Fan R. K. Chung

Spectral grouping using the Nystrom method

Spectral graph theoretic methods have recently shown great promise for the problem of image segmentation. However, due to the computational demands of these approaches, applications to large problems such as spatiotemporal data and high resolution imagery have been slow to appear. The contribution of this paper is a method that substantially reduces the computational requirements of grouping algorithms based on spectral partitioning making it feasible to apply them to very large grouping problems. Our approach is based on a technique for the numerical solution of eigenfunction problems known as the Nystrom method. This method allows one to extrapolate the complete grouping solution using only a small number of samples. In doing so, we leverage the fact that there are far fewer coherent groups in a scene than pixels.

A random graph model for massive graphs

We propose a random graph model which is a special case of sparse random graphs with given degree sequences. This model involves only a small number of parameters, called logsize and log-log growth rate. These parameters capture some universal characteristics of massive graphs. Furthermore, from these parameters, various properties of the graph can be derived. For example, for certain ranges of the parameters, we will compute the expected distribution of the sizes of the connected components which almost surely occur with high probability. We will illustrate the consistency of our model with the behavior of some massive graphs derived from data in telecommunications. We will also discuss the threshold function, the giant component, and the evolution of random graphs in this model.

Local Graph Partitioning using PageRank Vectors

A local graph partitioning algorithm finds a cut near a specified starting vertex, with a running time that depends largely on the size of the small side of the cut, rather than the size of the input graph. In this paper, we present a local partitioning algorithm using a variation of PageRank with a specified starting distribution. We derive a mixing result for PageRank vectors similar to that for random walks, and show that the ordering of the vertices produced by a PageRank vector reveals a cut with small conductance. In particular, we show that for any set C with conductance Phi and volume k, a PageRank vector with a certain starting distribution can be used to produce a set with conductance (O(radic(Phi log k)). We present an improved algorithm for computing approximate PageRank vectors, which allows us to find such a set in time proportional to its size. In particular, we can find a cut with conductance at most oslash, whose small side has volume at least 2b in time O(2 log m/(2b log2 m/oslash2) where m is the number of edges in the graph. By combining small sets found by this local partitioning algorithm, we obtain a cut with conductance oslash and approximately optimal balance in time O(m log4 m/oslash)

The average distances in random graphs with given expected degrees

Random graph theory is used to examine the “small-world phenomenon”; any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees the average distance is almost surely of order log n/log d̃, where d̃ is the weighted average of the sum of squares of the expected degrees. Of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1/kβ for some fixed exponent β. For the case of β > 3, we prove that the average distance of the power law graphs is almost surely of order log n/log d̃. However, many Internet, social, and citation networks are power law graphs with exponents in the range 2 < β < 3 for which the power law random graphs have average distance almost surely of order log log n, but have diameter of order log n (provided having some mild constraints for the average distance and maximum degree). In particular, these graphs contain a dense subgraph, which we call the core, having nc/log log n vertices. Almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core.

Connected Components in Random Graphs with Given Expected Degree Sequences

Abstract. We consider a family of random graphs with a given expected degree sequence. Each edge is chosen independently with probability proportional to the product of the expected degrees of its endpoints. We examine the distribution of the sizes/volumes of the connected components which turns out depending primarily on the average degree d and the second-order average degree d~. Here d~ denotes the weighted average of squares of the expected degrees. For example, we prove that the giant component exists if the expected average degree d is at least 1, and there is no giant component if the expected second-order average degree d~ is at most 1. Examples are given to illustrate that both bounds are best possible.

Quasi-random graphs

We introduce a large equivalence class of graph properties, all of which are shared by so-called random graphs. Unlike random graphs, however, it is often relatively easy to verify that a particular family of graphs possesses some property in this class.

A random graph model for power law graphs

We propose a random graph model which is a special case of sparserandom graphs with given degree sequences which satisfy a power law. This model involves only a small number of paramo eters, called logsize and log-log growth rate. These parameters capture some universal characteristics of massive graphs. From these parameters, various properties of the graph can be derived. For example, for certai n ranges of the parameters, we wi II compute the expected distribution of the sizes of the connected components which almost surely occur with high probability. We illustrate the consistency of our model with the behavior of some massive graphs derived from data in telecommunications. We also discuss the threshold function, the giant component, and the evolution of random graphs in this model.

Spectra of random graphs with given expected degrees

In the study of the spectra of power-law graphs, there are basically two competing approaches. One is to prove analogues of Wigners semicircle law, whereas the other predicts that the eigenvalues follow a power-law distribution. Although the semicircle law and the power law have nothing in common, we will show that both approaches are essentially correct if one considers the appropriate matrices. We will prove that (under certain mild conditions) the eigenvalues of the (normalized) Laplacian of a random power-law graph follow the semicircle law, whereas the spectrum of the adjacency matrix of a power-law graph obeys the power law. Our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants. Of interest are a number of (new) values for the exponent β, where phase transitions for eigenvalue distributions occur. The spectrum distributions have direct implications to numerous graph algorithms such as, for example, randomized algorithms that involve rapidly mixing Markov chains.

Duplication Models for Biological Networks

Are biological networks different from other large complex networks? Both large biological and nonbiological networks exhibit power-law graphs (number of nodes with degree k, N(k) approximately k(-beta)), yet the exponents, beta, fall into different ranges. This may be because duplication of the information in the genome is a dominant evolutionary force in shaping biological networks (like gene regulatory networks and protein-protein interaction networks) and is fundamentally different from the mechanisms thought to dominate the growth of most nonbiological networks (such as the Internet). The preferential choice models used for nonbiological networks like web graphs can only produce power-law graphs with exponents greater than 2. We use combinatorial probabilistic methods to examine the evolution of graphs by node duplication processes and derive exact analytical relationships between the exponent of the power law and the parameters of the model. Both full duplication of nodes (with all their connections) as well as partial duplication (with only some connections) are analyzed. We demonstrate that partial duplication can produce power-law graphs with exponents less than 2, consistent with current data on biological networks. The power-law exponent for large graphs depends only on the growth process, not on the starting graph.

Explicit construction of linear sized tolerant networks

For every @e>0 and every integer m>0, we construct explicitly graphs with O(m/@e) vertices and maximum degree O(1/@e^2), such that after removing any (1-@e) portion of their vertices or edges, the remaining graph still contains a path of length m. This settles a problem of Rosenberg, which was motivated by the study of fault tolerant linear arrays.