Abstract

Analyzing available FAO data from 176 countries over 21 years, we observe an increase of complexity in the international trade of maize, rice, soy, and wheat. A larger number of countries play a role as producers or intermediaries, either for trade or food processing. In consequence, we find that the trade networks become more prone to failure cascades caused by exogenous shocks. In our model, countries compensate for demand deficits by imposing export restrictions. To capture these, we construct higher-order trade dependency networks for the different crops and years. These networks reveal hidden dependencies between countries and provide an estimate of necessary stock reserves to protect countries from cascading export restrictions. They differ substantially from first order networks that do not take cascading effects into account. We find rice trade most prone to cascading export restrictions. A great number of Asian and African countries are most exposed to cascades. Noticeably, the main suppliers are similar for most of the crops: USA, Canada, Argentina, Brazil, and India. While shocks in the USA mainly affect South America and several Asian countries, the south of Africa is primarily dependent on American and Asian exporters. The north of Africa depends strongly on Europe, in particular via wheat imports.

Abstract

How big is the risk that a few initial failures of networked nodes amplify to large cascades that endanger the functioning of the system? Common answers refer to the average final cascade size. Two analytic approaches allow its computation: a) (heterogeneous) mean field approximation and b) belief propagation. The former applies to (infinitely) large locally tree-like networks, while the latter is exact on finite trees. Yet, cascade sizes can have broad and multi-modal distributions that are not well represented by their average. Full distribution information is essential to identify likely events and to estimate the tail risk, i.e. the probability of extreme events. We therefore present an efficient message passing algorithm that calculates the cascade size distribution in finite networks. It is exact on finite trees and for a large class of cascade processes. An approximate version applies to any network structure and performs well on locally tree-like networks, as we show with several examples.

Abstract

How big is the risk that a few initial failures of nodes in a network amplify to large cascades that span a substantial share of all nodes? Predicting the final cascade size is critical to ensure the functioning of a system as a whole. Yet, this task is hampered by uncertain or changing parameters and missing information. In infinitely large networks, the average cascade size can often be well estimated by established approaches building on local tree approximations and mean field approximations. Yet, as we demonstrate, in finite networks, this average does not even need to be a likely outcome. Instead, we find broad and even bimodal cascade size distributions. This phenomenon persists for system sizes up to 10 7 and different cascade models, i.e. it is relevant for most real systems. To show this, we derive explicit closed-form solutions for the full probability distribution of the final cascade size. We focus on two topological limit cases, the complete network representing a dense network with a very narrow degree distribution, and the star network representing a sparse network with a inhomogeneous degree distribution. Those topologies are of great interest, as they either minimize or maximize the average cascade size and are common motifs in many real world networks.

Abstract

Two node variables determine the evolution of cascades in random networks: a node's degree and threshold. Correlations between both fundamentally change the robustness of a network, yet, they are disregarded in standard analytic methods as local tree or heterogeneous mean field approximations because of the bad tractability of order statistics. We show how they become tractable in the thermodynamic limit of infinite network size. This enables the analytic description of node attacks that are characterized by threshold allocations based on node degree. Using two examples, we discuss possible implications of irregular phase transitions and different speeds of cascade evolution for the control of cascades.

Abstract

We present a framework to calculate the cascade size evolution for a large class of cascade models on random network ensembles in the limit of infinite network size. Our method is exact and applies to network ensembles with almost arbitrary degree distribution, degree-degree correlations and, in case of threshold models, for arbitrary threshold distribution. With our approach, we shift the perspective from the known branching process approximations to the iterative update of suitable probability distributions. Such distributions are key to capture cascade dynamics that involve possibly continuous quantities and that depend on the cascade history, e.g. if load is accumulated over time. As a proof of concept, we provide two examples: (a) Constant load models that cover many of the analytically tractable casacade models, and, as a highlight, (b) a fiber bundle model that was not tractable by branching process approximations before. Our derivations cover the whole cascade dynamics, not only their steady state. This allows to include interventions in time or further model complexity in the analysis.

Abstract

The authors develop an agent-based model to reproduce the size distribution of R&D alliances of firms. Agents are uniformly selected to initiate an alliance and to invite collaboration partners. These decide about acceptance based on an individual threshold that is compared with the utility expected from joining the current alliance. The benefit of alliances results from the fitness of the agents involved. Fitness is obtained from an empirical distribution of agent's activities. The cost of an alliance reflects its coordination effort. Two free parameters ac and a1 scale the costs and the individual threshold. If initiators receive R rejections of invitations, the alliance formation stops and another initiator is selected. The three free parameters (ac; a1; R) are calibrated against a large scale data set of about 15,000 firms engaging in about 15,000 R&D alliances over 26 years. For the validation of the model the authors compare the empirical size distribution with the theoretical one, using confidence bands, to find a very good agreement. As an asset of our agent-based model, they provide an analytical solution that allows to reduce the simulation effort considerably. The analytical solution applies to general forms of the utility of alliances. Hence, the model can be extended to other cases of alliance formation. While no information about the initiators of an alliance is available, the results indicate that mostly firms with high fitness are able to attract newcomers and to establish larger alliances.

Abstract

We study the influence of risk diversification on cascading failures in weighted complex networks, where weighted directed links represent exposures between nodes. These weights result from different diversification strategies and their adjustment allows us to reduce systemic risk significantly by topological means. As an example, we contrast a classical exposure diversification (ED) approach with a damage diversification (DD) variant. The latter reduces the loss that the failure of high degree nodes generally inflict to their network neighbors and thus hampers the cascade amplification. To quantify the final cascade size and obtain our results, we develop a branching process approximation taking into account that inflicted losses cannot only depend on properties of the exposed, but also of the failing node. This analytic extension is a natural consequence of the paradigm shift from individual to system safety. To deepen our understanding of the cascade process, we complement this systemic perspective by a mesoscopic one: an analysis of the failure risk of nodes dependent on their degree. Additionally, we ask for the role of these failures in the cascade amplification.

Abstract

We study cascades on a two-layer multiplex network, with asymmetric feedback that depends on the coupling strength between the layers. Based on an analytical branching process approximation, we calculate the systemic risk measured by the final fraction of failed nodes on a reference layer. The results are compared with the case of a single layer network that is an aggregated representation of the two layers. We find that systemic risk in the two-layer network is smaller than in the aggregated one only if the coupling strength between the two layers is small. Above a critical coupling strength, systemic risk is increased because of the mutual amplification of cascades in the two layers. We even observe sharp phase transitions in the cascade size that are less pronounced on the aggregated layer. Our insights can be applied to a scenario where firms decide whether they want to split their business into a less risky core business and a more risky subsidiary business. In most cases, this may lead to a drastic increase of systemic risk, which is underestimated in an aggregated approach.