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The statistical mechanical approach to complex networks is the dominant paradigm in describing natural and societal complex systems. The study of network properties, and their implications on dynamical processes, mostly focus on locally defined quantities of nodes and edges, such as node degrees, edge weights and -more recently- correlations between(More)
We show that the symmetric product of a flat affine scheme over a commutative ring can be embedded into the quotient by the general linear group of the scheme of commuting matrices. We also prove that the symmetric product of the affine space is isomorphic to the above quotient when the base ring is a characteristic zero field. Over an infinite field of(More)
Networks, as efficient representations of complex systems, have appealed to scientists for a long time and now permeate many areas of science, including neuroimaging (Bullmore and Sporns 2009 Nat. Rev. Neurosci. 10, 186-198. (doi:10.1038/nrn2618)). Traditionally, the structure of complex networks has been studied through their statistical properties and(More)
Let k be a commutative ring and let R be a commutative k−algebra. The aim of this paper is to define and discuss some connection morphisms between schemes associated to the representation theory of a (non necessarily commutative) R−algebra A. We focus on the scheme Rep n A //GLn of the n−dimensional representations of A, on the Hilbert scheme Hilb n A(More)
A multifiltration is a functor indexed by N r that maps any mor-phism to a monomorphism. The goal of this paper is to describe in an explicit and combinatorial way the natural N r-graded R[x 1 ,. .. , xr]-module structure on the homology of a multifiltration of simplicial complexes. To do that we study multifiltrations of sets and vector spaces. We prove in(More)