Critical random graphs: limiting constructions and distributional properties

We consider the Erdos-Renyi random graph G(n,p) inside the critical window, where p = 1/n + lambda * n^{-4/3} for some lambda in R. We proved in a previous paper (arXiv:0903.4730) that considering the connected components of G(n,p) as a sequence of metric spaces with the graph distance rescaled by n^{-1/3} and letting n go to infinity yields a non-trivial sequence of limit metric spaces C = (C_1, C_2, ...). These limit metric spaces can be constructed from certain random real trees with vertex-identifications. For a single such metric space, we give here two equivalent constructions, both of which are in terms of more standard probabilistic objects. The first is a global construction using Dirichlet random variables and Aldous' Brownian continuum random tree. The second is a recursive construction from an inhomogeneous Poisson point process on R_+. These constructions allow us to characterize the distributions of the masses and lengths in the constituent parts of a limit component when it is decomposed according to its cycle structure. In particular, this strengthens results of Luczak, Pittel and Wierman by providing precise distributional convergence for the lengths of paths between kernel vertices and the length of a shortest cycle, within any fixed limit component.Comment: 30 pages, 4 figure

Critical random graphs : limiting constructions and distributional properties

We consider the Erdos-Renyi random graph G(n, p) inside the critical window, where p = 1/n + lambda n(-4/3) for some lambda is an element of R. We proved in Addario-Berry et al. [2009+] that considering the connected components of G(n, p) as a sequence of metric spaces with the graph distance rescaled by n(-1/3) and letting n -> infinity yields a non-trivial sequence of limit metric spaces C = (C-1, C-2,...). These limit metric spaces can be constructed from certain random real trees with vertex-identifications. For a single such metric space, we give here two equivalent constructions, both of which are in terms of more standard probabilistic objects. The first is a global construction using Dirichlet random variables and Aldous' Brownian continuum random tree. The second is a recursive construction from an inhomogeneous Poisson point process on R+. These constructions allow us to characterize the distributions of the masses and lengths in the constituent parts of a limit component when it is decomposed according to its cycle structure. In particular, this strengthens results of Luczak et al. [1994] by providing precise distributional convergence for the lengths of paths between kernel vertices and the length of a shortest cycle, within any fixed limit component

Statistical Mechanics of Steiner trees

The Minimum Weight Steiner Tree (MST) is an important combinatorial optimization problem over networks that has applications in a wide range of fields. Here we discuss a general technique to translate the imposed global connectivity constrain into many local ones that can be analyzed with cavity equation techniques. This approach leads to a new optimization algorithm for MST and allows to analyze the statistical mechanics properties of MST on random graphs of various types

Exact asymptotics of the freezing transition of a logarithmically correlated random energy model

We consider a logarithmically correlated random energy model, namely a model for directed polymers on a Cayley tree, which was introduced by Derrida and Spohn. We prove asymptotic properties of a generating function of the partition function of the model by studying a discrete time analogy of the KPP-equation - thus translating Bramson's work on the KPP-equation into a discrete time case. We also discuss connections to extreme value statistics of a branching random walk and a rescaled multiplicative cascade measure beyond the critical point

Inverting the cut-tree transform

We consider fragmentations of an R-tree T driven by cuts arriving according to a Poisson process on T × [0, ∞), where the first co-ordinate specifies the location of the cut and the second the time at which it oc- curs. The genealogy of such a fragmentation is encoded by the so-called cut-tree, which was introduced by Bertoin and Miermont [17] for a fragmentation of the Brownian continuum random tree. The cut-tree was generalised by Dieuleveut [25] to a fragmentation of the α-stable trees, α ∈ (1, 2), and by Broutin and Wang [19] to the inhomoge- neous continuum random trees of Aldous and Pitman [11]. In the first two cases, the projections of the forest-valued fragmentation processes onto the sequence of masses of their constituent subtrees yield an important family of ex- amples of Bertoin’s self-similar fragmentations [14]; in the first case the time-reversal of the fragmentation gives an additive coalescent. Remarkably, in all of these cases, the law of the cut-tree is the same as that of the original R-tree. In this paper, we develop a clean general framework for the study of cut-trees of R-trees. We then focus particularly on the problem of reconstruction: how to recover the original R-tree from its cut-tree. This has been studied in the setting of the Brownian CRT by Broutin and Wang [20], who prove that it is possible to reconstruct the original tree in distribution. We describe an enrichment of the cut-tree transformation, which endows the cut-tree with information we call a consistent collection of routings. We show this procedure is well-defined under minimal conditions on the R-trees. We then show that, for the case of the Brownian CRT and the α-stable trees with α ∈ (1, 2), the original tree and the Poisson process of cuts thereon can both be almost surely reconstructed from the enriched cut-trees. For the latter results, our methods make essential use of the self-similarity and re-rooting invariance of these trees

Reconceptualizing Massachusetts’ Public High School Mentoring Programs

Abstract The goals of most secondary school mentoring programs are to advance the skills and competencies of beginning teachers while simultaneously reducing attrition and improving student learning. In Massachusetts, there is a lack of widespread efficacy in existing public high school mentoring programs. The purpose of this study was to examine how to reconceptualize such programs to increase efficacy. The study employed an explanatory sequential mixed methods research design. An online survey compiled quantitative and qualitative data from 38 mentor teachers across Massachusetts. Six of the 38 online survey respondents were interviewed to garner additional, in-depth, qualitative data. Both the quantitative and qualitative instruments addressed three guiding research questions: (a) What do experienced public high school mentor teachers understand about the practice of mentoring? (b)What do experienced public high school mentor teachers report are effective educational mentoring practices? (c) What do experienced public high school mentor teachers identify as factors and conditions that contribute to and inhibit educational mentoring? Adhering to phenomenological research, significant statements were disaggregated from the online survey and interview transcripts and coded for meaning. Codes were categorized and analyzed for themes which resulted in five findings. The findings revealed that, according to mentor teacher perceptions, current mentoring programs need significant revision and augmentation. Recommendations include mentoring programs based on explicitly stated goals, a focus on beginning teacher self-care, increased time for mentors and mentees to collaborate, and the formation of a mentoring program committee to oversee the complete cycle of school mentoring programs. Key words: mentor, mentee, beginning teacher, mentoring program, mentorshi

Acyclic dominating partitions,

Abstract Given a graph G = (V, E), let P be a partition of V . We say that P is dominating if, for each part P of P, the set V \ P is a dominating set in G (equivalently, if every vertex has a neighbour of a different colour from its own). We say that P is acyclic if for any parts P, P of P, the bipartite subgraph G [P, P ] consisting of the edges between P and P in P contains no cycles. The acyclic dominating number ad(G) of G is the least number of parts in any partition of V that is both acyclic and dominating; and we shall denote by ad(d) the maximum over all graphs G of maximum degree at most d of ad(G). In this paper, we prove that a

Slowdown for time inhomogeneous branching Brownian motion

We consider the maximal displacement of one dimensional branching Brownian motion with (macroscopically) time varying profiles. For monotone decreasing variances, we show that the correction from linear displacement is not logarithmic but rather proportional to $T^{1/3}$. We conjecture that this is the worse case correction possible