Collaborators: Mark Schubert.

**Background:*** The full exploitation of natural materials is often limited due to their inherent structural and functional complexity. Our research focus is on the development and use of machine learning techniques together with experimental analysis in order to achieve a more comprehensive understanding of such complex systems. We aim to make use of the predictive capacity of our algorithms in order to tailor the materials under study for specified tasks and purposes.*

Emergence is a process related to complex systems by which non-trivial patterns and structures arise through self-organisation out of a multiplicity of relatively simple interactions. We use wood and its hierarchical structure as an example of a complex system, and we show that it is possible to predict its macroscopic mechanical properties by accessing the information stored in the emergent fibre patterns. This is done using computer vision techniques on a large sample of wood lamellae images. We demonstrate that the accuracy of such predictions depends on the spatial scale examined. This leads to a method for characterizing better the complex structure of wood by quantifying the emergent properties manifested on its surface through the self-organization of its hierarchical components.

Collaborators: __Annunziato Siviglia__, Markus Holzner, François-Gaël Michalec.

**Background:*** It is still not well understood how micro-organisms in streams and rivers are able to maintain their position and avoid being swept downstream by the flow. We propose the use of diffusivity as an indicator of the ability of a population to persist in a localised region. Earlier studies showed that benthic copepods indeed move frequently upstream depending on the average flow velocity and that they also tend to hide in the sediment bed.*

We study the movement and behaviour of fresh water benthic copepods in a controlled laboratory setup in the form of a channel with the aim of mimicking as much as possible the natural river environment. The upper part of the channel consist of fresh water flowing with a logarithmic velocity profile whilst the lower part consists of a sediment bed of several layers of transparent PolyAcrylAmide grains with a diameter of 8-12mm. The copepods are tracked as they move through a section of the channel using the three-dimensional particle tracking velocimetry technique (3D-PTV). In partic- ular, we analyse the trajectories of a population of copepods and calculate the mean square displacement (MSD) over a set of samples. When moving inside the sediment, preliminary results show that the MSD grows linearly with time, indicating normal diffusion. However, in the upper part and especially in the direction of the flow, the MSD is found to grow sub- linearly with time, suggesting sub-diffusive motion that is probably the result of continuous anti-persistent motion of the copepods.

Collaborators: Erik A. Martens

**Background:*** Complex networks such as those used for ***communications, resource delivery and transportation** are ubiquitous in nature and society. From the internet and urban traffic to the intimate networks of the **circulatory systems** in our bodies, understanding how their **topology and structure** relates to their **function and efficiency** is an essential first step in their management.

We are trying to understand the role of loop-like structures in the mammalian microvasculature (capillaries) and how they affect the efficiency with respect to energy dissipation. Recent results suggest that the vascular network begins with a tree-like structure at the larger diameters and changes to a heterogeneous loopy structure at the level of arterioles and capillaries, just like the veins on the leaves of trees. We want to understand at what branching level such a change occurs and why.

Collaborators: __Tatiana A. Amor__, José S. Andrade Jr. & Hans J. Herrmann

**Background:*** What ***visual search strategy** do we employ when searching for a familiar face in a crowd and how would it change if we had to find a friend in a well-organized choir? For instance, do we explore each face sequentially or do we pick them randomly until reaching our target. Along these lines, it is not at all obvious whether there exists a characteristic strategy that is related to the scene content. Being able to identify objects efficiently in different scenes is important for the survival of an organism.

By tracking the eye motion of many **individuals performing various search tasks in an experiment**, we were able to identify different search strategies, ranging from systematic to entirely random, that emerge from exploring the same scene with no salient features. Motivated by these findings, **we developed a visual search model** that quantifies the **global persistent behaviour** and the overall strategy employed while looking for a hidden target. The parameters of the model define the **saccadic orientation distribution**, which has been experimentally proven to provide information regarding the strategy and the scene content. Identifying the model parameters allows us to quantify the strategy employed in terms of ensemble averages, characterizing each experimental cohort. In this way, we can discern with high sensitivity the relation between the visual landscape and the average strategy, disclosing how small variations in the image induce changes in the strategy (Amor et al. 2017). Our next goal is to understand how the **distribution of fixation times** depends on the features and random structure of the scene.

Collaborators: __Caio P. De Castro__ & Hans Herrmann

**Background:***Random landscapes have been used as the basis for modelling a vast range of properties of different natural systems such as the sea surface temperature, ocean depth, height of land masses above sea level and plasma vorticity fields. Generally, such surfaces are correlated and in some cases they can have ***long-range correlations** that are characterized by **the Hurst exponent, H**.

It has recently been shown that the iso-height lines taken at the percolation threshold of a long-range correlated random surface are scale-invariant with a fractal dimension df that depends on H. What still remains elusive is whether these curves have **a richer symmetry in the form of conformal invariance** (de Castro et al. 2018). We are also interested in how different distributions of Fourier phases of such surfaces affect these symmetries (de Castro et al. 2017).

Collaborators: Giorgos Tsironis & Patrick Navez

**Background:***Mechanical, electrical or optical devices never perfectly conserve energy and even when some input power compensates the losses, time reversal symmetry is not restored. Nevertheless, a global symmetry of the system can be restored in the form of ***a combined parity and time invariance**, where the parity operation refers to **switches between gain and loss states**. The fulfilment of such PT (parity-time) symmetry can lead to extraordinary properties and improve the efficiency of these systems.

In order to deepen our understanding of the influence of such combined symmetry, we focused on a damped harmonic oscillator that is reactivated with some gain input at appropriate time intervals. We showed that **in the stochastic gain-loss oscillator a PT-like symmetry is nevertheless preserved at the ensemble level via the probability distributions when the average gain and loss durations are balanced**. Beyond a certain damping threshold however, this symmetry is broken and the stability of the oscillator is recovered when loss dominates gain over time. This concept could be applied in the stabilization of light propagation in metamaterials (optical fibres) with random regions of asymmetric active and passive media (Lukovic et al. 2016).

Collaborators: Stephan Eule & Theo Geisel

**Background:***The idea that ***Levy walks** might give animals a slight evolutionary advantage over variants of the normal random walk while foraging in resource-scarce environments was first put forward by Shlesinger and Klafter in 1986. The probability of returning to a previously visited site is smaller than for a Gaussian distribution, thus avoiding oversampling. There is an ongoing debate within the scientific community as to whether there truly exist cases in nature where Levy walks are used as the **foraging strategy**.

My main focus was on the **first passage time** properties and the **geometry of random walks** that exhibit **anomalous diffusion**. Using the concept of subordination, I determined the exact analytical expressions for the average perimeter and area of the **convex hulls** for a class of **sub-diffusive non-Markovian processes**. As the convex hull is a simple measure to estimate the **home range of animals**, these results give analytical estimates for the home range of foraging animals that perform sub-diffusive search strategies. Such movement is observed with animals that ambush their prey and in some Mediterranean seabirds, whose search strategies have been influenced by human activity. I also applied our results to Levy flights where possible (Lukovic et al. 2013). Furthermore, I used the properties of convex hulls to develop a new method for **discriminating between Levy walks and simple Gasussian random walks on a plane**.

Collaborators: __Paolo Grigolini__

**Background:***In recent years, there has been intense activity to understand what makes a swarm of birds behave as a single unit. How does the swarm coordinate a change in direction in order to avoid an incoming predator? What makes it possible for a small subset of the swarm to convince the whole flock to pursue a specific course towards a resource?*

We studied a network of interacting two-state units as a model of **cooperative decision making**. Each unit in isolation uctuates between the two states with a constant rate g_{0}, i.e., the probability to switch remains constant in time. In this case the dynamics of each unit is Poissonian. When the units are coupled, the probability for individual switching is in uenced by the states of the neighbouring units, causing g_{0} to become time dependent. We showed that in that condition the units lose their statistical independence and if their number isnite this condition of **collective behaviour is reached when the cooperation strength has a value belonging to an extended interval**, with no need for the ne-tuning of ordinary phase transitions. We proved that the 'swarm cognition' is determined by the criticality-induced long-range correlation, a form of locality breakdown not requiring wave-like information transport (Vanni et al. 2011).

In a recent article (Luković et al. 2014) we proved that the extended critical region is responsible for the synchronization of two swarms and that this is generated in the extended time regime of regression to equilibrium. There is an intimate connection between 'swarm intelligence' and biological criticality and the results of this research are expected to give important contribution to the progress of this field.

Collaborators: __Paolo Grigolini__

**Background:***There are many examples of ***event-based random processes** in nature. Some obvious cases include the spiking of a neuron, the beating of the heart, blinking of quantum dots, occurrence of solar ares, etc. where the events can clearly be identied. While many of these processes are characterised by well-defined event rates, there are also cases in which the **inter-event time interval is scale-free**. Furthermore, it was shown that such processes do not respond to weak perturbations that have a characteristic time scale, a significant deviation from predictions of ordinary statistical physics that led some researchers to interpret the phenomenon as **'death of the linear response'**. We are interested in the properties of such systems and how information can be conveyed to them.

*
**Stochastic process characterised by ***renewal events** separated by non-exponentially distributed waiting times are said to exhibit aging effects. From an ensemble point of view, this is manifested as a continuous decrease in the event rate until the process reaches its stationary state. In cases where the rate change continues indenitely, we speak of **perennial aging**. One way of dealing with such processes is to consider them as being **subordinated** to an exponential process via a time transformation.

By analysing the dichotomous time series produced by a two-state non-Poissonian system, I showed that **1/f noise** is produced by systems that lie on the border that separates ergodic systems from **weakly-ergodic** ones. By weakly-ergodic, we intend non-ergodic systems that during a single realisation spend more time in one state than in the other. The key ingredient for understanding this result is the perennial aging property which accounts for the breakdown of ergodicity. We also showed how to extend the Wiener-Khintchine theorem to non-stationary processes (Luković & Grigolini 2008).

The study showed that it is possible to determine whether the truncation of the waiting time distribution of a stochastic process stems from an intrinsic physical effect or due to thenite size of the observed time series. Furthermore, these results were used **to developed a theory of 1/f noise of human cognition to explain the experimental observations** whereby increasing the diffcultly of cognitive tasks accelerates the transition from observed 1/f noise to white noise during the process of **decision-making** (Grigolini et al. 2009).

The principal object of our study was a stochastic system with inter-event time intervals that obey an **inverse power law** of the form 1/τ^{α} with 1 < α < 2. Such a system, which is non-ergodic, has no characteristic time scale and hence does not respond when coupled to a system with afixed event rate. We showed that the information transfer between two weakly coupled event-dominated systems is maximised when they have the same **temporal complexity**, which is closely related to α. We extended this idea to study how information could efficiently be transferred between two weakly interacting complex networks (Turalska et al. 2009 & Vanni et al. 2011). We also explored 8 the role of the breakdown of ergodicity and its recovery in the transmission of information within a network of coupled stochastic systems (Luković et al. 2014).