In this work, we investigate the existence and effect of percolation in training deep Neural Networks (NNs) with dropout. Dropout methods are regularisation techniques for training NNs, first introduced by G. Hinton et al. (2012). These methods temporarily remove connections in the NN, randomly at each stage of training, and update the remaining subnetwork with Stochastic Gradient Descent (SGD). The process of removing connections from a network at random is similar to percolation, a paradigm model of statistical physics. If dropout were to remove enough connections such that there is no path between the input and output of the NN, then the NN could not make predictions informed by the data. We study new percolation models that mimic dropout in NNs and characterise the relationship between network topology and this path problem. The theory shows the existence of a percolative effect in dropout. We also show that this percolative effect can cause a breakdown when training NNs without biases with dropout; and we argue heuristically that this breakdown extends to NNs with biases.

Generative modeling is typically framed as learning mapping rules, but from an observer's perspective without access to these rules, the task becomes disentangling the geometric support from the probability distribution. We propose that continuum percolation is uniquely suited to this support analysis, as the sampling process effectively projects high-dimensional density estimation onto a geometric counting problem on the support. In this work, we establish a rigorous correspondence between the topological phase transitions of random geometric graphs and the underlying data manifold in high-dimensional space. By analyzing the relationship between our proposed Percolation Shift metric and FID, we show that this metric captures structural pathologies, such as implicit mode collapse, where standard statistical metrics fail. Finally, we translate this topological phenomenon into a differentiable loss function that guides training. Experimental results confirm that this approach not only prevents manifold shrinkage but also fosters a form of synergistic improvement, where topological stability becomes a prerequisite for sustained high fidelity in both static generation and sequential decision making.

Analogical reasoning, the transfer of relational structures across contexts (e.g., planet is to sun as electron is to nucleus), is fundamental to scientific discovery. Yet human insight is often constrained by domain expertise and surface-level biases, limiting access to deeper, structure-driven analogies both within and across disciplines. Large language models (LLMs), trained on vast cross-domain data, present a promising yet underexplored tool for analogical reasoning in science. Here, we demonstrate that LLMs can generate novel battery materials by (1) retrieving cross-domain analogs and analogy-guided exemplars to steer exploration beyond conventional dopant substitutions, and (2) constructing in-domain analogical templates from few labeled examples to guide targeted exploitation. These explicit analogical reasoning strategies yield candidates outside established compositional spaces and outperform standard prompting baselines. Our findings position LLMs as interpretable, expert-like hypothesis generators that leverage analogy-driven generalization for scientific innovation.

In this paper we present a neural network-based method for the automatic detection of phase transitions and classification of hidden percolation patterns in a (1+1)-dimensional replication process. The proposed network model is based on the combination of CNN, TCN and GRU networks, which are trained directly on raw configurations without any manual feature extraction. The network reproduces the phase diagram and assigns phase labels to configurations. It shows that deep architectures are capable of extracting hierarchical structures from the raw data of numerical experiments.

In this paper, we derive theoretical bounds for the long-term influence of a node in an Independent Cascade Model (ICM). We relate these bounds to the spectral radius of a particular matrix and show that the behavior is sub-critical when this spectral radius is lower than 1. More specifically, we point out that, in general networks, the sub-critical regime behaves in O( n) where n is the size of the network, and that this upper bound is met for star-shaped networks. We apply our results to epidemiology and percolation on arbitrary networks, and derive a bound for the critical value beyond which a giant connected component arises. Finally, we show empirically the tightness of our bounds for a large family of networks.

A common assumption in the literature on information diffusion is that populations are homogeneous regarding individuals' information acquisition and propagation process: Individuals update their informed and actively communicating state either through imitation (simple contagion) or peer influence (complex contagion). Here, we study the impact of the mixing and placement of individuals with different update processes on how information cascades in social networks. We consider Simple Spreaders, which take information from a random neighbor and communicate it, and Threshold-based Spreaders, which require a threshold number of active neighbors to change their state to active communication. Even though, in a population made exclusively of Simple Spreaders, information reaches all elements of any (connected) network, we show that, when Simple and Threshold-based Spreaders coexist and occupy random positions in a social network, the number of Simple Spreaders systematically amplifies the cascades only in degree heterogeneous networks (exponential and scale-free). In random and modular structures, this cascading effect originated by Simple Spreaders only exists above a critical mass of these individuals. In contrast, when Threshold-based Spreaders are assorted preferentially in the nodes with a higher degree, the cascading effect of Simple Spreaders vanishes, and the spread of information is drastically impaired. Overall, the study highlights the significance of the strategic placement of different roles in networked structures, with Simple Spreaders driving widespread cascades in heterogeneous networks and Threshold-based Spreaders playing a critical regulatory role in information spread with a tunable effect based on the threshold value. These effects have consequences to our understanding of social phenomena, such as the spread of innovations in heterogeneous social systems with the presence of eager (Simple Spreaders) versus averse (Threshold-based Spreaders) adopters, but also to information warfare on social media where Simple Spreaders can be seen as embedded agents (e.g., bots) used to amplify the virality of ill-intended content and, oppositely, Threshold-based Spreaders as an essential self-regulatory element of social systems operating as information filters.

Deep neural networks exhibit empirical neural scaling laws, with error decreasing as a power law with increasing model or data size, across a wide variety of architectures, tasks, and datasets. This universality suggests that scaling laws may result from general properties of natural learning tasks. We develop a mathematical model intended to describe natural datasets using percolation theory. Two distinct criticality regimes emerge, each yielding optimal power-law neural scaling laws. These regimes, corresponding to power-law-distributed discrete subtasks and a dominant data manifold, can be associated with previously proposed theories of neural scaling, thereby grounding and unifying prior works. We test the theory by training regression models on toy datasets derived from percolation theory simulations. We suggest directions for quantitatively predicting language model scaling.

Bootstrap percolation, introduced by Chalupa, Leath, and Reich in their 1979 work [4], serves as a simplified model for ferromagnetic dynamics. Since then, it has been applied in numerous fields of physics and has become a significant topic of interest in mathematics. The process starts with an initial set of "infected" vertices in a graph G, where at each step, any vertex with at least r infected neighbors also becomes infected. A key problem in this framework is determining the minimum size of an initial set that results in the entire graph becoming infected, known as the percolating set. This minimum is denoted by m(G, r).

We extend the concept of loss landscape mode connectivity to the input space of deep neural networks. Mode connectivity was originally studied within parameter space, where it describes the existence of low-loss paths between different solutions (loss minimizers) obtained through gradient descent. We present theoretical and empirical evidence of its presence in the input space of deep networks, thereby highlighting the broader nature of the phenomenon. We observe that different input images with similar predictions are generally connected, and for trained models, the path tends to be simple, with only a small deviation from being a linear path. Our methodology utilizes real, interpolated, and synthetic inputs created using the input optimization technique for feature visualization. We conjecture that input space mode connectivity in high-dimensional spaces is a geometric effect that takes place even in untrained models and can be explained through percolation theory. We exploit mode connectivity to obtain new insights about adversarial examples and demonstrate its potential for adversarial detection. Additionally, we discuss applications for the interpretability of deep networks.

In this paper, we derive theoretical bounds for the long-term influence of a node in an Independent Cascade Model (ICM). We relate these bounds to the spectral radius of a particular matrix and show that the behavior is sub-critical when this spectral radius is lower than 1. More specifically, we point out that, in general networks, the sub-critical regime behaves in O( n) where n is the size of the network, and that this upper bound is met for star-shaped networks. We apply our results to epidemiology and percolation on arbitrary networks, and derive a bound for the critical value beyond which a giant connected component arises. Finally, we show empirically the tightness of our bounds for a large family of networks.