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.

On the problem of approximating the number of bases of a matroid

This paper addresses the problem of virtual circuit switching in bounded degree expander graphs. We study the static and dynamic versions of this problem. Our solutions are baaed on the rapidly mixing properties of random walks on expander graphs. In the static version of the problem an algorithm is required to route a path between each of K pairs of vertices so that no edge is used by more than g paths. A natural approach to this problem is through a multicommodity flow reduction. However, we show that the random walk approach leads to significantly stronger results than those recently obtained by Leighton and Rao [10] using the multi-commodity flow setup. In the dynamic version of the problem connection requests are continuously injected into the network, Once a connection is established it utilizes a path (a virtual circuit) for a certain time until the communication terminates and the pat h is deleted. Again each edge in the network should not be used by more than g paths at once. The dynamic version is a better model for the practical use of communication networks. Our random walk approach gives a simple and fully distributed solution for this problem. We show that if the injection to the network and the duration of connections are both controlled by Poisson processes then our algorithm achieves ●Digital Systems Research Center, 130 Lytton Ave, Palo Alto, CA 943o1 t Department of Mathematics, Carnegie-Mellon University. A portion of this work was done while the author was visiting Digital SRC. Supported in part by NSF grants CCR-9225008 and CCR9530974. t IBM Almaden Research Center, San Jose, CA 95120, and Department of Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel. Pumission 10nmkc digllalflmd topics ofnll or pflll ot’thismxtcrinlfhr pemmal or clmsnmm usc is gr:mtcd ivilhoul k pro!fidcd 111:11 the copies arc not mode or distrihltcd t’orprolit or conmwrciu I adwmtagc, Ihe copyrighl notice. Ihc Iitle ol”thc puldicoliol) :In(i ils tialc appcw. and nolicc is gi\&ll that LX)pyrigh(i, b) pNllli\,iOll (>~tht :’!Vhi. ill~. “[’0LOp\ Othtr\! ist. to republish. 10 post on wrvers or 10 rcdis!r}l>tjlc IO 1ists. requires speci Iic penniwion andfor kc ,$770( ‘ 97 1:1 1’,,so. ‘1’c\m 1 ‘s:\ Copyrighl 11)97 ,-\Ckl0-XtJ7’)I-XXX-(V97,05 .,$3 5[) a steady state utilization of the network which is similar to the utilization achieved in the static case situation.

Static and dynamic path selection on expander graphs (preliminary version): a random walk approach

A family of permutations F ⊆ Sn (the symmetric group) is called min-wise independent if for any set X ⊆ [n] and any x ∈ X, when a permutation π is chosen at random in F we have Pr(min{π(X)} = π(x) = 1/|X|.

In other words we require that all the elements of any fixed set X have an equal chance to become the minimum element of the image of X under π.

The rigorous study of such families was instigated by the fact that such a family (under some relaxations) is essential to the algorithm used by the AltaVista Web indexing software to detect and filter near-duplicate documents. The insights gained from theoretical investigations led to practical changes, which in turn inspired new mathematical inquiries and results.

This talk will review the current research in this area and will trace the interplay of theory and practice that motivated it.

Min-wise Independent Permutations: Theory and Practice

The non-English Web is growing at phenomenal speed, but available language processing tools and resources are predominantly English-based. Taxonomies are a case in point: while there are plenty of commercial and non-commercial taxonomies for the English Web, taxonomies for other languages are either not available or of arguable quality. Given that building comprehensive taxonomies for each language is prohibitively expensive, it is natural to ask whether existing English taxonomies can be leveraged, possibly via machine translation, to enable text processing tasks in other languages. Our experimental results confirm that the answer is affirmative with respect to at least one task. In this study we focus on query classification, which is essential for understanding the user intent both in Web search and in online advertising. We propose a robust method for classifying non-English queries into an English taxonomy, using an existing English text classifier and off-the-shelf machine translation systems. In particular, we show that by considering the Web search results in the query's original language as additional sources of information, we can alleviate the effect of erroneous machine translation. Empirical evaluation on query sets in languages as diverse as Chinese and Russian yields very encouraging results; consequently, we believe that our approach is also applicable to many additional languages.

/pdf/cross-language-query-classification-using-web-search-for-2b4s8rkt1k.pdf

Andrei Z. Broder

Papers

On the problem of approximating the number of bases of a matroid

Static and dynamic path selection on expander graphs (preliminary version): a random walk approach

Min-wise Independent Permutations: Theory and Practice

Cross-language query classification using web search for exogenous knowledge

A derandomization using min-wise independent permutations