Transportation networks play a crucial role in human mobility, the exchange of goods and the spread of invasive species. With 90 per cent of world trade carried by sea, the global network of merchant ships provides one of the most important modes of transportation. Here, we use information about the itineraries of 16 363 cargo ships during the year 2007 to construct a network of links between ports. We show that the network has several features that set it apart from other transportation networks. In particular, most ships can be classified into three categories: bulk dry carriers, container ships and oil tankers. These three categories do not only differ in the ships' physical characteristics, but also in their mobility patterns and networks. Container ships follow regularly repeating paths whereas bulk dry carriers and oil tankers move less predictably between ports. The network of all ship movements possesses a heavy-tailed distribution for the connectivity of ports and for the loads transported on the links with systematic differences between ship types. The data analysed in this paper improve current assumptions based on gravity models of ship movements, an important step towards understanding patterns of global trade and bioinvasion.

We study the role of frustration in excitable systems that allow for oscillations either by construction or in an induced way. We first generalize the notion of frustration to systems whose dynamical equations do not derive from a Hamiltonian. Their couplings can be directed or undirected; they should come in pairs of opposing effects like attractive and repulsive, or activating and repressive, ferromagnetic and antiferromagnetic. As examples we then consider bistable frustrated units as elementary building blocks of our motifs of coupled units. Frustration can be implemented in these systems in various ways: on the level of a single unit via the coupling of a self-loop of positive feedback to a negative feedback loop, on the level of coupled units via the topology or via the type of coupling which may be repressive or activating. In comparison to systems without frustration, we analyze the impact of frustration on the type and number of attractors and observe a considerable enrichment of phase space, ranging from stable fixed-point behavior over different patterns of coexisting options for phase-locked motion to chaotic behavior. In particular we find multistable behavior even for the smallest motifs as long as they are frustrated. Therefore we confirm an enrichment of phase space here for excitable systems with their many applications in biological systems, a phenomenon that is familiar from frustrated spin systems and less known from frustrated phase oscillators. So the enrichment of phase space seems to be a generic effect of frustration in dynamical systems. For a certain range of parameters our systems may be realized in cell tissues. Our results point therefore on a possible generic origin for dynamical behavior that is flexible and functionally stable at the same time, since frustrated systems provide alternative paths for the same set of parameters and at the same "energy costs."

A simple flow network model of biological signal transduction is investigated. Networks with prescribed signal processing functions, robust against random node or link removals, are designed through an evolutionary optimization process. Statistical properties of large ensembles of such networks, including their characteristic motif distributions, are determined. Our analysis suggests that robustness against link removals plays the principal role in the architecture of real signal transduction networks and developmental genetic transcription networks.

Statistical properties of large ensembles of networks, all designed to have the same functions of signal processing, but robust against different kinds of perturbations, are analyzed. We find that robustness against noise and random local damage plays a dominant role in determining motif distributions of networks and may underlie their classification into network superfamilies.High robustness against local damage and noise is a fundamental property of biological networks [1]. In gene knock-out experiments where effects of genes are studied by making inoperative one gene per experiment, it has been found that only 13% of genes in yeast are essential, so that their absence is lethal [2]. Because many subsystems of the cell are small, they are subject to strong stochastic fluctuations, but this also does not prevent regular cell functioning.To suppress fluctuations and increase stability of dynamical systems, negative feedback loops can be used (cf. [3]) and such mechanisms of noise reduction are indeed employed by the cells. Alternatively, robustness against random damage can be achieved through the use of self-correcting network architectures. Can small self-correcting networks with only tens of elements, typical for functional modules of biological cells, be constructed? What are characteristic statistical properties of functional networks capable of self-correction? Here, these questions are addressed for a class of networks that should produce predefined responses following activation of different input elements. In the realm of biology, this class includes not only signal-transduction networks of the cell, but also genetic networks. Moreover, the entire nervous system of primitive mul-2 ticellular organisms may also be viewed as belonging to such a class. In our analysis, an approach that can be described as constructive biology is chosen.Instead of investigating real biological networks, we consider a simple flow distribution model of signal processing where nonlinear feedbacks are excluded.By running evolutionary optimization algorithms, large ensembles of networks with the same size and the same output patterns, but robust against different kinds of local damage or distributed noise, are designed. This allows us to analyze how statistical properties of self-correcting networks depend on the kind of random perturbations against which they are robust. Special attention is paid to motif distributions of networks [4]. We discover that our designed networks, robust against random link deletions, have motif distributions characteristic for the second superfamily previously identified by U. Alon and coworkers [4], which includes signal transduction and genetic developmental networks of multicellular organisms and the neural system of the nematode Caenorhabditid elegans.Completely different motif distributions, that belong instead to the fourth superfamily, are characteristic for networks designed to be robust against node deletions or against of noise.

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