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32
Towards identity anonymization on graphs
 In Proceedings of ACM SIGMOD
, 2008
"... The proliferation of network data in various application domains has raised privacy concerns for the individuals involved. Recent studies show that simply removing the identities of the nodes before publishing the graph/social network data does not guarantee privacy. The structure of the graph itsel ..."
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Cited by 72 (4 self)
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The proliferation of network data in various application domains has raised privacy concerns for the individuals involved. Recent studies show that simply removing the identities of the nodes before publishing the graph/social network data does not guarantee privacy. The structure of the graph itself, and in its basic form the degree of the nodes, can be revealing the identities of individuals. To address this issue, we study a specific graphanonymization problem. We call a graph kdegree anonymous if for every node v, there exist at least k−1 other nodes in the graph with the same degree as v. This definition of anonymity prevents the reidentification of individuals by adversaries with a priori knowledge of the degree of certain nodes. We formally define the graphanonymization problem that, given a graph G, asks for the kdegree anonymous graph that stems from G with the minimum number of graphmodification operations. We devise simple and efficient algorithms for solving this problem. Our algorithms are based on principles related to the realizability of degree sequences. We apply our methods to a large spectrum of synthetic and real datasets and demonstrate their efficiency and practical utility.
Efficient and simple generation of random simple connected graphs with prescribed degree sequence
 in The Eleventh International Computing and Combinatorics Conference, Aug. 2005, kumming
, 2005
"... degree sequence ..."
Mathematics and the Internet: A Source of Enormous Confusion and Great Potential
"... For many mathematicians and physicists, the Internet has become a popular realworld domain for the application and/or development of new theories related to the organization and behavior of largescale, complex, and dynamic systems. In some cases, the Internet has served both as inspiration and just ..."
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Cited by 23 (4 self)
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For many mathematicians and physicists, the Internet has become a popular realworld domain for the application and/or development of new theories related to the organization and behavior of largescale, complex, and dynamic systems. In some cases, the Internet has served both as inspiration and justification for the popularization of new models and mathematics within the scientific enterprise. For example, scalefree network models of the preferential attachment type [8] have been claimed to describe the Internet’s connectivity structure, resulting in surprisingly general and strong claims about the network’s resilience to random failures of its components and its vulnerability to targeted attacks against its infrastructure [2]. These models have, as their trademark, powerlaw type node degree distributions that drastically distinguish them from the classical ErdősRényi type random graph models [13]. These “scalefree ” network models have attracted significant attention within the scientific community and have been partly responsible for launching and fueling the new field of network science [42, 4]. To date, the main role that mathematics has played in network science has been to put the physicists’ largely empirical findings on solid grounds Walter Willinger is at AT&T LabsResearch in Florham Park, NJ. His email address is walter@research.att. com.
A sequential importance sampling algorithm for generating random graphs with prescribed degrees
, 2006
"... Random graphs with a given degree sequence are a useful model capturing several features absent in the classical ErdősRényi model, such as dependent edges and nonbinomial degrees. In this paper, we use a characterization due to Erdős and Gallai to develop a sequential algorithm for generating a ra ..."
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Cited by 23 (0 self)
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Random graphs with a given degree sequence are a useful model capturing several features absent in the classical ErdősRényi model, such as dependent edges and nonbinomial degrees. In this paper, we use a characterization due to Erdős and Gallai to develop a sequential algorithm for generating a random labeled graph with a given degree sequence. The algorithm is easy to implement and allows surprisingly efficient sequential importance sampling. Applications are given, including simulating a biological network and estimating the number of graphs with a given degree sequence. 1. Introduction. Random
Independence Free Graphs and Vertex Connectivity Augmentation
, 2001
"... Given an undirected graph G and a positive integer k, the kvertexconnectivity augmentation problem is to nd a smallest set F of new edges for which G+F is kvertexconnected. Polynomial algorithms for this problem have been found only for k 4 and a major open question in graph connectivity is w ..."
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Cited by 19 (0 self)
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Given an undirected graph G and a positive integer k, the kvertexconnectivity augmentation problem is to nd a smallest set F of new edges for which G+F is kvertexconnected. Polynomial algorithms for this problem have been found only for k 4 and a major open question in graph connectivity is whether this problem is solvable in polynomial time in general. In this
Fast generation of random connected graphs with prescribed degrees
 In Proc. 11th International Computing and Combinatorics Conference
, 2005
"... ..."
Nonseparable detachments of graphs
 EGRES REPORT SERIES
, 2001
"... Let G = (V, E) be a graph and r: V → Z+. An rdetachment of G is a graph H obtained by ‘splitting ’ each vertex v ∈ V into r(v) vertices, called the pieces of v in H. Every edge uv ∈ E corresponds to an edge of H connecting some piece of u to some piece of v. An rdegree specification for G is a f ..."
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Cited by 3 (2 self)
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Let G = (V, E) be a graph and r: V → Z+. An rdetachment of G is a graph H obtained by ‘splitting ’ each vertex v ∈ V into r(v) vertices, called the pieces of v in H. Every edge uv ∈ E corresponds to an edge of H connecting some piece of u to some piece of v. An rdegree specification for G is a function f on V, such that, for each vertex v ∈ V, f(v) is a partition of d(v) into r(v) positive integers. An fdetachment of G is an rdetachment H in which the degrees in H of the pieces of each v ∈ V are given by f(v). Crispin NashWilliams [3] obtained necessary and sufficient conditions for a graph to have a kedgeconnected rdetachment or fdetachment. We solve a problem posed by NashWilliams in [2] by obtaining analogous results for nonseparable detachments of graphs.
Synchronizability and Connectivity of Discrete Complex Systems
, 2006
"... 1.1.1 Properties of graphs....................... 2 1.1.2 Types of graphs.......................... 4 ..."
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1.1.1 Properties of graphs....................... 2 1.1.2 Types of graphs.......................... 4
On graphs with a Zipfian degree distribution”, in preparation
"... We investigate nvertex graphs in which Pr(degree(v) = d) = c/d for appropriate constant c, that is, the degree sequence exhibits a socalled Zipfian distribution. We apply several well known theorems to show the existence of such graphs. Introduction: The probability distribution on a set S = { ..."
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Cited by 1 (1 self)
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We investigate nvertex graphs in which Pr(degree(v) = d) = c/d for appropriate constant c, that is, the degree sequence exhibits a socalled Zipfian distribution. We apply several well known theorems to show the existence of such graphs. Introduction: The probability distribution on a set S = { 1, 2,..., n} defined by Pr(k) = 1/(Hnk), where Hn in the nth harmonic number, is commonly called a Zipfian distribution, after George Zipf (1902 1950), who observed it in relation to word frequency in English. It has been observed [5] and [4] that the worldwide web, modeled as a directed graph, has a degree
Construction of Combinatorial Objects
, 1995
"... Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, u ..."
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Cited by 1 (1 self)
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Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, using an algorithmic version of Burnside's table of marks, computing double coset representatives, and computing Sylow subgroups of automorphism groups can be explained in this way. The exposition is based on graph theoretic concepts to give an easy explanation of data structures for group actions.