Abstract
We describe a data structure that maintains the number of triangles in a dynamic undirected graph, subject to insertions and deletions of edges and of degree-zero vertices. More generally it can be used to maintain the number of copies of each possible three-vertex subgraph in time O(h) per update, where h is the h-index of the graph, the maximum number such that the graph contains $h$ vertices of degree at least h. We also show how to maintain the h-index itself, and a collection of h high-degree vertices in the graph, in constant time per update. Our data structure has applications in social network analysis using the exponential random graph model (ERGM); its bound of O(h) time per edge is never worse than the Theta(sqrt m) time per edge necessary to list all triangles in a static graph, and is strictly better for graphs obeying a power law degree distribution. In order to better understand the behavior of the h-index statistic and its implications for the performance of our algorithms, we also study the behavior of the h-index on a set of 136 real-world networks.
Highlights
The exponential random graph model (ERGM, or p∗ model) [17, 29, 34] is a general technique for assigning probabilities to graphs that can be used both to generate simulated data for social network analysis and to perform probabilistic reasoning on realworld data
We study the behavior of the h-index, both on scale-free graph models and on a set of real-world graphs used in social network analysis
We begin by describing a data structure for the following problem, which generalizes that of maintaining h-indexes of dynamic graphs
Summary
The exponential random graph model (ERGM, or p∗ model) [17, 29, 34] is a general technique for assigning probabilities to graphs that can be used both to generate simulated data for social network analysis and to perform probabilistic reasoning on realworld data. In order to generate graphs in an ERG model or to perform other forms of probabilistic reasoning with the model, one typically uses a Markov Chain Monte Carlo method [30] in which one performs a large sequence of small changes to sample graphs, updates after each change the counts of the number of features of each type and the sum of the weights of each feature, and uses the updated values to determine whether to accept or reject each change. We show that for scale-free graphs, the h-index scales as a power of n, less than its square root, while in the realworld graphs we studied the scaling exponent appears to have a bimodal distribution
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
