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.

Provided is a computerized method for monitoring the content of documents. A set of documents is stored in memories of server computers. The server computers can be connected to each other by a network such as the Internet. Entries are generated in a search engine for each document of the set. The search engine is also connected to the Internet. The entries are in the form of a full word index of the set of documents. The search engine also maintains a first abstract for each document that is indexed. The abstract is highly dependent on the content of each document. For example, the abstract is in the form of a sketch or a feature vector. Periodically a query is submitted to the search engine. The query locates a result set of documents that satisfy the query. A second abstract is generated for each document member of the result set. The first and second abstracts are compared to identify documents that have changed between the time the set of documents were indexed and the time the result set is generated.

System and method for monitoring web pages by comparing generated abstracts

A data processing system and method particularly useful for network address lookup in interconnected local area networks uses a family of hashing algorithms that allow implementation of a dictionary that is particularly advantageous when the underlying hardware allows parallel memory reads in different memory banks. The system and method requires exactly one memory cycle for deletes and lookup and constant expected amortized cost for insertions. The system and method uses easy-to-compute hash functions and makes no unproven assumptions about their randomness properties, or about any property of keys drawn from a universe U. The system and method makes it possible to build a dictionary that is able to answer 20 parallel searches among 64,000 entries in less than 5 μs using relatively inexpensive hardware.

System with a plurality of hash tables each using different adaptive hashing functions

Given an expander graph $G=(V,E)$ and a set of $q$ disjoint pairs of vertices in $V$, the authors are interested in finding for each pair $(a_i, b_i)$ a path connecting $a_i$ to $b_i$ such that the set of $q$ paths so found is edge disjoint. (For general graphs the related decision problem is NP complete.) 
The authors prove sufficient conditions for the existence of edge-disjoint paths connecting any set of $q\leq n/(\log n)^\kappa$ disjoint pairs of vertices on any $n$ vertex bounded degree expander, where $\kappa$ depends only on the expansion properties of the input graph, and not on $n$. Furthermore, a randomized $o(n^3)$ time algorithm, and a random $\cal NC$ algorithm for constructing these paths is presented. (Previous existence proofs and construction algorithms allowed only up to $n^\epsilon$ pairs, for some $\epsilon\ll \frac{1}{3}$, and strong expanders [D. Peleg and E. Upfal, Combinatorica, 9 (1989), pp.~289--313.].) 
In passing, an algorithm is developed for splitting a sufficiently strong expander into two edge-disjoint spanning expanders.

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Existence and Construction of Edge-Disjoint Pathson Expander Graphs

In a distributed network, client computers are connected to server computers. The server computers store a plurality of Web pages. The Web pages are partitioned into sets, where each set includes Web pages that are substantially similar in content. A preset compression dictionary is generated for each set of Web pages. In addition, a fingerprint is generated for each preset dictionary. The fingerprints uniquely identify each of the preset dictionaries. When one of the client computers requests one of the Web pages, a compressed form of the Web page is sent along with the fingerprint of the dictionary that was used to compress the Web page. The client computer can then request the preset dictionary in order to decompress the Web page when the client does not have a copy of the preset dictionary.

Compression protocol with multiple preset dictionaries

We consider the problem of estimating the size of a collection of documents using only a standard query interface. Our main idea is to construct an unbiased and low-variance estimator that can closely approximate the size of any set of documents defined by certain conditions, including that each document in the set must match at least one query from a uniformly sampleable query pool of known size, fixed in advance.Using this basic estimator, we propose two approaches to estimating corpus size. The first approach requires a uniform random sample of documents from the corpus. The second approach avoids this notoriously difficult sample generation problem, and instead uses two fairly uncorrelated sets of terms as query pools; the accuracy of the second approach depends on the degree of correlation among the two sets of terms.Experiments on a large TREC collection and on three major search engines demonstrates the effectiveness of our algorithms.

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Andrei Z. Broder

Papers

System and method for monitoring web pages by comparing generated abstracts

System with a plurality of hash tables each using different adaptive hashing functions

Existence and Construction of Edge-Disjoint Pathson Expander Graphs

Compression protocol with multiple preset dictionaries

Estimating corpus size via queries