7,691 research outputs found
Spectra of Modular Random Graphs
We compute spectra of symmetric random matrices defined on graphs exhibiting
a modular structure. Modules are initially introduced as fully connected
sub-units of a graph. By contrast, inter-module connectivity is taken to be
incomplete. Two different types of inter-module connectivity are considered,
one where the number of intermodule connections per-node diverges, and one
where this number remains finite in the infinite module-size limit. In the
first case, results can be understood as a perturbation of a superposition of
semicircular spectral densities one would obtain for uncoupled modules. In the
second case, matters can be more involved, and depend in detail on inter-module
connectivities. For suitable parameters we even find near-triangular shaped
spectral densities, similar to those observed in certain scale-free networks,
in a system of consisting of just two coupled modules. Analytic results are
presented for the infinite module-size limit; they are well corroborated by
numerical simulations.Comment: 16 pages, 4 figures. to appear in J. Phys.
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
Denying Access to Legal Representation: The Attack on the Tulane Law Clinic
When people require assistance to advance public interests, rather than private interests, the lack of legal representation is even more severe—less than .001% of lawyers in the legal profession are public interest lawyers. “Although recent data are unavailable, the best available estimates suggest that the number of full-time public interest lawyers is less than one attorney for every 240,000 Americans.” Citizens advancing issues of public concern often have no choice but to turn for free assistance from law school professors or one of the nation’s law school clinics
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