Mira Gonen

A geographic directed preferential Internet topology model

The goal of this work is to model the peering arrangements between autonomous systems (ASes). Most existing models of the AS-graph assume an undirected graph. However, peering arrangements are mostly asymmetric customer-provider arrangements, which are better modeled as directed edges. Furthermore, it is well known that the AS-graph, and in particular its clustering structure, is influenced by geography. We introduce a new model that describes the AS-graph as a directed graph, with an edge going from the customer to the provider, but also models symmetric peer-to-peer arrangements. In addition, our model takes geography into account. We are able to mathematically analyze its power-law exponent and number of leaves. Beyond the analysis, we have implemented our model as a synthetic network generator called GDNG. Experimentation with GDNG shows that the networks it produces are more realistic than those generated by other network generators, in terms of its power-law exponent, fractions of customer-provider and symmetric peering arrangements, and the size of its dense core. We believe that our model is the first to manifest realistic regional dense cores that have a clear geographic flavor. Our synthetic networks also exhibit path inflation effects that are similar to those observed in the real AS graph.

Counting Stars and Other Small Subgraphs in Sublinear-Time

Detecting and counting the number of copies of certain subgraphs (also known as <i>network motifs</i> or <i>graphlets</i>), is motivated by applications in a variety of areas ranging from Biology to the study of the World-Wide-Web. Several polynomial-time algorithms have been suggested for counting or detecting the number of occurrences of certain network motifs. However, a need for more efficient algorithms arises when the input graph is very large, as is indeed the case in many applications of motif counting. In this paper we design <i>sublinear-time</i> algorithms for approximating the number of copies of certain constant-size subgraphs in a graph <i>G</i>. That is, our algorithms do not read the whole graph, but rather query parts of the graph. Specifically, we consider algorithms that may query the degree of any vertex of their choice and may ask for any neighbor of any vertex of their choice. The main focus of this work is on the basic problem of counting the number of length-2 paths and more generally on counting the number of stars of a certain size. Specifically, we design an algorithm that, given an approximation parameter 0 < ε < 1 and query access to a graph <i>G</i>, outputs an estimate <i>vC</i>_<i>s</i> such that with high constant probability, (1-ε)<i>v</i>_<i>s</i>(<i>G</i>) ≤ v_<i>s</i> ≤ (1 + ε)<i>v</i>_<i>s</i>(<i>G</i>), where <i>v</i>_<i>s</i>(<i>G</i>) denotes the number of stars of size <i>s</i> + 1 in the graph. The expected query complexity and running time of the algorithm are [EQUATION] poly (log <i>n</i>, 1/ε). We also prove lower bounds showing that this algorithm is tight up to polylogarithmic factors in <i>n</i> and the dependence on ε. Our work extends the work of Feige (<i>SIAM Journal on Computing, 2006</i>) and Goldreich and Ron (<i>Random Structures and Algorithms, 2008</i>) on approximating the number of edges (or average degree) in a graph. Combined with these results, our result can be used to obtain an estimate on the variance of the degrees in the graph and corresponding higher moments. In addition, we give some (negative) results on approximating the number of triangles and on approximating the number of length-3-paths in sublinear time.

Generalized trade reduction mechanisms

When designing a mechanism there are several desirable properties tomaintain such as incentive compatibility (IC), individual rationality (IR), and budget balance (BB). It is well known [15] that it is impossible for a mechanism to maximize social welfare whilst also being IR, IC, and BB. There have been several attempts to circumvent [15] by trading welfare for BB, e.g.,in domains such as double-sided auctions [13], distributed markets [3] and supply chain problems [2, 4]. In this paper we provide a procedure called a Generalized Trade Reduction (GTR) for single-value players, which given an IR and IC mechanism, outputs a mechanism which is IR, IC and BB with a loss of welfare. We bound the welfare achieved by our procedure for a wide range of domains. In particular, our resultsimprove on existing solutions for problems such as double sidedmarkets with homogenous goods, distributed markets and several kinds of supply chains. Furthermore, our solution provides budget balanced mechanisms for several open problems such as combinatorial double-sided auctions and distributed markets with strategic transportation edges.

Approximating the Number of Network Motifs

World Wide Web, the Internet, coupled biological and chemical systems, neural networks, and social interacting species, are only a few examples of systems composed by a large number of highly interconnected dynamical units. These networks contain characteristic patterns, termed network motifs, which occur far more often than in randomized networks with the same degree sequence. Several algorithms have been suggested for counting or detecting the number of induced or non-induced occurrences of network motifs in the form of trees and bounded treewidth subgraphs of size O(logn), and of size at most 7 for some motifs. In addition, counting the number of motifs a node is part of was recently suggested as a method to classify nodes in the network. The promise is that the distribution of motifs a node participate in is an indication of its function in the network. Therefore, counting the number of network motifs a node is part of provides a major challenge. However, no such practical algorithm exists. We present several algorithms with time complexity

On the Benefits of Adaptivity in Property Testing of Dense Graphs

O\left(e^{2k}k\cdot n \cdot |E|\cdot \right.

BOUNDING THE BIAS OF TREE-LIKE SAMPLING IN IP TOPOLOGIES

\left.\log\frac{1}{\delta}/{\epsilon^2}\right)

Finding a dense-core in Jellyfish graphs

that, for the first time, approximate for every vertex the number of non-induced occurrences of the motif the vertex is part of, for k-length cycles, k-length cycles with a chord, and (k − 1)-length paths, where k = O(logn), and for all motifs of size of at most four. In addition, we show algorithms that approximate the total number of non-induced occurrences of these network motifs, when no efficient algorithm exists. Some of our algorithms use the color coding technique.

On interval and circular-arc covering problems

We consider the question of whether adaptivity can improve the complexity of property testing algorithms in the dense graphsmodel. It is known that there can be at most a quadratic gap between adaptive and non-adaptive testers in this model, but it was not known whether any gap indeed exists. In this work we reveal such a gap. Specifically, we focus on the well studied property of bipartiteness. Bogdanov and Trevisan (IEEE Symposium on Computational Complexity, 2004) proved a lower bound of i¾?(1/i¾?2) on the query complexity of non-adaptivetesting algorithms for bipartiteness. This lower bound holds for graphs with maximum degree O(i¾?n). Our main result is an adaptivetesting algorithm for bipartiteness of graphs with maximum degree O(i¾?n) whose query complexity is

On nonlinear multi-covering problems

\tilde{O}(1/\epsilon^{3/2})