GG-Mapper: Learning a Cover in the Mapper Construction

The Mapper algorithm is a visualization technique in topological data analysis (TDA) that outputs a graph reflecting the structure of a given dataset. The Mapper algorithm requires tuning several parameters in order to generate a "nice" Mapper graph. The paper focuses on selecting the cover parameter. We present an algorithm that optimizes the cover of a Mapper graph by splitting a cover repeatedly according to a statistical test for normality. Our algorithm is based on $G$-means clustering which searches for the optimal number of clusters in $k$-means by conducting iteratively the Anderson-Darling test. Our splitting procedure employs a Gaussian mixture model in order to choose carefully the cover based on the distribution of a given data. Experiments for synthetic and real-world datasets demonstrate that our algorithm generates covers so that the Mapper graphs retain the essence of the datasets

Towards Directed Collapsibility

In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy equivalent to that of a single path; we call this the trivial space of directed paths. Directed spaces that are topologically trivial may have non-trivial spaces of directed paths, which means that information is lost when the direction of these topological spaces is ignored. We define a notion of directed collapsibility in the setting of a directed Euclidean cubical complex using the spaces of directed paths of the underlying directed topological space relative to an initial or a final vertex. In addition, we give sufficient conditions for a directed Euclidean cubical complex to have a contractible or a connected space of directed paths from a fixed initial vertex. We also give sufficient conditions for the path space between two vertices in a Euclidean cubical complex to be disconnected. Our results have applications to speeding up the verification process of concurrent programming and to understanding partial executions in concurrent programs

A SHAPE-CONTEXT MODEL FOR MATCHING PLACENTAL CHORIONIC SURFACE VASCULAR NETWORKS

Placental chorionic surface vascular networks (PCSVNs) are essential high-capacitance, low-resistance distribution and drainage networks, and are hence important to placental function and to fetal and newborn health. It was hypothesized that variations in the PCSVN structure may reflect both the overall effects of genetic and environmentally regulated variations in branching morphogenesis within the conceptus and the fetus’s vital organs. A critical step in PCSVN analysis is the extraction of blood vessel structure, which has only been done manually through a laborious process, making studies in large cohorts and applications in clinical settings nearly impossible. The large variation in the shape, color, and texture of the placenta presents significant challenges to both machine and human to accurately extract PCSVNs. To increase the visibility of the vessels, colored paint can be injected into the vascular networks of placentas, allowing PCSVNs to be manually traced with a high level of accuracy. This paper provides a proof-of-concept study to explain the geometric differences between manual tracings of paint-injected and un-manipulated PCSVNs under the framework of a shape-context model. Under this framework, paint-injected and un-manipulated tracings of PCSVNs can be matched with nearly 100% accuracy. The implication of our results is that the manual tracing protocol yields faithful PCSVN representations modulo a set of affine transformations, making manual tracing a reliable method for studying PCSVNs. Our work provides assurance to a new pre-processing approach for studying vascular networks by ways of dye-injection in medical imaging problems

Inhibition of cytokine-mediated JNK signalling by purinergic P2Y11 receptors, a novel protective mechanism in endothelial cells

Previous research from our laboratory has demonstrated a novel phenomenon whereby GPCRs play a role in inhibiting cytokine-mediated c-Jun N-terminal kinase (JNK) signalling. So far this novel phenomenon seems to have been vastly overlooked, with little research in the area. Therefore, in this study we explored this further; by assessing the potential of P2YRs to mediate inhibition of cytokine-mediated JNK signalling and related functional outcomes in human endothelial cells. We utilised primary endothelial cells, and employed the use of endogenous activators of P2YRs and well characterised pharmacological inhibitors, to assess signalling parameters mediated by P2YRs, Interleukin-1β (IL-1β), TNFα and JNK. Activation of P2YRs with adenosine tri-phosphate (ATP) resulted in a time- and concentration-dependent inhibition of IL-1β-mediated phosphorylation of JNK and associated kinase activity. The effect was specific for cytokine-mediated JNK signalling, as ATP was without effect on JNK induced by other non-specific activators (e.g. sorbitol, anisomycin), nor effective against other MAPK pathways such as p38 and the canonical NFκB cascade. Pharmacological studies demonstrated a role for the P2Y11 receptor in mediating this effect, but not the P2Y1 nor the adenosine receptors (A1, A2A, A2B & A3). The novel Gαq/11 inhibitor YM254890 and a protein kinase A (PKA) inhibitor H89 both partially reversed ATP-mediated inhibition of IL-1β-stimulated JNK indicating involvement of both Gαq/11 and Gαs mediated pathways. ATP also partially reversed IL-1β-mediated induction of cyclo‑oxygenase-2 (COX-2) and E-selectin. Collectively, these studies indicate the potential for activation of purinergic receptors to protect the endothelium from inflammatory driven JNK activation and may be a new target for inflammatory disease therapy

Combinatorial Conditions for Directed Collapsing

The purpose of this article is to study directed collapsibility of directed Euclidean cubical complexes. One application of this is in the nontrivial task of verifying the execution of concurrent programs. The classical definition of collapsibility involves certain conditions on a pair of cubes of the complex. The direction of the space can be taken into account by requiring that the past links of vertices remain homotopy equivalent after collapsing. We call this type of collapse a link-preserving directed collapse. In this paper, we give combinatorially equivalent conditions for preserving the topology of the links, allowing for the implementation of an algorithm for collapsing a directed Euclidean cubical complex. Furthermore, we give conditions for when link-preserving directed collapses preserve the contractability and connectedness of directed path spaces, as well as examples when link-preserving directed collapses do not preserve the number of connected components of the path space between the minimum and a given vertex.Comment: 23 pages, 11 figure