Abstract: An improved system and method is provided for identifying web communities from seed sets of web pages. A seed set of web pages may be represented as a set of seed vertices of a graph representing a collection of web pages. An initial probability distribution may be constructed on vertices of the graph by assigning a nonzero value to the vertices belonging to the seed set. Then a sequence of probability distributions may be produced on the vertices of the graph by modifying the probability distribution over a series of one-step walks of the probability distribution over the vertices of the graph. For each probability distribution produced in the sequence, level sets of vertices may be generated, and a level set with minimal conductance may be selected for each probability distribution. The level set with the least conductance may then be output representing a community of web pages.

Abstract: Methods and apparatus for locating a dense and isolated sub-graph from a weighted graph having multiple nodes and multiple weighted edges are described. Each node in the weighted graph represents an object. Each weighted edge in the weighted graph connects two nodes and represents the relationship between the two objects represented by the two corresponding nodes. To located the sub-graph, first, an auxiliary weighted graph is constructed using the weighted graph and three coefficients: ?, ?, and ?, where ?, ?, and ? are greater than 0, ? influences the number of nodes inside the sub-graph, ? influences the sum of the weights associated with the edges connecting a node inside the sub-graph and a node outside the sub-graph, and ? influences the sum of the weights associated with the edges connecting two nodes both inside the sub-graph, and by adding a source node s and a sink node t. Next, the auxiliary weighted graph is partitioned into two parts using the s-t minimum cut algorithm.

Abstract: Methods and apparatus for locating a dense and isolated sub-graph from a weighted graph having multiple nodes and multiple weighted edges are described. Each node in the weighted graph represents an object. Each weighted edge in the weighted graph connects two nodes and represents the relationship between the two objects represented by the two corresponding nodes. To located the sub-graph, first, an auxiliary weighted graph is constructed using the weighted graph and three coefficients: ?, ?, and ?, where ?, ?, and ? are greater than 0, ? influences the number of nodes inside the sub-graph, ? influences the sum of the weights associated with the edges connecting a node inside the sub-graph and a node outside the sub-graph, and ? influences the sum of the weights associated with the edges connecting two nodes both inside the sub-graph, and by adding a source node s and a sink node t. Next, the auxiliary weighted graph is partitioned into two parts using the s-t minimum cut algorithm.

Abstract: An improved system and method is provided for clustering objects using indexes for a matrix representing a collection of objects. Objects to be clustered may be represented as a rectangular matrix. An index may be created for accessing the rows of the matrix and an inverted index may be created for accessing the columns of the matrix based upon the connectivity of the edges between rows and columns of the matrix. Each node represented by a row may be joined to a nearest node represented by another row to produce disjoint sets of nodes. The disjoint sets of nodes may represent clusters that may then be output for use by an application. Moreover, the objects to be clustered may be clusters of objects that may be correlated into a hierarchy of clusters of objects.

Abstract: An improved system and method is provided for identifying web communities from seed sets of web pages. A seed set of web pages may be represented as a set of seed vertices of a graph representing a collection of web pages. An initial probability distribution may be constructed on vertices of the graph by assigning a nonzero value to the vertices belonging to the seed set. Then a sequence of probability distributions may be produced on the vertices of the graph by modifying the probability distribution over a series of one-step walks of the probability distribution over the vertices of the graph. For each probability distribution produced in the sequence, level sets of vertices may be generated, and a level set with minimal conductance may be selected for each probability distribution. The level set with the least conductance may then be output representing a community of web pages.