Discovering the structure of complex networks: Implementation and Complexity of the heuristic MACH

Getting a labeling of vertices close to the structure of the graph has been proven to be of interest in many applications e.g., to follow signals indexed by the vertices of the network. This question can be related to a graph labeling problem known as the cyclic bandwidth sum problem. It consists in finding a labeling of the vertices of an undirected and unweighted graph with distinct integers such that the sum of (cyclic) difference of labels of adjacent vertices is minimized. Although theoretical results exist that give optimal value of cyclic bandwidth sum for standard graphs, there are neither results in the general case, nor explicit methods to reach this optimal result. In addition to this lack of theoretical knowledge, only a few methods have been proposed to approximately solve this problem. This report describes the implementation and the complexity of the heuristic MACH, developed to find an approximate solution for the cyclic bandwidth sum problem, by following the structure of the graph. The heuristic is a two-step algorithm: the first step consists of traversing the graph to find a set of paths which follow the structure of the graph, using a similarity criterion based on the Jaccard index to jump from one vertex to the next one. The second step is the merging of all obtained paths, based on a greedy approach that extends a partial solution by inserting a new path at the position that minimizes the cyclic bandwidth sum. 1 Description of the heuristic MACH The aim of the heuristic, introduced in [1], is to build a labeling by traversing the graph to discover its structure. The heuristic we propose consists of a two-step algorithm. The first step performs local searches in order to find a collection of independent paths with respect to the local structure of the graph, while the second step determines the best way to arrange the paths such that the objective function of the Cyclic Bandwidth Sum is minimized

Gradient Coding from Cyclic MDS Codes and Expander Graphs

Gradient coding is a technique for straggler mitigation in distributed learning. In this paper we design novel gradient codes using tools from classical coding theory, namely, cyclic MDS codes, which compare favorably with existing solutions, both in the applicable range of parameters and in the complexity of the involved algorithms. Second, we introduce an approximate variant of the gradient coding problem, in which we settle for approximate gradient computation instead of the exact one. This approach enables graceful degradation, i.e., the $\ell_2$ error of the approximate gradient is a decreasing function of the number of stragglers. Our main result is that normalized adjacency matrices of expander graphs yield excellent approximate gradient codes, which enable significantly less computation compared to exact gradient coding, and guarantee faster convergence than trivial solutions under standard assumptions. We experimentally test our approach on Amazon EC2, and show that the generalization error of approximate gradient coding is very close to the full gradient while requiring significantly less computation from the workers