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We analyze an agent-based model to estimate how the costs and benefits of users in an online social network (OSN) impact the robustness of the OSN. Benefits are measured in terms of relative reputation that users receives from their followers. They can be increased by direct and indirect reciprocity in following each other, which leads to a core-periphery structure of the OSN. Costs relate to the effort to login, to maintain the profile, etc. and are assumed as constant for all users. The robustness of the OSN depends on the entry and exit of users over time. Intuitively, one would expect that higher costs lead to more users leaving and hence to a less robust OSN. We demonstrate that an optimal cost level exists, which maximizes both the performance of the OSN, measured by means of the long-term average benefit of its users, and the robustness of the OSN, measured by means of the life-time of the core of the OSN. Our mathematical and computational analyses unfold how changes in the cost level impact reciprocity and subsequently the core-periphery structure of the OSN, to explain the optimal cost level.

We provide an agent-based model to explain the emergence of collective opinions not based on feedback between different opinions, but based on emotional interactions between agents. The driving variable is the emotional state of agents, characterized by their valence and their arousal. Both determine their emotional expression, from which collective emotional information is generated. This information feeds back on the dynamics of emotional states and of individual opinions in a non-linear manner. We derive the critical conditions for emotional interactions to obtain either consensus or polarization of opinions. Stochastic agent-based simulations and formal analyses of the model explain our results. Possible ways to validate the model are discussed.

The mixing time of Markovian dissipative evolutions of open quantum many-body systems can be bounded using optimal constants of certain quantum functional inequalities, such as the modified logarithmic Sobolev constant. For classical spin systems, the positivity of such constants follows from a mixing condition for the Gibbs measure, via quasi-factorization results for the entropy. Inspired by the classical case, we present a strategy to derive the positivity of the modified logarithmic Sobolev constant associated to the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian. In particular we show that for the heat-bath dynamics for 1D systems, the modified logarithmic Sobolev constant is positive under the assumptions of a mixing condition on the Gibbs state and a strong quasi-factorization of the relative entropy., Comment: 28 pages, 4 figures. Included some additional comments and updated results in light of recent advances in the literature

Coupling among neural rhythms is one of the most important mechanisms at the basis of cognitive processes in the brain. In this study we consider a neural mass model, rigorously obtained from the microscopic dynamics of an inhibitory spiking network with exponential synapses, able to autonomously generate collective oscillations (COs). These oscillations emerge via a super-critical Hopf bifurcation, and their frequencies are controlled by the synaptic time scale, the synaptic coupling and the excitability of the neural population. Furthermore, we show that two inhibitory populations in a master-slave configuration with different synaptic time scales can display various collective dynamical regimes: namely, damped oscillations towards a stable focus, periodic and quasi-periodic oscillations, and chaos. Finally, when bidirectionally coupled the two inhibitory populations can exhibit different types of theta-gamma cross-frequency couplings (CFCs): namely, phase-phase and phase-amplitude CFC. The coupling between theta and gamma COs is enhanced in presence of a external theta forcing, reminiscent of the type of modulation induced in Hippocampal and Cortex circuits via optogenetic drive., Comment: 14 pages, 10 figures

Tensor network states are used extensively as a mathematically convenient description of physically relevant states of many-body quantum systems. Those built on regular lattices, i.e. matrix product states (MPS) in dimension 1 and projected entangled pair states (PEPS) in dimension 2 or higher, are of particular interest in condensed matter physics. The general goal of this work is to characterize which features of MPS and PEPS are generic and which are, on the contrary, exceptional. This problem can be rephrased as follows: given an MPS or PEPS sampled at random, what are the features that it displays with either high or low probability? One property which we are particularly interested in is that of having either rapidly decaying or long-range correlations. In a nutshell, our main result is that translation-invariant MPS and PEPS typically exhibit exponential decay of correlations at a high rate. We have two distinct ways of getting to this conclusion, depending on the dimensional regime under consideration. Both yield intermediate results which are of independent interest, namely: the parent Hamiltonian and the transfer operator of such MPS and PEPS typically have a large spectral gap. In all these statements, our aim is to get a quantitative estimate of the considered quantity (generic correlation length or spectral gap), which has the best possible dependency on the physical and bond dimensions of the random MPS or PEPS., Comment: 53 pages, 7 figures. v2: a bug in Section 4 has been fixed and a few changes have been made to the presentation. v3: published version

Projected Entangled Pair States (PEPS) are used in practice as an efficient parametrization of the set of ground states of quantum many body systems. The aim of this paper is to present, for a broad mathematical audience, some mathematical questions about PEPS., Comment: Notes associated to the Santal\'o Lecture 2017, Universidad Complutense de Madrid (UCM), minor typos corrected

We study a network of spiking neurons with heterogeneous excitabilities connected via inhibitory delayed pulses. For globally coupled systems the increase of the inhibitory coupling reduces the number of firing neurons by following a Winner Takes All mechanism. For sufficiently large transmission delay we observe the emergence of collective oscillations in the system beyond a critical coupling value. Heterogeneity promotes neural inactivation and asynchronous dynamics and its effect can be counteracted by considering longer time delays. In sparse networks, inhibition has the counterintuitive effect of promoting neural reactivation of silent neurons for sufficiently large coupling. In this regime, current fluctuations are on one side responsible for neural firing of sub-threshold neurons and on the other side for their desynchronization. Therefore, collective oscillations are present only in a limited range of coupling values, which remains finite in the thermodynamic limit. Out of this range the dynamics is asynchronous and for very large inhibition neurons display a bursting behaviour alternating periods of silence with periods where they fire freely in absence of any inhibition., Comment: 13 pages, 13 Figures, submitted to Physical review E

We study matrix integration over the classical Lie groups $U(N),Sp(2N),O(2N)$ and $O(2N+1)$, using symmetric function theory and the equivalent formulation in terms of determinants and minors of Toeplitz$\pm$Hankel matrices. We establish a number of factorizations and expansions for such integrals, also with insertions of irreducible characters. As a specific example, we compute both at finite and large $N$ the partition functions, Wilson loops and Hopf links of Chern-Simons theory on $S^{3}$ with the aforementioned symmetry groups. The identities found for the general models translate in this context to relations between observables of the theory. Finally, we use character expansions to evaluate averages in random matrix ensembles of Chern-Simons type, describing the spectra of solvable fermionic models with matrix degrees of freedom., Comment: 32 pages, v2: Several improvements, including a Conclusions and Outlook section, added. 36 pages

The proposed Circular statistics-based Inter-Microphone Phase difference estimation Localizer (CIMPL) method is tailored toward binaural hearing aid systems with microphone arrays in each unit. The method utilizes the circular statistics (circular mean and circular variance) of inter-microphone phase difference (IPD) across different microphone pairs. These IPDs are firstly mapped to time delays through a variance-weighted linear fit, then mapped to azimuth direction-of-arrival (DoA) and lastly information of different microphone pairs is combined. The variance is carried through the different transformations and acts as a reliability index of the estimated angle. Both the resulting angle and variance are fed into a wrapped Kalman filter, which provides a smoothed estimate of the DoA. The proposed method improves the accuracy of the tracked angle of a single moving source compared with the benchmark method provided by the LOCATA challenge, and it runs approximately 75 times faster., Comment: In Proceedings of the LOCATA Challenge Workshop - a satellite event of IWAENC 2018 (arXiv:1811.08482 )

Gliomas are the most common primary brain malignancies, with different degrees of aggressiveness, variable prognosis and various heterogeneous histologic sub-regions, i.e., peritumoral edematous/invaded tissue, necrotic core, active and non-enhancing core. This intrinsic heterogeneity is also portrayed in their radio-phenotype, as their sub-regions are depicted by varying intensity profiles disseminated across multi-parametric magnetic resonance imaging (mpMRI) scans, reflecting varying biological properties. Their heterogeneous shape, extent, and location are some of the factors that make these tumors difficult to resect, and in some cases inoperable. The amount of resected tumor is a factor also considered in longitudinal scans, when evaluating the apparent tumor for potential diagnosis of progression. Furthermore, there is mounting evidence that accurate segmentation of the various tumor sub-regions can offer the basis for quantitative image analysis towards prediction of patient overall survival. This study assesses the state-of-the-art machine learning (ML) methods used for brain tumor image analysis in mpMRI scans, during the last seven instances of the International Brain Tumor Segmentation (BraTS) challenge, i.e., 2012-2018. Specifically, we focus on i) evaluating segmentations of the various glioma sub-regions in pre-operative mpMRI scans, ii) assessing potential tumor progression by virtue of longitudinal growth of tumor sub-regions, beyond use of the RECIST/RANO criteria, and iii) predicting the overall survival from pre-operative mpMRI scans of patients that underwent gross total resection. Finally, we investigate the challenge of identifying the best ML algorithms for each of these tasks, considering that apart from being diverse on each instance of the challenge, the multi-institutional mpMRI BraTS dataset has also been a continuously evolving/growing dataset., Comment: The International Multimodal Brain Tumor Segmentation (BraTS) Challenge

We propose the following definition of topological quantum phases valid for mixed states: two states are in the same phase if there exists a time independent, fast and local Lindbladian evolution driving one state into the other. The underlying idea, motivated by K\"onig and Pastawski in 2013, is that it takes time to create new topological correlations, even with the use of dissipation. We show that it is a good definition in the following sense: (1) It divides the set of states into equivalent classes and it establishes a partial order between those according to their level of "topological complexity". (2) It provides a path between any two states belonging to the same phase where observables behave smoothly. We then focus on pure states to relate the new definition in this particular case with the usual definition for quantum phases of closed systems in terms of the existence of a gapped path of Hamiltonians connecting both states in the corresponding ground state path. We show first that if two pure states are in the same phase in the Hamiltonian sense, they are also in the same phase in the Lindbladian sense considered here. We then turn to analyse the reverse implication, where we point out a very different behaviour in the case of symmetry protected topological (SPT) phases in 1D. Whereas at the Hamiltonian level, phases are known to be classified with the second cohomology group of the symmetry group, we show that symmetry cannot give any protection in 1D in the Lindbladian sense: there is only one SPT phase in 1D independently of the symmetry group. We finish analysing the case of 2D topological quantum systems. There we expect that different topological phases in the Hamiltonian sense remain different in the Lindbladian sense. We show this formally only for the $\mathbb{Z}_n$ quantum double models., Comment: 46 pages, v3: published version

The spectral gap problem - determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a continuous range of low-energy excitations - pervades quantum many-body physics. Recently, this important problem was shown to be undecidable for quantum spin systems in two (or more) spatial dimensions: there exists no algorithm that determines in general whether a system is gapped or gapless, a result which has many unexpected consequences for the physics of such systems. However, there are many indications that one dimensional spin systems are simpler than their higher-dimensional counterparts: for example, they cannot have thermal phase transitions or topological order, and there exist highly-effective numerical algorithms such as DMRG - and even provably polynomial-time ones - for gapped 1D systems, exploiting the fact that such systems obey an entropy area-law. Furthermore, the spectral gap undecidability construction crucially relied on aperiodic tilings, which are not possible in 1D. So does the spectral gap problem become decidable in 1D? In this paper we prove this is not the case, by constructing a family of 1D spin chains with translationally-invariant nearest neighbour interactions for which no algorithm can determine the presence of a spectral gap. This not only proves that the spectral gap of 1D systems is just as intractable as in higher dimensions, but also predicts the existence of qualitatively new types of complex physics in 1D spin chains. In particular, it implies there are 1D systems with constant spectral gap and non-degenerate classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a gapless behaviour with dense spectrum., Comment: 7 figures