976 research outputs found
Forman-Ricci flow for change detection in large dynamic data sets
We present a viable solution to the challenging question of change detection
in complex networks inferred from large dynamic data sets. Building on Forman's
discretization of the classical notion of Ricci curvature, we introduce a novel
geometric method to characterize different types of real-world networks with an
emphasis on peer-to-peer networks. Furthermore we adapt the classical Ricci
flow that already proved to be a powerful tool in image processing and
graphics, to the case of undirected and weighted networks. The application of
the proposed method on peer-to-peer networks yields insights into topological
properties and the structure of their underlying data.Comment: Conference paper, accepted at ICICS 2016. (Updated version
Discrete curvature on graphs from the effective resistance
This article introduces a new approach to discrete curvature based on the
concept of effective resistances. We propose a curvature on the nodes and links
of a graph and present the evidence for their interpretation as a curvature.
Notably, we find a relation to a number of well-established discrete curvatures
(Ollivier, Forman, combinatorial curvature) and show evidence for convergence
to continuous curvature in the case of Euclidean random graphs. Being both
efficient to calculate and highly amenable to theoretical analysis, these
resistance curvatures have the potential to shed new light on the theory of
discrete curvature and its many applications in mathematics, network science,
data science and physics.Comment: 37 pages, 7 figures. Updates in this version: Section 3.2 added,
Appendix B added, Figure 3 extended, Proof of Proposition 2 correcte
Exploration of Chemical Space: Formal, chemical and historical aspects
Starting from the observation that substances and reactions are the central entities of chemistry, I have structured chemical knowledge into a formal space called a directed hypergraph, which arises when substances are connected by their reactions. I call this hypernet chemical space. In this thesis, I explore different levels of description of this space: its evolution over time, its curvature, and categorical models of its compositionality.
The vast majority of the chemical literature focuses on investigations of particular aspects of some substances or reactions, which have been systematically recorded in comprehensive databases such as Reaxys for the last 200 years. While complexity science has made important advances in physics, biology, economics, and many other fields, it has somewhat neglected chemistry. In this work, I propose to take a global view of chemistry and to combine complexity science tools, modern data analysis techniques, and geometric and compositional theories to explore chemical space. This provides a novel view of chemistry, its history, and its current status.
We argue that a large directed hypergraph, that is, a model of directed relations between sets, underlies chemical space and that a systematic study of this structure is a major challenge for chemistry. Using the Reaxys database as a proxy for chemical space, we search for large-scale changes in a directed hypergraph model of chemical knowledge and present a data-driven approach to navigate through its history and evolution. These investigations focus on the mechanistic features by which this space has been expanding: the role of synthesis and extraction in the production of new substances, patterns in the selection of starting materials, and the frequency with which reactions reach new regions of chemical space. Large-scale patterns that emerged in the last two centuries of chemical history are detected, in particular, in the growth of chemical knowledge, the use of reagents, and the synthesis of products, which reveal both conservatism and sharp transitions in the exploration of the space. Furthermore, since chemical similarity of substances arises from affinity patterns in chemical reactions, we quantify the impact of changes in the diversity of the space on the formulation of the system of chemical elements.
In addition, we develop formal tools to probe the local geometry of the resulting directed hypergraph and introduce the Forman-Ricci curvature for directed and undirected hypergraphs. This notion of curvature is characterized by applying it to social and chemical networks with higher order interactions, and then used for the investigation of the structure and dynamics of chemical space.
The network model of chemistry is strongly motivated by the observation that the compositional nature of chemical reactions must be captured in order to build a model of chemical reasoning. A step forward towards categorical chemistry, that is, a formalization of all the flavors of compositionality in chemistry, is taken by the construction of a categorical model of directed hypergraphs. We lifted the structure from a lineale (a poset version of a symmetric monoidal closed category) to a category of Petri nets, whose wiring is a bipartite directed graph equivalent to a directed hypergraph. The resulting construction, based on the Dialectica categories introduced by Valeria De Paiva, is a symmetric monoidal closed category with finite products and coproducts, which provides a formal way of composing smaller networks into larger in such a way that the algebraic properties of the components are preserved in the resulting network. Several sets of labels, often used in empirical data modeling, can be given the structure of a lineale, including: stoichiometric coefficients in chemical reaction networks, reaction rates, inhibitor arcs, Boolean interactions, unknown or incomplete data, and probabilities. Therefore, a wide range of empirical data types for chemical substances and reactions can be included in our model
The ALMA Discovery of the Rotating Disk and Fast Outflow of Cold Molecular Gas in NGC 1275
We present ALMA Band 6 observations of the CO(2-1), HCN(3-2), and
HCO(3-2) lines in the nearby radio galaxy / brightest cluster galaxy
(BCG) of NGC 1275 with the spatial resolution of pc. In the previous
observations, CO(2-1) emission was detected as radial filaments lying in the
east-west direction. We resolved the inner filament and found that the filament
cannot be represented by a simple infalling stream both morphologically and
kinematically. The observed complex nature of the filament resembles the cold
gas structure predicted by recent numerical simulations of cold chaotic
accretion. A crude estimate suggests that the accretion rate of the cold gas
can be higher than that of hot gas. Within the central 100 pc, we detected a
rotational disk of the molecular gas whose mass is \sim10^{8} M_{\sun}. This
is the first evidence of the presence of massive cold gas disk on this spatial
scale for BCGs. The disk rotation axis is approximately consistent with the
axis of the radio jet on subpc scales. This probably suggests that the cold gas
disk is physically connected to the innermost accretion disk which is
responsible for jet launching. We also detected absorption features in the
HCN(3-2) and HCO(3-2) spectra against the radio continuum emission mostly
radiated by -pc size jet. The absorption features are blue-shifted
from the systemic velocity by 300-600~km~s, which suggests the
presence of outflowing gas from the active galactic nucleus (AGN). We discuss
the relation of the AGN feeding with cold accretion, the origin of blue-shifted
absorption, and estimate of black hole mass using the molecular gas dynamics.Comment: Version 2 (accepted version). 18 pages, 16 figures. Accepted for
publication in Ap
\{kappa}HGCN: Tree-likeness Modeling via Continuous and Discrete Curvature Learning
The prevalence of tree-like structures, encompassing hierarchical structures
and power law distributions, exists extensively in real-world applications,
including recommendation systems, ecosystems, financial networks, social
networks, etc. Recently, the exploitation of hyperbolic space for tree-likeness
modeling has garnered considerable attention owing to its exponential growth
volume. Compared to the flat Euclidean space, the curved hyperbolic space
provides a more amenable and embeddable room, especially for datasets
exhibiting implicit tree-like architectures. However, the intricate nature of
real-world tree-like data presents a considerable challenge, as it frequently
displays a heterogeneous composition of tree-like, flat, and circular regions.
The direct embedding of such heterogeneous structures into a homogeneous
embedding space (i.e., hyperbolic space) inevitably leads to heavy distortions.
To mitigate the aforementioned shortage, this study endeavors to explore the
curvature between discrete structure and continuous learning space, aiming at
encoding the message conveyed by the network topology in the learning process,
thereby improving tree-likeness modeling. To the end, a curvature-aware
hyperbolic graph convolutional neural network, \{kappa}HGCN, is proposed, which
utilizes the curvature to guide message passing and improve long-range
propagation. Extensive experiments on node classification and link prediction
tasks verify the superiority of the proposal as it consistently outperforms
various competitive models by a large margin.Comment: KDD 202
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