Phase transition in a class of non-linear random networks

We discuss the complex dynamics of a non-linear random networks model, as a function of the connectivity k between the elements of the network. We show that this class of networks exhibit an order-chaos phase transition for a critical connectivity k = 2. Also, we show that both, pairwise correlation and complexity measures are maximized in dynamically critical networks. These results are in good agreement with the previously reported studies on random Boolean networks and random threshold networks, and show once again that critical networks provide an optimal coordination of diverse behavior.Comment: 9 pages, 3 figures, revised versio

Quantum entanglement, unitary braid representation and Temperley-Lieb algebra

Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate {\it directly}, from separable basis states, important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.Comment: 5 pages, no figur

Quantum entanglement: The unitary 8-vertex braid matrix with imaginary rapidity

We study quantum entanglements induced on product states by the action of 8-vertex braid matrices, rendered unitary with purely imaginary spectral parameters (rapidity). The unitarity is displayed via the "canonical factorization" of the coefficients of the projectors spanning the basis. This adds one more new facet to the famous and fascinating features of the 8-vertex model. The double periodicity and the analytic properties of the elliptic functions involved lead to a rich structure of the 3-tangle quantifying the entanglement. We thus explore the complex relationship between topological and quantum entanglement.Comment: 4 pages in REVTeX format, 2 figure

Stable and unstable attractors in Boolean networks

Boolean networks at the critical point have been a matter of debate for many years as, e.g., scaling of number of attractor with system size. Recently it was found that this number scales superpolynomially with system size, contrary to a common earlier expectation of sublinear scaling. We here point to the fact that these results are obtained using deterministic parallel update, where a large fraction of attractors in fact are an artifact of the updating scheme. This limits the significance of these results for biological systems where noise is omnipresent. We here take a fresh look at attractors in Boolean networks with the original motivation of simplified models for biological systems in mind. We test stability of attractors w.r.t. infinitesimal deviations from synchronous update and find that most attractors found under parallel update are artifacts arising from the synchronous clocking mode. The remaining fraction of attractors are stable against fluctuating response delays. For this subset of stable attractors we observe sublinear scaling of the number of attractors with system size.Comment: extended version, additional figur

Stress rapidly suppresses in vivo LH pulses and increases activation of RFRP-3 neurons in male mice

Restraint stress is a psychosocial stressor that suppresses reproductive status, including LH pulsatile secretion, but the neuroendocrine mechanisms underlying this inhibition remains unclear. Reproductive neural populations upstream of gonadotropin-releasing hormone (GnRH) neurons, such as kisspeptin, neurokinin B and RFRP-3 (GnIH) neurons, are possible targets for psychosocial stress to inhibit LH pulses, but this has not been well examined, especially in mice in which prior technical limitations prevented assessment of in vivo LH pulse secretion dynamics. Here, we examined whether one-time acute restraint stress alters in vivo LH pulsatility and reproductive neural populations in male mice, and what the time-course is for such alterations. We found that endogenous LH pulses in castrated male mice are robustly and rapidly suppressed by one-time, acute restraint stress, with suppression observed as quickly as 12–18 min. This rapid LH suppression parallels with increased in vivo corticosterone levels within 15 min of restraint stress. Although Kiss1, Tac2 and Rfrp gene expression in the hypothalamus did not significantly change after 90 or 180 min restraint stress, arcuate Kiss1 neural activation was significantly decreased after 180 min. Interestingly, hypothalamic Rfrp neuronal activation was strongly increased at early times after restraint stress initiation, but was attenuated to levels lower than controls by 180 min of restraint stress. Thus, the male neuroendocrine reproductive axis is quite sensitive to short-term stress exposure, with significantly decreased pulsatile LH secretion and increased hypothalamic Rfrp neuronal activation occurring rapidly, within minutes, and decreased Kiss1 neuronal activation also occurring after longer stress durations

Self-organization of heterogeneous topology and symmetry breaking in networks with adaptive thresholds and rewiring

We study an evolutionary algorithm that locally adapts thresholds and wiring in Random Threshold Networks, based on measurements of a dynamical order parameter. A control parameter $p$ determines the probability of threshold adaptations vs. link rewiring. For any $p < 1$, we find spontaneous symmetry breaking into a new class of self-organized networks, characterized by a much higher average connectivity $\bar{K}_{evo}$ than networks without threshold adaptation ($p =1$). While $\bar{K}_{evo}$ and evolved out-degree distributions are independent from $p$ for $p <1$, in-degree distributions become broader when $p \to 1$, approaching a power-law. In this limit, time scale separation between threshold adaptions and rewiring also leads to strong correlations between thresholds and in-degree. Finally, evidence is presented that networks converge to self-organized criticality for large $N$.Comment: 4 pages revtex, 6 figure

Evolutionary dynamics on strongly correlated fitness landscapes

We study the evolutionary dynamics of a maladapted population of self-replicating sequences on strongly correlated fitness landscapes. Each sequence is assumed to be composed of blocks of equal length and its fitness is given by a linear combination of four independent block fitnesses. A mutation affects the fitness contribution of a single block leaving the other blocks unchanged and hence inducing correlations between the parent and mutant fitness. On such strongly correlated fitness landscapes, we calculate the dynamical properties like the number of jumps in the most populated sequence and the temporal distribution of the last jump which is shown to exhibit a inverse square dependence as in evolution on uncorrelated fitness landscapes. We also obtain exact results for the distribution of records and extremes for correlated random variables

Two-dimensional projections of an hypercube

We present a method to project a hypercube of arbitrary dimension on the plane, in such a way as to preserve, as well as possible, the distribution of distances between vertices. The method relies on a Montecarlo optimization procedure that minimizes the squared difference between distances in the plane and in the hypercube, appropriately weighted. The plane projections provide a convenient way of visualization for dynamical processes taking place on the hypercube.Comment: 4 pages, 3 figures, Revtex

Unsigned state models for the Jones polynomial

It is well a known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs and show that the Jones polynomial of any link can be expressed as a vertex model of an unsigned embedded graph. In the process of deriving this result, we show that for every diagram of a link in the 3-sphere there exists a diagram of an alternating link in a thickened surface (and an alternating virtual link) with the same Kauffman bracket. We also recover two recent results in the literature relating the Jones and Bollobas-Riordan polynomials and show they arise from two different interpretations of the same embedded graph.Comment: Minor corrections. To appear in Annals of Combinatoric

Growing networks with two vertex types

Growing networks are introduced in which the vertices are allocated one of two possible growth rates; type A with probability p(t), or type B with probability 1−p(t). We investigate the networks using rate equations to obtain their degree distributions. In the first model (I), the network is constructed by connecting an arriving vertex to either a type A vertex of degree k with rate μk, where μ0, or to a type B vertex of degree k with rate k. We study several p(t), starting with p(t) as a constant and then considering networks where p(t) depends on network parameters that change with time. We find the degree distributions to be power laws with exponents mostly in the range 2γ3. In the second model (II), the network is constructed in the same way but with growth rate k for type A vertices and 1 for type B vertices. We analyse the case p(t)=c, where 0c1 is a constant, and again find a power-law degree distribution with an exponent 2γ3