🍩 Database of Original & Non-Theoretical Uses of Topology
(found 7 matches in 0.002s)
Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems (2020)Sarah Tymochko, Elizabeth Munch, Firas A. Khasawneh
AbstractBifurcations in dynamical systems characterize qualitative changes in the system behavior. Therefore, their detection is important because they can signal the transition from normal system operation to imminent failure. While standard persistent homology has been used in this setting, it usually requires analyzing a collection of persistence diagrams, which in turn drives up the computational cost considerably. Using zigzag persistence, we can capture topological changes in the state space of the dynamical system in only one persistence diagram. Here we present Bifurcations using ZigZag (BuZZ), a one-step method to study and detect bifurcations using zigzag persistence. The BuZZ method is successfully able to detect this type of behavior in two synthetic examples as well as an example dynamical system.
On the Nonlinear Statistics of Optical Flow (2019)Henry Adams, Johnathan Bush, Brittany Carr, Lara Kassab, Joshua Mirth
The Importance of Forgetting: Limiting Memory Improves Recovery of Topological Characteristics From Neural Data (2018)Samir Chowdhury, Bowen Dai, Facundo Mémoli
AbstractWe develop of a line of work initiated by Curto and Itskov towards understanding the amount of information contained in the spike trains of hippocampal place cells via topology considerations. Previously, it was established that simply knowing which groups of place cells fire together in an animal’s hippocampus is sufficient to extract the global topology of the animal’s physical environment. We model a system where collections of place cells group and ungroup according to short-term plasticity rules. In particular, we obtain the surprising result that in experiments with spurious firing, the accuracy of the extracted topological information decreases with the persistence (beyond a certain regime) of the cell groups. This suggests that synaptic transience, or forgetting, is a mechanism by which the brain counteracts the effects of spurious place cell activity.
Stable Signatures for Dynamic Graphs and Dynamic Metric Spaces via Zigzag Persistence (2018)Woojin Kim, Facundo Memoli
AbstractWhen studying flocking/swarming behaviors in animals one is interested in quantifying and comparing the dynamics of the clustering induced by the coalescence and disbanding of animals in different groups. In a similar vein, studying the dynamics of social networks leads to the problem of characterizing groups/communities as they form and disperse throughout time. Motivated by this, we study the problem of obtaining persistent homology based summaries of time-dependent data. Given a finite dynamic graph (DG), we first construct a zigzag persistence module arising from linearizing the dynamic transitive graph naturally induced from the input DG. Based on standard results, we then obtain a persistence diagram or barcode from this zigzag persistence module. We prove that these barcodes are stable under perturbations in the input DG under a suitable distance between DGs that we identify. More precisely, our stability theorem can be interpreted as providing a lower bound for the distance between DGs. Since it relies on barcodes, and their bottleneck distance, this lower bound can be computed in polynomial time from the DG inputs. Since DGs can be given rise by applying the Rips functor (with a fixed threshold) to dynamic metric spaces, we are also able to derive related stable invariants for these richer class of dynamic objects. Along the way, we propose a summarization of dynamic graphs that captures their time-dependent clustering features which we call formigrams. These set-valued functions generalize the notion of dendrogram, a prevalent tool for hierarchical clustering. In order to elucidate the relationship between our distance between two DGs and the bottleneck distance between their associated barcodes, we exploit recent advances in the stability of zigzag persistence due to Botnan and Lesnick, and to Bjerkevik.
Modelling Topological Features of Swarm Behaviour in Space and Time With Persistence Landscapes (2017)P. Corcoran, C. B. Jones
AbstractThis paper presents a model of swarm behavior that encodes the spatial-temporal characteristics of topological features, such as holes and connected components. Specifically, the persistence of topological features with respect to time is computed using zig-zag persistent homology. This information is in turn modelled as a persistence landscape, which forms a normed vector space and facilitates the application of statistical and data mining techniques. Validation of the proposed model is performed using a real data set corresponding to a swarm of fish. It is demonstrated that the proposed model may be used to perform retrieval and clustering of swarm behavior in terms of topological features. In fact, it is discovered that clustering returns clusters corresponding to the swarm behaviors of flock, torus, and disordered. These are the most frequently occurring types of behavior exhibited by swarms in general.
Evasion Paths in Mobile Sensor Networks (2015)Henry Adams, Gunnar Carlsson
AbstractSuppose that ball-shaped sensors wander in a bounded domain. A sensor does not know its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving intruder can avoid detection. In ‘Coordinate-free coverage in sensor networks with controlled boundaries via homology', Vin de Silva and Robert Ghrist give a necessary condition, depending only on the time-varying connectivity data of the sensors, for an evasion path to exist. Using zigzag persistent homology, we provide an equivalent condition that moreover can be computed in a streaming fashion. However, no method with time-varying connectivity data as input can give necessary and sufficient conditions for the existence of an evasion path. Indeed, we show that the existence of an evasion path depends not only on the fibrewise homotopy type of the region covered by sensors but also on its embedding in spacetime. For planar sensors that also measure weak rotation and distance information, we provide necessary and sufficient conditions for the existence of an evasion path.