Eckehard Olbrich

Max Planck Institute for Mathematics in the Sciences | MIS · Geometry and Complex Systems

Sleep spindles are traditionally defined as 10-15Hz thalamo-cortical oscillations typical of NREM sleep. While substantial heterogeneity in the appearance or spatio-temporal dynamics of spindle events is well recognised, the physiological relevance of the underlying fundamental property-the oscillatory strength-has not been studied. Here we introdu...

Understanding how different networks relate to each other is key for obtaining a greater insight into complex systems. Here, we introduce an intuitive yet powerful framework to characterise the relationship between two networks, comprising the same nodes. We showcase our framework by decomposing the shortest paths between nodes as being contributed...

Force-directed layout algorithms are ubiquitously used tools for network visualization. However, existing algorithms either lack clear interpretation, or they are based on techniques of dimensionality reduction which simply seek to preserve network-immanent topological features, such as geodesic distance. We propose an alternative layout algorithm....

The zero-temperature Ising model is known to reach a fully ordered ground state in sufficiently dense random graphs. In sparse random graphs, the dynamics gets absorbed in disordered local minima at magnetization close to zero. Here, we find that the nonequilibrium transition between the ordered and the disordered regime occurs at an average degree...

A growing number of papers on style transfer for texts rely on information decomposition. The performance of the resulting systems is usually assessed empirically in terms of the output quality or requires laborious experiments. This paper suggests a straightforward information theoretical framework to assess the quality of information decompositio...

What are the mechanisms by which groups with certain opinions gain public voice and force others holding a different view into silence? Furthermore, how does social media play into this? Drawing on neuroscientific insights into the processing of social feedback, we develop a theoretical model that allows us to address these questions. In repeated i...

The zero-temperature Ising model is known to reach a fully ordered ground state in sufficiently dense graphs. In sparse random graphs, the dynamics gets absorbed in disordered local minima at magnetization close to zero. Here we find that the non-equilibrium transition between the ordered and the disordered regime occurs at an average degree that i...

Information decompositions quantify how the Shannon information about a given random variable is distributed among several other random variables. Various requirements have been proposed that such a decomposition should satisfy, leading to different candidate solutions. Curiously, however, only two of the original requirements that determined the S...

[This corrects the article DOI: 10.3389/fdata.2021.731349.].

We demonstrate how a systematic theory of complexity emerges from information theoretical concepts. The complexity of a structure may refer to the difficulty of its description, the encoding of its regularities or the relations between its elements, components or parts. All such measures can be and usually are quantified with the help of informatio...

Force-directed layout algorithms are ubiquitously-used tools for network visualization across a variety of scientific disciplines. However, they lack theoretical grounding which allows to interpret their outcomes rigorously. We propose an approach building on latent space network models, which assume that the probability of nodes forming a tie depe...

The paper explores the notion of a reconfiguration of political space in the context of the rise of populism and its effects on the political system. We focus on Germany and the appearance of the new right wing party “Alternative for Germany” (AfD). The idea of a political space is closely connected to the ubiquitous use of spatial metaphors in pol...

The paper explores the notion of a reconfiguration of political space in the context of the rise of populism and its effects on the political system. We focus on Germany and the appearance of the new right wing party "Alternative for Germany" (AfD). Many scholars of politics discuss the rise of the new populism in Western Europe and the US with res...

We present an open-source interface for scientists to explore Twitter data through interactive network visualizations. Combining data collection, transformation and visualization in one easily accessible framework, the twitter explorer connects distant and close reading of Twitter data through the interactive exploration of interaction networks and...

This article analyses public debate on Twitter via network representations of retweets and replies. We argue that tweets observable on Twitter have both a direct and mediated effect on the perception of public opinion. Through the interplay of the two networks, it is possible to identify potentially misleading representations of public opinion on t...

This multi-level model of opinion formation considers that attitudes on different issues are usually not independent. In the model, agents exchange beliefs regarding a series of facts. A cognitive structure of evaluative associations links different (partially overlapping) sets of facts on different political issues and determines agents’ attitudinal...

This article analyses public debate on Twitter via network representations of retweets and replies. We argue that tweets observable on Twitter have both a direct and mediated effect on the perception of public opinion. On this basis, we show that through the interplay of the two network representations, it is possible to investigate which opinion g...

We consider biological individuality in terms of information theoretic and graphical principles. Our purpose is to extract through an algorithmic decomposition system-environment boundaries supporting individuality. We infer or detect evolved individuals rather than assume that they exist. Given a set of consistent measurements over time, we discov...

What are the mechanisms by which groups with certain opinions gain public voice and force others holding a different view into silence? And how does social media play into this? Drawing on recent neuro-scientific insights into the processing of social feedback, we develop a theoretical model that allows to address these questions. The model capture...

We present an open-source interface for scientists to explore Twitter data through interactive network visualizations. Combining data collection, transformation and visualization in one easily accessible framework, the twitter explorer connects distant and close reading of Twitter data through the interactive exploration of interaction networks and...

Modelling efforts in opinion dynamics have to a large extent ignored that opinion exchange between individuals can also have an effect on how willing they are to express their opinion publicly. Here, we introduce a model of public opinion expression. Two groups of agents with different opinion on an issue interact with each other, changing the will...

Falling asleep is a gradually unfolding process. We investigated the role of various oscillatory activities including sleep spindles and alpha and delta oscillations at sleep onset (SO) by automatically detecting oscillatory events. We used two datasets of healthy young males, eight with four baseline recordings, and eight with a baseline and recov...

We explore a new mechanism to explain polarization phenomena in opinion dynamics. The model is based on the idea that agents evaluate alternative views on the basis of the social feedback obtained on expressing them. A high support of the favored and therefore expressed opinion in the social environment, is treated as a positive social feedback whi...

The unique information ($UI$) is an information measure that quantifies a deviation from the Blackwell order. We have recently shown that this quantity is an upper bound on the one-way secret key rate. In this paper, we prove a triangle inequality for the $UI$, which implies that the $UI$ is never greater than one of the best known upper bounds on...

A multi-level model of opinion formation is presented which takes into account that attitudes on different issues are usually not independent. In the model, agents exchange beliefs regarding a series of facts. A cognitive structure of evaluative associations links different (partially overlapping) sets of facts to different political issues and det...

Given two channels that convey information about the same random variable, we introduce two measures of the unique information of one channel with respect to the other. The two quantities are based on the notion of generalized weighted Le Cam deficiencies and differ on whether one channel can approximate the other by a randomization at either its i...

When the strict rationality underlying the Nash equilibria in game theory is relaxed, one arrives at the quantal response equilibria introduced by McKelvey and Palfrey. Here, the players are assigned parameters measuring their degree of rationality, and the resulting equilibria are Gibbs type distribution. This brings us into the realm of the expon...

Although quantitative analysis of the sleep electroencephalogram (EEG) has uncovered important aspects of brain activity during sleep in adolescents and adults, similar findings from preschool-age children remain scarce. This study utilized our time-frequency method to examine sleep oscillations as characteristic features of human sleep EEG. Data w...

We study the approach towards equilibrium in a dynamic Ising model, the Q2R cellular automaton, with microscopic reversibility and conserved energy for an infinite one-dimensional system. Starting from a low-entropy state with positive magnetisation, we investigate how the system approaches equilibrium characteristics given by statistical mechanics...

We consider the problem of quantifying the information shared by a pair of random variables $X_{1},X_{2}$ about another variable $S$. We propose a new measure of shared information, called extractable shared information that is left monotonic; that is, the information shared about $S$ is bounded from below by the information shared about $f(S)$ for...

Suppose we have a pair of information channels, $\kappa_{1},\kappa_{2}$ with a common input. The Blackwell order is a partial order over channels that compares $\kappa_{1}$ and $\kappa_{2}$ by the maximal expected utility an agent can obtain when decisions are based on the outputs of $\kappa_{1}$ and $\kappa_{2}$. Equivalently, $\kappa_{1}$ is said...

Supplementary Figures S1 to S8: Top: Spectrograms, oscillatory events and hypnograms of individual children (A-H) recorded longitudinally at 2Y (a), 3Y (b) and 5Y (c). For details, see Figure 1. Bottom: Mean power density spectra and event ratios of individual children (A-H) recorded longitudinally. For details, see Figure 3.

In this paper, we develop an agent-based version of the Diamond search equilibrium model - also called Coconut Model. In this model, agents are faced with production decisions that have to be evaluated based on their expectations about the future utility of the produced entity which in turn depends on the global production level via a trading mecha...

Because the dynamics of complex systems is the result of both decisive local events and reinforced global effects, the prediction of such systems could not do without a genuine multilevel approach. This paper proposes to found such an approach on information theory. Starting from a complete microscopic description of the system dynamics, we are loo...

As Physics did in previous centuries, there is currently a common dream of extracting generic laws of nature in economics, sociology, neuroscience, by focalising the description of phenomena to a minimal set of variables and parameters, linked together by causal equations of evolution whose structure may reveal hidden principles. This requires a hu...

Quantifying behaviors of robots which were generated autonomously from task-independent objective functions is an important prerequisite for objective comparisons of algorithms and movements of animals. The temporal sequence of such a behavior can be considered as a time series and hence complexity measures developed for time series are natural can...

Recently, a series of papers addressed the problem of decomposing the information of two random variables into shared information, unique information and synergistic information. Several measures were proposed, although still no consensus has been reached. Here, we compare these proposals with an older approach to define synergistic information bas...

The information that two random variables $Y$, $Z$ contain about a third random variable $X$ can have aspects of shared information (contained in both $Y$ and $Z$), of complementary information (only available from $(Y,Z)$ together) and of unique information (contained exclusively in either $Y$ or $Z$). Here, we study measures $\widetilde{SI}$ of s...

We quantify the relationship between the dynamics of a time-discrete dynamical system, the tent map T and its iterations T(m), and the induced dynamics at a symbolical level in information theoretical terms. The symbol dynamics, given by a binary string s of length m, is obtained by choosing a partition point [Formula: see text] and lumping togethe...

Levels of a complex system are characterized by the fact that they admit a closed functional description in terms of concepts and quantities intrinsic to that level. Several ideas have come up so far in order to make the notion of a closed description precise. In this paper, we present four of these approaches and investigate their mutual relations...

What is the "value of information" in non-cooperative games with imperfect information? To answer this question, we propose to quantify information using concepts from Shannon's information theory. We then relate quantitative changes to the information structure of a game to changes in the expected utility of the players. Our approach is based on t...

The human sleep electroencephalogram (EEG) is characterized by the occurrence of distinct oscillatory events such as delta waves, sleep spindles and alpha activity. We applied a previously proposed algorithm for the detection of such events and investigated their incidence and frequency in baseline and recovery sleep after 40 h of sustained wakeful...

We propose new measures of shared information, unique information and synergistic information that can be used to decompose the multi-information of a pair of random variables $(Y,Z)$ with a third random variable $X$. Our measures are motivated by an operational idea of unique information which suggests that shared information and unique informatio...

Sleep encompasses approximately a third of our lifetime, yet its purpose and biological function are not well understood. Without sleep optimal brain functioning such as responsiveness to stimuli, information processing, or learning may be impaired. Such observations suggest that sleep plays a crucial role in organizing or reorganizing neuronal net...

How can the information that a set {X 1,…,X n } of random variables contains about another random variable S be decomposed? To what extent do different subgroups provide the same, i.e. shared or redundant, information, carry unique information or interact for the emergence of synergistic information? Recently Williams and Beer proposed such a decom...

We adapt the method used by Jaynes to derive the equilibria of statistical physics to instead derive equilibria of bounded rational game theory. We analyze the dependence of these equilibria on the parameters of the underlying game, focusing on hysteresis effects. In particular, we show that by gradually imposing individual-specific tax rates on th...

'Complexity science' is a rapidly developing research direction with applications in a multitude of fields that study complex systems consisting of a number of nonlinear elements with interesting dynamics and mutual interactions. This Theme Issue 'The complexity of sleep' aims at fostering the application of complexity science to sleep research, be...

A particular property of the sleeping brain is that it exhibits dynamics on very different time scales ranging from the typical sleep oscillations such as sleep spindles and slow waves that can be observed in electroencephalogram (EEG) segments of several seconds duration over the transitions between the different sleep stages on a time scale of mi...

We develop a geometric approach to complexity based on the principle that complexity requires interactions at different scales of description. Complex systems are more than the sum of their parts of any size and not just more than the sum of their elements. Using information geometry, we therefore analyze the decomposition of a system in terms of a...

We analyze the time honored subject of emergence with modern tools from theoretical computer science, dynamical systems and complex systems theory. The features we identify are regularities that allow for a reduced description at a higher scale, coordination between the parts that enable a system to explore new regions of its state space, and the c...

We investigate exponential families of random graph distributions as a framework for systematic quantification of structure in networks. In this paper we restrict ourselves to undirected unlabeled graphs. For these graphs, the counts of subgraphs with no more than k links are a sufficient statistics for the exponential families of graphs with inter...

The response to a knockout of a node is a characteristic feature of a networked dynamical system. Knockout resilience in the dynamics of the remaining nodes is a sign of robustness. Here we study the effect of knockouts for binary state sequences and their implementations in terms of Boolean threshold networks. Besides random sequences with biologi...

No abstract received.

We evaluate information-theoretic quantities that quantify complexity in terms of kth-order statistical dependences that cannot be reduced to interactions among k-1 random variables. Using symbolic dynamics of coupled maps and cellular automata as model systems, we demonstrate that these measures are able to identify complex dynamical regimes.

The sleep electroencephalogram (EEG) is characterized by typical oscillatory patterns such as sleep spindles and slow waves. Recently, we proposed a method to detect and analyze these patterns using linear autoregressive models for short (approximately 1 s) data segments. We analyzed the temporal organization of sleep spindles and discuss to what e...

. We study how statistical complexity depends on the system size and how the complexity of the whole system relates to the complexity of its subsystems. We study this size dependence for two well-known complexity measures, the excess entropy of Grassberger and the neural complexity introduced by Tononi, Sporns and Edelman. We compare these results...

Measures of complexity are of immediate interest for the field of autonomous robots both as a means to classify the behavior and as an objective function for the autonomous development of robot behavior. In the present paper we consider predictive information in sensor space as a measure for the behavioral complexity of a two-wheel embodied robot...

We present a tentative proposal for a quantitative measure of autonomy. This is something that, surprisingly, is rarely found in the literature, even though autonomy is considered to be a basic concept in many disciplines, including artificial life. We work in an information theoretic setting for which the distinction between system and environment...

The use of memory kernels stemming from a Mori-Zwanzig approach to time series analysis is discussed. We show that despite its success in determining properties from an analytical model, the kernel itself is not easily interpreted. We consider a recently introduced discretization of the kernel and show that its properties can be quite different fro...

We investigate a recently proposed method for the analysis of oscillatory patterns in EEG data, with respect to its capacity of further quantifying processes on slower time scales. The method is based on modeling the EEG time series by linear autoregressive (AR) models with time dependent parameters. Systems described by such linear models can be i...

We develop an analysis of complex systems in terms of statistical correlations between the dynamics of its subsystems as a formal framework within which to understand processes of system differentiation.

Zusammenfassung Ein einflussreiches Paradigma in der Untersuchung komplexer Systeme versucht biologische und soziale Systeme als »quasi-physikalische« Systeme zu verstehen, d.h. Systeme, in denen die Elemente nach Regeln interagieren, die wie physikalische Gesetze behandelt werden können, und die durch Prozesse der Selbstorganisation zur Emergenz v...

A typical time series analysis task is to extract knowledge from the past in order to make predictions about the future. Such an endeavor relies on the presence of correlations in time. We present concepts, methods, and algorithms for this task. Special emphasis is laid on nonlinear stochastic processes, probabilistic predictions, and their verific...

The notion of closure plays a prominent role in systems theory where it is used to identify or define the system in distinction from its environment and to explain the autonomy of the system. Here, we present a quantitative measure, as opposed to the already existing qualitative notions, of closure. We shall elaborate upon the observation that cogn...

We develop a unifying approach for complexity measures, based on the principle that complexity requires interactions at different scales of description. Complex systems are more than the sum of their parts of any size, and not just more than the sum of their elements. We therefore analyze the decomposition of a system in terms of an interaction hie...

The different brain states during sleep are characterized by the occurrence of distinct oscillatory patterns such as spindles or delta waves. Using a new algorithm to detect oscillatory events in the electroencephalogram (EEG), we studied their properties and changes throughout the night. The present approach was based on the idea that the EEG may...

A new algorithm for the detection of oscillatory events in the EEG is presented. By estimating autoregressive (AR) models on short segments the EEG is described as a superposition of harmonic oscillators with damping and frequencies varying in time. Oscillatory events are detected, whenever the damping of one or more frequencies falls below a prede...

Several investigators of EEG time series reported a rejection of the null hypothesis of linear stochastic dynamics for epochs longer than . We examine whether this rejection is related to nonlinearity or to nonstationarity. Our approach is a combination of autoregressive (AR-) modeling and surrogate data testing. It is shown that the fraction of su...

The relevance of the dimensional complexity (DC) for the analysis of sleep EEG data is investigated and compared to linear measures. We calculated DC of artifact-free 1 min segments of all-night sleep EEG recordings of 4 healthy young subjects. Non-linearity was tested by comparing with DC values of surrogate data. Linear properties of the segments...

Objective: The relevance of the dimensional complexity (DC) for the analysis of sleep EEG data is investigated and compared to linear measures.Methods: We calculated DC of artifact-free 1 min segments of all-night sleep EEG recordings of 4 healthy young subjects. Non-linearity was tested by comparing with DC values of surrogate data. Linear propert...

Parameter variations in the equations of motion of dynamical systems are identified by time series analysis. The information contained in time series data is transformed and compressed to feature vectors. The space of feature vectors is an embedding for the unobserved parameters of the system. We show that the smooth variation of d system parameter...