arXiv subjects

Jason W. Rocks

Publications and source records attributed to Jason W. Rocks.

17 recordsLinked to original sources

arXiv2024

A universal niche geometry governs the response of ecosystems to environmental perturbations

How ecosystems respond to environmental perturbations is a fundamental question in ecology, made especially challenging due to the strong coupling between species and their environment. Here, we introduce a theoretical framework for calculating the steady-state response of ecosystems to environmental perturbations in generalized consumer-resource. Our construction is applicable to a wide class of systems, including models with non-reciprocal interactions, cross-feeding, and non-linear growth/consumption rates. Within our framework, all ecological variables are embedded into four distinct vector spaces and ecological interactions are represented by geometric transformations between these spaces. We show that near a steady state, such geometric transformations directly map environmental perturbations - in resource availability and mortality rates - to shifts in niche structure. We illustrate these ideas in a variety of settings including a minimal model for pH-induced toxicity in bacterial denitrification. We end by discussing the biological implications of our framework. In particular, we show that it is extremely difficult to distinguish cooperative and competitive interactions by measuring species' responses to external perturbations.

q-bio.PE↗

arXiv2024

Topological characterization of the continuum of allosteric response

Allosteric regulation in proteins is often accompanied by conformational changes that facilitate transmission of mechanical signals between distant ligand binding sites. Typically, these deformations are classified in terms of specific archetypes, including various types of hinge mechanisms or allosteric pathways localized to sequences of amino acids.However, many allosteric deformations resist such strict categorization. Here, we introduce a quantitative topological description of allosteric deformation, unifying all archetypal mechanisms into a single framework. The topological description aligns with two key structural features often associated with allosteric deformations, namely hinge domains and allosteric pathways, enabling us to quantify the significance of each of these features. To develop the analysis, we tune computer-generated mechanical networks to perform allostery-like functions, obtaining an ensemble of networks that establish a range of possible allosteric deformations. The analysis shows that these networks' allosteric mechanisms cannot be described in terms of discrete archetypes - they fall on a continuum. We then apply the same analysis to a collection of allosteric proteins with similar results, showing that our framework encompasses these proteins as well as designed allosteric networks. Our results provide a new picture for allostery, demonstrating not only how it can be described quantitatively, but also giving insight into how it emerges as a collective property.

cond-mat.soft↗

arXiv2023

Phase transition to chaos in complex ecosystems with non-reciprocal species-resource interactions

Non-reciprocal interactions between microscopic constituents can profoundly shape the large-scale properties of complex systems. Here, we investigate the effects of non-reciprocity in the context of theoretical ecology by analyzing a generalization of MacArthur's consumer-resource model with asymmetric interactions between species and resources. Using a mixture of analytic cavity calculations and numerical simulations, we show that such ecosystems generically undergo a phase transition to chaotic dynamics as the amount of non-reciprocity is increased. We analytically construct the phase diagram for this model and show that the emergence of chaos is controlled by a single quantity: the ratio of surviving species to surviving resources. We also numerically calculate the Lyapunov exponents in the chaotic phase and carefully analyze finite-size effects. Our findings show how non-reciprocal interactions can give rise to complex and unpredictable dynamical behaviors even in the simplest ecological consumer-resource models.

physics.bio-ph↗

arXiv2023

Double Descent Demystified: Identifying, Interpreting & Ablating the Sources of a Deep Learning Puzzle

Double descent is a surprising phenomenon in machine learning, in which as the number of model parameters grows relative to the number of data, test error drops as models grow ever larger into the highly overparameterized (data undersampled) regime. This drop in test error flies against classical learning theory on overfitting and has arguably underpinned the success of large models in machine learning. This non-monotonic behavior of test loss depends on the number of data, the dimensionality of the data and the number of model parameters. Here, we briefly describe double descent, then provide an explanation of why double descent occurs in an informal and approachable manner, requiring only familiarity with linear algebra and introductory probability. We provide visual intuition using polynomial regression, then mathematically analyze double descent with ordinary linear regression and identify three interpretable factors that, when simultaneously all present, together create double descent. We demonstrate that double descent occurs on real data when using ordinary linear regression, then demonstrate that double descent does not occur when any of the three factors are ablated. We use this understanding to shed light on recent observations in nonlinear models concerning superposition and double descent. Code is publicly available.

cs.LG↗

arXiv2023

Emergent competition shapes the ecological properties of multi-trophic ecosystems

Ecosystems are commonly organized into trophic levels -- organisms that occupy the same level in a food chain (e.g., plants, herbivores, carnivores). A fundamental question in theoretical ecology is how the interplay between trophic structure, diversity, and competition shapes the properties of ecosystems. To address this problem, we analyze a generalized Consumer Resource Model with three trophic levels using the zero-temperature cavity method and numerical simulations. We find that intra-trophic diversity gives rise to ``emergent competition'' between species within a trophic level due to feedbacks mediated by other trophic levels. This emergent competition gives rise to a crossover from a regime of top-down control (populations are limited by predators) to a regime of bottom-up control (populations are limited by primary producers) and is captured by a simple order parameter related to the ratio of surviving species in different trophic levels. We show that our theoretical results agree with empirical observations, suggesting that the theoretical approach outlined here can be used to understand complex ecosystems with multiple trophic levels.

q-bio.PE↗

arXiv2022

Integrating local energetics into Maxwell-Calladine constraint counting to design mechanical metamaterials

The Maxwell-Calladine index theorem plays a central role in our current understanding of the mechanical rigidity of discrete materials. By considering the geometric constraints each material component imposes on a set of underlying degrees of freedom, the theorem relates the emergence of rigidity to constraint counting arguments. However, the Maxwell-Calladine paradigm is significantly limited -- its exclusive reliance on the geometric relationships between constraints and degrees of freedom completely neglects the actual energetic costs of deforming individual components. To address this limitation, we derive a generalization of the Maxwell-Calladine index theorem based on susceptibilities that naturally incorporate local energetic properties such as stiffness and prestress. Using this extended framework, we investigate how local energetics modify the classical constraint counting picture to capture the relationship between deformations and external forces. We then combine this formalism with group representation theory to design mechanical metamaterials where differences in symmetry between local energy costs and structural geometry are exploited to control responses to external forces.

cond-mat.soft↗

arXiv2022

Bias-variance decomposition of overparameterized regression with random linear features

In classical statistics, the bias-variance trade-off describes how varying a model's complexity (e.g., number of fit parameters) affects its ability to make accurate predictions. According to this trade-off, optimal performance is achieved when a model is expressive enough to capture trends in the data, yet not so complex that it overfits idiosyncratic features of the training data. Recently, it has become clear that this classic understanding of the bias-variance must be fundamentally revisited in light of the incredible predictive performance of "overparameterized models" -- models that avoid overfitting even when the number of fit parameters is large enough to perfectly fit the training data. Here, we present results for one of the simplest examples of an overparameterized model: regression with random linear features (i.e. a two-layer neural network with a linear activation function). Using the zero-temperature cavity method, we derive analytic expressions for the training error, test error, bias, and variance. We show that the linear random features model exhibits three phase transitions: two different transitions to an interpolation regime where the training error is zero, along with an additional transition between regimes with large bias and minimal bias. Using random matrix theory, we show how each transition arises due to small nonzero eigenvalues in the Hessian matrix. Finally, we compare and contrast the phase diagram of the random linear features model to the random nonlinear features model and ordinary regression, highlighting the new phase transitions that result from the use of linear basis functions.

stat.ML↗

arXiv2022

Memorizing without overfitting: Bias, variance, and interpolation in over-parameterized models

The bias-variance trade-off is a central concept in supervised learning. In classical statistics, increasing the complexity of a model (e.g., number of parameters) reduces bias but also increases variance. Until recently, it was commonly believed that optimal performance is achieved at intermediate model complexities which strike a balance between bias and variance. Modern Deep Learning methods flout this dogma, achieving state-of-the-art performance using "over-parameterized models" where the number of fit parameters is large enough to perfectly fit the training data. As a result, understanding bias and variance in over-parameterized models has emerged as a fundamental problem in machine learning. Here, we use methods from statistical physics to derive analytic expressions for bias and variance in two minimal models of over-parameterization (linear regression and two-layer neural networks with nonlinear data distributions), allowing us to disentangle properties stemming from the model architecture and random sampling of data. In both models, increasing the number of fit parameters leads to a phase transition where the training error goes to zero and the test error diverges as a result of the variance (while the bias remains finite). Beyond this threshold, the test error of the two-layer neural network decreases due to a monotonic decrease in \emph{both} the bias and variance in contrast with the classical bias-variance trade-off. We also show that in contrast with classical intuition, over-parameterized models can overfit even in the absence of noise and exhibit bias even if the student and teacher models match. We synthesize these results to construct a holistic understanding of generalization error and the bias-variance trade-off in over-parameterized models and relate our results to random matrix theory.

stat.ML↗

arXiv2021

Supervised learning in physical networks: From machine learning to learning machines

Materials and machines are often designed with particular goals in mind, so that they exhibit desired responses to given forces or constraints. Here we explore an alternative approach, namely physical coupled learning. In this paradigm, the system is not initially designed to accomplish a task, but physically adapts to applied forces to develop the ability to perform the task. Crucially, we require coupled learning to be facilitated by physically plausible learning rules, meaning that learning requires only local responses and no explicit information about the desired functionality. We show that such local learning rules can be derived for any physical network, whether in equilibrium or in steady state, with specific focus on two particular systems, namely disordered flow networks and elastic networks. By applying and adapting advances of statistical learning theory to the physical world, we demonstrate the plausibility of new classes of smart metamaterials capable of adapting to users' needs in-situ.

cond-mat.soft↗

arXiv2021

The Geometry of Over-parameterized Regression and Adversarial Perturbations

Classical regression has a simple geometric description in terms of a projection of the training labels onto the column space of the design matrix. However, for over-parameterized models -- where the number of fit parameters is large enough to perfectly fit the training data -- this picture becomes uninformative. Here, we present an alternative geometric interpretation of regression that applies to both under- and over-parameterized models. Unlike the classical picture which takes place in the space of training labels, our new picture resides in the space of input features. This new feature-based perspective provides a natural geometric interpretation of the double-descent phenomenon in the context of bias and variance, explaining why it can occur even in the absence of label noise. Furthermore, we show that adversarial perturbations -- small perturbations to the input features that result in large changes in label values -- are a generic feature of biased models, arising from the underlying geometry. We demonstrate these ideas by analyzing three minimal models for over-parameterized linear least squares regression: without basis functions (input features equal model features) and with linear or nonlinear basis functions (two-layer neural networks with linear or nonlinear activation functions, respectively).

stat.ML↗

arXiv2020

The Perturbative Resolvent Method: spectral densities of random matrix ensembles via perturbation theory

We present a simple, perturbative approach for calculating spectral densities for random matrix ensembles in the thermodynamic limit we call the Perturbative Resolvent Method (PRM). The PRM is based on constructing a linear system of equations and calculating how the solutions to these equation change in response to a small perturbation using the zero-temperature cavity method. We illustrate the power of the method by providing simple analytic derivations of the Wigner Semi-circle Law for symmetric matrices, the Marchenko-Pastur Law for Wishart matrices, the spectral density for a product Wishart matrix composed of two square matrices, and the Circle and elliptic laws for real random matrices.

cond-mat.dis-nn↗

arXiv2020

Correlation of plastic events with local structure in jammed packings across spatial dimensions

In jammed packings, it is usually thought that local structure only plays a significant role in specific regimes. The standard deviation of the relative excess coordination, $σ_Z/ Z_\mathrm{c}$, decays like $1/\sqrt{d}$, so that local structure should play no role in high spatial dimensions. Furthermore, in any fixed dimension $d \geq 2$, there are diverging length scales as the pressure vanishes approaching the unjamming transition, again suggesting that local structure should not be sufficient to describe response. Here we challenge the assumption that local structure does not matter in these cases. In simulations of jammed packings under athermal, quasistatic shear, we use machine learning to identify a local structural variable, softness, that correlates with rearrangements in dimensions $d=2$ to $d=5$. We find that softness - and even just the coordination number $Z$ - are quite predictive of rearrangements over a wide range of pressures, all the way down to unjamming, in all $d$ studied. This result provides direct evidence that local structure can play a role in higher spatial dimensions.

cond-mat.soft↗

arXiv2020

Learning-based approach to plasticity in athermal sheared amorphous packings: Improving softness

The plasticity of amorphous solids undergoing shear is characterized by quasi-localized rearrangements of particles. While many models of plasticity exist, the precise relationship between plastic dynamics and the structure of a particle's local environment remains an open question. Previously, machine learning was used to identify a structural predictor of rearrangements, called "softness." Although softness has been shown to predict which particles will rearrange with high accuracy, the method can be difficult to implement in experiments where data is limited and the combinations of descriptors it identifies are often difficult to interpret physically. Here we address both of these weaknesses, presenting two major improvements to the standard softness method. First, we present a natural representation of each particle's observed mobility, allowing for the use of statistical models which are both simpler and provide greater accuracy in limited data sets. Second, we employ persistent homology as a systematic means of identifying simple, topologically-informed, structural quantities that are easy to interpret and measure experimentally. We test our methods on two-dimensional athermal packings of soft spheres under quasi-static shear. We find that the same structural information which predicts small variations in the response is also predictive of where plastic events will localize. We also find that excellent accuracy is achieved in athermal sheared packings using simply a particle's species and the number of nearest neighbor contacts.

cond-mat.soft↗

arXiv2019

The hidden topological structure of flow network functionality

The ability to reroute and control flow is vital to the function of venation networks across a wide range of organisms. By modifying individual edges in these networks, either by adjusting edge conductances or creating and destroying edges, organisms can robustly control the propagation of inputs to perform specific tasks. However, a fundamental disconnect exists between the structure and function of these networks: networks with different local architectures can perform the same functions. Here we answer the question of how structural changes at the microscopic level are able to collectively create functionality at the scale of an entire network. Using persistent homology, we analyze networks tuned to perform complex multifunctional tasks. We find that the responses of such networks encode a hidden topological structure composed of sectors of uniform pressure. Although these sectors are not apparent in the underlying network architectures, we find that they nonetheless correlate strongly with the tuned function. We conclude that the connectivity of these sectors, rather than that of the individual nodes, provides a quantitative relationship between structure and function in flow networks. Finally, we use this topological description to place a bound on the limits of task complexity.

cond-mat.soft↗

arXiv2019

Revealing structure-function relationships in functional flow networks via persistent homology

Complex networks encountered in biology are often characterized by significant structural diversity. Whether it be differences in the three-dimensional structure of allosteric proteins, or the variation among the micro-scale structures of organisms' cerebral vasculature systems, identifying relationships between structure and function often poses a difficult challenge. Here we showcase an approach to characterizing structure-function relationships in complex networks applied in the context of flow networks tuned to perform specific functions. Using persistent homology, we analyze flow networks tuned to perform complex multifunctional tasks, answering the question of how local changes in the network structure coordinate to create functionality at at the scale of the entire network. We find that the response of such networks encodes hidden topological features - sectors of uniform pressure - that are not apparent in the underlying network architectures, Regardless of differences in local connectivity, these features provide a universal topological description for all networks that perform these types of functions. We show that these features correlate strongly with the tuned response, providing a clear topological relationship between structure and function and structural insight into the limits of multifunctionality.

physics.soc-ph↗

arXiv2018

The limits of multifunctionality in tunable networks

Nature is rife with networks that are functionally optimized to propagate inputs in order to perform specific tasks. Whether via genetic evolution or dynamic adaptation, many networks create functionality by locally tuning interactions between nodes. Here we explore this behavior in two contexts: strain propagation in mechanical networks and pressure redistribution in flow networks. By adding and removing links, we are able to optimize both types of networks to perform specific functions. We define a single function as a tuned response of a single "target" link when another, predetermined part of the network is activated. Using network structures generated via such optimization, we investigate how many simultaneous functions such networks can be programmed to fulfill. We find that both flow and mechanical networks display qualitatively similar phase transitions in the number of targets that can be tuned, along with the same robust finite-size scaling behavior. We discuss how these properties can be understood in the context of a new class of constraint-satisfaction problems.

cond-mat.soft↗

arXiv2016

Designing allostery-inspired response in mechanical networks

Recent advances in designing meta-materials have demonstrated that global mechanical properties of disordered spring networks can be tuned by selectively modifying only a small subset of bonds. Here, using a computationally-efficient approach, we extend this idea in order to tune more general properties of networks. With nearly complete success, we are able to produce a strain between any pair of target nodes in a network in response to an applied source strain on any other pair of nodes by removing only ~1% of the bonds. We are also able to control multiple pairs of target nodes, each with a different individual response, from a single source, and to tune multiple independent source/target responses simultaneously into a network. We have fabricated physical networks in macroscopic two- and three-dimensional systems that exhibit these responses. This targeted behavior is reminiscent of the long-range coupled conformational changes that often occur during allostery in proteins. The ease with which we create these responses may give insight into why allostery is a common means for the regulation of activity in biological molecules.

cond-mat.soft↗