Common adversaries form alliances: modelling complex networks via anti-transitivity

Anti-transitivity captures the notion that enemies of enemies are friends, and arises naturally in the study of adversaries in social networks and in the study of conflicting nation states or organizations. We present a simplified, evolutionary model for anti-transitivity influencing link formation in complex networks, and analyze the model's network dynamics. The Iterated Local Anti-Transitivity (or ILAT) model creates anti-clone nodes in each time-step, and joins anti-clones to the parent node's non-neighbor set. The graphs generated by ILAT exhibit familiar properties of complex networks such as densification, short distances (bounded by absolute constants), and bad spectral expansion. We determine the cop and domination number for graphs generated by ILAT, and finish with an analysis of their clustering coefficients. We interpret these results within the context of real-world complex networks and present open problems

Unveiling residual, spontaneous recovery from subtle hemispatial neglect three years after stroke

A common and disabling consequence of stroke is the difficulty in processing contralesional space (i.e., hemispatial neglect). According to paper-and-pencil tests, neglect remits or stabilizes in severity within a few months after a brain injury. This arbitrary temporal limit, however, is at odds with neglect's well-known dependency on task-sensitivity. The present study tested the hypothesis that the putative early resolution of neglect might be due to the insensitivity of testing methods rather than to the lack of spontaneous recovery at later stages. A right hemisphere stroke patient was studied longitudinally for 3 years. According to paper-and-pencil tests the patient showed no symptom of hemispatial neglect 1 month post stroke. Awareness of spatially lateralized visual targets was then assessed by means of computer based single-and dual tasks requiring an additional top-down deployment of attention for the parallel processing of visual or auditory stimuli. Errorless performance at computer-based tasks was reached at month 12 and maintained until month 29 after stroke. A bottom-up manipulation was then implemented by reducing target diameter. Following this change, more than 50% of contralesional targets were omitted, mostly under dual-tasking. At months 40 and 41 the same task revealed a significant (but not complete) reduction in the number of contralesional omissions. lpsilesional targets were, in contrast, still errorless detected. The coupling of a bottom-up (target change) and a top-down (dual-tasking) manipulation revealed the presence of a long-lasting spontaneous recovery from contralesional spatial awareness deficits. In contrast, neither manipulation was effective when implemented separately. After having excluded the potential confound of practice effects, it was concluded that not only the presence but also the time course of hemispatial neglect strongly depends on the degree of attentional engagement required by the task

Characterizations and algorithms for generalized Cops and Robbers games

We propose a definition of generalized Cops and Robbers games where there are two players, the Pursuer and the Evader, who each move via prescribed rules. If the Pursuer can ensure that the game enters into a fixed set of final positions, then the Pursuer wins; otherwise, the Evader wins. A relational characterization of the games where the Pursuer wins is provided. A precise formula is given for the length of the game, along with an algorithm for computing if the Pursuer has a winning strategy whose complexity is a function of the parameters of the game. For games where the position of one player does not affect the available moves of he other, a vertex elimination ordering characterization, analogous to a cop-win ordering, is given for when the Pursuer has a winning strategy

Distinguishing homomorphisms of infinite graphs

We supply an upper bound on the distinguishing chromatic number of certain infinite graphs satisfying an adjacency property. Distinguishing proper $n$-colourings are generalized to the new notion of distinguishing homomorphisms. We prove that if a graph $G$ satisfies the connected existentially closed property and admits a homomorphism to $H$, then it admits continuum-many distinguishing homomorphisms from $G$ to $H$ join $K_2.$ Applications are given to a family universal $H$-colourable graphs, for $H$ a finite core

Adaptive tracking of a time-varying field with a quantum sensor

Sensors based on single spins can enable magnetic field detection with very high sensitivity and spatial resolution. Previous work has concentrated on sensing of a constant magnetic field or a periodic signal. Here, we instead investigate the problem of estimating a field with non-periodic variation described by a Wiener process. We propose and study, by numerical simulations, an adaptive tracking protocol based on Bayesian estimation. The tracking protocol updates the probability distribution for the magnetic field, based on measurement outcomes, and adapts the choice of sensing time and phase in real time. By taking the statistical properties of the signal into account, our protocol strongly reduces the required measurement time. This leads to a reduction of the error in the estimation of a time-varying signal by up to a factor 4 compared to protocols that do not take this information into account.Comment: 10 pages, 6 figure