Small cycles, generalized prisms and Hamiltonian cycles in the Bubble-sort graph

The Bubble-sort graph $BS_n,\,n\geqslant 2$, is a Cayley graph over the symmetric group $Sym_n$ generated by transpositions from the set $\{(1 2), (2 3),\ldots, (n-1 n)\}$. It is a bipartite graph containing all even cycles of length $\ell$, where $4\leqslant \ell\leqslant n!$. We give an explicit combinatorial characterization of all its $4$- and $6$-cycles. Based on this characterization, we define generalized prisms in $BS_n,\,n\geqslant 5$, and present a new approach to construct a Hamiltonian cycle based on these generalized prisms.Comment: 13 pages, 7 figure

Speeding up non-Markovian First Passage Percolation with a few extra edges

One model of real-life spreading processes is First Passage Percolation (also called SI model) on random graphs. Social interactions often follow bursty patterns, which are usually modelled with i.i.d.~heavy-tailed passage times on edges. On the other hand, random graphs are often locally tree-like, and spreading on trees with leaves might be very slow, because of bottleneck edges with huge passage times. Here we consider the SI model with passage times following a power law distribution $\mathbb{P}(\xi>t)\sim t^{-\alpha}$, with infinite mean. For any finite connected graph $G$ with a root $s$, we find the largest number of vertices $\kappa(G,s)$ that are infected in finite expected time, and prove that for every $k \leq \kappa(G,s)$, the expected time to infect $k$ vertices is at most $O(k^{1/\alpha})$. Then, we show that adding a single edge from $s$ to a random vertex in a random tree $\mathcal{T}$ typically increases $\kappa(\mathcal{T},s)$ from a bounded variable to a fraction of the size of $\mathcal{T}$, thus severely accelerating the process. We examine this acceleration effect on some natural models of random graphs: critical Galton-Watson trees conditioned to be large, uniform spanning trees of the complete graph, and on the largest cluster of near-critical Erd\H{o}s-R\'enyi graphs. In particular, at the upper end of the critical window, the process is already much faster than exactly at criticality.Comment: 35 pages, 4 figure

Identifying exogenous and endogenous activity in social media

The occurrence of new events in a system is typically driven by external causes and by previous events taking place inside the system. This is a general statement, applying to a range of situations including, more recently, to the activity of users in Online social networks (OSNs). Here we develop a method for extracting from a series of posting times the relative contributions of exogenous, e.g. news media, and endogenous, e.g. information cascade. The method is based on the fitting of a generalized linear model (GLM) equipped with a self-excitation mechanism. We test the method with synthetic data generated by a nonlinear Hawkes process, and apply it to a real time series of tweets with a given hashtag. In the empirical dataset, the estimated contributions of exogenous and endogenous volumes are close to the amounts of original tweets and retweets respectively. We conclude by discussing the possible applications of the method, for instance in online marketing.Comment: 5 figure

Context Query Simulation for Smart Carparking Scenarios in the Melbourne CDB

The rapid growth in Internet of Things (IoT) has ushered in the way for better context-awareness enabling more smarter applications. Although for the growth in the number of IoT devices, Context Management Platforms (CMPs) that integrate different domains of IoT to produce context information lacks scalability to cater to a high volume of context queries. Research in scalability and adaptation in CMPs are of significant importance due to this reason. However, there is limited methods to benchmarks and validate research in this area due to the lack of sizable sets of context queries that could simulate real-world situations, scenarios, and scenes. Commercially collected context query logs are not publicly accessible and deploying IoT devices, and context consumers in the real-world at scale is expensive and consumes a significant effort and time. Therefore, there is a need to develop a method to reliably generate and simulate context query loads that resembles real-world scenarios to test CMPs for scale. In this paper, we propose a context query simulator for the context-aware smart car parking scenario in Melbourne Central Business District in Australia. We present the process of generating context queries using multiple real-world datasets and publicly accessible reports, followed by the context query execution process. The context query generator matches the popularity of places with the different profiles of commuters, preferences, and traffic variations to produce a dataset of context query templates containing 898,050 records. The simulator is executable over a seven-day profile which far exceeds the simulation time of any IoT system simulator. The context query generation process is also generic and context query language independent

From Traditional Adaptive Data Caching to Adaptive Context Caching: A Survey

Context data is in demand more than ever with the rapid increase in the development of many context-aware Internet of Things applications. Research in context and context-awareness is being conducted to broaden its applicability in light of many practical and technical challenges. One of the challenges is improving performance when responding to large number of context queries. Context Management Platforms that infer and deliver context to applications measure this problem using Quality of Service (QoS) parameters. Although caching is a proven way to improve QoS, transiency of context and features such as variability, heterogeneity of context queries pose an additional real-time cost management problem. This paper presents a critical survey of state-of-the-art in adaptive data caching with the objective of developing a body of knowledge in cost- and performance-efficient adaptive caching strategies. We comprehensively survey a large number of research publications and evaluate, compare, and contrast different techniques, policies, approaches, and schemes in adaptive caching. Our critical analysis is motivated by the focus on adaptively caching context as a core research problem. A formal definition for adaptive context caching is then proposed, followed by identified features and requirements of a well-designed, objective optimal adaptive context caching strategy.Comment: This paper is currently under review with ACM Computing Surveys Journal at this time of publishing in arxiv.or

Evaluation of thermal properties of insulation systems in pitched roofs

The article outlines the basic requirements for pitched roof insulation systems. The analysis of the properties of thermal insulation materials used in insulation systems was conducted. It is substantiated that the thermal resistance of such structures on the surface of the roof is formed taking into account the thermal conductivity of thermal insulation, thermal conductivity of wooden rafters and heat loss through the leakiness of joints and mounting devices. An assessment was made of the effect of loads of various types on the heat-insulating layer, namely: the effect of the air flow in the ventilated gap; the movement of the vapor-air mixture in the material; condensation of water vapor and penetration of drip liquid; exfiltration of air at the joints of the plates and on the surfaces of contact with the supporting structures. The expediency of using products on the basis of unstitched polyethylene foam in the construction of pitched roofs with a wooden roof system, taking into account the advantages and features of this material, as well as taking into account the possibility of creating a seamless insulating shell, is substantiated