Random graph model with power-law distributed triangle subgraphs

Clustering is well-known to play a prominent role in the description and understanding of complex networks, and a large spectrum of tools and ideas have been introduced to this end. In particular, it has been recognized that the abundance of small subgraphs is important. Here, we study the arrangement of triangles in a model for scale-free random graphs and determine the asymptotic behavior of the clustering coefficient, the average number of triangles, as well as the number of triangles attached to the vertex of maximum degree. We prove that triangles are power-law distributed among vertices and characterized by both vertex and edge coagulation when the degree exponent satisfies $2<\beta<2.5$; furthermore, a finite density of triangles appears as $\beta=2+1/3$.Comment: 4 pages, 2 figure; v2: major conceptual change

CO2-crystal wettability in potassic magmas. Implications for eruptive dynamics in light of experimental evidence for heterogeneous nucleation.

The volatile content in magmas is fundamental for the triggering and style of volcanic eruptions. Carbon dioxide, the second most abundant volatile component in magmas after H2O, is the first to reach saturation upon ascent and depressurization. We investigate experimentally CO2-bubble nucleation in trachybasalt and trachyte melts at high temperature and high pressure (HT and HP) through wetting-angle measurements on different (sialic, mafic or oxide) phenocryst phases. The presence of crystals lowers the supersaturation required for CO2- bubble nucleation up to 37 per cent (heterogeneous nucleation, HeN), with a minor role of mineral chemistry. Different from H2O-rich systems, feldspar crystals are effective in reducing required supersaturation for bubble nucleation. Our data suggest that leucite, the dominant liquidus phase in ultrapotassic systems at shallow depth (i.e. <100 MPa), facilitates late-stage, extensive magma vesiculation through CO2 HeN, which may explain the shifting of CO2-rich eruptive systems towards an apparently anomalous explosive behaviour