Dynamic Digraph Research Articles

Discretized Distributed Optimization Over Dynamic Digraphs

Discretized Distributed Optimization Over Dynamic Digraphs

Mixing time trichotomy in regenerating dynamic digraphs

We study the convergence to stationarity for random walks on dynamic random digraphs with given degree sequences. The digraphs undergo full regeneration at independent geometrically distributed random time intervals with parameter α. Relaxation to stationarity is the result of an interplay of regeneration and mixing on the static digraph. When the number of vertices n tends to infinity and the parameter α tends to zero, we find three scenarios according to whether αlogn converges to zero, infinity or to some finite positive value: when the limit is zero, relaxation to stationarity occurs in two separate stages, the first due to mixing on the static digraph, and the second due to regeneration; when the limit is infinite, there is not enough time for the static digraph to mix and the relaxation to stationarity is dictated by the regeneration only; finally, when the limit is a finite positive value we find a mixed behavior interpolating between the two extremes. A crucial ingredient of our analysis is the control of suitable approximations for the unknown stationary distribution.

A Distributed Algorithm for Resource Allocation Over Dynamic Digraphs

This paper studies a distributed resource allocation problem for a multiagent network with a time-varying digraph. Each agent in the network is associated with a local variable (resource) and a convex cost function. The goal is to collectively minimize the total cost in a distributed fashion, subject to individual resource constraints, and collective equality constraints. The main challenge of the problem is due to the local information structure imposed by the time-varying digraph that should be considered as part of the problem formulation. This paper develops a nonnegative surplus-based distributed optimization algorithm. It is shown that the proposed distributed algorithm converges to the global minimizer provided that the time-varying digraph is jointly strongly connected. Also, all the parameters used in the proposed algorithm rely only on local knowledge.

Visual analytics of large dynamic digraphs

In this article, we investigate the problem of visually representing and analyzing large dynamic directed graphs that consist of many vertices, edges, and time steps. With this work we do not primarily focus on graph details but more on achieving an overview about long graph sequences with the major focus to be scalable in vertex, edge, and time dimensions. To reach this goal, we first map each graph to a bipartite layout with vertices in the same order for each graph supporting a preservation of the viewer’s mental map. A sequence of graphs is placed in a left-to-right reading direction. To further reduce link crossings, we draw partial links with user-definable lengths and finally apply edge splatting as a concept to emphasize graph structures by color coding the generated density fields. Time-varying visual patterns can be recognized by inspecting the changes in the color coding in certain regions in the display. We illustrate the usefulness of the approach in two case studies investigating call graphs changing during software development with 21 releases which is a rather short graph sequence but contains several thousand vertices and edges. Visual scalability in the time dimension is shown with more than 1000 graphs from a dynamic social network dataset consisting of face-to-face contacts acquired during the Hypertext 2009 conference recorded by radio-frequency identification badges.

Groupwise Image Registration Guided by a Dynamic Digraph of Images

For groupwise image registration, graph theoretic methods have been adopted for discovering the manifold of images to be registered so that accurate registration of images to a group center image can be achieved by aligning similar images that are linked by the shortest graph paths. However, the image similarity measures adopted to build a graph of images in the extant methods are essentially pairwise measures, not effective for capturing the groupwise similarity among multiple images. To overcome this problem, we present a groupwise image similarity measure that is built on sparse coding for characterizing image similarity among all input images and build a directed graph (digraph) of images so that similar images are connected by the shortest paths of the digraph. Following the shortest paths determined according to the digraph, images are registered to a group center image in an iterative manner by decomposing a large anatomical deformation field required to register an image to the group center image into a series of small ones between similar images. During the iterative image registration, the digraph of images evolves dynamically at each iteration step to pursue an accurate estimation of the image manifold. Moreover, an adaptive dictionary strategy is adopted in the groupwise image similarity measure to ensure fast convergence of the iterative registration procedure. The proposed method has been validated based on both simulated and real brain images, and experiment results have demonstrated that our method was more effective for learning the manifold of input images and achieved higher registration accuracy than state-of-the-art groupwise image registration methods.

Containment control of discrete-time general linear multi-agent systems under dynamic digraph based on trajectory analysis

This paper devotes itself to the containment tracking problem of general linear time-varying discrete-time multi-agent systems (MASs). The convergence analysis is based on trajectory analysis rather than Lyapunov method. Nonnegative matrix theory, in particular the row-stochastic matrix properties, together with the relation between matrix and graph topology, are explored to handle the containment control problem. Based on a technical lemma concerning norm estimation of infinite product of row-stochastic matrices, we show that the followers can exponentially tend to the dynamical convex hull spanned by the leaders, i.e. the containment can be achieved under the very relaxed conditions: strictly instability of the individual-system of each agent (without interactions with other agents) is allowable and the graph topology is only required to be jointly having directed spanning forest frequently enough. Moreover, the least convergence rate is explicitly specified. Simulation is also provided to demonstrate the effectiveness of our result.

Calculation of the seismic first-break time field and its ray path distribution using a minimum traveltime tree algorithm

A grid search method, analogous to Huygens' Principle, is employed to find the first-break seismic traveltime field in a seismic model. By defining a set of directions in space, a dynamic directed graph (digraph) containing the first-break time information can be constructed during the minimum traveltime tree-searching process. The dynamic di-graph has far fewer edges than the global graph suggested by previous researchers. An efficient sorting algorithm (heapsort) is adapted to the traveltime data structure by employing an indirect heap strategy. The computational speed can thus be improved several times over other approaches. The minimum traveltime tree algorithm is an efficient and flexible method to simultaneously calculate the first-break time field and the corresponding ray paths. It produces a robust and global result in comparison with traditional ray tracing methods. Later seismic phases can also be handled by imposing constraints on the ray paths. A subgrid technique is better than the normal grid technique in terms of accuracy and efficiency. The construction of the di-graph uses the local directional information of ray-paths and therefore local anisotropic information can be naturely incorporated. The first-break time field and its ray paths in an anisotropic model can be calculated as easily as those for an isotropic model.