16,497 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Vertex Sparsification for Edge Connectivity in Polynomial Time

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    Non-Abelian homology and homotopy colimit of classifying spaces for a diagram of groups

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    The paper considers non-Abelian homology groups for a diagram of groups introduced as homotopy groups of a simplicial replacement. It is proved that the non-Abelian homology groups of the group diagram are isomorphic to the homotopy groups of the homotopy colimit of the diagram of classifying spaces, with a dimension shift of 1. As an application, a method is developed for finding a nonzero homotopy group of least dimension for a homotopy colimit of classifying spaces. For a group diagram over a free category with a zero colimit, a criterion for the isomorphism of the first non-Abelian and first Abelian homology groups is obtained. It is established that the non-Abelian homology groups are isomorphic to the cotriple derived functors of the colimit functor defined on the category of group diagrams.Comment: 32 page

    Non-perturbative renormalization group analysis of nonlinear spiking networks

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    The critical brain hypothesis posits that neural circuits may operate close to critical points of a phase transition, which has been argued to have functional benefits for neural computation. Theoretical and computational studies arguing for or against criticality in neural dynamics largely rely on establishing power laws or scaling functions of statistical quantities, while a proper understanding of critical phenomena requires a renormalization group (RG) analysis. However, neural activity is typically non-Gaussian, nonlinear, and non-local, rendering models that capture all of these features difficult to study using standard statistical physics techniques. Here, we overcome these issues by adapting the non-perturbative renormalization group (NPRG) to work on (symmetric) network models of stochastic spiking neurons. By deriving a pair of Ward-Takahashi identities and making a ``local potential approximation,'' we are able to calculate non-universal quantities such as the effective firing rate nonlinearity of the network, allowing improved quantitative estimates of network statistics. We also derive the dimensionless flow equation that admits universal critical points in the renormalization group flow of the model, and identify two important types of critical points: in networks with an absorbing state there is Directed Percolation (DP) fixed point corresponding to a non-equilibrium phase transition between sustained activity and extinction of activity, and in spontaneously active networks there is a \emph{complex valued} critical point, corresponding to a spinodal transition observed, e.g., in the Lee-Yang ϕ3\phi^3 model of Ising magnets with explicitly broken symmetry. Our Ward-Takahashi identities imply trivial dynamical exponents z∗=2z_\ast = 2 in both cases, rendering it unclear whether these critical points fall into the known DP or Ising universality classes

    A Dynamical Graph Prior for Relational Inference

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    Relational inference aims to identify interactions between parts of a dynamical system from the observed dynamics. Current state-of-the-art methods fit a graph neural network (GNN) on a learnable graph to the dynamics. They use one-step message-passing GNNs -- intuitively the right choice since non-locality of multi-step or spectral GNNs may confuse direct and indirect interactions. But the \textit{effective} interaction graph depends on the sampling rate and it is rarely localized to direct neighbors, leading to local minima for the one-step model. In this work, we propose a \textit{dynamical graph prior} (DYGR) for relational inference. The reason we call it a prior is that, contrary to established practice, it constructively uses error amplification in high-degree non-local polynomial filters to generate good gradients for graph learning. To deal with non-uniqueness, DYGR simultaneously fits a ``shallow'' one-step model with shared graph topology. Experiments show that DYGR reconstructs graphs far more accurately than earlier methods, with remarkable robustness to under-sampling. Since appropriate sampling rates for unknown dynamical systems are not known a priori, this robustness makes DYGR suitable for real applications in scientific machine learning

    (Almost) Ruling Out SETH Lower Bounds for All-Pairs Max-Flow

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    The All-Pairs Max-Flow problem has gained significant popularity in the last two decades, and many results are known regarding its fine-grained complexity. Despite this, wide gaps remain in our understanding of the time complexity for several basic variants of the problem. In this paper, we aim to bridge this gap by providing algorithms, conditional lower bounds, and non-reducibility results. Our main result is that for most problem settings, deterministic reductions based on the Strong Exponential Time Hypothesis (SETH) cannot rule out n4−o(1)n^{4-o(1)} time algorithms under a hypothesis called NSETH. In particular, to obtain our result for the setting of undirected graphs with unit node-capacities, we design a new randomized O(m2+o(1))O(m^{2+o(1)}) time combinatorial algorithm, improving on the recent O(m11/5+o(1))O(m^{11/5+o(1)}) time algorithm [Huang et al., STOC 2023] and matching their m2−o(1)m^{2-o(1)} lower bound (up to subpolynomial factors), thus essentially settling the time complexity for this setting of the problem. More generally, our main technical contribution is the insight that stst-cuts can be verified quickly, and that in most settings, stst-flows can be shipped succinctly (i.e., with respect to the flow support). This is a key idea in our non-reducibility results, and it may be of independent interest

    Differences in well-being:the biological and environmental causes, related phenotypes, and real-time assessment

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    Well-being is a complex, and multifaceted construct that includes feeling good and functioning well. There is a growing global recognition of well-being as an important research topic and public policy goal. Well-being is related to less behavioral and emotional problems, and is associated with many positive aspects of daily life, including longevity, higher educational achievement, happier marriage, and more productivity at work. People differ in their levels of well-being, i.e., some people are in general happier or more satisfied with their lives than others. These individual differences in well-being can arise from many different factors, including biological (genetic) influences and environmental influences. To enhance the development of future mental health prevention and intervention strategies to increase well-being, more knowledge about these determinants and factors underlying well-being is needed. In this dissertation, I aimed to increase the understanding of the etiology in a series of studies using different methods, including systematic reviews, meta-analyses, twin designs, and molecular genetic designs. In part I, we brought together all published studies on the neural and physiological factors underlying well-being. This overview allowed us to critically investigate the claims made about the biology involved in well-being. The number of studies on the neural and physiological factors underlying well-being is increasing and the results point towards potential correlates of well-being. However, samples are often still small, and studies focus mostly on a single biomarker. Therefore, more well-powered, data-driven, and integrative studies across biological categories are needed to better understand the neural and physiological pathways that play a role in well-being. In part II, we investigated the overlap between well-being and a range of other phenotypes to learn more about the etiology of well-being. We report a large overlap with phenotypes including optimism, resilience, and depressive symptoms. Furthermore, when removing the genetic overlap between well-being and depressive symptoms, we showed that well-being has unique genetic associations with a range of phenotypes, independently from depressive symptoms. These results can be helpful in designing more effective interventions to increase well-being, taking into account the overlap and possible causality with other phenotypes. In part III, we used the extreme environmental change during the COVID-19 pandemic to investigate individual differences in the effects of such environmental changes on well-being. On average, we found a negative effect of the pandemic on different aspects of well-being, especially further into the pandemic. Whereas most previous studies only looked at this average negative effect of the pandemic on well-being, we focused on the individual differences as well. We reported large individual differences in the effects of the pandemic on well-being in both chapters. This indicates that one-size-fits-all preventions or interventions to maintain or increase well-being during the pandemic or lockdowns will not be successful for the whole population. Further research is needed for the identification of protective factors and resilience mechanisms to prevent further inequality during extreme environmental situations. In part IV, we looked at the real-time assessment of well-being, investigating the feasibility and results of previous studies. The real-time assessment of well-being, related variables, and the environment can lead to new insights about well-being, i.e., results that we cannot capture with traditional survey research. The real-time assessment of well-being is therefore a promising area for future research to unravel the dynamic nature of well-being fluctuations and the interaction with the environment in daily life. Integrating all results in this dissertation confirmed that well-being is a complex human trait that is influenced by many interrelated and interacting factors. Future directions to understand individual differences in well-being will be a data-driven approach to investigate the complex interplay of neural, physiological, genetic, and environmental factors in well-being

    Attribute network models, stochastic approximation, and network sampling and ranking algorithms

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    We analyze dynamic random network models where younger vertices connect to older ones with probabilities proportional to their degrees as well as a propensity kernel governed by their attribute types. Using stochastic approximation techniques we show that, in the large network limit, such networks converge in the local weak sense to randomly stopped multitype branching processes whose explicit description allows for the derivation of asymptotics for a wide class of network functionals. These asymptotics imply that while degree distribution tail exponents depend on the attribute type (already derived by Jordan (2013)), Page-rank centrality scores have the \emph{same} tail exponent across attributes. Moreover, the mean behavior of the limiting Page-rank score distribution can be explicitly described and shown to depend on the attribute type. The limit results also give explicit formulae for the performance of various network sampling mechanisms. One surprising consequence is the efficacy of Page-rank and walk based network sampling schemes for directed networks in the setting of rare minorities. The results also allow one to evaluate the impact of various proposed mechanisms to increase degree centrality of minority attributes in the network, and to quantify the bias in inferring about the network from an observed sample. Further, we formalize the notion of resolvability of such models where, owing to propagation of chaos type phenomenon in the evolution dynamics for such models, one can set up a correspondence to models driven by continuous time branching process dynamics.Comment: 48 page

    Reconfiguration of Digraph Homomorphisms

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    For a fixed graph H, the H-Recoloring problem asks whether, given two homomorphisms from a graph G to H, one homomorphism can be transformed into the other by changing the image of a single vertex in each step and maintaining a homomorphism to H throughout. The most general algorithmic result for H-Recoloring so far has been proposed by Wrochna in 2014, who introduced a topological approach to obtain a polynomial-time algorithm for any undirected loopless square-free graph H. We show that the topological approach can be used to recover essentially all previous algorithmic results for H-Recoloring and that it is applicable also in the more general setting of digraph homomorphisms. In particular, we show that H-Recoloring admits a polynomial-time algorithm i) if H is a loopless digraph that does not contain a 4-cycle of algebraic girth 0 and ii) if H is a reflexive digraph that contains no triangle of algebraic girth 1 and no 4-cycle of algebraic girth 0
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