David F. Gleich

Higher-order organization of complex networks

Resolving a network of hubs Graphs are a pervasive tool for modeling and analyzing network data throughout the sciences. Benson et al. developed an algorithmic framework for studying how complex networks are organized by higher-order connectivity patterns (see the Perspective by Pržulj and Malod-Dognin). Motifs in transportation networks reveal hubs and geographical elements not readily achievable by other methods. A motif previously suggested as important for neuronal networks is part of a “rich club” of subnetworks. Science, this issue p. 163; see also p. 123 A mathematical framework for clustering reveals organizational features of a variety of networks. Networks are a fundamental tool for understanding and modeling complex systems in physics, biology, neuroscience, engineering, and social science. Many networks are known to exhibit rich, lower-order connectivity patterns that can be captured at the level of individual nodes and edges. However, higher-order organization of complex networks—at the level of small network subgraphs—remains largely unknown. Here, we develop a generalized framework for clustering networks on the basis of higher-order connectivity patterns. This framework provides mathematical guarantees on the optimality of obtained clusters and scales to networks with billions of edges. The framework reveals higher-order organization in a number of networks, including information propagation units in neuronal networks and hub structure in transportation networks. Results show that networks exhibit rich higher-order organizational structures that are exposed by clustering based on higher-order connectivity patterns.

Vertex neighborhoods, low conductance cuts, and good seeds for local community methods

The communities of a social network are sets of vertices with more connections inside the set than outside. We theoretically demonstrate that two commonly observed properties of social networks, heavy-tailed degree distributions and large clustering coefficients, imply the existence of vertex neighborhoods (also known as egonets) that are themselves good communities. We evaluate these neighborhood communities on a range of graphs. What we find is that the neighborhood communities can exhibit conductance scores that are as good as the Fiedler cut. Also, the conductance of neighborhood communities shows similar behavior as the network community profile computed with a personalized PageRank community detection method. Neighborhood communities give us a simple and powerful heuristic for speeding up local partitioning methods. Since finding good seeds for the PageRank clustering method is difficult, most approaches involve an expensive sweep over a great many starting vertices. We show how to use neighborhood communities to quickly generate a small set of seeds.

Overlapping community detection using seed set expansion

Community detection is an important task in network analysis. A community (also referred to as a cluster) is a set of cohesive vertices that have more connections inside the set than outside. In many social and information networks, these communities naturally overlap. For instance, in a social network, each vertex in a graph corresponds to an individual who usually participates in multiple communities. One of the most successful techniques for finding overlapping communities is based on local optimization and expansion of a community metric around a seed set of vertices. In this paper, we propose an efficient overlapping community detection algorithm using a seed set expansion approach. In particular, we develop new seeding strategies for a personalized PageRank scheme that optimizes the conductance community score. The key idea of our algorithm is to find good seeds, and then expand these seed sets using the personalized PageRank clustering procedure. Experimental results show that this seed set expansion approach outperforms other state-of-the-art overlapping community detection methods. We also show that our new seeding strategies are better than previous strategies, and are thus effective in finding good overlapping clusters in a graph.

Rank aggregation via nuclear norm minimization

The process of rank aggregation is intimately intertwined with the structure of skew symmetric matrices. We apply recent advances in the theory and algorithms of matrix completion to skew-symmetric matrices. This combination of ideas produces a new method for ranking a set of items. The essence of our idea is that a rank aggregation describes a partially filled skew-symmetric matrix. We extend an algorithm for matrix completion to handle skew-symmetric data and use that to extract ranks for each item. Our algorithm applies to both pairwise comparison and rating data. Because it is based on matrix completion, it is robust to both noise and incomplete data. We show a formal recovery result for the noiseless case and present a detailed study of the algorithm on synthetic data and Netflix ratings.

Algorithms for Large, Sparse Network Alignment Problems

We propose a new distributed algorithm for sparse variants of the network alignment problem, which occurs in a variety of data mining areas including systems biology, database matching, and computer vision. Our algorithm uses a belief propagation heuristic and provides near optimal solutions for this NP-hard combinatorial optimization problem. We show that our algorithm is faster and outperforms or ties existing algorithms on synthetic problems, a problem in bioinformatics, and a problem in ontology matching. We also provide a unified framework for studying and comparing all network alignment solvers.

Heat kernel based community detection

The heat kernel is a type of graph diffusion that, like the much-used personalized PageRank diffusion, is useful in identifying a community nearby a starting seed node. We present the first deterministic, local algorithm to compute this diffusion and use that algorithm to study the communities that it produces. Our algorithm is formally a relaxation method for solving a linear system to estimate the matrix exponential in a degree-weighted norm. We prove that this algorithm stays localized in a large graph and has a worst-case constant runtime that depends only on the parameters of the diffusion, not the size of the graph. On large graphs, our experiments indicate that the communities produced by this method have better conductance than those produced by PageRank, although they take slightly longer to compute. On a real-world community identification task, the heat kernel communities perform better than those from the PageRank diffusion.

Tracking the random surfer: empirically measured teleportation parameters in PageRank

PageRank computes the importance of each node in a directed graph under a random surfer model governed by a teleportation parameter. Commonly denoted alpha, this parameter models the probability of following an edge inside the graph or, when the graph comes from a network of web pages and links, clicking a link on a web page. We empirically measure the teleportation parameter based on browser toolbar logs and a click trail analysis. For a particular user or machine, such analysis produces a value of alpha. We find that these values nicely fit a Beta distribution with mean edge-following probability between 0.3 and 0.7, depending on the site. Using these distributions, we compute PageRank scores where PageRank is computed with respect to a distribution as the teleportation parameter, rather than a constant teleportation parameter. These new metrics are evaluated on the graph of pages in Wikipedia.

Overlapping Community Detection Using Neighborhood-Inflated Seed Expansion

Community detection is an important task in network analysis. A community (also referred to as a cluster) is a set of cohesive vertices that have more connections inside the set than outside. In many social and information networks, these communities naturally overlap. For instance, in a social network, each vertex in a graph corresponds to an individual who usually participates in multiple communities. In this paper, we propose an efficient overlapping community detection algorithm using a seed expansion approach. The key idea of our algorithm is to find good seeds, and then greedily expand these seeds based on a community metric. Within this seed expansion method, we investigate the problem of how to determine good seed nodes in a graph. In particular, we develop new seeding strategies for a personalized PageRank clustering scheme that optimizes the conductance community score. An important step in our method is the neighborhood inflation step where seeds are modified to represent their entire vertex neighborhood. Experimental results show that our seed expansion algorithm outperforms other state-of-the-art overlapping community detection methods in terms of producing cohesive clusters and identifying ground-truth communities. We also show that our new seeding strategies are better than existing strategies, and are thus effective in finding good overlapping communities in real-world networks.

An Inner-Outer Iteration for Computing PageRank

We present a new iterative scheme for PageRank computation. The algorithm is applied to the linear system formulation of the problem, using inner-outer stationary iterations. It is simple, can be easily implemented and parallelized, and requires minimal storage overhead. Our convergence analysis shows that the algorithm is effective for a crude inner tolerance and is not sensitive to the choice of the parameters involved. The same idea can be used as a preconditioning technique for nonstationary schemes. Numerical examples featuring matrices of dimensions exceeding 100,000,000 in sequential and parallel environments demonstrate the merits of our technique. Our code is available online for viewing and testing, along with several large scale examples.

Tall and skinny QR factorizations in MapReduce architectures

The QR factorization is one of the most important and useful matrix factorizations in scientific computing. A recent communication-avoiding version of the QR factorization trades flops for messages and is ideal for MapReduce, where computationally intensive processes operate locally on subsets of the data. We present an implementation of the tall and skinny QR (TSQR) factorization in the MapReduce framework, and we provide computational results for nearly terabyte-sized datasets. These tasks run in just a few minutes under a variety of parameter choices.