Oxidation of an oil rich in docosahexaenoic acid compared to linoleic acid in lactating women

Background: We studied the oxidation of an oil rich in docosahexaenoic acid (DHA; DHASCO(R)) in lactating mothers receiving a dietary DHA supplement or a placebo. The results were compared with the oxidation of linoleic acid. Methods: Breast-feeding mothers received a dietary supplement (DHASCO; 200 mg DHA/day, n = 5) or a placebo (n = 5) for 14 days. Six weeks post partum all 10 mothers received a single dose of 2 mg/kg body weight uniformly C-13-labeled DHASCO. In a previously reported study 6 mothers received 1 mg/kg body weight uniformly C-13-labeled linoleic acid. Breath samples were collected over 48 h after tracer application. The total CO2 production was measured by indirect calorimetry and the C-13 isotopic enrichment of labeled CO2 by isotopic ratio mass spectrometry. Results: The oxidation of C-13-labeled DHASCO in the supplemented and placebo groups was similar. Maximal C-13 enrichment was reached earlier in the group receiving C-13-DHASCO (median 1.0 vs. 3.0 h in the linoleic acid group). The cumulative C-13 recovery in breath was higher in the DHASCO versus the linoleic acid group until 10 h after tracer application and comparable thereafter. Conclusions: The difference in oxidation of DHASCO versus linoleic acid after tracer ingestion might be partly due to a faster absorption and oxidation of shorter chain saturated fatty acids contained in DHASCO. The cumulative oxidation of DHASCO and linoleic acid 24 and 48 h after tracer ingestion is similar. Copyright (C) 2000 S. Karger AG, Basel

Random walks on dynamic graphs: Mixing times, hitting times, and return probabilities

We establish and generalise several bounds for various random walk quantities including the mixing time and the maximum hitting time. Unlike previous analyses, our derivations are based on rather intuitive notions of local expansion properties which allows us to capture the progress the random walk makes through t-step probabilities. We apply our framework to dynamically changing graphs, where the set of vertices is fixed while the set of edges changes in each round. For random walks on dynamic connected graphs for which the stationary distribution does not change over time, we show that their behaviour is in a certain sense similar to static graphs. For example, we show that the mixing and hitting times of any sequence of d-regular connected graphs is O(n^2), generalising a well-known result for static graphs. We also provide refined bounds depending on the isoperimetric dimension of the graph, matching again known results for static graphs. Finally, we investigate properties of random walks on dynamic graphs that are not always connected: we relate their convergence to stationarity to the spectral properties of an average of transition matrices and provide some examples that demonstrate strong discrepancies between static and dynamic graphs

On coalescence time in graphs: When is coalescing as fast as meeting?

Coalescing random walks is a fundamental stochastic process, where a set of particles perform independent discrete-time random walks on an undirected graph. Whenever two or more particles meet at a given node, they merge and continue as a single random walk. The coalescence time is defined as the expected time until only one particle remains, starting from one particle at every node. Despite recent progress the coalescence time for graphs such as binary trees, d-dimensional tori, hypercubes and more generally, vertex-transitive graphs, remains unresolved. We provide a powerful toolkit that results in tight bounds for various topologies including the aforementioned ones. The meeting time is defined as the worst-case expected time required for two random walks to arrive at the same node at the same time. As a general result, we establish that for graphs whose meeting time is only marginally larger than the mixing time (a factor of log^2 n), the coalescence time of n random walks equals the meeting time up to constant factors. This upper bound is complemented by the construction of a graph family demonstrating that this result is the best possible up to constant factors. For almost-regular graphs, we bound the coalescence time by the hitting time, resolving the discrete-time variant of a conjecture by Aldous for this class of graphs. Finally, we prove that for any graph the coalescence time is bounded by O(n^3) (which is tight for the Barbell graph); surprisingly even such a basic question about the coalescing time was not answered before this work. By duality, our results give bounds on the voter model and therefore give bounds on the consensus time in arbitrary undirected graphs. We also establish a new bound on the hitting time and cover time of regular graphs, improving and tightening previous results by Broder and Karlin, as well as those by Aldous and Fill

Bounds on the satisfiability threshold for power law distributed random SAT

Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. The worst-case hardness of SAT lies at the core of computational complexity theory. The averagecase analysis of SAT has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures. Despite a long line of research and substantial progress, nearly all theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a scale-free distribution of the variables, which results in distributions closer to industrial SAT instances. We study random k-SAT on n variables, m = ϵ(n) clauses, and a power law distribution on the variable occurrences with exponent β. We observe a satisfiability threshold at β≤(2k-1)/(k-1). This threshold is tight in the sense that instances with β ≥ (2k-1)/(k-1)-ϵ for any constant ϵ > 0 are unsatisfiable with high probability (w. h. p.). For β > (2k-1)/(k-1)+ ϵ, the picture is reminiscent of the uniform case: instances are satisfiable w. h. p. for sufficiently small constant clause-variable ratios m/n; they are unsatisfiable above a ratio m/n that depends on β

Intersection and mixing times for reversible chains

© 2017, University of Washington. All rights reserved. We consider two independent Markov chains on the same finite state space, and study their intersection time, which is the first time that the trajectories of the two chains intersect. We denote by tI the expectation of the intersection time, maximized over the starting states of the two chains. We show that, for any reversible and lazy chain, the total variation mixing time is O(tI). When the chain is reversible and transitive, we give an expression for tI using the eigenvalues of the transition matrix. In this case, we also show that tI is of order √nE[I], where I is the number of intersections of the trajectories of the two chains up to the uniform mixing time, and n is the number of states. For random walks on trees, we show that tI and the total variation mixing time are of the same order. Finally, for random walks on regular expanders, we show that tI is of order √n

A simple approach for adapting continuous load balancing processes to discrete settings

We consider the neighbourhood load balancing problem. Given a network of processors and an arbitrary distribution of tasks over the network, the goal is to balance load by exchanging tasks between neighbours. In the continuous model, tasks can be arbitrarily divided and perfectly balanced state can always be reached. This is not possible in the discrete model where tasks are non-divisible. In this paper we consider the problem in a very general setting, where the tasks can have arbitrary weights and the nodes can have different speeds. Given a continuous load balancing algorithm that balances the load perfectly in (Formula presented.) rounds, we convert the algorithm into a discrete version. This new algorithm is deterministic and balances the load in  (Formula presented.) rounds so that the difference between the average and the maximum load is at most (Formula presented.) , where d is the maximum degree of the network and  (Formula presented.) is the maximum weight of any task. For general graphs, these bounds are asymptotically lower compared to the previous results. The proposed conversion scheme can be applied to a wide class of continuous processes, including first and second order diffusion, dimension exchange, and random matching processes. For the case of identical tasks, we present a randomized version of our algorithm that balances the load up to a discrepancy of  (Formula presented.) provided that the initial load on every node is large enough.Hoda Akbari, Petra Berenbrink and Thomas Sauerwald work was supported by an NSERC Discovery Grant “Analysis of Randomized Algorithms”

Communication Complexity of Quasirandom Rumor Spreading

We consider rumor spreading on random graphs and hypercubes in the quasirandom phone call model. In this model, every node has a list of neighbors whose order is specified by an adversary. In step i every node opens a channel to its ith neighbor (modulo degree) on that list, beginning from a randomly chosen starting position. Then, the channels can be used for bi-directional communication in that step. The goal is to spread a message efficiently to all nodes of the graph.For random graphs (with sufficiently many edges) we present an address-oblivious algorithm with runtime O(logn) that uses at most O(nloglogn) message transmissions. For hypercubes of dimension logn we present an address-oblivious algorithm with runtime O(logn) that uses at most O(n(loglogn)2) message transmissions.Together with a result of Elsässer (Proc. of SPAA’06, pp. 148–157, 2006), our results imply that for random graphs the communication complexity of the quasirandom phone call model is significantly smaller than that of the standard phone call model

Randomized rumor spreading in dynamic graphs

International audienceWe consider the well-studied rumor spreading model in which nodes contact a random neighbor in each round in order to push or pull the rumor. Unlike most previous works which focus on static topologies, we look at a dynamic graph model where an adversary is allowed to rewire the connections between vertices before each round, giving rise to a sequence of graphs, G1, G2, . . . Our first result is a bound on the rumor spreading time in terms of the conductance of those graphs. We show that if the degree of each node does not change much during the protocol (that is, by at most a constant factor), then the spread completes within t rounds for some t such that the sum of conductances of the graphs G1 up to Gt is O(log n). This result holds even against an adaptive adversary whose decisions in a round may depend on the set of informed vertices before the round, and implies the known tight bound with conductance for static graphs. Next we show that for the alternative expansion measure of vertex expansion, the situation is different. An adaptive adversary can delay the spread of rumor significantly even if graphs are regular and have high expansion, unlike in the static graph case where high expansion is known to guarantee fast rumor spreading. However, if the adversary is oblivious, i.e., the graph sequence is decided before the protocol begins, then we show that a bound close to the one for the static case holds for any sequence of regular graphs

Faster rumor spreading with multiple calls

We consider the random phone call model introduced by Demers et al., which is a well-studied model for information dissemination in networks. One basic protocol in this model is the so-called Push protocol that proceeds in synchronous rounds. Starting with a single node which knows of a rumor, every informed node calls in each round a random neighbor and informs it of the rumor. The Push-Pull protocol works similarly, but additionally every uninformed node calls a random neighbor and may learn the rumor from it. It is well-known that both protocols need Θ(log n) rounds to spread a rumor on a complete network with n nodes. Here we are interested in how much the spread can be speeded up by enabling nodes to make more than one call in each round. We propose a new model where the number of calls of a node is chosen independently according to a probability distribution R. We provide both lower and upper bounds on the rumor spreading time depending on statistical properties of R such as the mean or the variance (if they exist). In particular, if R follows a power law distribution with exponent β ∈ (2, 3), we show that the Push-Pull protocol spreads a rumor in Θ(log log n) rounds. Moreover, when β = 3, the Push- Pull protocol spreads a rumor in Θ(formula presented) rounds