We introduce a general framework for models of cascade and contagion processes on networks, to identify their commonalities and differences. In particular, models of social and financial cascades, as well as the fiber bundle model, the voter model, and models of epidemic spreading are recovered as special cases. To unify their description, we define the net fragility of a node, which is the difference between its fragility and the threshold that determines its failure. Nodes fail if their net fragility grows above zero and their failure increases the fragility of neighbouring nodes, thus possibly triggering a cascade. In this framework, we identify three classes depending on the way the fragility of a node is increased by the failure of a neighbour. At the microscopic level, we illustrate with specific examples how the failure spreading pattern varies with the node triggering the cascade, depending on its position in the network and its degree. At the macroscopic level, systemic risk is measured as the nal fraction of failed nodes, X ∗ , and for each of the three classes we derive a recursive equation to compute its value. The phase diagram of X ∗ as a function of the initial conditions, thus allows for a prediction of the systemic risk as well as a comparison of the three different model classes. We could identify which model class lead to a first-order phase transition in systemic risk, i.e. situations where small changes in the initial conditions may lead to a global failure. Eventually, we generalize our framework to encompass stochastic contagion models. This indicates the potential for further generalizations.

We examine the emergent field of economic networks and explore its ability to shed light on the global and volatile economy where credit, ownership, innovation, investment, and virtually every other economic activity is carried at a scale and scope that respects no geographical, organizational, or political boundaries. In this context, the study of economic networks and their dynamics must reflect the vast complexity of the interaction patterns and integrate it with a realistic account of the incentives and information that govern agents’ behavior. The interplay of both has been shown to produce metastabilities, system crashes, and emergent structures in ways that are yet only poorly understood. Meeting this exciting scientific challenge requires a combination of time series analysis, complexity theory, and simulation with the analytical tools that have been developed by game theory, as well as graph and matrix theories. We argue that this will help achieving a better integration of theory and data models and provide a better understanding of the potentials and risks of modern economic systems. 1

The last 20 years have witnessed different forms of col-laborative relationships that have been theorized, for example, in terms of innovation systems, innovation

This work introduces a new model to investigate the efficiency and evolution of networks of firms exchanging knowledge in R&D partnerships. We first examine the efficiency of a given network structure in terms of the maximization of total profits in the industry. We show that the efficient network structure depends on the marginal cost of collaboration. When the marginal cost is low, the complete graph is efficient. However, a high marginal cost implies that the efficient network is sparser and has a core-periphery structure. Next, we examine the evolution of the network structure when the decision on collaborating partners is decentralized. We show the existence of multiple equilibrium structures which are in general inefficient. This is due to (i) the path dependent character of the partner selection process, (ii) the presence of knowledge externalities and (iii) the presence of severance costs involved in link deletion. Finally, we study the properties of the emerging equilibrium networks and we show that they are coherent with the stylized facts of R&D networks.

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This work introduces a new model to investigate the efficiency and evolution of networks of firms exchanging knowledge in R&D partnerships. We first examine the efficiency of a given network structure in terms of the maximization of total profits in the industry. We show that the efficient network structure depends on the marginal cost of collaboration. When the marginal cost is low, the complete graph is efficient. However, a high marginal cost implies that the efficient network is sparser and has a core-periphery structure. Next, we examine the evolution of the network struc-ture when the decision on collaborating partners is decentralized. We show the existence of mul-tiple equilibrium structures which are in general inefficient. This is due to (i) the path dependent character of the partner selection process, (ii) the presence of knowledge externalities and (iii) the presence of severance costs involved in link deletion. Finally, we study the properties of the emerg-ing equilibrium networks and we show that they are coherent with the stylized facts of R&D net-works.

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