Spectral Renormalization Group for the Gaussian model and ψ4\psi^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 $\psi^4$ perturbation up to second order. However, on generalized diamond lattices, non-Gaussian fixed points arise for $2<\tilde{d}<4$.Comment: 16 pages, 14 figures, 5 tables. The paper has been extended to include a $\psi^4$ interactions and higher spectral dimension

Hamiltonian model for multidimensional epistasis

We propose and solve a Hamiltonian model for multidimensional epistastatic interactions between beneficial mutations. The model is able to give rise either to a phase transition between two equilibrium states, without any coexistence, or exhibits a state where hybrid species can coexist, with gradual passage from one wild type to another. The transition takes place as a function of the "tolerance" of the environment, which we define as the amount of noise in the system.Comment: 3 pages, 2 figures (in seperate files) spelling corrected and a reference adde

İçerik-temelli ağlar üzerinde analitik hesaplar

Content-based networks have been proposed (Balcan and Erzan, 2004; Mungan et al., 2005) to model the topological properties of complex networks built on the principle of information sharing, where the interactions between system components assume the simultaneous fulfillment of a series of constraints (Mezard et al., 2002). In content-based networks, the constraint-satisfaction problem is realized by means of a sequence-matching rule between sequences associated with the nodes of a network. In the case of transcriptional gene regulation, the transcription factors recognize special subsequences of DNA and bind them. This is one instance of constraint-satisfaction, which can be realized with a sequence-matching rule between two different classes of sequences (Balcan et al., 2006). Another example is the so called the RNA interference (Balcan and Erzan, 2004), where sequence-specific gene silencing occurs at the level of post-transcriptional gene regulation. In our content-based networks, n linear codes are associated with each node of the network. For n=2, one of the sequences associated with the node represents the key-sequence through which the node recognizes other nodes, whereas the second sequence represents the lock-sequence through which the same node is recognized. An interaction between a pair of nodes is established if the key-sequence associated with the first node is repeated as an uninterrupted subsequence in the lock-sequence associated with the second node. Thus, the length distributions of these sequences are the most important parameters determining the topological properties of the content-based networks. In this article we will discuss the validity of analytical calculations performed on the topological properties of content-based networks in the mean-field approximation (Balcan and Erzan, 2007), by means of two examples. In this mean field approach (Mungan et al., 2005) the pair-wise connectivity probabilities are only functions of the respective lengths of the sequences which must satisfy an inclusion requirement, and of the size r of the alphabet from which the symbols are drawn. This approximation ignores the correlations between the overlapping subsequences within a sequence. Moreover the fluctuations in the information content of finite sequences are neglected. In Balcan and Erzan (2007), the correlations between the edges co-incident on the same node were also ignored. In the first example, the key- sequences of unit length (thus, they consist of single letters) are searched in lock-sequences of an arbitrary fixed length. Via this simple example it is possible to show that the probability that lock-sequences will be recognized by a key-sequence depends not only on the length of the lock-sequence but also on the number of distinct subsequences embedded in it. At this point the coarse grained approximation neglecting the fluctuations in the information content of the finite lock sequences about their mean information content, misses the behavior of the in-degree distribution. This error is in fact identical to neglecting the correlations between edges incident upon a given node. In the second example, the lengths of the key sequences are fixed at an arbitrary value l, and the lock-sequences are chosen to be of length k=l+1, one character longer than the key-sequences. In this example, it is clear that the correlations between the two subsequences of length l cannot be neglected. It has already been shown (Guibas and Odlyzko, 1981; Mungan et al., 2005; Mungan, 2007; Bilge et al., 2004) that the connection probability of a key-sequence depends on the ?shift-match number? which measures the auto-correlations within a sequence, in other words, the degree to which successive subsequences are correlated with each other. We show here by an explicit and rather transparent calculation that, neglecting this correlation yields out- and in-degree distributions that are totally in error. The mean-field approximations used in the calculation of the topological properties of the double-string model (Balcan and Erzan, 2007) yield results that are in good agreement with the simulations, since i) the lengths k of the lock sequences far exceed r, ii) the number of distinct substrings contained in any given lock string is large ( k-l >> rl ) and iii) the fine structure of the topological properties are determined by the fact that there is a disribution of lock- and key-string lengths. Keywords: Complex networks, content-based networks, mean-field approach.Bu makalede i&ccedil;erik-temelli ağlar &uuml;zerinde, ağın topolojik &ouml;zelliklerini belirlemek i&ccedil;in, ortalama-alan yaklaşımlarıyla yapılan analitik hesapların g&uuml;venilirliği tartışılacaktır. İ&ccedil;erik-temelli ağları, &ldquo;tanıma ve bağlanma&rdquo; mekanizmalarının belirleyici olduğu kontrol &ccedil;izgelerinin topolojik &ouml;zelliklerini tasvir etmek i&ccedil;in &ouml;nermiştik. Bir&ccedil;ok karmaşık ağ yapısının bu t&uuml;r enformasyon paylaşımına dayalı bir prensibe g&ouml;re inşa edildiğini s&ouml;yleyebiliriz. &Ouml;rneğin gen ifadesinin d&uuml;zenlemesinde, anahtar/kilit olarak niteleyebileceğimiz elemanların &ouml;zelleşmiş etkileşimleri s&ouml;z konusudur. Bu sebeple modelimizin biyolojik &ccedil;izgeler de dahil olmak &uuml;zere, bir&ccedil;ok ger&ccedil;ek ağ yapısının tasviri i&ccedil;in uygun olduğunu d&uuml;ş&uuml;n&uuml;yoruz. İ&ccedil;erik-temelli ağımızda, ağın d&uuml;ğ&uuml;mlerini bir ya da birden fazla rastgele dizi ile eşleştirip, d&uuml;ğ&uuml;mler arasındaki etkileşimleri onlara atanan dizilerin birbirleri i&ccedil;inde tekrarlanma koşulu altında inşa ediyoruz. B&ouml;ylece, bu dizilerin uzunlukları ve i&ccedil;erikleri, ortaya &ccedil;ıkacak olan &ccedil;izgenin t&uuml;m topolojik &ouml;zelliklerini belirlemektedir. D&uuml;ğ&uuml;m &ccedil;iftleri arasındaki bağlanma olasılıklarının hesabında yapılan ortalama-alan yaklaşımlarının ise, dizilerin uzunluk dağılımlarına bağlı olarak, varılan sonu&ccedil;larda ağın ger&ccedil;ek &ouml;zelliklerinden &ouml;nemli farklılaşmalara yol a&ccedil;abileceği g&ouml;r&uuml;l&uuml;yor. Bu yaklaşımlarda, dizilerin farklı enformasyon i&ccedil;erikleri ihmal edilmekte ve olasılıklar sadece dizilerin uzunlukları cinsinden elde edilmektedir. Halbuki her sonlu dizi i&ccedil;in, dizinin i&ccedil;erdiği farklı sembol sayısı ek bir enformasyon i&ccedil;ermektedir. Burada sergilemeye &ccedil;alışacağımız, kabalaştırılmış ortalama-alan t&uuml;r&uuml;nden yaklaşımların, belli ekstrem durumlarda, tasvir etmeyi ama&ccedil;ladıkları ağın &ouml;zelliklerinden uzak sonu&ccedil;lar verebileceğidir. Ancak ger&ccedil;ek biyolojik ağ yapılarının modellenmesinde karşımıza &ccedil;ıkan uzunluk dağılımlarında ortaya &ccedil;ıkan hatalar hi&ccedil;bir zaman burada sergileyeceğimiz &ouml;rneklerde olduğu kadar b&uuml;y&uuml;k olmamış, bilakis ortalama-alan &nbsp;yaklaşımı sim&uuml;lasyon sonu&ccedil;larına olduk&ccedil;a yakın sonu&ccedil;lar vermiştir.&nbsp;Anahtar Kelimeler: Karmaşık ağ yapıları, i&ccedil;erik-temelli ağlar, ortalama-alan yaklaşımı

Novel Position-Space Renormalization Group for Bond Directed Percolation in Two Dimensions

A new position-space renormalization group approach is investigated for bond directed percolation in two dimensions. The threshold value for the bond occupation probabilities is found to be $p_c=0.6443$. Correlation length exponents on time (parallel) and space (transverse) directions are found to be $\nu_\parallel=1.719$ and $\nu_\perp=1.076$, respectively, which are in very good agreement with the best known series expansion results.Comment: Latex - Revtex, 5 pages with 6 figures, to be appeared in Physica

Analytical Solution of a Stochastic Content Based Network Model

We define and completely solve a content-based directed network whose nodes consist of random words and an adjacency rule involving perfect or approximate matches, for an alphabet with an arbitrary number of letters. The analytic expression for the out-degree distribution shows a crossover from a leading power law behavior to a log-periodic regime bounded by a different power law decay. The leading exponents in the two regions have a weak dependence on the mean word length, and an even weaker dependence on the alphabet size. The in-degree distribution, on the other hand, is much narrower and does not show scaling behavior. The results might be of interest for understanding the emergence of genomic interaction networks, which rely, to a large extent, on mechanisms based on sequence matching, and exhibit similar global features to those found here.Comment: 13 pages, 5 figures. Rewrote conclusions regarding the relevance to gene regulation networks, fixed minor errors and replaced fig. 4. Main body of paper (model and calculations) remains unchanged. Submitted for publicatio

Monte Carlo Renormalization Group for Entanglement Percolation

We use a large cell Monte Carlo Renormalization procedure, to compute the critical exponents of a system of growing linear polymers. We simulate the growth of non-intersecting chains in large MC cells. Dense regions where chains get in each others' way, give rise to connected clusters under coarse graining. At each time step, the fraction of occupied bonds is determined in both the original and the coarse grained configurations, and averaged over many realizations. Our results for the fractal dimension on three dimensional lattices are consistent with the percolation value.Comment: 5 pages, 5 figure

Glassy Dynamics of Protein Folding

A coarse grained model of a random polypeptide chain, with only discrete torsional degrees of freedom and Hookean springs connecting pairs of hydrophobic residues is shown to display stretched exponential relaxation under Metropolis dynamics at low temperatures with the exponent $\beta\simeq 1/4$, in agreement with the best experimental results. The time dependent correlation functions for fluctuations about the native state, computed in the Gaussian approximation for real proteins, have also been found to have the same functional form. Our results indicate that the energy landscape exhibits universal features over a very large range of energies and is relatively independent of the specific dynamics.Comment: RevTeX, 4 pages, multicolumn, including 5 figures; larger computations performed, error bars improve

Content based network model with duplication and divergence

We construct a minimal content-based realization of the duplication and divergence model of genomic networks introduced by Wagner [A. Wagner, Proc. Natl. Acad. Sci. {\bf 91}, 4387 (1994)] and investigate the scaling properties of the directed degree distribution and clustering coefficient. We find that the content based network exhibits crossover between two scaling regimes, with log-periodic oscillations for large degrees. These features are not present in the original gene duplication model, but inherent in the content based model of Balcan and Erzan. The scaling exponents $\gamma_1$ and $\gamma_2=\gamma_1-1/2$ of the Balcan-Erzan model turn out to be robust under duplication and point mutations, but get modified in the presence of splitting and merging of strings. The clustering coefficient as a function of the degree, $C(d)$, is found, for the Balcan-Erzan model, to behave in a way qualitatively similar to the out-degree distribution, however with a very small exponent $\alpha_1= 1-\gamma_1$ and an envelope for the oscillatory part, which is essentially flat, thus $\alpha_2= 0$. Under duplication and mutations including splitting and merging of strings, $C(d)$ is found to decay exponentially.Comment: 12 pages, 6 figure

Delocalization Transition of a Rough Adsorption-Reaction Interface

We introduce a new kinetic interface model suitable for simulating adsorption-reaction processes which take place preferentially at surface defects such as steps and vacancies. As the average interface velocity is taken to zero, the self- affine interface with Kardar-Parisi-Zhang like scaling behaviour undergoes a delocalization transition with critical exponents that fall into a novel universality class. As the critical point is approached, the interface becomes a multi-valued, multiply connected self-similar fractal set. The scaling behaviour and critical exponents of the relevant correlation functions are determined from Monte Carlo simulations and scaling arguments.Comment: 4 pages with 6 figures, new comment