Behavioral clusters in dynamic graphs

Abstract

a b s t r a c tThis paper contributes a method for combining sparse parallel graph algorithms with dense parallel linear algebra algorithms in order to understand dynamic graphs including the temporal behavior of vertices. Our method is the first to cluster vertices in a dynamic graph based on arbitrary temporal behaviors. In order to successfully implement this method, we develop a feature based pipeline for dynamic graphs and apply Nonnegative Matrix Factorization (NMF) to these features. We demonstrate these steps with a sample of the Twitter mentions graph as well as a CAIDA network traffic graph. We contribute and analyze a parallel NMF algorithm presenting both theoretical and empirical studies of performance. This work can be leveraged by graph/network analysts to understand the temporal behavior cluster structure and segmentation structure of dynamic graphs.Ó 2015 Published by Elsevier B.V.

MPI-FAUN: An MPI-Based Framework for Alternating-Updating Nonnegative Matrix Factorization

“…Fairbanks et al [Fairbanks et al 2015] present a parallel NMF algorithm designed for multicore machines. To demonstrate the importance of minimizing communication, we consider this approach to parallelizing an alternating-updating NMF algorithm in distributed memory (see Section 5.1).…”

“…In this section we present a naive parallelization of NMF algorithms, which has previously appeared in the context of a shared-memory parallel platform [Fairbanks et al 2015]. Each NLS problem with multiple right-hand sides can be parallelized on the observation that the problems for multiple righthand sides are independent from each other.…”

“…We summarize the algorithmic costs of Algorithm 2 (derived in the following subsections) in Table III. This naive algorithm [Fairbanks et al 2015] has three main drawbacks: (1) it requires storing two copies of the data matrix (one in row distribution and one in column distribution) and both full factor matrices locally, (2) it does not parallelize the computation of HH T and W T W (each processor computes it redundantly), and (3) as we will see in Section 5.2, it communicates more data than necessary. the algorithm-specific Local Update Computations for m/p rows of W. Likewise, the computation at line 6 consists of computing W T A i , W T W, and performing the Local Update Computations for n/p columns of H. In the dense case, this amounts to 4mnk/p + (m + n)k 2 + F(m/p, n/p, k) flops.…”

A high-performance parallel algorithm for nonnegative matrix factorization

“…collect W on each processor using all-gather 6: In this section we present a naive parallelization of NMF algorithms [5]. Each NLS problem with multiple right-hand sides can be parallelized on the observation that the problems for multiple right-hand sides are independent from each other.…”

“…Fairbanks et al [5] present a parallel NMF algorithm designed for multicore machines. To demonstrate the importance of minimizing communication, we consider this approach to parallelizing an alternating NMF algorithm in distributed memory.…”