Self-adjoint symmetry operators connected with the magnetic Heisenberg ring

We consider symmetry operators a from the group ring C[S_N] which act on the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites. We investigate such symmetry operators a which are self-adjoint (in a sence defined in the paper) and which yield consequently observables of the Heisenberg model. We prove the following results: (i) One can construct a self-adjoint idempotent symmetry operator from every irreducible character of every subgroup of S_N. This leads to a big manifold of observables. In particular every commutation symmetry yields such an idempotent. (ii) The set of all generating idempotents of a minimal right ideal R of C[S_N] contains one and only one idempotent which ist self-adjoint. (iii) Every self-adjoint idempotent e can be decomposed into primitive idempotents e = f_1 + ... + f_k which are also self-adjoint and pairwise orthogonal. We give a computer algorithm for the calculation of such decompositions. Furthermore we present 3 additional algorithms which are helpful for the calculation of self-adjoint operators by means of discrete Fourier transforms of S_N. In our investigations we use computer calculations by means of our Mathematica packages PERMS and HRing.Comment: 13 page

Encoding dynamics for multiscale community detection: Markov time sweeping for the Map equation

The detection of community structure in networks is intimately related to finding a concise description of the network in terms of its modules. This notion has been recently exploited by the Map equation formalism (M. Rosvall and C.T. Bergstrom, PNAS, 105(4), pp.1118--1123, 2008) through an information-theoretic description of the process of coding inter- and intra-community transitions of a random walker in the network at stationarity. However, a thorough study of the relationship between the full Markov dynamics and the coding mechanism is still lacking. We show here that the original Map coding scheme, which is both block-averaged and one-step, neglects the internal structure of the communities and introduces an upper scale, the `field-of-view' limit, in the communities it can detect. As a consequence, Map is well tuned to detect clique-like communities but can lead to undesirable overpartitioning when communities are far from clique-like. We show that a signature of this behavior is a large compression gap: the Map description length is far from its ideal limit. To address this issue, we propose a simple dynamic approach that introduces time explicitly into the Map coding through the analysis of the weighted adjacency matrix of the time-dependent multistep transition matrix of the Markov process. The resulting Markov time sweeping induces a dynamical zooming across scales that can reveal (potentially multiscale) community structure above the field-of-view limit, with the relevant partitions indicated by a small compression gap.Comment: 10 pages, 6 figure

Approximating Spectral Impact of Structural Perturbations in Large Networks

Determining the effect of structural perturbations on the eigenvalue spectra of networks is an important problem because the spectra characterize not only their topological structures, but also their dynamical behavior, such as synchronization and cascading processes on networks. Here we develop a theory for estimating the change of the largest eigenvalue of the adjacency matrix or the extreme eigenvalues of the graph Laplacian when small but arbitrary set of links are added or removed from the network. We demonstrate the effectiveness of our approximation schemes using both real and artificial networks, showing in particular that we can accurately obtain the spectral ranking of small subgraphs. We also propose a local iterative scheme which computes the relative ranking of a subgraph using only the connectivity information of its neighbors within a few links. Our results may not only contribute to our theoretical understanding of dynamical processes on networks, but also lead to practical applications in ranking subgraphs of real complex networks.Comment: 9 pages, 3 figures, 2 table

Finding community structure in very large networks

The discovery and analysis of community structure in networks is a topic of considerable recent interest within the physics community, but most methods proposed so far are unsuitable for very large networks because of their computational cost. Here we present a hierarchical agglomeration algorithm for detecting community structure which is faster than many competing algorithms: its running time on a network with n vertices and m edges is O(m d log n) where d is the depth of the dendrogram describing the community structure. Many real-world networks are sparse and hierarchical, with m ~ n and d ~ log n, in which case our algorithm runs in essentially linear time, O(n log^2 n). As an example of the application of this algorithm we use it to analyze a network of items for sale on the web-site of a large online retailer, items in the network being linked if they are frequently purchased by the same buyer. The network has more than 400,000 vertices and 2 million edges. We show that our algorithm can extract meaningful communities from this network, revealing large-scale patterns present in the purchasing habits of customers

Transmission Dynamics of Highly Pathogenic Avian Influenza at Lake Constance (Europe) During the Outbreak of Winter 2005-2006

Highly pathogenic avian influenza virus (HPAI) H5N1 poses a serious threat to domestic animals. Despite the large number of studies on influenza A virus in waterbirds, little is still known about the transmission dynamics, including prevalence, behavior, and spread of these viruses in the wild waterbird population. From January to April 2006, the HPAI H5N1 virus was confirmed in 82 dead wild waterbirds at the shores of Lake Constance. In this study, we present simple mathematical models to examine this outbreak and to investigate the transmission dynamics of HPAI in wild waterbirds. The population dynamics model of wintering birds was best represented by a sinusoidal function. This model was considered the most adequate to represent the susceptible compartment of the SIR model. The three transmission models predict a basic reproduction ratio (R 0) with value of approximately 1.6, indicating a small epidemic, which ended with the migration of susceptible wild waterbirds at the end of the winter. With this study, we quantify for the first time the transmission of HPAI H5N1 virus at Lake Constance during the outbreak of winter 2005-2006. It is a step toward the improvement of the knowledge of transmission of the virus among wild waterbird

Nitrogen Spatial Heterogeneity Influences Diversity Following Restoration in a Ponderosa Pine Forest, Montana

The resource heterogeneity hypothesis (RHH) is frequently cited in the ecological literature as an important mechanism for maintaining species diversity. The RHH has rarely been evaluated in the context of restoration ecology in which a commonly cited goal is to restore diversity. In this study we focused oil the spatial heterogeneity of total inorganic nitrogen (TIN) following restoration treatments in a ponderosa pine (Pinus ponderosa)/Douglas-fir (Pseudotsuga inenziesii) forest in western Montana, USA. Our objective was to evaluate relationships between understory species richness and TIN heterogeneity following mechanical thinning (thin-only), prescribed burning (burn-only), and mechanical thinning with prescribed burning (thin/burn) to discern the ecological and management implications of these restoration approaches. We employed a randomized block design, with three 9-ha replicates of each treatment and an untreated control. Within each treatment, we randomly established a 20 X 50 in (1000 m(2)) Plot in which we measured species richness across the entire plot and in 12 I-m 2 quadrats randomly placed within each larger plot. Additionally, we measured TIN from a grid consisting of 112 soil samples (0-5 cm) in each plot and computed standard deviations as a measure of heterogeneity. We found a correlation between the net increase in species richness and the TIN standard deviations one and two years following restoration treatments, supporting RHH. Using nonmetric multidimensional scaling ordination and chi-squared analysis, we found that high and low TIN quadrats contained different understory communities in 2003 and 2004, further supporting RHH. A comparison of restoration treatments demonstrated that thin/burn and burn-only treatments created higher N heterogeneity relative to the control. We also found that within prescribed burn treatments, TIN heterogeneity was positively correlated with fine-fuel consumption, a variable reflecting burn severity. These findings may lead to more informed restoration decisions that consider treatment effects on understory diversity in ponderosa pine/Douglas-fir ecosystems