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.

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Statistical mechanics of complex networks

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.

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The Structure and Function of Complex Networks

A number of recent studies have focused on the statistical properties of networked systems such as social networks and the Worldwide Web. Researchers have concentrated particularly on a few properties that seem to be common to many networks: the small-world property, power-law degree distributions, and network transitivity. In this article, we highlight another property that is found in many networks, the property of community structure, in which network nodes are joined together in tightly knit groups, between which there are only looser connections. We propose a method for detecting such communities, built around the idea of using centrality indices to find community boundaries. We test our method on computer-generated and real-world graphs whose community structure is already known and find that the method detects this known structure with high sensitivity and reliability. We also apply the method to two networks whose community structure is not well known—a collaboration network and a food web—and find that it detects significant and informative community divisions in both cases.

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Community structure in social and biological networks

Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

Machine learning

We propose and study a set of algorithms for discovering community structure in networks-natural divisions of network nodes into densely connected subgroups. Our algorithms all share two definitive features: first, they involve iterative removal of edges from the network to split it into communities, the edges removed being identified using any one of a number of possible "betweenness" measures, and second, these measures are, crucially, recalculated after each removal. We also propose a measure for the strength of the community structure found by our algorithms, which gives us an objective metric for choosing the number of communities into which a network should be divided. We demonstrate that our algorithms are highly effective at discovering community structure in both computer-generated and real-world network data, and show how they can be used to shed light on the sometimes dauntingly complex structure of networked systems.

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Finding and evaluating community structure in networks.

We analyze the pure literal rule heuristic for computing a satisfying assignment to a random 3-CNF formula with n variables. We show that the pure literal rule by itself finds satisfying assignments for almost all 3-CNF formulas with up to 1.63n clauses, but it fails for more than 1.7n clauses. As an aside we show that the value of maximum satisfiability for random 3-CNF formulas is tightly concentrated around its mean.

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On the satisfiability and maximum satisfiability of random 3-CNF formulas

A method is described for identifying related pages among a plurality of pages in a linked database such as the World Wide Web. An initial page is selected from the plurality of pages. Pages linked to the initial page are represented as a graph in a memory. The pages represented in the graph are scored on content, and a set of pages is selected, the selected set of pages having scores greater than a first predetermined threshold. The selected set of pages is scored on connectivity, and a subset of the set of pages that have scores greater than a second predetermined threshold are selected as related pages.

Method for identifying related pages in a hyperlinked database

Disclosed is a system architecture, components and a searching technique for an Unstructured Information Management System (UIMS). The UIMS may be provided as middleware for the effective management and interchange of unstructured information over a wide array of information sources. The architecture generally includes a search engine, data storage, analysis engines containing pipelined document annotators and various adapters. The searching technique makes use of a two-level searching technique. Also disclosed is system, method and computer program product to process document data. The method includes inputting a document and operating at least one text analysis engine that comprises a plurality of coupled annotators for tokenizing document data for identifying and annotating a particular type of semantic content. Operating the at least one text analysis engine generates a plurality of views of a document, where each of the plurality of views are derived from a different tokenization of the document. The method further includes storing the plurality of views in a common data structure associated with the document.

System, method and computer program product for performing unstructured information management and automatic text analysis, and providing multiple document views derived from different document tokenizations

The rapid growth of the web has been noted and tracked extensively. Recent studies have however documented the dual phenomenon: web pages have small half lives, and thus the web exhibits rapid death as well. Consequently, page creators are faced with an increasingly burdensome task of keeping links up-to-date, and many are falling behind. In addition to just individual pages, collections of pages or even entire neighborhoods of the web exhibit significant decay, rendering them less effective as information resources. Such neighborhoods are identified only by frustrated searchers, seeking a way out of these stale neighborhoods, back to more up-to-date sections of the web; measuring the decay of a page purely on the basis of dead links on the page is too naive to reflect this frustration. In this paper we formalize a strong notion of a decay measure and present algorithms for computing it efficiently. We explore this measure by presenting a number of validations, and use it to identify interesting artifacts on today's web. We then describe a number of applications of such a measure to search engines, web page maintainers, ontologists, and individual users.

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Sic transit gloria telae: towards an understanding of the web's decay

For barely a decade now the Web graph (the network formed by Web pages and their hyperlinks) has been the focus of scientific study. In that short a time, this study has made a significant impact on research in physics, computer science and mathematics. It has focussed the attention of the scientific community on all the different kinds of networks that have arisen through technology and human activity; some speak of a "new science of networks". It has brought the computational and deductive power of computer science to the study of the complex social networks formed by inter-human relationships. And, it has given birth to new branches of research in different areas of mathematics, most notably graph theory and probability.

Andrei Z. Broder

Papers

On the satisfiability and maximum satisfiability of random 3-CNF formulas

Method for identifying related pages in a hyperlinked database

System, method and computer program product for performing unstructured information management and automatic text analysis, and providing multiple document views derived from different document tokenizations

Sic transit gloria telae: towards an understanding of the web's decay

Workshop on Algorithms and Models for the Web Graph