3$ This note shows that $G$ has a maximum matching which includes all but an exponentially small fraction of the vertices, $O((d-1)^{-g/2})$ Specifically, in a maximum matching of $G$, the number of unmatched vertices is at most $n/n_0(d,g)$, where $n_0(d,g)$ is the number of vertices in a ball of radius $\lfloor(g-1)/2\rfloor$ around a vertex, for odd values of $g$, and around an edge, for even values of $g$ This result is tight if $n

Expansion and Lack Thereof in Randomly Perturbed Graphs

Abstract: This paper studies the expansion properties of randomly perturbed graphs. These graphs are formed by, for example, adding a random $1{\text{-out}}$ or very sparse Erdős-Renyi graph to an arbitrary connected graph. The central results show that there exists a constant ?such that when any connected n-vertex base graph $\bar{G}$ is perturbed by adding a random 1-out then, with high probability, the resulting graph has $e(S,\bar S) \geq \delta |S|$ for all S? Vwith $|S| \leq \frac34 n$. When $\bar{G}$ is perturbed by adding a random Erdős-Renyi graph, $\mathbb{G}_{n,\epsilon/n}$, the expansion of the perturbed graph depends on the structure of the base graph. A necessary condition for the base graph is given under which the resulting graph is an expander with high probability. The proof techniques are also applied to study rapid mixing in the small worlds graphs described by Watts and Strogatz in [Nature 292(1998), 440---442] and by Kleinberg in [Proc. of 32nd ACM Symposium on Theory of Computing(2000), 163---170]. Analysis of Kleinberg's model shows that the graph stops being an expander exactly at the point where a decentralized algorithm is effective in constructing a short path. The proofs of expansion rely on a way of summing over subsets of vertices which allows an argument based on the First Moment Method to succeed.

The diameter of randomly perturbed digraphs and some applications

Abstract: The central observation of this paper is that if en random arcs are added to any n-node strongly connected digraph with bounded degree then the resulting graph has diameter O(lnn) with high probability. We apply this to smoothed analysis of algorithms and property testing. Smoothed Analysis: Recognizing strongly connected digraphs is a basic computational task in graph theory. Even for digraphs with bounded degree, it is NL-complete. By XORing an arbitrary bounded degree digraph with a sparse random digraph R ∼ Dn,e/n we obtain a “smoothed” instance. We show that, with high probability, a log-space algorithm will correctly determine if a smoothed instance is strongly connected. We also show that if NL n almost-L then no heuristic can recognize similarly perturbed instances of (s,t)-connectivity. Property Testing: A digraph is called k-linked if, for every choice of 2k distinct vertices s1,…,sk,t1,…,tk, the graph contains k vertex disjoint paths joining sr to tr for r = 1,…,k. Recognizing k-linked digraphs is NP-complete for k ≥ 2. We describe a polynomial time algorithm for bounded degree digraphs, which accepts k-linked graphs with high probability, and rejects all graphs that are at least en arcs away from being k-linked. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007

Adversarial Deletion in a Scale-Free Random Graph Process

Abstract: We study a dynamically evolving random graph which adds vertices and edges using preferential attachment and is ‘attacked by an adversary’. At time $t$, we add a new vertex $x_t$ and $m$ random edges incident with $x_t$, where $m$ is constant. The neighbours of $x_t$ are chosen with probability proportional to degree. After adding the edges, the adversary is allowed to delete vertices. The only constraint on the adversarial deletions is that the total number of vertices deleted by time $n$ must be no larger than $\delta n$, where $\delta$ is a constant. We show that if $\delta$ is sufficiently small and $m$ is sufficiently large then with high probability at time $n$ the generated graph has a component of size at least $n/30$.

Bias reduction in traceroute sampling: towards a more accurate map of the Internet

Abstract: Traceroute sampling is an important technique in exploring the internet router graph and the autonomous system graph. Although it is one of the primary techniques used in calculating statistics about the internet, it can introduce bias that corrupts these estimates. This paper reports on a theoretical and experimental investigation of a new technique to reduce the bias of traceroute sampling when estimating the degree distribution. We develop a new estimator for the degree of a node in a traceroute-sampled graph; validate the estimator theoretically in Erdos-Renyi graphs and, through computer experiments, for a wider range of graphs; and apply it to produce a new picture of the degree distribution of the autonomous system graph.

On the average case performance of some greedy approximation algorithms for the uncapacitated facility location problem

Abstract: In combinatorial optimization, a popular approach toNP-hard problems is the design of approximation algorithms. These algorithms typically run in polynomial time and are guaranteed to produce a solution which is within a known multiplicative factor of optimal. Unfortunately, the known factor is often known to be large in pathological instances. Conventional wisdom holds that, in practice, approximation algorithms will produce solutions closer to optimal than their proven guarantees. In this paper, we use the rigorous-analysis-of-heuristics framework to investigate this conventional wisdom.We analyze the performance of 3 related approximation algorithms for the uncapacitated facility location problem (from [Jain, Mahdian, Markakis, Saberi, Vazirani, 2003] and [Mahdian, Ye, Zhang, 2002]) when each is applied to an instances created by placing n points uniformly at random in the unit square. We find that, with high probability, these 3 algorithms do not find asymptotically optimal solutions, and, also with high probability, a simple plane partitioning heuristic does find an asymptotically optimal solution.

Bias reduction in traceroute sampling - towards a more accurate map of the internet

Abstract: Traceroute sampling is an important technique in exploring the internet router graph and the autonomous system graph. Although it is one of the primary techniques used in calculating statistics about the internet, it can introduce bias that corrupts these estimates. This paper reports on a theoretical and experimental investigation of a new technique to reduce the bias of traceroute sampling when estimating the degree distribution. We develop a new estimator for the degree of a node in a traceroute-sampled graph; validate the estimator theoretically in Erdos-Renyi graphs and, through computer experiments, for a wider range of graphs; and apply it to produce a new picture of the degree distribution of the autonomous system graph.

The lower tail of the random minimum spanning tree

Abstract: Consider a complete graph $K_n$ where the edges have costs given by independent random variables, each distributed uniformly between 0 and 1. The cost of the minimum spanning tree in this graph is a random variable which has been the subject of much study. This note considers the large deviation probability of this random variable. Previous work has shown that the log-probability of deviation by $\varepsilon$ is $-\Omega(n)$, and that for the log-probability of $Z$ exceeding $\zeta(3)$ this bound is correct; $\log {\rm Pr}[Z \geq \zeta(3) + \varepsilon] = -\Theta(n)$. The purpose of this note is to provide a simple proof that the scaling of the lower tail is also linear, $\log {\rm Pr}[Z \leq \zeta(3) - \varepsilon] = -\Theta(n)$.