The t-stability number of a random graph

Given a graph G = (V,E), a vertex subset S is called t-stable (or t-dependent) if the subgraph G[S] induced on S has maximum degree at most t. The t-stability number of G is the maximum order of a t-stable set in G. We investigate the typical values that this parameter takes on a random graph on n vertices and edge probability equal to p. For any fixed 0 < p < 1 and fixed non-negative integer t, we show that, with probability tending to 1 as n grows, the t-stability number takes on at most two values which we identify as functions of t, p and n. The main tool we use is an asymptotic expression for the expected number of t-stable sets of order k. We derive this expression by performing a precise count of the number of graphs on k vertices that have maximum degree at most k. Using the above results, we also obtain asymptotic bounds on the t-improper chromatic number of a random graph (this is the generalisation of the chromatic number, where we partition of the vertex set of the graph into t-stable sets).Comment: 25 pages; v2 has 30 pages and is identical to the journal version apart from formatting and a minor amendment to Lemma 8 (and its proof on p. 21

Largest sparse subgraphs of random graphs

For the Erd\H{o}s-R\'enyi random graph G(n,p), we give a precise asymptotic formula for the size of a largest vertex subset in G(n,p) that induces a subgraph with average degree at most t, provided that p = p(n) is not too small and t = t(n) is not too large. In the case of fixed t and p, we find that this value is asymptotically almost surely concentrated on at most two explicitly given points. This generalises a result on the independence number of random graphs. For both the upper and lower bounds, we rely on large deviations inequalities for the binomial distribution.Comment: 15 page

Anxiety and Learning in Dynamic and Static Clock Game Experiments

In clock games, agents receive differently-timed private signals when an asset value is above its fundamental. The price crashes to the fundamental when K of N agents have decided to sell. If selling decisions are private, bubbles can be sustained because people delay selling, after receiving signals, knowing that others will delay too. Our results replicate the main features of the one previous experimental study of clock game (in two subject pools): Selling delays are shorter than predicted, but converge toward equilibrium predictions over repeated trials. We also find that delays are shorter in a dynamic game in which selling decisions unfold over time, compared to a static equivalent in which subjects precommit to selling decisions. A model of learning with growing anxiety after signal arrival can reproduce the empirical observations of shorter-than-predicted delay, smaller delay after later signal arrival, and shorter delays in dynamic games

The t-improper chromatic number of random graphs

We consider the $t$-improper chromatic number of the Erd{\H o}s-R{\'e}nyi random graph $G(n,p)$. The t-improper chromatic number $\chi^t(G)$ of $G$ is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most $t$. If $t = 0$, then this is the usual notion of proper colouring. When the edge probability $p$ is constant, we provide a detailed description of the asymptotic behaviour of $\chi^t(G(n,p))$ over the range of choices for the growth of $t = t(n)$.Comment: 12 page

Hypothetical and Real Choice Differentially Activate Common Valuation Areas

Hypothetical reports of intended behavior are commonly used to draw conclusions about real choices. A fundamental question in decision neuroscience is whether the same type of valuation and choice computations are performed in hypothetical and real decisions. We investigated this question using functional magnetic resonance imaging while human subjects made real and hypothetical choices about purchases of consumer goods. We found that activity in common areas of the orbitofrontal cortex and the ventral striatum correlated with behavioral measures of the stimulus value of the goods in both types of decision. Furthermore, we found that activity in these regions was stronger in response to the stimulus value signals in the real choice condition. The findings suggest that the difference between real and hypothetical choice is primarily attributable to variations in the value computations of the medial orbitofrontal cortex and the ventral striatum, and not attributable to the use of different valuation systems, or to the computation of stronger stimulus value signals in the hypothetical condition

Measured Anxiety Affects Choices in Experimental "Clock" Games

We measure anxiety by skin conductance response (SCR) in an economic setting. In “clock” games, six agents receive private signals when an asset's price exceeds its fundamental value. They can sell for immediate value or wait to sell at a higher value. Waiting is risky because the price crashes to a lower value when three agents sell. Anxiety could lead people to sell too quickly when the game is played dynamically over time, compared to a static version with precommitted selling. Empirically, delays are shorter in dynamic games than in payoff-equivalent static games, and are associated with anxiety as measured

Self-organization of charge under pressure in the organic conductor (TMTSF)2ReO4

(TMTSF)2ReO4 presents a phase coexistence between two anion orderings defined by their wave vectors q_2=(1/2,1/2,1/2) and q_3=(0,1/2,1/2) in a wide range of pressure (8-11kbar) and temperature. From the determination of the anisotropy of the conductivity and the superconducting transitions in this regime we were able to extract the texture which results from a self-organization of the orientations of the ReO4 anions in the sample. At the lowest pressures, the metallic parts, related to the q_3 order, form droplets elongated along the a-axis embedded in the semiconducting matrix associated with the q_2 order. Above 10kbar, filaments along the a-axis extend from one end of the sample to the other nearly up to the end of the coexistence regime. A mapping of the system into an anisotropic Ising lattice is satisfactory to analyze the data. satisfactory to analyze the data.Comment: 7 pages, 3 figures, EPL forma

Ligand chemistry of titania precursor affects transient photovoltaic behavior in inverted organic solar cells

The chemistry of the precursor from which charge transport layers are formed can significantly affect the device performance of organic solar cells. Here, we compare two common precursors that are used to generate titania electron transport layers and elucidate their effects on the transient characteristics of inverted bulk-heterojunction polymer solar cells comprising poly(3-hexyl hiophene) and [6,6]-phenyl-C-61-butyric acid methyl ester. Substituting the isopropyl ligands of titanium isopropoxide with 2-methoxyethanol leads to electron transport layers that require a shorter illumination time to fill shallow electron traps. Furthermore, organic solar cells with titania electron transport layers prepared with such pre-modified titania precursor exhibit higher power-conversion efficiencies stemming from lower trap densities. (C) 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4795287open4