Using persistent homology to recover spatial information from encounter traces

Abstract

In order to better understand human and animal mobility and its potential effects on Mobile Ad-Hoc networks and Delay-Tolerant Networks, many researchers have conducted experiments which collect encounter data. Most analyses of these data have focused on isolated statistical properties such as the distribution of node inter-encounter times and the degree distribution of the connectivity graph.On the other hand, new developments in computational topology, in particular persistent homology, have made it possible to compute topological invariants from noisy data. These homological methods provide a natural way to draw conclusions about global structure based on collections of local information.We use persistent homology techniques to show that in some cases encounter traces can be used to deduce information about the topology of the physical space the experiment was conducted in, and detect certain changes in the space. We also show that one can distinguish between simulated encounter traces generated on a bounded rectangular grid from traces generated on a grid with the opposite edges wrapped (a toroidal grid). Finally, we have found that nontrivial topological features also appear in real experimental encounter traces, and we speculate on types of node behavior that could produce these results. This demonstrates the ability of persistent homology to detect topological features in encounter data that could be difficult to describe using traditional statistical and geometric methods. Figure 1: The topology of the space affects the type of encounter patterns that are possible. If the space is like a line, node A cannot encounter node C without one of them encountering node B. If the space is like a loop, node A and C can encounter each other without encountering node B.

Topological mapping of unknown environments using an unlocalized robotic swarm

“…One way to reduce this complexity is to look for patterns created by tracing the encounters of the nodes instead of investigating the mobility data itself. As such, Walker [21] employed persistent homology to compute topological invariants from encounter data of the mobile nodes in Mobile Ad-Hoc networks in order to infer global information regarding the topology of a physical environment. However, the nodes are assumed to follow a simple mobility model.…”

“…Given a distance threshold , a (k − 1)-dimensional simplex is formed if there is a subset of k points in the graph that are within distance from each other. The Rips complex (also referred to as the Encounter complex [21] in this application) for this value is the collection of all such simplices. A filtration is obtained by varying the value of from 0 to the diameter of the weighted graph.…”

Beyond graphs: Capturing groups in networks

“…Each center has a number of tasks, each task comprising 3-7 researchers targeting a specific topic of research within the center's agenda. A researcher can be (and typically is) in more than one task 6 . Figure 5 shows the simplicial complex representation of one of the centers.…”

“…Applications to image This paper was presented as part of the Workshop on Network Science for Communication Networks (NetSciCom) 978-1-4577-0248-8/11/$26.00 ©2011 IEEE processing have been fairly well studied (see for example [3]). In communications networks, simplicial complexes have been used for studying sensor network coverage [4], [5], and the analysis of contact times in a mobile ad hoc network [6]. The problems studied in these papers are quite different from our problem, which is to capture groups.…”

“…The facets are (0,1,2),(2,3,4), and(1,4,5,6), and the faces (simplices) are all subsets of the facets, and the facets themselves. Note that(1,2,4) is not a face even though(1,2),(1,4) and(2,4) are faces.…”