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The evolution of many dynamical systems that describe relationships or interactions between objects can be effectively modeled by temporal networks, which are typically represented as a sequence of static network snapshots. In this paper,…
Networks are a powerful tool to model the structure and dynamics of complex systems across scales. Direct connections between system components are often represented as edges, while paths and walks capture indirect interactions. This…
Although higher-order interactions are known to affect the typical state of dynamical processes giving rise to new collective behavior, how they drive the emergence of rare events and fluctuations is still an open problem. We investigate…
Can the direction of time and the causal structure of space-time be inferred from operational principles? Causal models and tensor networks offer complementary perspectives: the former encodes cause-effect relations via directed graphs,…
Markov chains are simple yet powerful mathematical structures to model temporally dependent processes. They generally assume stationary data, i.e., fixed transition probabilities between observations/states. However, live, real-world…
Given observations of a physical system, identifying the underlying non-linear governing equation is a fundamental task, necessary both for gaining understanding and generating deterministic future predictions. Of most practical relevance…
Higher-order networks have emerged as a powerful framework to model complex systems and their collective behavior. Going beyond pairwise interactions, they encode structured relations among arbitrary numbers of units through representations…
Markov chains are fundamental models for stochastic dynamics, with applications in a wide range of areas such as population dynamics, queueing systems, reinforcement learning, and Monte Carlo methods. Estimating the transition matrix and…
We consider Markov processes, which describe e.g. queueing network processes, in a random environment which influences the network by determining random breakdown of nodes, and the necessity of repair thereafter. Starting from an explicit…
We introduce a set of techniques that allow for efficiently generating many independent random walks in the Massive Parallel Computation (MPC) model with space per machine strongly sublinear in the number of vertices. In this…
Contrary to the theory of Markov processes, no general theory exists for the so called nonlinear Markov processes. We study an example of "nonlinear Markov process" related to classical probability theory, merely to random walks. This model…
Empirical networks possess considerable heterogeneity of node connections, resulting in a small portion of nodes playing crucial roles in network structure and function. Yet, how to characterize nodes' influence and identify vital nodes is…
Many biological, social and man-made systems are better described in terms of temporal networks, i.e. networks whose links are only present at certain points in time, rather than by static ones. In particular, it has been found that…
Analyzing, understanding, and describing human behavior is advantageous in different settings, such as web browsing or traffic navigation. Understanding human behavior naturally helps to improve and optimize the underlying infrastructure or…
A new model of quantum random walks is introduced, on lattices as well as on finite graphs. These quantum random walks take into account the behavior of open quantum systems. They are the exact quantum analogues of classical Markov chains.…
How to enable efficient analytics over such data has been an increasingly important research problem. Given the sheer size of such social networks, many existing studies resort to sampling techniques that draw random nodes from an online…
We study various classes of random processes defined on the regular tree $T_d$ that are invariant under the automorphism group of $T_d$. Most important ones are factor of i.i.d. processes (randomized local algorithms), branching Markov…
A classical random walk $(S_t, t\in\mathbb{N})$ is defined by $S_t:=\displaystyle\sum_{n=0}^t X_n$, where $(X_n)$ are i.i.d. When the increments $(X_n)_{n\in\mathbb{N}}$ are a one-order Markov chain, a short memory is introduced in the…
Graph vertex embeddings based on random walks have become increasingly influential in recent years, showing good performance in several tasks as they efficiently transform a graph into a more computationally digestible format while…
We introduce a new interpretation of the attention matrix as a discrete-time Markov chain. Our interpretation sheds light on common operations involving attention scores such as selection, summation, and averaging in a unified framework. It…