arXiv subjects

Ayşe Erzan

Publications and source records attributed to Ayşe Erzan.

4 recordsLinked to original sources

arXiv2018

Explicit construction of the eigenvectors and eigenvalues of the graph Laplacian on the Cayley tree

A generalized Fourier analysis on arbitrary graphs calls for a detailed knowledge of the eigenvectors of the graph Laplacian. Using the symmetries of the Cayley tree, we recursively construct the family of eigenvectors with exponentially growing eigenspaces, associated with eigenvalues in the lower part of the spectrum. The spectral gap decays exponentially with the tree size, for large trees. The eigenvalues and eigenvectors obey recursion relations which arise from the nested geometry of the tree. Such analytical solutions for the eigenvectors of non-periodic networks are needed to provide a firm basis for the spectral renormalization group which we have proposed earlier [A. Tuncer and A. Erzan, Phys. Rev. E {\bf 92}, 022106 (2015)]. PACS Nos. 02.10.Ox Combinatorics; graph theory, 02.10.Ud Linear algebra, 02.30 Nw Fourier analysis

cond-mat.stat-mech↗

arXiv2015

Spectral Renormalization Group for the Gaussian model and $ψ^4$ theory on non-spatial networks

We implement the spectral renormalization group on different deterministic non-spatial networks without translational invariance. We calculate the thermodynamic critical exponents for the Gaussian model on the Cayley tree and the diamond lattice, and find that they are functions of the spectral dimension, $\tilde{d}$. The results are shown to be consistent with those from exact summation and finite size scaling approaches. At $\tilde{d}=2$, the lower critical dimension for the Ising universality class, the Gaussian fixed point is stable with respect to a $ψ^4$ perturbation up to second order. However, on generalized diamond lattices, non-Gaussian fixed points arise for $2<\tilde{d}<4$.

cond-mat.stat-mech↗

arXiv2015

Metanetworks of artificially evolved regulatory networks

We study metanetworks arising in genotype and phenotype spaces, in the context of a model population of Boolean graphs evolved under selection for short dynamical attractors. We define the adjacency matrix of a graph as its genotype, which gets mutated in the course of evolution, while its phenotype is its set of dynamical attractors. Metanetworks in the genotype and phenotype spaces are formed, respectively, by genetic proximity and by phenotypic similarity, the latter weighted by the sizes of the basins of attraction of the shared attractors. We find that populations of evolved networks form giant clusters in genotype space, have Poissonian degree distributions but exhibit hierarchically organized $κ$-core decompositions. Nevertheless, at large scales, they form tree-like expander graphs. Random populations of Boolean graphs are typically so far removed from each other genetically that they cannot form a metanetwork. In phenotype space, the metanetworks of evolved populations are super robust both under the elimination of weak connections and random removal of nodes.

physics.bio-ph↗

arXiv2014

Motif statistics of artificially evolved and biological networks

Topological features of gene regulatory networks can be successfully reproduced by a model population evolving under selection for short dynamical attractors. The evolved population of networks exhibit motif statistics, summarized by significance profiles, which closely match those of {\it E. coli, S. cerevsiae} and {\it B. subtilis}, in such features as the excess of linear motifs and feed-forward loops, and deficiency of feedback loops. The slow relaxation to stasis is a hallmark of a rugged fitness landscape, with independently evolving populations exploring distinct valleys strongly differing in network properties.

q-bio.MN↗