arXiv subjects

Sungmin Hwang

Publications and source records attributed to Sungmin Hwang.

13 recordsLinked to original sources

arXiv2023

Heterogeneous Mean First-Passage Time Scaling in Fractal Media

The mean first passage time~(MFPT) of random walks is a key quantity characterizing dynamic processes on disordered media. In a random fractal embedded in the Euclidean space, the MFPT is known to obey the power law scaling with the distance between a source and a target site with a universal exponent. We find that the scaling law for the MFPT is not determined solely by the distance between a source and a target but also by their locations. The role of a site in the first passage processes is quantified by the random walk centrality. It turns out that the site of highest random walk centrality, dubbed as a hub, intervenes in first passage processes. We show that the MFPT from a departure site to a target site is determined by a competition between direct paths and indirect paths detouring via the hub. Consequently, the MFPT displays a crossover scaling between a short distance regime, where direct paths are dominant, and a long distance regime, where indirect paths are dominant. The two regimes are characterized by power laws with different scaling exponents. The crossover scaling behavior is confirmed by extensive numerical calculations of the MFPTs on the critical percolation cluster in two dimensional square lattices.

cond-mat.stat-mech↗

arXiv2020

Distribution of the number of fitness maxima in Fisher's Geometric Model

Fisher's geometric model describes biological fitness landscapes by combining a linear map from the discrete space of genotypes to an $n$-dimensional Euclidean phenotype space with a nonlinear, single-peaked phenotype-fitness map. Genotypes are represented by binary sequences of length $L$, and the phenotypic effects of mutations at different sites are represented by $L$ random vectors drawn from an isotropic Gaussian distribution. Recent work has shown that the interplay between the genotypic and phenotypic levels gives rise to a range of different landscape topographies that can be characterised by the number of local fitness maxima. Extending our previous study of the mean number of local maxima, here we focus on the distribution of the number of maxima when the limit $L \to \infty$ is taken at finite $n$. We identify the typical scale of the number of maxima for general $n$, and determine the full scaled probability density and two point correlation function of maxima for the one-dimensional case. We also elaborate on the close relation of the model to the anti-ferromagnetic Hopfield model with $n$ random continuous pattern vectors, and show that many of our results carry over to this setting. More generally, we expect that our analysis can help to elucidate the fluctuation structure of metastable states in various spin glass problems.

q-bio.PE↗

arXiv2020

Force balance controls the relaxation time of the gradient descent algorithm in the satisfiable phase

We numerically study the relaxation dynamics of the single layer perceptron with the spherical constraint. This is the simplest model of neural networks and serves a prototypical mean-field model of both convex and non-convex optimization problems. The relaxation time of the gradient descent algorithm rapidly increases near the SAT-UNSAT transition point. We numerically confirm that the first non-zero eigenvalue of the Hessian controls the relaxation time. This first eigenvalue vanishes much faster upon approaching the SAT-UNSAT transition point than the prediction of Marchenko-Pastur law in random matrix theory derived under the assumption that the set of unsatisfied constraints are uncorrelated. This leads to a non-trivial critical exponent of the relaxation time in the SAT phase. Using a simple scaling analysis, we show that the isolation of this first eigenvalue from the bulk of spectrum is attributed to the force balance at the SAT-UNSAT transition point. Finally, we show that the estimated critical exponent of the relaxation time in the non-convex region agrees very well with that of frictionless spherical particles, which have been studied in the context of the jamming transition of granular materials.

cond-mat.dis-nn↗

arXiv2020

On the number of limit cycles in diluted neural networks

We consider the storage properties of temporal patterns, i.e. cycles of finite lengths, in neural networks represented by (generally asymmetric) spin glasses defined on random graphs. Inspired by the observation that dynamics on sparse systems have more basins of attractions than the dynamics of densely connected ones, we consider the attractors of a greedy dynamics in sparse topologies, considered as proxy for the stored memories. We enumerate them using numerical simulation and extend the analysis to large systems sizes using belief propagation. We find that the logarithm of the number of such cycles is a non monotonic function of the mean connectivity and we discuss the similarities with biological neural networks describing the memory capacity of the hippocampus.

cond-mat.dis-nn↗

arXiv2019

Inverse problems for structured datasets using parallel TAP equations and RBM

We propose an efficient algorithm to solve inverse problems in the presence of binary clustered datasets. We consider the paradigmatic Hopfield model in a teacher student scenario, where this situation is found in the retrieval phase. This problem has been widely analyzed through various methods such as mean-field approaches or the pseudo-likelihood optimization. Our approach is based on the estimation of the posterior using the Thouless-Anderson-Palmer (TAP) equations in a parallel updating scheme. At the difference with other methods, it allows to retrieve the exact patterns of the teacher and the parallel update makes it possible to apply it for large system sizes. We also observe that the Approximate Message Passing (AMP) equations do not reproduce the expected behavior in the direct problem, questioning the standard practice used to obtain time indexes coming from Belief Propagation (BP). We tackle the same problem using a Restricted Boltzmann Machine (RBM) and discuss the analogies between the two algorithms.

cond-mat.dis-nn↗

arXiv2019

Jamming in multilayer supervised learning models

Critical jamming transitions are characterized by an astonishing degree of universality. Analytic and numerical evidence points to the existence of a large universality class that encompasses finite and infinite dimensional spheres and continuous constraint satisfaction problems (CCSP) such as the non-convex perceptron and related models. In this paper we investigate multilayer neural networks (MLNN) learning random associations as models for CCSP which could potentially define different jamming universality classes. As opposed to simple perceptrons and infinite dimensional spheres, which are described by a single effective field in terms of which the constraints appear to be one-dimensional, the description of MLNN, involves multiple fields, and the constraints acquire a multidimensional character. We first study the models numerically and show that similarly to the perceptron, whenever jamming is isostatic, the sphere universality class is recovered, we then write the exact mean-field equations for the models and identify a dimensional reduction mechanism that leads to a scaling regime identical to one of the infinite dimensional spheres. We suggest that this mechanism could be general enough to explain finite dimensional universality.

cond-mat.dis-nn↗

arXiv2019

On the number of limit cycles in asymmetric neural networks

The comprehension of the mechanisms at the basis of the functioning of complexly interconnected networks represents one of the main goals of neuroscience. In this work, we investigate how the structure of recurrent connectivity influences the ability of a network to have storable patterns and in particular limit cycles, by modeling a recurrent neural network with McCulloch-Pitts neurons as a content-addressable memory system. A key role in such models is played by the connectivity matrix, which, for neural networks, corresponds to a schematic representation of the "connectome": the set of chemical synapses and electrical junctions among neurons. The shape of the recurrent connectivity matrix plays a crucial role in the process of storing memories. This relation has already been exposed by the work of Tanaka and Edwards, which presents a theoretical approach to evaluate the mean number of fixed points in a fully connected model at thermodynamic limit. Interestingly, further studies on the same kind of model but with a finite number of nodes have shown how the symmetry parameter influences the types of attractors featured in the system. Our study extends the work of Tanaka and Edwards by providing a theoretical evaluation of the mean number of attractors of any given length $L$ for different degrees of symmetry in the connectivity matrices.

cond-mat.dis-nn↗

arXiv2018

Poincaré invariance in NRQCD and pNRQCD revisited

We investigate how fields transform under the Poincaré group in nonrelativistic effective field theories of QCD. In constructing these transformations, we rely only on symmetries and field redefinitions to limit the number of allowed terms. By requiring invariance of the action under these transformations, nontrivial relations between Wilson coefficients for both nonrelativistic QCD and potential nonrelativistic QCD are derived. We show explicitly how the Poincaré algebra is satisfied, and how this gives complementary information on the Wilson coefficients. We also briefly discuss the implications of our results, as well as the possibility of applying this method to other types of effective field theories.

hep-th↗

arXiv2018

Universality classes of interaction structures for NK fitness landscapes

Kauffman's NK-model is a paradigmatic example of a class of stochastic models of genotypic fitness landscapes that aim to capture generic features of epistatic interactions in multilocus systems. Genotypes are represented as sequences of $L$ binary loci. The fitness assigned to a genotype is a sum of contributions, each of which is a random function defined on a subset of $k \le L$ loci. These subsets or neighborhoods determine the genetic interactions of the model. Whereas earlier work on the NK model suggested that most of its properties are robust with regard to the choice of neighborhoods, recent work has revealed an important and sometimes counter-intuitive influence of the interaction structure on the properties of NK fitness landscapes. Here we review these developments and present new results concerning the number of local fitness maxima and the statistics of selectively accessible (that is, fitness-monotonic) mutational pathways. In particular, we develop a unified framework for computing the exponential growth rate of the expected number of local fitness maxima as a function of $L$, and identify two different universality classes of interaction structures that display different asymptotics of this quantity for large $k$. Moreover, we show that the probability that the fitness landscape can be traversed along an accessible path decreases exponentially in $L$ for a large class of interaction structures that we characterize as locally bounded. Finally, we discuss the impact of the NK interaction structures on the dynamics of evolution using adaptive walk models.

q-bio.PE↗

arXiv2017

Genotypic complexity of Fisher's geometric model

Fisher's geometric model was originally introduced to argue that complex adaptations must occur in small steps because of pleiotropic constraints. When supplemented with the assumption of additivity of mutational effects on phenotypic traits, it provides a simple mechanism for the emergence of genotypic epistasis from the nonlinear mapping of phenotypes to fitness. Of particular interest is the occurrence of reciprocal sign epistasis, which is a necessary condition for multipeaked genotypic fitness landscapes. Here we compute the probability that a pair of randomly chosen mutations interacts sign epistatically, which is found to decrease with increasing phenotypic dimension $n$, and varies nonmonotonically with the distance from the phenotypic optimum. We then derive expressions for the mean number of fitness maxima in genotypic landscapes comprised of all combinations of $L$ random mutations. This number increases exponentially with $L$, and the corresponding growth rate is used as a measure of the complexity of the landscape. The dependence of the complexity on the model parameters is found to be surprisingly rich, and three distinct phases characterized by different landscape structures are identified. Our analysis shows that the phenotypic dimension, which is often referred to as phenotypic complexity, does not generally correlate with the complexity of fitness landscapes and that even organisms with a single phenotypic trait can have complex landscapes. Our results further inform the interpretation of experiments where the parameters of Fisher's model have been inferred from data, and help to elucidate which features of empirical fitness landscapes can be described by this model.

q-bio.PE↗

arXiv2017

Mutation supply and the repeatability of selection for antibiotic resistance

Whether evolution can be predicted is a key question in evolutionary biology. Here we set out to better understand the repeatability of evolution. We explored experimentally the effect of mutation supply and the strength of selective pressure on the repeatability of selection from standing genetic variation. Different sizes of mutant libraries of an antibiotic resistance gene, TEM-1 $β$-lactamase in Escherichia coli, were subjected to different antibiotic concentrations. We determined whether populations went extinct or survived, and sequenced the TEM gene of the surviving populations. The distribution of mutations per allele in our mutant libraries- generated by error-prone PCR- followed a Poisson distribution. Extinction patterns could be explained by a simple stochastic model that assumed the sampling of beneficial mutations was key for survival. In most surviving populations, alleles containing at least one known large-effect beneficial mutation were present. These genotype data also support a model which only invokes sampling effects to describe the occurrence of alleles containing large-effect driver mutations. Hence, evolution is largely predictable given cursory knowledge of mutational fitness effects, the mutation rate and population size. There were no clear trends in the repeatability of selected mutants when we considered all mutations present. However, when only known large-effect mutations were considered, the outcome of selection is less repeatable for large libraries, in contrast to expectations. Furthermore, we show experimentally that alleles carrying multiple mutations selected from large libraries confer higher resistance levels relative to alleles with only a known large-effect mutation, suggesting that the scarcity of high-resistance alleles carrying multiple mutations may contribute to the decrease in repeatability at large library sizes.

q-bio.PE↗

arXiv2015

Vacuum Persistence in Fierz-Pauli Theory on a Curved Background

By explicitly constructing the Hilbert space, Higuchi showed that there is a lower bound on the mass of a minimally-coupled free spin-2 field on a curved background \cite{HiguchiBound}. Using the vacuum persistence amplitude, we show that this bound is modified by taking into account additional terms not prohibited by symmetry in the case of a maximally symmetric spacetime. This result can further be generalized to the maximally symmetric space case, such as the FRW universe, and its corresponding bound of the deformation parameter is discussed.

hep-th↗

arXiv2014

Ground state energy of $q$-state Potts model: the minimum modularity

A wide range of interacting systems can be described by complex networks. A common feature of such networks is that they consist of several communities or modules, the degree of which may quantified as the \emph{modularity}. However, even a random uncorrelated network, which has no obvious modular structure, has a finite modularity due to the quenched disorder. For this reason, the modularity of a given network is meaningful only when it is compared with that of a randomized network with the same degree distribution. In this context, it is important to calculate the modularity of a random uncorrelated network with an arbitrary degree distribution. The modularity of a random network has been calculated [Phys. Rev. E \textbf{76}, 015102 (2007)]; however, this was limited to the case whereby the network was assumed to have only two communities, and it is evident that the modularity should be calculated in general with $q(\geq 2)$ communities. Here, we calculate the modularity for $q$ communities by evaluating the ground state energy of the $q$-state Potts Hamiltonian, based on replica symmetric solutions assuming that the mean degree is large. We found that the modularity is proportional to $\langle \sqrt{k} \rangle / \langle k \rangle$ regardless of $q$ and that only the coefficient depends on $q$. In particular, when the degree distribution follows a power law, the modularity is proportional to $\langle k \rangle^{-1/2}$. Our analytical results are confirmed by comparison with numerical simulations. Therefore, our results can be used as reference values for real-world networks.

cond-mat.stat-mech↗