Simplex polynomial in complex networks and its applications to compute the Euler characteristic

In this chapter we briefly introduce important concepts from homology theory and we highlight how some results from Chap. 3 can be understood through the lens of algebraic topology. This chapter accumulates in showing that the Euler characteristic can be understood by means of CW complexes, homology or intersection theory.

The workshop was conducted jointly with a workshop in statistical learning theory. There was substantial interaction between the two groups, both formally in terms of talks attended by members of both groups, as well as via informal discussions. The intellectual themes which were presented during the workshop are described below. Sensor nets and engineering applications: In the opening talk R. Ghrist spoke about the topology necessary to develop methods for determining intruders have entered a net of sensors, and for counting their number. Ghrist, jointly with V. de Silva and Y. Baryshnikov, has developed techniques based directly on homological calculations as well as on integrals over Euler characteristics which hold promise for implementable algorithms. In order for such algorithms to be maximally useful, one must develop error insensitive methods, which will require more probabilistic methods to be included within the algebraic topological framework. Combinatorial applications: Several presentations at the workshop elaborated on the subject of combinatorial algebraic topology. D. Kozlov has given a survey talk, which has set the accents on the subject, tying together structures, methods, and applications, as these are present at the current state of the development. Talks by R. Jardine and M. Raussen concerned the combinatorial and computational aspects of homotopy theory, finding applications of such abstract notions as Quillen's closed model category. K. Knudson gave an interesting account of connections between persistent homology and discrete morse theory. Finally, the talk of E. Babson dealt with more probabilistic aspects and served as a bridge to the presentations of M. Kahle and P. Bubenik. Dynamical systems: K. Mischaikow and S. Day spoke about the use of algebraic topology to understand the qualitative structure of dynamical systems. Mischaikow introduced his paradigm of building databases of dynamical systems based on choices of parameter values. His methods permit the construction of partitions of parameter space within which the qualitative structure remains the same. In addition, Conley index methods, or rather their computational versions, are used to prove the existence of fixed points, recurrent points, and invariant subsets within a given region in a spatial domain. Data analysis: G. Carlsson and V. de Silva spoke about applications of various kinds of diagrams to understand the qualitative geometric nature of data sets. For example, persistence diagrams allow one to recover Betti numbers of sublevel sets of a probability distribution, multidimensional persistence allows one to study sublevel sets of various functions as well, and the analysis of structure theorems for certain kinds of quivers permits one to extend the bootstrap methods to clustering, Betti numbers, as well as to perform dynamic clustering (i.e. clustering over time). There are now viable computational methods for all of these applications. Probabilistic methods: M. Kahle and P. Bubenik spoke about the beginnings of stochastic algebraic topology. Work at the level of zeroth Betti numbers has already been carried out by M. Penrose, under the heading of “geometric random graphs”. What is now needed is an extension of this work to higher dimensional homology groups, as well as to the barcodes which arise in persistent homology. Ultimately, precise results along these lines will open up the possibility of direct evaluation of significance of various qualitative observations given a null hypothesis. There were also several talks more centered at applications, such as vision recognition (J. Giesen) and material science (R. MacPherson). All things considered, the workshop was a great success in terms of scientific interaction, both within this group, as well as with the researchers in statistical learning theory, as was witnessed by many involved discussions, which often lasted well into the late evenings.

The purpose of these notes is to give a geometrical treatment of generalized homology and cohomology theories. The central idea is that of a 'mock bundle', which is the geometric cocycle of a general cobordism theory, and the main new result is that any homology theory is a generalized bordism theory. The book will interest mathematicians working in both piecewise linear and algebraic topology especially homology theory as it reaches the frontiers of current research in the topic. The book is also suitable for use as a graduate course in homology theory.

The Euler characteristic: A general topological descriptor for complex data

Data based identification and prediction of nonlinear and complex dynamical systems

The well-known Euler characteristic is an invariant of graphs defined by means of the vertex, edge and face numbers of a graph, to determine the genus of the underlying surface of the graph. By means of it, it is possible to determine the vertex, edge and face numbers of all possible graphs which can be drawn in a given orientable/non-orientable surface. In this paper, by means of a given degree sequence, a new number denoted by which is related to Euler characteristic and has several applications in Graph Theory is defined. This formula gives direct information compared with the Euler characteristic on the realizability, number of realizations, connectedness, being acyclic or cyclic, number of components, chords, loops, pendant edges, faces, bridges etc.

We prove a Lefschetz formula for graph endomorphisms , where G is a general finite simple graph and ℱ is the set of simplices fixed by T. The degree of T at the simplex x is defined as , a graded sign of the permutation of T restricted to the simplex. The Lefschetz number is defined similarly as in the continuum as , where is the map induced on the k th cohomology group of G. The theorem can be seen as a generalization of the Nowakowski-Rival fixed-edge theorem (Nowakowski and Rival in J. Graph Theory 3:339-350, 1979). A special case is the identity map T, where the formula reduces to the Euler-Poincaré formula equating the Euler characteristic with the cohomological Euler characteristic. The theorem assures that if is nonzero, then T has a fixed clique. A special case is the discrete Brouwer fixed-point theorem for graphs: if T is a graph endomorphism of a connected graph G, which is star-shaped in the sense that only the zeroth cohomology group is nontrivial, like for connected trees or triangularizations of star shaped Euclidean domains, then there is clique x which is fixed by T. If is the automorphism group of a graph, we look at the average Lefschetz number . We prove that this is the Euler characteristic of the chain and especially an integer. We also show that as a consequence of the Lefschetz formula, the zeta function is a product of two dynamical zeta functions and, therefore, has an analytic continuation as a rational function. This explicitly computable product formula involves the dimension and the signature of prime orbits. MSC:58J20, 47H10, 37C25, 05C80, 05C82, 05C10, 90B15, 57M15, 55M20.

The four-colour theorem is one of the famous problems of mathematics, that frustrated generations of mathematicians from its birth in 1852 to its solution (using substantial assistance from electronic computers) in 1976. The theorem asks whether four colours are sufficient to colour all conceivable maps, in such a way that countries with a common border are coloured with different colours. The book discusses various attempts to solve this problem, and some of the mathematics which developed out of these attempts. Much of this mathematics has developed a life of its own, and forms a fascinating part of the subject now known as graph theory. The book is designed to be self-contained, and develops all the graph-theoretical tools needed as it goes along. It includes all the elementary graph theory that should be included in an introduction to the subject, before concentrating on specific topics relevant to the four-colour problem. Part I covers basic graph theory, Euler's polyhedral formula, and the first published false `proof' of the four-colour theorem. Part II ranges widely through related topics, including map-colouring on surfaces with holes, the famous theorems of Kuratowski, Vizing, and Brooks, the conjectures of Hadwiger and Hajos, and much more besides. In Part III we return to the four-colour theorem, and study in detail the methods which finally cracked the problem.

The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely related to the homology of a space, as it can be expressed as the alternating sum of its Betti numbers, whenever the sum is well-defined. Thus, one says that homology categorifies the Euler characteristic. In his work on the generalisation of cardinality-like invariants, Leinster introduced the magnitude of a metric space, a real number that gives the number of of the space. Recently, Leinster and Shulman introduced a homology theory for metric spaces, called magnitude homology, which categorifies the magnitude of a space. When studying a metric space, one is often only interested in the metric space up to a rescaling of the distance of the points by a non-negative real number. The magnitude function describes how the effective number of points changes as one scales the distance, and it is completely encoded in the Euler characteristic of magnitude homology. When studying a finite metric space in topological data analysis using persistent homology, one approximates the space through a nested sequence of simplicial complexes so as to recover topological information about the space by studying the homology of this sequence. Here we relate magnitude homology and persistent homology as two different ways of computing homology of filtered simplicial sets.

Preface: Volume 81

Preface: Volume 39, Issue 2

Preface: Volume 52, Issue 2

Identifying coverage holes makes an important topic for optimization of quality service for wireless sensor network hosts. This paper introduces a new way to identify and describe how is the network's structure, its number of holes and its components, assuming there's a sensor covering an area where a network communication exists. The simplicial complex method and algebraic graph theory will be applied. Betti numbers and Euler characteristics will be used for a sensor network represented by a simplicial complex, and the Tutte polynomial will be used for describing visual graphs algebraically, for a complete identification.

The evolution of a simulated feed-forward neural network with recurrent excitatory connections and inhibitory forward connections is studied within the framework of algebraic topology. The dynamics includes pruning and strengthening of the excitatory connections. The invariants that we define are based on the connectivity structure of the underlying graph and its directed clique complex. The computation of this complex and of its Euler characteristic are related with the dynamical evolution of the network. As the network evolves dynamically, its network topology changes because of the pruning and strengthening of the onnections and algebraic topological invariants can be computed at different time steps providing a description of the process. We observe that the initial values of the topological invariant computed on the network before it evolves can predict the intensity of the activity.

Wolfram’s Elementary Cellular Automata (ECA) serve as fundamental models for studying discrete dynamical systems, yet their classification remains challenging under traditional statistical and heuristic methods. By leveraging tools from algebraic topology, homotopy theory and differential geometry, we establish a formal connection between topological invariants and ECA’s structural properties and evolution. We analyse the role of Betti numbers, Euler characteristics, edge complexity and persistent homology in achieving robust separation of the four ECA classes. Additionally, we apply coarse proximity theory and assessed the applicability of Poincaré duality, Nash embedding and Seifert–van Kampen theorems to quantify large-scale connectivity patterns. We find that Class 1 automata exhibit simple, contractible topological spaces, indicating minimal structural complexity, while Class 2 automata exhibit periodic fluctuations in their topological features, reflecting their cyclic structure and repeating patterns. Class 3 automata exhibit a higher variance in their structural properties with persistent topological features forming and dissolving across scales, a signature of chaotic evolution. Class 4 automata exhibit statistically significant increases in higher-dimensional topological voids, suggesting the appearance of stable formations. Edge complexity and fractal dimension emergd as the strongest predictors of increasing computational and topological complexity, confirming that self-similarity and structural complexity play a crucial role in distinguishing cellular automata classes. Further, we address the critical distinction between Class 3 and Class 4 automata, which holds paramount importance in practical applications. Our approach establishes a mathematical framework for automaton classification by identifying emergent structures, with potential applications in computational physics, artificial intelligence and theoretical biology.