MMDS 2008: Algorithmic and Statistical Challenges in Modern Large-Scale Data Analysis, Part II

Algorithmic Approaches to Networked Data In an algorithmic perspective on improved models for data, Milena Mihail of the Georgia Institute of Technology began by describing the recent development of a rich theory of power-law random graphs, i.e., graphs that are random conditioned on a specified input power-law degree distribution. With the increasingly wide range of large-scale social and information networks now available, however, generative models that are structurally or syntactically more flexible have become necessary. Mihail described two such extensions: one in which semantics on nodes is modeled by a feature vector, with edges added between nodes based on their semantic proximity, and another in which the phenomenon of associativity/disassociativity is modeled by fixing the probability that nodes of a given degree di tend to link to nodes of degree dj. A small extension in the parameters of a generative model, of course, can lead to a large increase in the observed properties of generated graphs. This observation raises interesting statistical questions about model overfitting, and argues for more refined and systematic methods for model parameterization. It also leads to some of the new algorithmic questions that were the topic of Mihail’s talk. Mihail posed the following algorithmic problem for the basic power-law random graph model: Given as input an N-vector specifying a degree sequence, determine whether a graph with that degree sequence exists and, if it does, efficiently generate one (perhaps approximately uniformly randomly from the ensemble of such graphs). Such realizability problems have a long history in graph theory and theoretical computer science. Because their solutions are intimately related to the theory of graph matchings, many generalizations of the basic problem can be addressed in a strict theoretical framework. For example, motivated by associative/disassociative networks, Mihail described recent progress on the jointdegree matrix realization problem: Given a partition of the node set into classes of vertices of the same degree, a vector specifying the degree of each class, and a matrix specifying the number of edges between any two classes, determine whether such a graph exists and, if it does, construct one. She also described extensions of this basic problem to connected graphs, to finding minimum cost realizations, and to finding a random graph satisfying those basic constraints.