Topological Features Of Graphs Research Articles

Quantum topological data analysis via the estimation of the density of states

We develop a quantum topological data analysis (QTDA) protocol based on the estimation of the density of states (DOS) of the combinatorial Laplacian. Computing topological features of graphs and simplicial complexes is crucial for analyzing data sets and building explainable artificial intelligence solutions. This task becomes computationally hard for simplicial complexes with over 60 vertices and high-degree topological features due to a combinatorial scaling. We propose to approach the task by embedding underlying hypergraphs as effective quantum Hamiltonians and evaluating their density of states from the time evolution. Specifically, we compose propagators as quantum circuits using the Cartan decomposition of effective Hamiltonians and sample overlaps of time-evolved states using multifidelity protocols. Next we develop various postprocessing routines and implement a Fourier-like transform to recover the rank (and kernel) of Hamiltonians. This enables us to estimate the Betti numbers, revealing the topological features of simplicial complexes. We test our protocol on noiseless and noisy quantum simulators and run examples on IBM quantum processors. We observe the resilience of the proposed QTDA approach to real-hardware noise even in the absence of error mitigation, showing the promise for near-term device implementations and highlighting the utility of global DOS-based estimators. Published by the American Physical Society 2024

Modular structures and the delivery of inpatient care in hospitals: a Network Science perspective on healthcare function and dysfunction

Reinforced by the COVID-19 pandemic, the capacity of health systems to cope with increasing healthcare demands has been an abiding concern of both governments and the public. Health systems are made up from non-identical human and physical components interacting in diverse ways in varying locations. It is challenging to represent the function and dysfunction of such systems in a scientific manner. We describe a Network Science approach to that dilemma. General hospitals with large emergency caseloads are the resource intensive components of health systems. We propose that the care-delivery services in such entities are modular, and that their structure and function can be usefully analysed by contemporary Network Science. We explore that possibility in a study of Australian hospitals during 2019 and 2020. We accessed monthly snapshots of whole of hospital administrative patient level data in two general hospitals during 2019 and 2020. We represented the organisations inpatient services as network graphs and explored their graph structural characteristics using the Louvain algorithm and other methods. We related graph topological features to aspects of observable function and dysfunction in the delivery of care. We constructed a series of whole of institution bipartite hospital graphs with clinical unit and labelled wards as nodes, and patients treated by units in particular wards as edges. Examples of the graphs are provided. Algorithmic identification of community structures confirmed the modular structure of the graphs. Their functional implications were readily identified by domain experts. Topological graph features could be related to functional and dysfunctional issues such as COVID-19 related service changes and levels of hospital congestion. Contemporary Network Science is one of the fastest growing areas of current scientific and technical advance. Network Science confirms the modular nature of healthcare service structures. It holds considerable promise for understanding function and dysfunction in healthcare systems, and for reconceptualising issues such as hospital capacity in new and interesting ways.

Spreading dynamics in heterogeneous graphs: Beyond the assortativity coefficient

Abstract We study spreading dynamics of a reaction–diffusion process in a special class of heterogeneous graphs with Poissonian degree distribution and composed of both local and long range links. The behavior of the spreading dynamics on such networks are investigated by relating them to the topological features of graphs. We find that the degree of assortativity can give just some indication about the large scale behavior of the spreading dynamics while a detailed description of the process can be addressed by introducing new, more appropriate, topological quantities linked to the distance between nodes.

Graphs with given diameter maximizing the algebraic connectivity

We propose a class of graphs G D ∗ ( n 1 , n 2 , … , n D + 1 ) , containing of a chain of D + 1 cliques K n 1 , K n 2 , … , K n D + 1 , where neighboring cliques are fully-interconnected. The class of graphs has diameter D and size N = ∑ 1 ⩽ i ⩽ D + 1 n i . We prove that this class of graphs can achieve the maximal number of links, the minimum average hopcount, and more interestingly, the maximal of any Laplacian eigenvalue among all graphs with N nodes and diameter D. The algebraic connectivity is the eigenvalue of the Laplacian that has been studied most, because it features many interesting properties. We determine the graph with the largest algebraic connectivity among graphs with N nodes and diameter D ⩽ 4 . For other diameters, numerically searching for the maximum of any eigenvalue is feasible, because (a) the searching space within the class G D ∗ ( n 1 , n 2 , … , n D + 1 ) is much smaller than within all graphs with N nodes and diameter D; (b) we reduce the calculation of the Laplacian spectrum from a N × N to a ( D + 1 ) × ( D + 1 ) matrix. The maximum of any Laplacian eigenvalue obtained either theoretically or by numerical searching is applied to (1) investigate the topological features of graphs that maximize different Laplacian eigenvalues; (2) study the correlation between the maximum algebraic connectivity a max ( N , D ) and N as well as D and (3) evaluate two upper bounds of the algebraic connectivity that are proposed in the literature.

Automatic Cel Painting in Computer-assisted Cartoon Production using Similarity Recognition

The task of painting each cel of an animated sequence has always been a tedious work using traditional methods. In order to simplify and speed up the process, the development of an automatic cel painting system is desired. This paper presents a method for assisting the production of cartoon animation using computers in order to reduce the production expense and to improve the overall quality. It signals a great improvement in the working environment where 2-D animation is produced. The basic idea of our automatic cel painting mechanism is to extract both statistical and topological features of regions from input animation drawings, and to search for similar corresponding regions in the previous drawing. If the similarity measures are within the accepted thresholds, the corresponding colour will be inherited. A new integrated labelling algorithm constructing statistical region and topological graph features is proposed. Furthermore, a similarity measure of regions is defined and an algorithm of partial matching of attribute graphs is presented. The experimental result shows the feasibility and practicability of the proposed approach. © 1997 John Wiley & Sons, Ltd.

Recognition of topological features of graphs and images in neural networks

The authors extend the architecture of the Hopfield network, such that it can recognise transformed versions of a set of learnt prototypes. As an example they construct a network which can generalise over all topologically equivalent representations of graphs or images. The construction is based on two coupled networks: a Hopfield network to store and retrieve patterns and a preprocessor to transform the input data.