The sub-editorial group which considered the interpretation papers in the following section consisted of Elizabeth Beckmann, who provided the introduction below, Pat Devlin and Stephen Wearing . Environmental interpretation occurs as part of the educational continuum that ranges from simple awareness-raising sought by promotional activities to the major attitudinal shifts often pursued in environmental lifestyle education. Interpretation has long been seen by natural resource managers and others not only as “an educational activity…to reveal meaning and relationships” (Tilden 1977) but also as a means of creating “a desire to contribute to environmental conservation” (Aldridge 1974). In 1996 how are we using interpretive theory, techniques and programs to contribute towards developing the cutting edge of environmental education?
The availability of large scale streaming network data has reinforced the ubiquity of power-law distributions in observations and enabled precision measurements of the distribution parameters. The increased accuracy of these measurements allows new underlying generative network models to be explored. The preferential attachment model is a natural starting point for these models. This work adds additional model components to account for observed phenomena in the distributions. In this model, preferential attachment is supplemented to provide a more accurate theoretical model of network traffic. Specifically, a probabilistic complex network model is proposed using preferential attachment as well as additional parameters to describe the newly observed prevalence of leaves and unattached nodes. Example distributions from this model are generated by considering random sampling of the networks created by the model in such a way that replicates the current data collection methods.
In this paper, we use the notion of twisted subgroups (i.e. subsets of group elements closed under the binary operation [Formula: see text]) to provide the first structural characterization of optimal play in the Explorer-Director game, introduced as the Magnus–Derek game by Nedev and Muthukrishnan and generalized to finite groups by Gerbner. In particular, we reduce the game to the problem of finding the largest proper twisted subgroup, and as a corollary we resolve the Explorer-Director game completely for all nilpotent groups.
Extending the notion of (random) $k$-out graphs, we consider when the $k$-out hypergraph is likely to have a perfect fractional matching. In particular, we show that for each $r$ there is a $k=k(r)$ such that the $k$-out $r$-uniform hypergraph on $n$ vertices has a perfect fractional matching with high probability (i.e., with probability tending to $1$ as $n\to \infty$) and prove an analogous result for $r$-uniform $r$-partite hypergraphs. This is based on a new notion of hypergraph expansion and the observation that sufficiently expansive hypergraphs admit perfect fractional matchings. As a further application, we give a short proof of a stopping-time result originally due to Krivelevich.
Abstract In this note we study the emergence of Hamiltonian Berge cycles in random r -uniform hypergraphs. For $r\geq 3$ we prove an optimal stopping time result that if edges are sequentially added to an initially empty r -graph, then as soon as the minimum degree is at least 2, the hypergraph with high probability has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erdős–Rényi random r -graph, and we also show that the 2 -out random r -graph with high probability has such a cycle. We obtain similar results for weak Berge cycles as well, thus resolving a conjecture of Poole.
Given a graph $G$ and some initial labelling $\sigma : V(G) \to \{Red, Blue\}$ of its vertices, the \textit{majority dynamics model} is the deterministic process where at each stage, every vertex simultaneously replaces its label with the majority label among its neighbors (remaining unchanged in the case of a tie). We prove---for a wide range of parameters---that if an initial assignment is fixed and we independently sample an Erdős--Renyi random graph, $G_{n,p}$, then after one step of majority dynamics, the number of vertices of each label follows a central limit law. As a corollary, we provide a strengthening of a theorem of Benjamini, Chan, O'Donnell, Tamuz, and Tan about the number of steps required for the process to reach unanimity when the initial assignment is also chosen randomly. Moreover, suppose there are initially three more red vertices than blue. In this setting, we prove that if we independently sample the graph $G_{n,1/2}$, then with probability at least $51\%$, the majority dynamics process will converge to every vertex being red. This improves a result of Tran and Vu who addressed the case that the initial lead is at least 10.
