Analysis of weighted networks

The connections in many networks are not merely binary entities, either present or not, but have associated weights that record their strengths relative to one another. Recent studies of networks have, by and large, steered clear of such weighted networks, which are often perceived as being harder to analyze than their unweighted counterparts. Here we point out that weighted networks can in many cases be analyzed using a simple mapping from a weighted network to an unweighted multigraph, allowing us to apply standard techniques for unweighted graphs to weighted ones as well. We give a number of examples of the method, including an algorithm for detecting community structure in weighted networks and a new and simple proof of the max-flow/min-cut theorem.Comment: 9 pages, 3 figure

(Quantum) Space-Time as a Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces

We develop a kind of pregeometry consisting of a web of overlapping fuzzy lumps which interact with each other. The individual lumps are understood as certain closely entangled subgraphs (cliques) in a dynamically evolving network which, in a certain approximation, can be visualized as a time-dependent random graph. This strand of ideas is merged with another one, deriving from ideas, developed some time ago by Menger et al, that is, the concept of probabilistic- or random metric spaces, representing a natural extension of the metrical continuum into a more microscopic regime. It is our general goal to find a better adapted geometric environment for the description of microphysics. In this sense one may it also view as a dynamical randomisation of the causal-set framework developed by e.g. Sorkin et al. In doing this we incorporate, as a perhaps new aspect, various concepts from fuzzy set theory.Comment: 25 pages, Latex, no figures, some references added, some minor changes added relating to previous wor

Inductive Construction of 2-Connected Graphs for Calculating the Virial Coefficients

In this paper we give a method for constructing systematically all simple 2-connected graphs with n vertices from the set of simple 2-connected graphs with n-1 vertices, by means of two operations: subdivision of an edge and addition of a vertex. The motivation of our study comes from the theory of non-ideal gases and, more specifically, from the virial equation of state. It is a known result of Statistical Mechanics that the coefficients in the virial equation of state are sums over labelled 2-connected graphs. These graphs correspond to clusters of particles. Thus, theoretically, the virial coefficients of any order can be calculated by means of 2-connected graphs used in the virial coefficient of the previous order. Our main result gives a method for constructing inductively all simple 2-connected graphs, by induction on the number of vertices. Moreover, the two operations we are using maintain the correspondence between graphs and clusters of particles.Comment: 23 pages, 5 figures, 3 table

Representations of Menger (2,n)(2,n)-semigroups by multiplace functions

Investigation of partial multiplace functions by algebraic methods plays an important role in modern mathematics were we consider various operations on sets of functions, which are naturally defined. The basic operation for $n$-place functions is an $(n+1)$-ary superposition $[ ]$, but there are some other naturally defined operations, which are also worth of consideration. In this paper we consider binary Mann's compositions \op{1},...,\op{n} for partial $n$-place functions, which have many important applications for the study of binary and $n$-ary operations. We present methods of representations of such algebras by $n$-place functions and find an abstract characterization of the set of $n$-place functions closed with respect to the set-theoretic inclusion

Complete shutdown of microvascular perfusion upon hepatic cryothermia is critically dependent on local tissue temperature

Since microvascular dysfunction with complete circulatory arrest and, thus, prolongation of tissue ischaemia is considered a potential mechanism for cell necrosis following hepatic cryosurgery, we determined the temperature necessary for induction of complete nutritive perfusion failure in cryothermia-treated rat livers. After localization of the cryoprobe with seven thermocouples and application of a single or double freeze–thaw cycle, in vivo fluorescence microscopy of the cryoinjured left lobe was performed over a 2-h period using a computer-controlled stepping motor, which guaranteed analysis of the identical liver tissue segments with exact allocation of the thermocouples and thus determination of tissue temperature. Cryothermia resulted in a central non-perfused part of injury, surrounded by a heterogeneously perfused peripheral zone. The non-perfused area after single and double freezing continuously increased over the first 90-min period due to a successive shutdown of perfusion within the peripheral border zone. Analysis of the thermocouples' temperature at the end of freezing revealed the 0°C-front at 11.7 mm (single freeze–thaw cycle) and 12.1 mm (double freeze–thaw cycle) distant from the centre of the cryoprobe, which exactly corresponds with the initial (30 min) expansion of the area with nutritive perfusion failure. The increased non-perfused tissue area at 2 h conformed a critical border temperature between 8.29 ± 1.63°C and 9.07 ± 0.24°C. From these findings, we conclude that freezing of liver tissue to temperatures of at least < 0°C causes complete/irreversible perfusion failure, which consequently will result in cell death and tissue necrosis, and may thus be supposed as a prerequisite for the safe and successful application of cryosurgery in hepatic tumour ablation. © 2000 Cancer Research Campaig

Local stimulation of articular cartilage repair by transplantation of encapsulated chondrocytes overexpressing human fibroblast growth factor 2 (FGF-2) in vivo

Background Defects of articular cartilage are an unsolved problem in orthopaedics. In the present study, we tested the hypothesis that gene transfer of human fibroblast growth factor 2 (FGF-2) via transplantation of encapsulated genetically modified articular chondrocytes stimulates chondrogenesis in cartilage defects in vivo. Methods Lapine articular chondrocytes overexpressing a lacZ or a human FGF-2 gene sequence were encapsulated in alginate and further characterized. The resulting lacZ or FGF-2 spheres were applied to cartilage defects in the knee joints of rabbits. In vivo, cartilage repair was assessed qualitatively and quantitatively at 3 and 14 weeks after implantation. Results In vitro, bioactive FGF-2 was secreted, leading to a significant increase in the cell numbers in FGF-2 spheres. In vivo, FGF-2 continued to be expressed for at least 3 weeks without leading to differences in FGF-2 concentrations in the synovial fluid between treatment groups. Histological analysis revealed no adverse pathologic effects on the synovial membrane at any time point. FGF-2 gene transfer enhanced type II collagen expression and individual parameters of chondrogenesis, such as the cell morphology and architecture of the new tissue. Overall articular cartilage repair was significantly improved at both time points in vivo. Conclusions The data suggest that localized overexpression of FGF-2 enhances the repair of cartilage defects via stimulation of chondrogenesis, without adverse effects on the synovial membrane. These results may lead to the development of safe gene-based therapies for human articular cartilage defects