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 graph-anonymization problem. We call a graph k-degree 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 re-identification of individuals by adversaries with a priori knowledge of the degree of certain nodes. We formally define the graph-anonymization problem that, given a graph G, asks for the k-degree anonymous graph that stems from G with the minimum number of graph-modification 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.

degree sequence

Random graphs with a given degree sequence are a useful model capturing several features absent in the classical Erdős-Rényi model, such as dependent edges and non-binomial 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

Given an undirected graph G and a positive integer k, the k-vertex-connectivity augmentation problem is to nd a smallest set F of new edges for which G+F is k-vertex-connected. 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

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 large-scale, 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, scale-free 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, power-law type node degree distributions that drastically distinguish them from the classical Erdős-Rényi type random graph models [13]. These “scale-free ” 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 Labs-Research in Florham Park, NJ. His email address is walter@research.att. com.

We address here the problem of generating random graphs uniformly from the set of simple connected graphs having a prescribed degree sequence. Our goal is to provide an algorithm designed for practical use both because of its ability to generate very large graphs (efficiency) and because it is easy to implement (simplicity). We focus on a family of heuristics for which we prove optimality conditions, and show how this optimality can be reached in practice. We then propose a different approach, specifically designed for typical real-world degree distributions, which outperforms the first one. Assuming a conjecture which we state and argue rigorously, we finally obtain an O(n log n) algorithm, which, in spite of being very simple, improves the best known complexity. 1

Let G = (V, E) be a graph and r: V → Z+. An r-detachment 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 r-degree 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 f-detachment of G is an r-detachment H in which the degrees in H of the pieces of each v ∈ V are given by f(v). Crispin Nash-Williams [3] obtained necessary and sufficient conditions for a graph to have a k-edge-connected rdetachment or f-detachment. We solve a problem posed by Nash-Williams in [2] by obtaining analogous results for non-separable detachments of graphs.

The generation of discrete structures up to isomorphism is interesting as well for theoretical as for practical purposes. Mathematicians want to look at and analyse structures and for example chemical industry uses mathematical generators of isomers for structure elucidation. The example chosen in this paper for explaining general generation methods is a relatively far reaching and fast graph generator which should serve as a basis for the next more powerful version of MOLGEN, our generator of chemical isomers. 1

We investigate n-vertex graphs in which Pr(degree(v) = d) = c/d for appropriate constant c, that is, the degree sequence exhibits a so-called 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

1.1.1 Properties of graphs....................... 2 1.1.2 Types of graphs.......................... 4