Effects of Economic Interactions on Credit Risk

We study a credit risk model which captures effects of economic interactions on a firm's default probability. Economic interactions are represented as a functionally defined graph, and the existence of both cooperative, and competitive, business relations is taken into account. We provide an analytic solution of the model in a limit where the number of business relations of each company is large, but the overall fraction of the economy with which a given company interacts may be small. While the effects of economic interactions are relatively weak in typical (most probable) scenarios, they are pronounced in situations of economic stress, and thus lead to a substantial fattening of the tails of loss distributions in large loan portfolios. This manifests itself in a pronounced enhancement of the Value at Risk computed for interacting economies in comparison with their non-interacting counterparts.Comment: 24 pages, 6 figure

Dynamic rewiring in small world networks

We investigate equilibrium properties of small world networks, in which both connectivity and spin variables are dynamic, using replicated transfer matrices within the replica symmetric approximation. Population dynamics techniques allow us to examine order parameters of our system at total equilibrium, probing both spin- and graph-statistics. Of these, interestingly, the degree distribution is found to acquire a Poisson-like form (both within and outside the ordered phase). Comparison with Glauber simulations confirms our results satisfactorily.Comment: 21 pages, 5 figure

Dynamical replica analysis of disordered Ising spin systems on finitely connected random graphs

We study the dynamics of macroscopic observables such as the magnetization and the energy per degree of freedom in Ising spin models on random graphs of finite connectivity, with random bonds and/or heterogeneous degree distributions. To do so we generalize existing implementations of dynamical replica theory and cavity field techniques to systems with strongly disordered and locally tree-like interactions. We illustrate our results via application to the dynamics of e.g. $\pm J$ spin-glasses on random graphs and of the overlap in finite connectivity Sourlas codes. All results are tested against Monte Carlo simulations.Comment: 4 pages, 14 .eps file

Cavity approach for real variables on diluted graphs and application to synchronization in small-world lattices

We study XY spin systems on small world lattices for a variety of graph structures, e.g. Poisson and scale-free, superimposed upon a one dimensional chain. In order to solve this model we extend the cavity method in the one pure-state approximation to deal with real-valued dynamical variables. We find that small-world architectures significantly enlarge the region in parameter space where synchronization occurs. We contrast the results of population dynamics performed on a truncated set of cavity fields with Monte Carlo simulations and find excellent agreement. Further, we investigate the appearance of replica symmetry breaking in the spin-glass phase by numerically analyzing the proliferation of pure states in the message passing equations.Comment: 10 pages, 3 figure

Replica symmetry breaking in the `small world' spin glass

We apply the cavity method to a spin glass model on a `small world' lattice, a random bond graph super-imposed upon a 1-dimensional ferromagnetic ring. We show the correspondence with a replicated transfer matrix approach, up to the level of one step replica symmetry breaking (1RSB). Using the scheme developed by M\'ezard & Parisi for the Bethe lattice, we evaluate observables for a model with fixed connectivity and $\pm J$ long range bonds. Our results agree with numerical simulations significantly better than the replica symmetric (RS) theory.Comment: 21 pages, 3 figure

Trading interactions for topology in scale-free networks

Scale-free networks with topology-dependent interactions are studied. It is shown that the universality classes of critical behavior, which conventionally depend only on topology, can also be explored by tuning the interactions. A mapping, $\gamma' = (\gamma - \mu)/(1-\mu)$, describes how a shift of the standard exponent $\gamma$ of the degree distribution $P(q)$ can absorb the effect of degree-dependent pair interactions $J_{ij} \propto (q_iq_j)^{-\mu}$. Replica technique, cavity method and Monte Carlo simulation support the physical picture suggested by Landau theory for the critical exponents and by the Bethe-Peierls approximation for the critical temperature. The equivalence of topology and interaction holds for equilibrium and non-equilibrium systems, and is illustrated with interdisciplinary applications.Comment: 4 pages, 5 figure