Scale-freeness for networks as a degenerate ground state: A Hamiltonian formulation

The origin of scale-free degree distributions in the context of networks is addressed through an analogous non-network model in which the node degree corresponds to the number of balls in a box and the rewiring of links to balls moving between the boxes. A statistical mechanical formulation is presented and the corresponding Hamiltonian is derived. The energy, the entropy, as well as the degree distribution and its fluctuations are investigated at various temperatures. The scale-free distribution is shown to correspond to the degenerate ground state, which has small fluctuations in the degree distribution and yet a large entropy. We suggest an implication of our results from the viewpoint of the stability in evolution of networks.Comment: 7 pages, 3 figures. To appear in Europhysics lette

Information and Control in Networks

Information and Control in Networks demonstrates the way in which system dynamics and information flows intertwine as they evolve, and the central role played by information in the control of complex networked systems. It is a milestone on the road to that convergence from traditionally independent development of control theory and information theory which has emerged strongly in the last fifteen years, and is now a very active research field. In addition to efforts in control and information theory, the text is witness to strong research in such diverse fields as computer science, mathematics, and statistics. Aspects that are given specialist treatment include: ·                 data-rate theorems; ·                 computation and control over communication networks; ·                 decentralized stochastic control; ·                 Gaussian networks and Gaussian–Markov random fields; and ·                 routability in information networks. Information and Control in Networks collects contributions from world-leading researchers in the area who came together for the Lund Center for Control of Complex Engineering Systems Workshop in Information and Control in Networks from 17th–19th October 2012; the workshop being the centrepiece of a five-week-long focus period on the same theme. A source of exciting cross-fertilization and new ideas for extensive future research, this volume will be of great interest to any researcher or graduate student interested in the interaction of control and information theory

Neutral theory of chemical reaction networks

To what extent do the characteristic features of a chemical reaction network reflect its purpose and function? In general, one argues that correlations between specific features and specific functions are key to understanding a complex structure. However, specific features may sometimes be neutral and uncorrelated with any system-specific purpose, function or causal chain. Such neutral features are caused by chance and randomness. Here we compare two classes of chemical networks: one that has been subjected to biological evolution (the chemical reaction network of metabolism in living cells) and one that has not (the atmospheric planetary chemical reaction networks). Their degree distributions are shown to share the very same neutral system-independent features. The shape of the broad distributions is to a large extent controlled by a single parameter, the network size. From this perspective, there is little difference between atmospheric and metabolic networks; they are just different sizes of the same random assembling network. In other words, the shape of the degree distribution is a neutral characteristic feature and has no functional or evolutionary implications in itself; it is not a matter of life and death.Comment: 13 pages, 8 figure

A theoretical study of the 1B2u and 1B1u vibronic bands in benzene

The two lowest bands, 1B2u and 1B1u, of the electronic spectrum of the benzene molecule have been studied theoretically using a new method to compute vibronic excitation energies and intensities. The complete active space (CAS) self-contained field (SCF) method (with six active π-orbitals) was used to compute harmonic force field for the ground state and the 1B2u and 1B1u electronic states. A linear approximation has been used for the transition dipole as a function of the nuclear displacement coordinates. Derivatives of the transition dipole were computed using a variant of the CASSCF state interaction method. Multiconfigurational second-order perturbation theory (CASPT2) was used to obtain absolute excitation energies (12 active π-orbitals). The results show that the approach works well. Vibrational progressions are well described in both bands and intensities, and energies are in agreement with experiment, in particular when CASPT2 derived geometries are used. One interesting result is that computed vertical energies fall about 0.1 eV on the high energy side of the band [email protected]

The Blind Watchmaker Network: Scale-freeness and Evolution

It is suggested that the degree distribution for networks of the cell-metabolism for simple organisms reflects an ubiquitous randomness. This implies that natural selection has exerted no or very little pressure on the network degree distribution during evolution. The corresponding random network, here termed the blind watchmaker network has a power-law degree distribution with an exponent gamma >= 2. It is random with respect to a complete set of network states characterized by a description of which links are attached to a node as well as a time-ordering of these links. No a priory assumption of any growth mechanism or evolution process is made. It is found that the degree distribution of the blind watchmaker network agrees very precisely with that of the metabolic networks. This implies that the evolutionary pathway of the cell-metabolism, when projected onto a metabolic network representation, has remained statistically random with respect to a complete set of network states. This suggests that even a biological system, which due to natural selection has developed an enormous specificity like the cellular metabolism, nevertheless can, at the same time, display well defined characteristics emanating from the ubiquitous inherent random element of Darwinian evolution. The fact that also completely random networks may have scale-free node distributions gives a new perspective on the origin of scale-free networks in general.Comment: 5 pages, 3 figure

CFT description of the Fully Frustrated XY model and phase diagram analysis

Following a suggestion given in Nucl. Phys. B 300 (1988)611,we show how the U(1)*Z_{2} symmetry of the fully frustrated XY (FFXY) model on a square lattice can be accounted for in the framework of the m-reduction procedure developed for a Quantum Hall system at "paired states" fillings nu =1 (cfr. Cristofano et al.,Mod. Phys. Lett. A 15 (2000)1679;Nucl. Phys. B 641 (2002)547). The resulting twisted conformal field theory (CFT) with central charge c=2 is shown to well describe the physical properties of the FFXY model. In particular the whole phase diagram is recovered by analyzing the flow from the Z_{2} degenerate vacuum of the c=2 CFT to the infrared fixed point unique vacuum of the c=3/2 CFT. The last theory is known to successfully describe the critical behavior of the system at the overlap temperature for the Ising and vortex-unbinding transitions.Comment: 18 pages, 1 figure, to appear in JSTA

Languages cool as they expand: Allometric scaling and the decreasing need for new words

We analyze the occurrence frequencies of over 15 million words recorded in millions of books published during the past two centuries in seven different languages. For all languages and chronological subsets of the data we confirm that two scaling regimes characterize the word frequency distributions, with only the more common words obeying the classic Zipf law. Using corpora of unprecedented size, we test the allometric scaling relation between the corpus size and the vocabulary size of growing languages to demonstrate a decreasing marginal need for new words, a feature that is likely related to the underlying correlations between words. We calculate the annual growth fluctuations of word use which has a decreasing trend as the corpus size increases, indicating a slowdown in linguistic evolution following language expansion. This ‘‘cooling pattern’’ forms the basis of a third statistical regularity, which unlike the Zipf and the Heaps law, is dynamical in nature

Scaling Laws in Human Language

Zipf's law on word frequency is observed in English, French, Spanish, Italian, and so on, yet it does not hold for Chinese, Japanese or Korean characters. A model for writing process is proposed to explain the above difference, which takes into account the effects of finite vocabulary size. Experiments, simulations and analytical solution agree well with each other. The results show that the frequency distribution follows a power law with exponent being equal to 1, at which the corresponding Zipf's exponent diverges. Actually, the distribution obeys exponential form in the Zipf's plot. Deviating from the Heaps' law, the number of distinct words grows with the text length in three stages: It grows linearly in the beginning, then turns to a logarithmical form, and eventually saturates. This work refines previous understanding about Zipf's law and Heaps' law in language systems.Comment: 6 pages, 4 figure

Statistical Laws Governing Fluctuations in Word Use from Word Birth to Word Death

We analyze the dynamic properties of 10^7 words recorded in English, Spanish and Hebrew over the period 1800--2008 in order to gain insight into the coevolution of language and culture. We report language independent patterns useful as benchmarks for theoretical models of language evolution. A significantly decreasing (increasing) trend in the birth (death) rate of words indicates a recent shift in the selection laws governing word use. For new words, we observe a peak in the growth-rate fluctuations around 40 years after introduction, consistent with the typical entry time into standard dictionaries and the human generational timescale. Pronounced changes in the dynamics of language during periods of war shows that word correlations, occurring across time and between words, are largely influenced by coevolutionary social, technological, and political factors. We quantify cultural memory by analyzing the long-term correlations in the use of individual words using detrended fluctuation analysis.Comment: Version 1: 31 pages, 17 figures, 3 tables. Version 2 is streamlined, eliminates substantial material and incorporates referee comments: 19 pages, 14 figures, 3 table

Size-Dependency of Income Distributions and Its Implications

This paper highlights the size-dependency of income distributions, i.e. the income distribution curves versus the population of a country systematically. By using the generalized Lotka-Volterra model to fit the empirical income data in the United States during 1996-2007, we found an important parameter $\lambda$ can scale with a $\beta$ power of the size (population) of U.S. in that year. We pointed out that the size-dependency of the income distributions, which is a very important property but seldom addressed by previous studies, has two non-trivial implications: (1) the allometric growth pattern, i.e. the power law relationship between population and GDP in different years, which can be mathematically derived from the size-dependent income distributions and also supported by the empirical data; (2) the connection with the anomalous scaling for the probability density function in critical phenomena since the re-scaled form of the income distributions has the exactly same mathematical expression for the limit distribution of the sum of many correlated random variables asymptotically.Comment: 4 figures, 4 page