A Weighted Correlation Index for Rankings with Ties

Understanding the correlation between two different scores for the same set of items is a common problem in information retrieval, and the most commonly used statistics that quantifies this correlation is Kendall's $\tau$. However, the standard definition fails to capture that discordances between items with high rank are more important than those between items with low rank. Recently, a new measure of correlation based on average precision has been proposed to solve this problem, but like many alternative proposals in the literature it assumes that there are no ties in the scores. This is a major deficiency in a number of contexts, and in particular while comparing centrality scores on large graphs, as the obvious baseline, indegree, has a very large number of ties in web and social graphs. We propose to extend Kendall's definition in a natural way to take into account weights in the presence of ties. We prove a number of interesting mathematical properties of our generalization and describe an $O(n\log n)$ algorithm for its computation. We also validate the usefulness of our weighted measure of correlation using experimental data

Excessive functions of continuous time Markov chains

AbstractWe consider transient continuous time Markov chains P(t) with P′ij(0)=qiΠij for i≠j and −qi for i=j. We assume 0<qi<∞ for all i. Then 1/qi is the mean time the process remains in state i, and Π is the transition matrix of the imbedded jump process. We let q be a diagonal matrix with diagonal entries qi.A non-negative function h is P(t)-excessive (invariant) if h≥P(t)h, (h=P(t) h) for all t. It is Π-superregular (regular) if h≥Πh (h=Πh). Our main results characterize the excessive functions of the minimal process in terms of q and Π. These results can also be used to characterize excessive functions of certain non-minimal processes

A Place Apart

The inaugural address pf Henry J. Copeland. 9th President of the College of Wooster. The program also includes an inaugural letter by John G. Kemeny, the President of Dartmouth Collegehttps://openworks.wooster.edu/presidents/1035/thumbnail.jp

Synchronous vs. asynchronous dynamics of diffusion-controlled reactions

An analytical method based on the classical ruin problem is developed to compute the mean reaction time between two walkers undergoing a generalized random walk on a 1d lattice. At each time step, either both walkers diffuse simultaneously with probability $p$ (synchronous event) or one of them diffuses while the other remains immobile with complementary probability (asynchronous event). Reaction takes place through same site occupation or position exchange. We study the influence of the degree of synchronicity $p$ of the walkers and the lattice size $N$ on the global reaction's efficiency. For odd $N$, the purely synchronous case ($p=1$) is always the most effective one, while for even $N$, the encounter time is minimized by a combination of synchronous and asynchronous events. This new parity effect is fully confirmed by Monte Carlo simulations on 1d lattices as well as for 2d and 3d lattices. In contrast, the 1d continuum approximation valid for sufficiently large lattices predicts a monotonic increase of the efficiency as a function of $p$. The relevance of the model for several research areas is briefly discussed.Comment: 21 pages (including 12 figures and 4 tables), uses revtex4.cls, accepted for publication in Physica

Inverted regions induced by geometric constraints on a classical encounter-controlled binary reaction

The efficiency of an encounter-controlled two-channel reaction between two independently-mobile reactants on a lattice is characterized by the mean number \rt of steps to reaction. The two reactants are distinguished by their mass with the "light" walker performing a jump to a nearest-neighbor site in each time step, while the "heavy" walker hops only with a probability $p$; we associate $p$ with the "temperature" of the system. Lattices subject to periodic and to confining boundary conditions are considered. For periodic lattices, depending on the initial state, the reaction time either falls off monotonically with $p$ or displays a local minimum with respect to $p$; occurrence of the latter signals a regime where the efficiency of the reaction effectively decreases with increasing temperature. Such behavior disappears if the jump probability of the light walker falls below a characteristic threshold value. In lattices subject to confining boundary conditions, the behavior is more complex. Depending on the initial conditions, the reaction time as a function of $p$ may increase monotonically, decrease monotonically, display a single maximum or even a maximum and minimum. These inverted regions are a consequence of a strictly classical interplay between excluded volume effects implicit in the specification of the two reaction channels, and the system's dimensionality and spatial extent. Our results highlight situations where the description of an encounter-controlled reactive event cannot be described by a single, effective diffusion coefficient. We also distinguish between the inversion region identified here and the Marcus inverted region which arises in electron transfer reactions.Comment: revtex4 manuscript, approx. 45 pages, contains 18 figures and 18 tables, uses placeins.sty fil

Evolution Equation of Phenotype Distribution: General Formulation and Application to Error Catastrophe

An equation describing the evolution of phenotypic distribution is derived using methods developed in statistical physics. The equation is solved by using the singular perturbation method, and assuming that the number of bases in the genetic sequence is large. Applying the equation to the mutation-selection model by Eigen provides the critical mutation rate for the error catastrophe. Phenotypic fluctuation of clones (individuals sharing the same gene) is introduced into this evolution equation. With this formalism, it is found that the critical mutation rate is sometimes increased by the phenotypic fluctuations, i.e., noise can enhance robustness of a fitted state to mutation. Our formalism is systematic and general, while approximations to derive more tractable evolution equations are also discussed.Comment: 22 pages, 2 figure

Lumpability Abstractions of Rule-based Systems

The induction of a signaling pathway is characterized by transient complex formation and mutual posttranslational modification of proteins. To faithfully capture this combinatorial process in a mathematical model is an important challenge in systems biology. Exploiting the limited context on which most binding and modification events are conditioned, attempts have been made to reduce the combinatorial complexity by quotienting the reachable set of molecular species, into species aggregates while preserving the deterministic semantics of the thermodynamic limit. Recently we proposed a quotienting that also preserves the stochastic semantics and that is complete in the sense that the semantics of individual species can be recovered from the aggregate semantics. In this paper we prove that this quotienting yields a sufficient condition for weak lumpability and that it gives rise to a backward Markov bisimulation between the original and aggregated transition system. We illustrate the framework on a case study of the EGF/insulin receptor crosstalk.Comment: In Proceedings MeCBIC 2010, arXiv:1011.005

Bereavement and marriage are associated with antibody response to influenza vaccination in the elderly

Stressful life events exposure including bereavement, an event commonly experienced by elderly people, social support, marital status and satisfaction were examined in relation to antibody response to the annual trivalent influenza vaccination in an elderly community sample (N = 184). Antibody response was assessed at baseline, and at one and 12 months following vaccination. Taking into account baseline antibody titer, overall life events exposure and social support were not associated with response to any of the influenza strains. However, bereavement in the year prior to vaccination was negatively associated with the one-month response to the A/Panama and B/Shangdong strains. Being married and having higher marital satisfaction was also associated with higher peak responses to the A/Panama influenza strain at one month. The positive association between marital satisfaction and A/Panama response was particularly evident in the younger half of the married sample. These associations largely withstood adjustment for potential confounders. Thus, in the elderly, peak antibody response was associated with bereavement and marriage, and not the more general factors, life events and social support, related to antibody response in student samples. This suggests the importance of taking a life course approach to examining relationships between psychosocial factors and immunity, and that interventions to modify the impact of these factors should address those most salient for each age group