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The problem at the heart of this tutorial consists in modeling the path choice behavior of network users. This problem has been extensively studied in transportation science, where it is known as the route choice problem. In this…

Traffic jams on roadways, echo chambers on social media, crowds of moving pedestrians, and opinion dynamics during elections are all complex social systems. These applications may seem disparate, but some of the questions that they motivate…

Identifying and understanding modular organizations is centrally important in the study of complex systems. Several approaches to this problem have been advanced, many framed in information-theoretic terms. Our treatment starts from the…

Multivariate categorical data are routinely collected in many application areas. As the number of cells in the table grows exponentially with the number of variables, many or even most cells will contain zero observations. This severe…

Many socio-economic and biological processes can be modeled as systems of interacting individuals. The behaviour of such systems can be often described within game-theoretic models. In these lecture notes, we introduce fundamental concepts…

Deep sequence models are receiving significant interest in current machine learning research. By representing probability distributions that are fit to data using maximum likelihood estimation, such models can model data on general…

A class of discrete distributions can be derived from stationary renewal processes. They have the useful property that the mean is a simple function of the model parameters. Thus regressions of the distribution mean on covariates can be…

Two symmetrically coupled logistic equations are proposed to mimic the competitive interaction between two species. The phenomena of coexistence, oscillations and chaos are present in this cubic discrete system. This work, together with two…

We consider two classes of computations which admit taking linear combinations of execution runs: probabilistic sampling and generalized animation. We argue that the task of program learning should be more tractable for these architectures…

In this paper, we introduce two deterministic models aimed at capturing the dynamics of congested Internet connections. The first model is a continuous-time model that combines a system of differential equations with a sudden change in one…

Literature involving preferences of artificial agents or human beings often assume their preferences can be represented using a complete transitive binary relation. Much has been written however on different models of preferences. We review…

The long-term behaviour of dynamic systems can be classified in two different regimes, regular or chaotic, depending on the values of the control parameters, which are kept constant during the time evolution. Starting from slightly…

This paper considers a population process on a dynamically evolving graph, which can be alternatively interpreted as a queueing network. The queues are of infinite-server type, entailing that at each node all customers present are served in…

We give a direct proof of well-posedness of solutions to general selection-mutation and structured population models with measures as initial data. This is motivated by the fact that some stationary states of these models are measures and…

We present a simple model for describing the dynamics of the interaction between a homogeneous population or society, and the natural resources and reserves that the society needs for its survival. The model is formulated in terms of…

A new delay equation is introduced to describe the punctuated evolution of complex nonlinear systems. A detailed analytical and numerical investigation provides the classification of all possible types of solutions for the dynamics of a…

The dynamics by iteration of a function on a compact metric space, sometimes called a cascade, can be extended to the dynamics of a closed relation on such a space. Here we apply this relation dynamics to study semiflows (and their relation…

We consider prediction theory for stationary stochastic processes in continuous time. We discuss prediction using the whole (infinite) past, and using only a finite section of the past. The solutions to both these classical problems have…

Different kinds of models are used to study various natural and technical phenomena. Usually, the researcher is limited to using a certain kind of model approach, not using others (or even not realizing the existence of other model…

Biological entities are inherently dynamic. As such, various ecological disciplines use mathematical models to describe temporal evolution. Typically, growth curves are modelled as sigmoids, with the evolution modelled by ordinary…