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Business innovations are often arising from new combinations of pre-existing systems into new ones, coherently assembling features from their sources. We propose an abstract mathematical concept from category theory, the presheaf, to efficiently represent such coherent feature combinations. Moreover, operations on presheaves will allow us to formally describe and analyze business innovations that arise from novel mergings of systems from different domains. Equipped with such tools we provide an example by analyzing a successful case of such type of recombinant innovation, the digital hub concept proposed by Steve Jobs. The example shows how our framework can be used to bring formal rigor while preserving a fundamentally qualitative reasoning style., Comment: 29 pages

This work introduces IsUMap, a novel manifold learning technique that enhances data representation by integrating aspects of UMAP and Isomap with Vietoris-Rips filtrations. We present a systematic and detailed construction of a metric representation for locally distorted metric spaces that captures complex data structures more accurately than the previous schemes. Our approach addresses limitations in existing methods by accommodating non-uniform data distributions and intricate local geometries. We validate its performance through extensive experiments on examples of various geometric objects and benchmark real-world datasets, demonstrating significant improvements in representation quality.

We introduce a formalism for constructing cohomological field theories (CohFT) out of nonlinear PDEs based on the first author's previous work (arXiv:2202.12425). We apply the formalism to the generalized Seiberg-Witten equations and show that the obtained CohFT functionals agree with the existing ones proposed by physicists. This leads to a unified perspective from which to view the full supersymmetric functionals of the Donaldson-Witten, Seiberg-Witten, and Kapustin-Witten theories and understand the relations between them. We also outline a quantization program for our framework and discuss its potential to produce manifold invariants and quantum cohomologies., Comment: 31 pages, 2 figures

In this article, we expand upon the concepts introduced in \cite{Spivak09} about the relationship between the category $\mathbf{UM}$ of uber metric spaces and the category $\mathbf{sFuz}$ of fuzzy simplicial sets. We show that fuzzy simplicial sets can be regarded as natural combinatorial generalizations of metric relations. Furthermore, we take inspiration from UMAP to apply the theory to dimension reduction (manifold learning) and data visualization, while refining some of their constructions to put the corresponding theory on a more solid footing. A generalization of the adjunction between $\mathbf{UM}$ and $\mathbf{sFuz}$ will allow us to view the adjunctions used in both publications as special cases. Moreover, we derive an explicit description of colimits in $\mathbf{UM}$ and the realization functor $\text{Re}:\mathbf{sFuz}\to\mathbf{UM}$, as well as rigorous definitions of functors that make it possible to recursively merge sets of fuzzy simplicial sets. We show that $\mathbf{UM}$ and the category of extended-pseudo metric spaces $\mathbf{EPMet}$ can be embedded into $\mathbf{sFuz}$ and provide a description of the adjunctions between the category of truncated fuzzy simplicial sets and $\mathbf{sFuz}$, which we relate to persistent homology. Combining those constructions, we can show a surprising connection between the well-known dimension reduction methods UMAP and Isomap and derive an alternative algorithm, which we call IsUMap, that combines some of the strengths of both methods. We compare it with UMAP and Isomap and provide explanations for observed differences.

Discrete curvatures are quantities associated to the nodes and edges of a graph that reflect the local geometry around them. These curvatures have a rich mathematical theory and they have recently found success as a tool to analyze networks across a wide range of domains. In this work, we consider the problem of constructing graphs with a prescribed set of discrete edge curvatures, and explore the space of such graphs. We address this problem in two ways: first, we develop an evolutionary algorithm to sample graphs with discrete curvatures close to a given set. We use this algorithm to explore how other network statistics vary when constrained by the discrete curvatures in the network. Second, we solve the exact reconstruction problem for the specific case of Forman-Ricci curvature. By leveraging the theory of Markov bases, we obtain a finite set of rewiring moves that connects the space of all graphs with a fixed discrete curvature., Comment: Second version. 24 pages, 10 figures

The characterization of complex networks with tools originating in geometry, for instance through the statistics of so-called Ricci curvatures, is a well established tool of network science. There exist various types of such Ricci curvatures, capturing different aspects of network geometry. In the present work, we investigate Bakry-\'Emery-Ricci curvature, a notion of discrete Ricci curvature that has been studied much in geometry, but so far has not been applied to networks. We explore on standard classes of artificial networks as well as on selected empirical ones to what the statistics of that curvature are similar to or different from that of other curvatures, how it is correlated to other important network measures, and what it tells us about the underlying network. We observe that most vertices typically have negative curvature. Random and small-world networks exhibit a narrow curvature distribution whereas other classes and most of the real-world networks possess a wide curvature distribution. When we compare Bakry-\'Emery-Ricci curvature with two other discrete notions of Ricci-curvature, Forman-Ricci and Ollivier-Ricci curvature for both model and real-world networks, we observe a high positive correlation between Bakry-\'Emery-Ricci and both Forman-Ricci and Ollivier-Ricci curvature, and in particular with the augmented version of Forman-Ricci curvature. Bakry-\'Emery-Ricci curvature also exhibits a high negative correlation with the vertex centrality measure and degree for most of the model and real-world networks. However, it does not correlate with the clustering coefficient. Also, we investigate the importance of vertices with highly negative curvature values to maintain communication in the network. The computational time for Bakry-\'Emery-Ricci curvature is shorter than that required for Ollivier-Ricci curvature but higher than for Augmented Forman-Ricci curvature., Comment: 34 pages, 9 main figures, 12 supplementary tables, 1 supplementary figure

We prove the local Lipschitz continuity of sub-elliptic harmonic maps between certain singular spaces, more specifically from the $n$-dimensional Heisenberg group into $CAT(0)$ spaces. Our main theorem establishes that these maps have the desired Lipschitz regularity, extending the H\"older regularity in this setting proven by Y. Gui et. al and obtaining same regularity as in the work of H-C. Zhang and X-P. Zhu for certain sub-Riemannian geometries, see also the works of A. Mondino and D. Semola, as well as N. Gigli, for generalisations to RCD spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group., Comment: 27 pages, no figures, comments are welcome. In version 2, a missing bibliographical information has been added

Commitment devices are powerful tools that can influence and incentivise certain behaviours by linking them to rewards or punishments. These devices are particularly useful in decision-making, as they can steer individuals towards specific choices. In the field of game theory, commitment devices can alter a player's payoff matrix, ultimately changing the game's Nash equilibria. Interestingly, agents, whom we term game manipulators and who can be external to the original game, can leverage such devices to extract fees from players by making them contingent offers that modify the payoffs of their actions. This can result in a different Nash equilibrium with potentially lower payoffs for the players compared to the original game. For this scheme to work, it is required that all commitments be binding, meaning that once an offer is made, it cannot be revoked. Consequently, we analyse binding contracts as the commitment mechanism that enables game manipulation scenarios. The main focus of this study is to formulate the logic of this setting, expand its scope to encompass more intricate schemes, and analyse the behaviour of regret-minimizing agents in scenarios involving game manipulation.

We investigate the canonical pseudo-Riemannian metrics associated with Jordan-analogues of the coadjoint orbits for pseudo-Euclidean Jordan superalgebras., Comment: 20 pages, 1 figure

High-order structures have been recognised as suitable models for systems going beyond the binary relationships for which graph models are appropriate. Despite their importance and surge in research on these structures, their random cases have been only recently become subjects of interest. One of these high-order structures is the oriented hypergraph, which relates couples of subsets of an arbitrary number of vertices. Here we develop the Erd\H{o}s-R\'enyi model for oriented hypergraphs, which corresponds to the random realisation of oriented hyperedges of the complete oriented hypergraph. A particular feature of random oriented hypergraphs is that the ratio between their expected number of oriented hyperedges and their expected degree or size is 3/2 for large number of vertices. We highlight the suitability of oriented hypergraphs for modelling large collections of chemical reactions and the importance of random oriented hypergraphs to analyse the unfolding of chemistry., Comment: 25 pages, 6 figures

Forman-Ricci curvature (FRC) is a potent and powerful tool for analysing empirical networks, as the distribution of the curvature values can identify structural information that is not readily detected by other geometrical methods. Crucially, FRC captures higher-order structural information of clique complexes of a graph or Vietoris-Rips complexes, which is not readily accessible to alternative methods. However, existing FRC platforms are prohibitively computationally expensive. Therefore, herein we develop an efficient set-theoretic formulation for computing such high-order FRC in simplicial complexes. Significantly, our set theory representation reveals previous computational bottlenecks and also accelerates the computation of FRC. Finally, We provide a pseudo-code, a software implementation coined FastForman, as well as a benchmark comparison with alternative implementations. We envisage that FastForman will be used in Topological and Geometrical Data analysis for high-dimensional complex data sets. Moreover, our development paves the way for future generalisations towards efficient computations of FRC on cell complexes.

Understanding how different networks relate to each other is key for obtaining a greater insight into complex systems. Here, we introduce an intuitive yet powerful framework to characterise the relationship between two networks comprising the same nodes. We showcase our framework by decomposing the shortest paths between nodes as being contributed uniquely by one or the other source network, or redundantly by either, or synergistically by the two together. Our approach takes into account the networks' full topology, and it also provides insights at multiple levels of resolution: from global statistics, to individual paths of different length. We show that this approach is widely applicable, from brains to the London public transport system. In humans and across 123 other mammalian species, we demonstrate that reliance on unique contributions by long-range white matter fibers is a conserved feature of mammalian structural brain networks. Across species, we also find that efficient communication relies on significantly greater synergy between long-range and short-range fibers than expected by chance, and significantly less redundancy. Our framework may find applications to help decide how to trade-off different desiderata when designing network systems, or to evaluate their relative presence in existing systems, whether biological or artificial.

Synchronization phenomena on networks have attracted much attention in studies of neural, social, economic, and biological systems, yet we still lack a systematic understanding of how relative synchronizability relates to underlying network structure. Indeed, this question is of central importance to the key theme of how dynamics on networks relate to their structure more generally. We present an analytic technique to directly measure the relative synchronizability of noise-driven time-series processes on networks, in terms of the directed network structure. We consider both discrete-time auto-regressive processes and continuous-time Ornstein-Uhlenbeck dynamics on networks. Our technique builds on computation of the network covariance matrix in the space orthogonal to the synchronized state, enabling it to be more general than previous work in not requiring either symmetric (undirected) or diagonalizable connectivity matrices, and allowing arbitrary self-link weights. More importantly, our approach quantifies the relative synchronisation specifically in terms of the contribution of process motif (walk) structures. We demonstrate that in general the relative abundance of process motifs with convergent directed walks (including feedback and feedforward loops) hinders synchronizability. We also reveal subtle differences between the motifs involved for discrete or continuous-time dynamics. Our insights analytically explain several known general results regarding synchronizability of networks, including that small-world and regular networks are less synchronizable than random networks., Comment: 20 pages, 5 figures

The more then hundred years old Bernstein inequality states that the supremum norm of the derivative of a trigonometric polynomial of fixed degree can be bounded from above by supremum norm of the polynomial itself. The reversed Bernstein inequality, that we prove in this note, says that the reverse inequality holds for functions in the orthogonal complement of the space of polynomials of fixed degree. In fact, we derived a more general result for the lower bounds on higher derivatives. These bounds are better then those obtained by applying bound for the first derivative successively several times., Comment: 8 pages, additional references in v2

Cheeger-type inequalities in which the decomposability of a graph and the spectral gap of its Laplacian mutually control each other play an important role in graph theory and network analysis, in particular in the context of expander theory. The natural problem, however, to extend such inequalities to simplicial complexes and their higher order Eckmann Laplacians has been open for a long time. The question is not only to prove an inequality, but also to identify the right Cheeger-type constant in the first place. Here, we solve this problem. Our solution involves and combines constructions from simplicial topology, signed graphs, Gromov filling radii and an interpolation between the standard 2-Laplacians and the analytically more difficult 1-Laplacians, for which, however, the inequalities become equalities. It is then natural to develop a general theory for $p$-Laplacians on simplicial complexes and investigate the related Cheeger-type inequalities., Comment: Comments are welcome

The extensive surviving corpus of the ancient scholar Plutarch of Chaeronea (ca. 45-120 CE) also contains several texts which, according to current scholarly opinion, did not originate with him and are therefore attributed to an anonymous author Pseudo-Plutarch. These include, in particular, the work Placita Philosophorum (Quotations and Opinions of the Ancient Philosophers), which is extremely important for the history of ancient philosophy. Little is known about the identity of that anonymous author and its relation to other authors from the same period. This paper presents a BERT language model for Ancient Greek. The model discovers previously unknown statistical properties relevant to these literary, philosophical, and historical problems and can shed new light on this authorship question. In particular, the Placita Philosophorum, together with one of the other Pseudo-Plutarch texts, shows similarities with the texts written by authors from an Alexandrian context (2nd/3rd century CE).

Social dilemmas are situations in which collective welfare is at odds with individual gain. One widely studied example, due to the conflict it poses between human behaviour and game theoretic reasoning, is the Traveler's Dilemma. The dilemma relies on the players' incentive to undercut their opponent at the expense of losing a collective high payoff. Such individual incentive leads players to a systematic mutual undercutting until the lowest possible payoff is reached, which is the game's unique Nash equilibrium. However, if players were satisfied with a high payoff -- that is not necessarily higher than their opponent's -- they would both be better off individually and collectively. Here, we explain how it is possible to converge to this cooperative high payoff equilibrium. Our analysis focuses on decomposing the dilemma into a local and a global game. We show that players need to escape the local maximisation and jump to the global game, in order to reach the cooperative equilibrium. Using a dynamic approach, based on evolutionary game theory and learning theory models, we find that diversity, understood as the presence of suboptimal strategies, is the general mechanism that enables the jump towards cooperation., Comment: 10 pages, 4 figures

As a contribution to metaphor analysis, we introduce a statistical, data-based investigation with empirical analysis of long-standing conjectures and a first-ever empirical exploration of the systematic features of metaphors. Conversely, this also makes metaphor theory available as a basis of meaning emergence that can be quantitatively explored and integrated into the framework of NLP., Comment: 25 pages, 7 figures, 2 tables

For a homotopy class $[u]$ of maps between a closed Riemannian manifold $M$ and a general manifold $N$, we want to find a Dirac-harmonic map with map component in the given homotopy class. Most known results require the index to be nontrivial. When the index is trivial, the few known results are all constructive and produce uncoupled solutions. In this paper, we define a new quantity. As a byproduct of proving the homotopy invariance of this new quantity, we find a new simple proof for the fact that all Dirac-harmonic spheres in surfaces are uncoupled. More importantly, by using the homotopy invariance of this new quantity, we prove the existence of Dirac-harmonic maps from manifolds in the trivial index case. In particular, when the domain is a closed Riemann surface, we prove the short-time existence of the $\alpha$-Dirac-harmonic map flow in the trivial index case. Together with the density of the minimal kernel, we get an existence result for Dirac-harmonic maps from closed Riemann surfaces to K\"ahler manifolds, which extends the previous result of the first and third authors. This establishes a general existence theory for Dirac-harmonic maps in the context of trivial index., Comment: arXiv admin note: substantial text overlap with arXiv:1912.11268; text overlap with arXiv:1705.08935 by other authors

Centrality, in some sense, captures the extent to which a vertex controls the flow of information in a network. Here, we propose Local Detour Centrality as a novel centrality-based betweenness measure that captures the extent to which a vertex shortens paths between neighboring vertices as compared to alternative paths. After presenting our measure, we demonstrate empirically that it differs from other leading central measures, such as betweenness, degree, closeness, and the number of triangles. Through an empirical case study, we provide a possible interpretation for Local Detour Centrality as a measure that captures the extent to which a word is characterized by contextual diversity within a semantic network. We then examine the relationship between our measure and the accessibility to knowledge stored in memory. To do so, we show that words that occur in several different and distinct contexts are significantly more effective in facilitating the retrieval of subsequent words than are words that lack this contextual diversity.

We introduce the notion of smooth parametric model of normal positive linear functionals on possibly infinite-dimensional W*-algebras generalizing the notions of parametric models used in classical and quantum information geometry. We then use the Jordan product naturally available in this context in order to define a Riemannian metric tensor on parametric models satsfying suitable regularity conditions. This Riemannian metric tensor reduces to the Fisher-Rao metric tensor, or to the Fubini-Study metric tensor, or to the Bures-Helstrom metric tensor when suitable choices for the W*-algebra and the models are made., Comment: 24 pages. Comments are welcome!

Information decompositions quantify how the Shannon information about a given random variable is distributed among several other random variables. Various requirements have been proposed that such a decomposition should satisfy, leading to different candidate solutions. Curiously, however, only two of the original requirements that determined the Shannon information have been considered, namely monotonicity and normalization. Two other important properties, continuity and additivity, have not been considered. In this contribution, we focus on the mutual information of two finite variables $Y,Z$ about a third finite variable $S$ and check which of the decompositions satisfy these two properties. While most of them satisfy continuity, only one of them is both continuous and additive., Comment: 17 pages

We introduce a non-backtracking Laplace operator for graphs and we investigate its spectral properties. With the use of both theoretical and computational techniques, we show that the spectrum of this operator captures several structural properties of the graph in a more precise way than the classical operators that have been studied so far in the literature, including the non-backtracking matrix.

In [1], we have shown that the dynamics of an interconnected population of excitatory and inhibitory spiking neurons wandering around a Bogdanov-Takens (BT)bifurcation point can generate the observed scale-free avalanches at the population level and the highly variable spike patterns of individual neurons. These characteristics match experimental findings for spontaneous intrinsic activity in the brain. In this paper, we address the mechanisms causing the system to get and remain near this BT point. We propose an effective stochastic neural field model which captures the dynamics of the mean-field model. We show how the network tunes itself through local long-term synaptic plasticity by STDP and short-term synaptic depression to be close to this bifurcation point. The mesoscopic model that we derive matches the directed percolation model at the absorbing state phase transition.

We investigate spontaneous critical dynamics of excitatory and inhibitory (EI) sparsely connected populations of spiking leaky integrate-and-fire neurons with conductance-based synapses. We use a bottom-up approach to derive a single neuron gain function and a linear Poisson neuron approximation which we use to study mean-field dynamics of the EI population and its bifurcations. In the low firing rate regime, the quiescent state loses stability due to saddle-node or Hopf bifurcations. In particular, at the Bogdanov-Takens (BT) bifurcation point which is the intersection of the Hopf bifurcation and the saddle-node bifurcation lines of the 2D dynamical system, the network shows avalanche dynamics with power-law avalanche size and duration distributions. This matches the characteristics of low firing spontaneous activity in the cortex. By linearizing gain functions and excitatory and inhibitory nullclines, we can approximate the location of the BT bifurcation point. This point in the control parameter phase space corresponds to the internal balance of excitation and inhibition and a slight excess of external excitatory input to the excitatory population. Due to the tight balance of average excitation and inhibition currents, the firing of the individual cells is fluctuation-driven. Around the BT point, the spiking of neurons is a Poisson process and the population average membrane potential of neurons is approximately at the middle of the operating interval $[V_{Rest}, V_{th}]$. Moreover, the EI network is close to both oscillatory and active-inactive phase transition regimes.

Topological data analysis asks when balls in a metric space $(X,d)$ intersect. Geometric data analysis asks how much balls have to be enlarged to intersect. We connect this principle to the traditional core geometric concept of curvature. This enables us, on one hand, to reconceptualize curvature and link it to the geometric notion of hyperconvexity. On the other hand, we can then also understand methods of topological data analysis from a geometric perspective.

Jordan algebras arise naturally in (quantum) information geometry, and we want to understand their role and their structure within that framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide a similar construction in the case of real Jordan algebras. Given a real, finite-dimensional, formally real Jordan algebra ${\mathcal J}$, we exploit the generalized distribution determined by the Jordan product on the dual ${\mathcal J}^{\star}$ to induce a pseudo-Riemannian metric tensor on the leaves of the distribution. In particular, these leaves are the orbits of a Lie group, which is the structure group of ${\mathcal J}$, in clear analogy with what happens for coadjoint orbits. However, this time in contrast with the Lie-algebraic case, we prove that not all points in ${\mathcal J}^{*}$ lie on a leaf of the canonical Jordan distribution. When the leaves are contained in the cone of positive linear functionals on ${\mathcal J}$, the pseudo-Riemannian structure becomes Riemannian and, for appropriate choices of ${\mathcal J}$, it coincides with the Fisher-Rao metric on non-normalized probability distributions on a finite sample space, or with the Bures-Helstrom metric for non-normalized, faithful quantum states of a finite-level quantum system, thus showing a direct link between the mathematics of Jordan algebras and both classical and quantum information geometry.

We prove that, for any connected graph on $N\geq 3$ vertices, the spectral gap from the value $1$ with respect to the normalized Laplacian is at most $1/2$. Moreover, we show that equality is achieved if and only if the graph is either a petal graph (for $N$ odd) or a book graph (for $N$ even). This implies that $(\frac12,\frac32)$ is a maximal gap interval for the normalized Laplacian on connected graphs. This is closely related to the Alon-Boppana bound on regular graphs and a recent result by Koll\'ar and Sarnak on cubic graphs. Our result also provides a sharp bound for the convergence rate of some eigenvalues of the Laplacian on neighborhood graphs., Comment: 32 pages

We characterize Forman curvature lower bounds via contractivity of the Hodge Laplacian semigroup. We prove that Ollivier and Forman curvature coincide on edges when maximizing the Forman curvature over the choice of 2-cells. To this end, we translate between 2-cells and transport plans. Moreover, we give improved diameter bounds. We explicitly warn the reader that our Forman curvature notion does not coincide with Forman's original definition, but can be seen as generalization of the latter one.