Konstantin M. Zuev

Emergence of soft communities from geometric preferential attachment.

All real networks are different, but many have some structural properties in common. There seems to be no consensus on what the most common properties are, but scale-free degree distributions, strong clustering, and community structure are frequently mentioned without question. Surprisingly, there exists no simple generative mechanism explaining all the three properties at once in growing networks. Here we show how latent network geometry coupled with preferential attachment of nodes to this geometry fills this gap. We call this mechanism geometric preferential attachment (GPA), and validate it against the Internet. GPA gives rise to soft communities that provide a different perspective on the community structure in networks. The connections between GPA and cosmological models, including inflation, are also discussed.

Asymptotically Independent Markov Sampling: A New MCMC Scheme for Bayesian Inference

In Bayesian statistics, many problems can be expressed as the evaluation of the expectation of a quantity of interest with respect to the posterior distribution. Standard Monte Carlo method is often not applicable because the encountered posterior distributions cannot be sampled directly. In this case, the most popular strategies are the importance sampling method, Markov chain Monte Carlo, and annealing. In this paper, we introduce a new scheme for Bayesian inference, called Asymptotically Independent Markov Sampling (AIMS), which is based on the above methods. We derive important ergodic properties of AIMS. In particular, it is shown that, under certain conditions, the AIMS algorithm produces a uniformly ergodic Markov chain. The choice of the free parameters of the algorithm is discussed and recommendations are provided for this choice, both theoretically and heuristically based. The efficiency of AIMS is demonstrated with three numerical examples, which include both multi-modal and higher-dimensional target posterior distributions.

A formal Frobenius theorem and argument shift

A formal Frobenius theorem, which is an analog of the classical integrability theorem for smooth distributions, is proved and applied to generalize the argument shift method of A. S. Mishchenko and A. T. Fomenko to finite-dimensional Lie algebras over any field of characteristic zero. A completeness criterion for a commutative set of polynomials constructed by the formal argument shift method is obtained.

Exponential Random Simplicial Complexes

Exponential random graph models have attracted significant research attention over the past decades. These models are maximum-entropy ensembles subject to the constraints that the expected values of a set of graph observables are equal to given values. Here we extend these maximum-entropy ensembles to random simplicial complexes, which are more adequate and versatile constructions to model complex systems in many applications. We show that many random simplicial complex models considered in the literature can be casted as maximum-entropy ensembles under certain constraints. We introduce and analyze the most general random simplicial complex ensemble with statistically independent simplices. Our analysis is simplified by the observation that any distribution on any collection of objects including graphs and simplicial complexes, is maximum-entropy subject to the constraint that the expected value of is equal to the entropy of the distribution. With the help of this observation, we prove that ensemble is maximum-entropy subject to the two types of constraints which fix the expected numbers of simplices and their boundaries.

Hamiltonian Dynamics of Preferential Attachment

Prediction and control of network dynamics are grand-challenge problems in network science. The lack of understanding of fundamental laws driving the dynamics of networks is among the reasons why many practical problems of great significance remain unsolved for decades. Here we study the dynamics of networks evolving according to preferential attachment, known to approximate well the large-scale growth dynamics of a variety of real networks. We show that this dynamics is Hamiltonian, thus casting the study of complex networks dynamics to the powerful canonical formalism, in which the time evolution of a dynamical system is described by Hamiltons equations. We derive the explicit form of the Hamiltonian that governs network growth in preferential attachment. This Hamiltonian turns out to be nearly identical to graph energy in the configuration model, which shows that the ensemble of random graphs generated by preferential attachment is nearly identical to the ensemble of random graphs with scale-free degree distributions. In other words, preferential attachment generates nothing but random graphs with power-law degree distribution. The extension of the developed canonical formalism for network analysis to richer geometric network models with non-degenerate groups of symmetries may eventually lead to a system of equations describing network dynamics at small scales.

The maximum number of 3- and 4-cliques within a planar maximally filtered graph

Planar Maximally Filtered Graphs (PMFG) are an important tool for filtering the most relevant information from correlation based networks such as stock market networks. One of the main characteristics of a PMFG is the number of its 3- and 4-cliques. Recently in a few high impact papers it was stated that, based on heuristic evidence, the maximum number of 3- and 4-cliques that can exist in a PMFG with n vertices is 3n−8 and n−4 respectively. In this paper, we prove that this is indeed the case.

Gaussian process hyper-parameter estimation using Parallel Asymptotically Independent Markov Sampling

Gaussian process emulators of computationally expensive computer codes provide fast statistical approximations to model physical processes. The training of these surrogates depends on the set of design points chosen to run the simulator. Due to computational cost, such training set is bound to be limited and quantifying the resulting uncertainty in the hyper-parameters of the emulator by uni-modal distributions is likely to induce bias. In order to quantify this uncertainty, this paper proposes a computationally efficient sampler based on an extension of Asymptotically Independent Markov Sampling, a recently developed algorithm for Bayesian inference. Structural uncertainty of the emulator is obtained as a by-product of the Bayesian treatment of the hyper-parameters. Additionally, the user can choose to perform stochastic optimisation to sample from a neighbourhood of the Maximum a Posteriori estimate, even in the presence of multimodality. Model uncertainty is also acknowledged through numerical stabilisation measures by including a nugget term in the formulation of the probability model. The efficiency of the proposed sampler is illustrated in examples where multi-modal distributions are encountered. For the purpose of reproducibility, further development, and use in other applications the code used to generate the examples is freely available for download at https://github.com/agarbuno/paims_codes.

Navigability of Random Geometric Graphs in the Universe and Other Spacetimes

Random geometric graphs in hyperbolic spaces explain many common structural and dynamical properties of real networks, yet they fail to predict the correct values of the exponents of power-law degree distributions observed in real networks. In that respect, random geometric graphs in asymptotically de Sitter spacetimes, such as the Lorentzian spacetime of our accelerating universe, are more attractive as their predictions are more consistent with observations in real networks. Yet another important property of hyperbolic graphs is their navigability, and it remains unclear if de Sitter graphs are as navigable as hyperbolic ones. Here we study the navigability of random geometric graphs in three Lorentzian manifolds corresponding to universes filled only with dark energy (de Sitter spacetime), only with matter, and with a mixture of dark energy and matter as in our universe. We find that these graphs are navigable only in the manifolds with dark energy. This result implies that, in terms of navigability, random geometric graphs in asymptotically de Sitter spacetimes are as good as random hyperbolic graphs. It also establishes a connection between the presence of dark energy and navigability of the discretized causal structure of spacetime, which provides a basis for a different approach to the dark energy problem in cosmology. Introduction Random geometric graphs1–3 formalize the notion of “discretization” of a continuous geometric space or manifold. Nodes in these graphs are points, sprinkled randomly at constant sprinkling density, over the manifold, thus representing “atoms” of space, while links encode geometry—two nodes are connected if they happen to lie close in the space. These graphs are also a central object in algebraic topology since their clique complexes4 are Rips complexes5, 6 whose topology is known to converge to the manifold topology under very mild assumptions7. In network science and applied mathematics, random geometric graphs have attracted increasing attention over recent years8–41, since it was shown that if the space defining these graphs is not Euclidean but negatively curved, i.e., hyperbolic, then these graphs provide a geometric explanation of many common structural and dynamical properties of many real networks, including scale-free degree distributions, strong clustering, community structure, and network growth dynamics42–44. Yet more interestingly, these graphs also explain the optimality of many network functions related to finding paths in the network without global knowledge of the network structure45, 46. Random hyperbolic graphs appear to be optimal, that is, maximally efficient, with respect to the greedy path finding strategy that uses only spatial geometry to navigate through a complex network structure by moving at each step from a current node to its neighbor closest to the destination in the space37, 42. The efficiency of this process is called network navigability47. High navigability of random hyperbolic graphs led to practically viable applications, including the design of efficient routing in the future Internet48, 49, and demonstration that the spatiostructural organization of the human brain is nearly as needed for optimal information routing between different parts of the brain50. Yet if random hyperbolic graphs are truly geometric, meaning that if the sprinkling density is indeed constant with respect to the hyperbolic volume form, then the exponent γ of the distribution P(k)∼ k−γ of node degrees k in the resulting graphs is exactly γ = 342. In contrast, in random geometric graphs in de Sitter spacetime, which is asymptotically the spacetime of our accelerating universe, or indeed in the spacetime representing the exact large-scale Lorentzian geometry of our universe, this exponent asymptotically approaches γ = 251, as in many real networks52. Yet it remains unclear if these random Lorentzian graphs are as navigable as random hyperbolic graphs. In physics, random geometric graphs in Lorentzian spacetimes are known as causal sets, a central object in the causal set approach to quantum gravity53. A seemingly unrelated big, if not the biggest unsolved problem in cosmology is the dark energy puzzle54. What is dark energy? Why is its density orders of magnitude smaller than one would expect from high-energy physics? Causal sets provide one of the simplest explanation attempts to date55, but there are many other attempts56–62, none commonly considered to be the final answer. Here we study the navigability of random geometric graphs in three Lorentzian manifolds. One manifold is de Sitter 1 ar X iv :1 70 3. 09 05 7v 1 [ gr -q c] 1 6 M ar 2 01 7 spacetime, corresponding to a universe filled with dark energy only, and no matter. Another manifold is the other extreme, a universe filled only with dust matter, and no dark energy. The third manifold is a universe like ours, containing both matter and dark energy. This last manifold interpolates between the other two. At early times and small graph sizes, it is matter-dominated and “looks” like the dust-only spacetime. At later times and large graph sizes, it is dark-energy-dominated and “looks” increasingly more like de Sitter spacetime. We find that random geometric graphs only in manifolds with dark energy are navigable. Specifically, if there is no dark energy, that is, in the dust-only spacetime, there is a finite fraction of paths for which geometric path finding fails, and that this fraction is constant—it does not depend on the cutoff time, i.e., the present cosmological time in the universe, if the average degree in the graph is kept constant. In contrast, in spacetimes with dark energy, i.e., de Sitter spacetime and the spacetime of our universe, the fraction of unsuccessful paths quickly approaches zero as the cutoff time increases. For network science this finding implies that in terms of navigability, random geometric graphs in Lorentzian spacetimes with dark energy are as good as random hyperbolic graphs. For physics, this finding establishes a connection between the presence of dark energy and navigability of the discretized causal structure of spacetime, which provides a basis for a different approach to the dark energy problem. Results Lorentzian Manifolds While Riemannian manifolds are manifolds with positive-definite metric tensors gi j defining geodesic distances ds by ds2 = ∑i, j=1 gi j dxi dx j, where d is the manifold dimension, Lorentzian manifolds are manifolds whose metric tensors gμν , μ,ν = {0,1, . . . ,d}, have signature (−++ . . .+), meaning that if diagonalized by a proper choice of the coordinate system, these tensors have one negative entry on the diagonal, while all other entries are positive. In general relativity, Lorentzian manifolds represent relativistic spacetimes, which are solutions of Einstein’s equations. Typically, the dimension of a Lorentzian manifold is denoted by d+1, with the “+1” referring to the temporal (zeroth) dimension, while the other d dimensions are spatial. In this paper we consider only (3+1)-dimensional Lorentzian manifolds, that is, manifolds of dimension equal to the dimension of our universe63. The Lorentzian metric structure naturally defines spacetime’s causal structure: timelike intervals with ∆s2 < 0 connect pairs of causally related events, i.e., timelike-separated points on a manifold. Einstein’s equations are a set of ten coupled non-linear partial differential equations: Rμν − 1 2 Rgμν +Λgμν = 8πTμν , (1) where we use the natural units G = c = 1. The Ricci curvature tensor Rμν and Ricci scalar R measure the manifold curvature, the cosmological constant Λ is proportional to the dark energy density in the spacetime, and the stress-energy tensor Tμν represents the matter content. Spacetimes which are homogeneous and isotropic are called Friedmann-Robertson-Walker (FRW) spacetimes, which have a metric of the form ds2 =−dt2 +a(t)2dΣ2. The time-dependent function a(t) in front of the spatial metric dΣ is called the scale factor. This function characterizes the expansion of the volume form in a spatial hypersurface with respect to time; it alone tells whether there is a “Big Bang” at t = 0, i.e., whether a(0) = 0. The scale factor is derived explicitly as a solution to the 00-component (μ = ν = 0) of (1), known as the first Friedmann equation: ( ȧ a )2 = Λ 3 − K a 2 + c a3g . (2) The variable g represents the type of matter in the spacetime: in this work we use the values g = {0,1} to indicate no matter and dust matter, respectively. The spatial curvature of the spacetime is captured by K: K = {+1,0,−1} implies positive, zero, or negative spatial curvature, respectively. Motivated by the observation that our universe is nearly flat64, in this work we use K = 0, which significantly simplifies the calculations below. In the flat case, the spatial metric, hereafter using dimensionless spherical coordinates, becomes dΣ2 = dr2 + r2 dθ 2 + r2 sin2 θ dφ 2. Finally, c is a constant proportional to the density of matter in the universe. The total energy density in the universe is known to come from four sources: the matter (dark and baryonic) density ρM , the dark energy density ρΛ, the radiation energy density ρR, and the curvature K. The densities may be rescaled by a critical density: Ω≡ ρ/ρc, where ρc ≡ 3H2 0/8π; H0 ≡ ȧ0/a0 is the Hubble constant and a0 ≡ a(t0), i.e., the scale factor at the present time. Similarly, the curvature density parameter may be written as ΩK ≡ −K/(a0H0) so that we obtain the state equation ΩM +ΩΛ +ΩR +ΩK = 1. This allows us to rewrite (2) in the integral form65 H0t = ∫ a/a0 0 dx x √ ΩΛ +ΩKx−2 +ΩMx−3 +ΩRx−4 . (3)Random geometric graphs in hyperbolic spaces explain many common structural and dynamical properties of real networks, yet they fail to predict the correct values of the exponents of power-law degree distributions observed in real networks. In that respect, random geometric graphs in asymptotically de Sitter spacetimes, such as the Lorentzian spacetime of our accelerating universe, are more attractive as their predictions are more consistent with observations in real networks. Yet another important property of hyperbolic graphs is their navigability, and it remains unclear if de Sitter graphs are as navigable as hyperbolic ones. Here we study the navigability of random geometric graphs in three Lorentzian manifolds corresponding to universes filled only with dark energy (de Sitter spacetime), only with matter, and with a mixture of dark energy and matter. We find these graphs are navigable only in the manifolds with dark energy. This result implies that, in terms of navigability, random geometric graphs in asymptotically de Sitter spacetimes are as good as random hyperbolic graphs. It also establishes a connection between the presence of dark energy and navigability of the discretized causal structure of spacetime, which provides a basis for a different approach to the dark energy problem in cosmology.

Transitional annealed adaptive slice sampling for Gaussian process hyper-parameter estimation

Surrogate models have become ubiquitous in science and engineering for their capability of emulating expensive computer codes, necessary to model and investigate complex phenomena. Bayesian emulators based on Gaussian processes adequately quantify the uncertainty that results from the cost of the original simulator, and thus the inability to evaluate it on the whole input space. However, it is common in the literature that only a partial Bayesian analysis is carried out, whereby the underlying hyper-parameters are estimated via gradient-free optimisation or genetic algorithms, to name a few methods. On the other hand, maximum a posteriori (MAP) estimation could discard important regions of the hyper-parameter space. In this paper, we carry out a more complete Bayesian inference, that combines Slice Sampling with some recently developed Sequential Monte Carlo samplers. The resulting algorithm improves the mixing in the sampling through delayed-rejection, the inclusion of an annealing scheme akin to Asymptotically Independent Markov Sampling and parallelisation via Transitional Markov Chain Monte Carlo. Examples related to the estimation of Gaussian process hyper-parameters are presented. For the purpose of reproducibility, further development, and use in other applications, the code to generate the examples in this paper is freely available for download at this http URL

Investors' behavior on s&p 500 index during periods of market crashes: A visibility graph approach

Investors’ behavior in the market is highly related to the properties that financial time series capture. Particularly, nowadays the availability of high frequency datasets provides a reliable source for the better understanding of investors’ psychology. The main aim of this chapter is to identify changes in the persistency as well as in the local degree of irreversibility of S&P 500 price-index time series. Thus, by considering the US stock market from 1996 to 2010, we investigate how the Dot.com as well as the Subprime crashes affected investors’ behavior. Our results provide evidences that Efficient Market Hypothesis does not hold as the high frequency S&P 500 data can be better modeled by using a fractional Brownian motion. In addition, we report that both crises only temporary effect investors’ behavior, and interestingly, before the occurrence of these two major events, the index series exhibited a kind of “nervousness” on behalf of the investors.