Network Topology Mapping From Partial Virtual Coordinates and Graph Geodesics

Abstract

For many important network types (e.g., sensor networks in complex harsh environments and social networks) physical coordinate systems (e.g., Cartesian), and physical distances (e.g., Euclidean), are either difficult to discern or inapplicable. Accordingly, coordinate systems and characterizations based on hop-distance measurements, such as Topology Preserving Maps (TPMs) and Virtual-Coordinate (VC) systems are attractive alternatives to Cartesian coordinates for many network algorithms. Herein, we present an approach to recover geometric and topological properties of a network with a small set of distance measurements. In particular, our approach is a combination of shortest path (often called geodesic) recovery concepts and low-rank matrix completion, generalized to the case of hop-distances in graphs. Results for sensor networks embedded in 2-D and 3-D spaces, as well as a social networks, indicates that the method can accurately capture the network connectivity with a small set of measurements. TPM generation can now also be based on various context appropriate measurements or VC systems, as long as they characterize different nodes by distances to small sets of random nodes (instead of a set of global anchors). The proposed method is a significant generalization that allows the topology to be extracted from a random set of graph shortest paths, making it applicable in contexts such as social networks where VC generation may not be possible.

Bounded Manifold Completion

“…Specifically, we will utilize BMC technique to capture and characterize network topology of social networks such as Facebook, Arxiv, and Enron e-mail. We will also utilize BMC scheme to map network topology of 2-D and 3-D sensor networks from partial virtual coordinates to graph geodesics [28]. In addition, we are interested in DR of collective motion such as schools of fish and flocks of birds.…”

Virtual Coordinate System Based on a Circulant Topology for Routing in Networks-On-Chip

“…We also note that the approach that we call Topological is widespread; it is used in various areas where communication networks are established [30] since it is convenient to study the topology properties of networks using graph theory (for example, as in [30,31], dedicated to supercomputers, or as in [32,33], dedicated to decision-making in communication networks). Network information transmission delay is related to the diameter and MPL of the graph, while network fault tolerance is connected to the connectivity of the graph [34][35][36]. In the following sections, we will rely on the Topological approach, considering NoC and routing algorithms in it at a high level of its topology and abstracting from the NoC physical implementation for simplicity.…”

“…diameter and MPL of the graph, while network fault tolerance is connected to the connectivity of the graph [34][35][36]. In the following sections, we will rely on the Topological approach, considering NoC and routing algorithms in it at a high level of its topology and abstracting from the NoC physical implementation for simplicity.…”

Reconstruction of Agents’ Corrupted Trajectories of Collective Motion Using Low-rank Matrix Completion

“…We compare the performance of HDA against a generalized version of the benchmark low-rank matrix completion (LMC) technique given in [18]. We presented in our previous work [28,29], that LMC exhibit superior performance in topology recovery of both partially observed directed graphs and partially observed uncorrected graphs. In a similar vein, we have also shown that LMC methods have significant performance in reconstruction of fragmented trajectories because the coordinate matrix representing agents' trajectories of collective motion is often low-rank due to mutual interactions and dependencies between the agents [7].…”