Random Binary Matrix Research Articles

Random dense bipartite graphs and directed graphs with specified degrees

AbstractLet s and t be vectors of positive integers with the same sum. We study the uniform distribution on the space of simple bipartite graphs with degree sequence s in one part and t in the other; equivalently, binary matrices with row sums s and column sums t. In particular, we find precise formulae for the probabilities that a given bipartite graph is edge‐disjoint from, a subgraph of, or an induced subgraph of a random graph in the class. We also give similar formulae for the uniform distribution on the set of simple directed graphs with out‐degrees s and in‐degrees t. In each case, the graphs or digraphs are required to be sufficiently dense, with the degrees varying within certain limits, and the subgraphs are required to be sufficiently sparse. Previous results were restricted to spaces of sparse graphs. Our theorems are based on an enumeration of bipartite graphs avoiding a given set of edges, proved by multidimensional complex integration. As a sample application, we determine the expected permanent of a random binary matrix with row sums s and column sums t. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009

On the Size and Recovery of Submatrices of Ones in a Random Binary Matrix

Binary matrices, and their associated submatrices of 1s, play a central role in the study of random bipartite graphs and in core data mining problems such as frequent itemset mining (FIM). Motivated by these connections, this paper addresses several statistical questions regarding submatrices of 1s in a random binary matrix with independent Bernoulli entries. We establish a three-point concentration result, and a related probability bound, for the size of the largest square submatrix of 1s in a square Bernoulli matrix, and extend these results to non-square matrices and submatrices with fixed aspect ratios. We then consider the noise sensitivity of frequent itemset mining under a simple binary additive noise model, and show that, even at small noise levels, large blocks of 1s leave behind fragments of only logarithmic size. As a result, standard FIM algorithms, which search only for submatrices of 1s, cannot directly recover such blocks when noise is present. On the positive side, we show that an error-tolerant frequent itemset criterion can recover a submatrix of 1s against a background of 0s plus noise, even when the size of the submatrix of 1s is very small.

Detecting Unknown Worms Using Randomness Check

From the introduction of CodeRed and Slammer worms, it has been learned that the early detection of worm epidemics is important in order to reduce the damage resulting from outbreaks. A prominent characteristic of Internet worms is the random selection of subsequent targets. In this paper, we propose a new worm detection mechanism by checking the random distribution of destination addresses in network traffic. The proposed mechanism constructs a matrix from network traffic and checks the rank of the matrix in order to detect the spreading of Internet worms. From the fact that a random binary matrix holds a high rank value, ADUR (Anomaly Detection Using Randomness check) is proposed for detecting unknown worms based on the rank of the matrix. From experiments on various environments, it is demonstrated that the ADUR mechanism effectively detects the spread of new worms in the early stages, even when there is only a single host infected in a monitoring network. Also, we show that ADUR is highly sensitive so that the worm epidemic can be detectable quickly, e.g., three times earlier than the infection of 90% vulnerable hosts.