Statistical Mechanics of Community Detection

Starting from a general \textit{ansatz}, we show how community detection can be interpreted as finding the ground state of an infinite range spin glass. Our approach applies to weighted and directed networks alike. It contains the \textit{at hoc} introduced quality function from \cite{ReichardtPRL} and the modularity $Q$ as defined by Newman and Girvan \cite{Girvan03} as special cases. The community structure of the network is interpreted as the spin configuration that minimizes the energy of the spin glass with the spin states being the community indices. We elucidate the properties of the ground state configuration to give a concise definition of communities as cohesive subgroups in networks that is adaptive to the specific class of network under study. Further we show, how hierarchies and overlap in the community structure can be detected. Computationally effective local update rules for optimization procedures to find the ground state are given. We show how the \textit{ansatz} may be used to discover the community around a given node without detecting all communities in the full network and we give benchmarks for the performance of this extension. Finally, we give expectation values for the modularity of random graphs, which can be used in the assessment of statistical significance of community structure

Is a multiple excitation of a single atom equivalent to a single excitation of an ensemble of atoms?

Recent technological advances have enabled to isolate, control and measure the properties of a single atom, leading to the possibility to perform statistics on the behavior of single quantum systems. These experiments have enabled to check a question which was out of reach previously: Is the statistics of a repeatedly excitation of an atom N times equivalent to a single excitation of an ensemble of N atoms? We present a new method to analyze quantum measurements which leads to the postulation that the answer is most probably no. We discuss the merits of the analysis and its conclusion.Comment: 3 pages, 3 figure

Finite size effects and error-free communication in Gaussian channels

The efficacy of a specially constructed Gallager-type error-correcting code to communication in a Gaussian channel is being examined. The construction is based on the introduction of complex matrices, used in both encoding and decoding, which comprise sub-matrices of cascading connection values. The finite size effects are estimated for comparing the results to the bounds set by Shannon. The critical noise level achieved for certain code-rates and infinitely large systems nearly saturates the bounds set by Shannon even when the connectivity used is low

Effect of Nigella sativa L. on heart rate and some haematological values of alloxan-induced diabetic rabbits

This study was designed to investigate the effect of an extract of Nigella sativa L. on the heart rate and  some haematological values in alloxan-induced diabetic rabbits. Fifteen New Zealand male rabbits were  divided into three experimental groups: control, diabetic and N. sativa L.-treated diabetic. At the end of the  experimental period (2 months), animals in all three groups were fasted for 12 hours and blood samples  were taken for the determination of glucose levels, RBC and WBC (red and white blood cell) counts,  packed cell volume (PCV), and haemoglobin (Hb) concentration. Heart rates were also measured by a  direct-writing electrocardiograph before the blood withdrawals. It was found that N. sativa L. treatment  increased the lowered RBC and WBC counts, PCV and neutrophil percentage in diabetic rabbits. However,  the WBC count of the N. sativa L. treated diabetic group was still lower than the control. N. sativa L.  treatment also decreased the elevated heart rate and glucose concentration of diabetic rabbits. It is concluded  that oral N. sativa L. treatment might decrease the diabetes-induced disturbances of heart rate and some  haematological parameters of alloxan-induced diabetic rabbits.

Distribution of mast cells in lung tissues of rats exposed to biomass smoke

This study was designed to evaluate the distribution of mast cells in the lung tissues of rats exposed to biomass  smoke. Fifty six female Wistar albino adult rats were used. They were divided into two experimental groups  (control and biomass smoke-treated), each containing 28 animals. Control rats were not exposed to the  biomass smoke at any time during the experiment. Rats in the treatment group were exposed daily (one hour)  to biomass smoke for 3, 6 or 9 months. Lung tissues samples were obtained under deep anesthesia from the  randomly selected 7 animals in both groups. Lung tissues were fixed in Mota’s fixative (BLA) for 24 h and  embedded in paraffin. Sections of 6 μm thickness were cut and stained with 0.5% toluidine blue in 0.5 N  hydrochloric acid at pH 0.5 for 30 min. The numbers of mast cell in lung tissues of the animals exposed to  the biomass for 6 or 9 months were significantly (P<0.05) higher than controls. This study showed that long  term exposure to biomass smoke was associated with the increased number of mast cells in the lung.

Mean Field Behavior of Cluster Dynamics

The dynamic behavior of cluster algorithms is analyzed in the classical mean field limit. Rigorous analytical results below $T_c$ establish that the dynamic exponent has the value $z_{sw}=1$ for the Swendsen-Wang algorithm and $z_{uw}=0$ for the Wolff algorithm. An efficient Monte Carlo implementation is introduced, adapted for using these algorithms for fully connected graphs. Extensive simulations both above and below $T_c$ demonstrate scaling and evaluate the finite-size scaling function by means of a rather impressive collapse of the data.Comment: Revtex, 9 pages with 7 figure

Partitioning and modularity of graphs with arbitrary degree distribution

We solve the graph bi-partitioning problem in dense graphs with arbitrary degree distribution using the replica method. We find the cut-size to scale universally with . In contrast, earlier results studying the problem in graphs with a Poissonian degree distribution had found a scaling with ^1/2 [Fu and Anderson, J. Phys. A: Math. Gen. 19, 1986]. The new results also generalize to the problem of q-partitioning. They can be used to find the expected modularity Q [Newman and Grivan, Phys. Rev. E, 69, 2004] of random graphs and allow for the assessment of statistical significance of the output of community detection algorithms.Comment: Revised version including new plots and improved discussion of some mathematical detail

Microscopic Study of the Superconducting State of the Iron Pnictide RbFe_2As_2

A study of the temperature and field dependence of the penetration depth \lambda of the superconductor RbFe_2As_2 (T_c=2.52 K) was carried out by means of muon-spin rotation measurements. In addition to the zero temperature value of the penetration depth \lambda(0)=267(5) nm, a determination of the upper critical field B_c2(0)=2.6(2) T was obtained. The temperature dependence of the superconducting carrier concentration is discussed within the framework of a multi-gap scenario. Compared to the other "122" systems which exhibit much higher Fermi level, a strong reduction of the large gap BCS ratio 2\Delta/k_B T_c is observed. This is interpreted as a consequence of the absence of interband processes. Indications of possible pair-breaking effect are also discussed.Comment: 5 pages, 4 figure

Multilayer neural networks with extensively many hidden units

The information processing abilities of a multilayer neural network with a number of hidden units scaling as the input dimension are studied using statistical mechanics methods. The mapping from the input layer to the hidden units is performed by general symmetric Boolean functions whereas the hidden layer is connected to the output by either discrete or continuous couplings. Introducing an overlap in the space of Boolean functions as order parameter the storage capacity if found to scale with the logarithm of the number of implementable Boolean functions. The generalization behaviour is smooth for continuous couplings and shows a discontinuous transition to perfect generalization for discrete ones.Comment: 4 pages, 2 figure

Training a perceptron in a discrete weight space

On-line and batch learning of a perceptron in a discrete weight space, where each weight can take $2 L+1$ different values, are examined analytically and numerically. The learning algorithm is based on the training of the continuous perceptron and prediction following the clipped weights. The learning is described by a new set of order parameters, composed of the overlaps between the teacher and the continuous/clipped students. Different scenarios are examined among them on-line learning with discrete/continuous transfer functions and off-line Hebb learning. The generalization error of the clipped weights decays asymptotically as $exp(-K \alpha^2)$/$exp(-e^{|\lambda| \alpha})$ in the case of on-line learning with binary/continuous activation functions, respectively, where $\alpha$ is the number of examples divided by N, the size of the input vector and $K$ is a positive constant that decays linearly with 1/L. For finite $N$ and $L$, a perfect agreement between the discrete student and the teacher is obtained for $\alpha \propto \sqrt{L \ln(NL)}$. A crossover to the generalization error $\propto 1/\alpha$, characterized continuous weights with binary output, is obtained for synaptic depth $L > O(\sqrt{N})$.Comment: 10 pages, 5 figs., submitted to PR