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Keren Erez

Bio: Keren Erez is an academic researcher from Bar-Ilan University. The author has contributed to research in topics: Markov process & Markov chain. The author has an hindex of 4, co-authored 6 publications receiving 3261 citations.

Papers
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Journal ArticleDOI
TL;DR: This work shows analytically and numerically that for alpha0.99.
Abstract: A common property of many large networks, including the Internet, is that the connectivity of the various nodes follows a scale-free power-law distribution, P(k) = ck(-alpha). We study the stability of such networks with respect to crashes, such as random removal of sites. Our approach, based on percolation theory, leads to a general condition for the critical fraction of nodes, p(c), that needs to be removed before the network disintegrates. We show analytically and numerically that for alpha 0.99.

2,199 citations

Journal ArticleDOI
TL;DR: It is argued that, near criticality, the average distance between sites in the spanning (largest) cluster scales with its mass, M, as square root of [M], rather than as log (k)M, as expected for random networks away from criticality.
Abstract: We study the tolerance of random networks to intentional attack, whereby a fraction $p$ of the most connected sites is removed. We focus on scale-free networks, having connectivity distribution $P(k)\ensuremath{\sim}{k}^{\ensuremath{-}\ensuremath{\alpha}}$, and use percolation theory to study analytically and numerically the critical fraction ${p}_{c}$ needed for the disintegration of the network, as well as the size of the largest connected cluster. We find that even networks with $\ensuremath{\alpha}\ensuremath{\le}3$, known to be resilient to random removal of sites, are sensitive to intentional attack. We also argue that, near criticality, the average distance between sites in the spanning (largest) cluster scales with its mass, $M$, as $\sqrt{M}$, rather than as ${log}_{k}M$, as expected for random networks away from criticality.

1,316 citations

Proceedings ArticleDOI
01 Aug 2007
TL;DR: An information theoretic co-clustering algorithm is presented which can be interpreted as a block-matrix approximation of the Markov transition matrix and the performance of the suggested approach is demonstrated on real recorded data.
Abstract: In this study we analyze monkeys' hand movement; our strategy is compositional, division of complex movement into basic simple components-primitives. Representing each trajectory segment as vectors of directions, we model the movement trajectory as a large Markov process where each state is related with an average trajectory pattern. In the next step, in order to find the movements primitives, we cluster the Markov states according to their probabilistic similarity. We present an information theoretic co-clustering algorithm which can be interpreted as a block-matrix approximation of the Markov transition matrix. The performance of the suggested approach is demonstrated on real recorded data.

6 citations


Cited by
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Journal ArticleDOI
TL;DR: In this paper, a simple model based on the power-law degree distribution of real networks was proposed, which was able to reproduce the power law degree distribution in real networks and to capture the evolution of networks, not just their static topology.
Abstract: The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology.

18,415 citations

Journal ArticleDOI
TL;DR: Developments in this field are reviewed, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
Abstract: Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

17,647 citations

Journal ArticleDOI
TL;DR: The major concepts and results recently achieved in the study of the structure and dynamics of complex networks are reviewed, and the relevant applications of these ideas in many different disciplines are summarized, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.

9,441 citations

Journal ArticleDOI
TL;DR: A thorough exposition of community structure, or clustering, is attempted, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists.
Abstract: The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e. g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks.

9,057 citations

Journal ArticleDOI
TL;DR: A thorough exposition of the main elements of the clustering problem can be found in this paper, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks.

8,432 citations