Effect of initial infection size on a network susceptible-infected-recovered model.

We consider the effect of a nonvanishing fraction of initially infected nodes (seeds) on the susceptible-infected-recovered epidemic model on random networks. This is relevant when the number of arriving infected individuals is large, or to the spread of ideas with publicity campaigns. This model is frequently studied by mapping to a bond percolation problem, in which edges are occupied with the probability p of eventual infection along an edge. This gives accurate measures of the final size of the infection and epidemic threshold in the limit of a vanishingly small seed fraction. We show, however, that when the initial infection occupies a nonvanishing fraction, f, of the network, this method yields ambiguous results, as the correspondence between edge occupation and contagion transmission no longer holds. We propose instead to measure the giant component of recovered individuals within the original contact network. We derive exact equations for the size of the epidemic and the epidemic threshold in the infinite size limit in heterogeneous sparse random networks, and we confirm them with numerical results. We observe that the epidemic threshold correctly depends on f, decreasing as f increases. When the seed fraction tends to zero, we recover the standard results.

Hidden transition in multiplex networks.

Weak multiplex percolation generalizes percolation to multi-layer networks, represented as networks with a common set of nodes linked by multiple types (colors) of edges. We report a novel discontinuous phase transition in this problem. This anomalous transition occurs in networks of three or more layers without unconnected nodes, [Formula: see text]. Above a critical value of a control parameter, the removal of a tiny fraction [Formula: see text] of nodes or edges triggers a failure cascade which ends either with the total collapse of the network, or a return to stability with the system essentially intact. The discontinuity is not accompanied by any singularity of the giant component, in contrast to the discontinuous hybrid transition which usually appears in such problems. The control parameter is the fraction of nodes in each layer with a single connection, [Formula: see text]. We obtain asymptotic expressions for the collapse time and relaxation time, above and below the critical point [Formula: see text], respectively. In the limit [Formula: see text] the total collapse for [Formula: see text] takes a time [Formula: see text], while there is an exponential relaxation below [Formula: see text] with a relaxation time [Formula: see text].

Filtering Statistics on Networks.

Compression, filtering, and cryptography, as well as the sampling of complex systems, can be seen as processing information. A large initial configuration or input space is nontrivially mapped to a smaller set of output or final states. We explored the statistics of filtering of simple patterns on a number of deterministic and random graphs as a tractable example of such information processing in complex systems. In this problem, multiple inputs map to the same output, and the statistics of filtering is represented by the distribution of this degeneracy. For a few simple filter patterns on a ring, we obtained an exact solution of the problem and numerically described more difficult filter setups. For each of the filter patterns and networks, we found three key numbers that essentially describe the statistics of filtering and compared them for different networks. Our results for networks with diverse architectures are essentially determined by two factors: whether the graphs structure is deterministic or random and the vertex degree. We find that filtering in random graphs produces much richer statistics than in deterministic graphs, reflecting the greater complexity of such graphs. Increasing the graph's degree reduces this statistical richness, while being at its maximum at the smallest degree not equal to two. A filter pattern with a strong dependence on the neighbourhood of a node is much more sensitive to these effects.

Exotic critical behavior of weak multiplex percolation.

We describe the critical behavior of weak multiplex percolation, a generalization of percolation to multiplex or interdependent networks. A node can determine its active or inactive status simply by referencing neighboring nodes. This is not the case for the more commonly studied generalization of percolation to multiplex networks, the mutually connected clusters, which requires an interconnecting path within each layer between any two vertices in the giant mutually connected component. We study the emergence of a giant connected component of active nodes under the weak percolation rule, finding several nontypical phenomena. In two layers, the giant component emerges with a continuous phase transition, but with quadratic growth above the critical threshold. In three or more layers, a discontinuous hybrid transition occurs, similar to that found in the giant mutually connected component. In networks with asymptotically powerlaw degree distributions, defined by the decay exponent γ, the discontinuity vanishes but at γ=1.5 in three layers, more generally at γ=1+1/(M-1) in M layers.

Avalanche collapse of interdependent networks.

We reveal the nature of the avalanche collapse of the giant viable component in multiplex networks under perturbations such as random damage. Specifically, we identify latent critical clusters associated with the avalanches of random damage. Divergence of their mean size signals the approach to the hybrid phase transition from one side, while there are no critical precursors on the other side. We find that this discontinuous transition occurs in scale-free multiplex networks whenever the mean degree of at least one of the interdependent networks does not diverge.

Heterogeneous k-core versus bootstrap percolation on complex networks.

We introduce the heterogeneous k-core, which generalizes the k-core, and contrast it with bootstrap percolation. Vertices have a threshold r(i), that may be different at each vertex. If a vertex has fewer than r(i) neighbors it is pruned from the network. The heterogeneous k-core is the subgraph remaining after no further vertices can be pruned. If the thresholds r(i) are 1 with probability f, or k ≥ 3 with probability 1-f, the process can be thought of as a pruning process counterpart to ordinary bootstrap percolation, which is an activation process. We show that there are two types of transitions in this heterogeneous k-core process: the giant heterogeneous k-core may appear with a continuous transition and there may be a second discontinuous hybrid transition. We compare critical phenomena, critical clusters, and avalanches at the heterogeneous k-core and bootstrap percolation transitions. We also show that the network structure has a crucial effect on these processes, with the giant heterogeneous k-core appearing immediately at a finite value for any f>0 when the degree distribution tends to a power law P(q)~q(-γ) with γ<3.

Bootstrap percolation on complex networks.

We consider bootstrap percolation on uncorrelated complex networks. We obtain the phase diagram for this process with respect to two parameters: f, the fraction of vertices initially activated, and p, the fraction of undamaged vertices in the graph. We observe two transitions: the giant active component appears continuously at a first threshold. There may also be a second, discontinuous, hybrid transition at a higher threshold. Avalanches of activations increase in size as this second critical point is approached, finally diverging at this threshold. We describe the existence of a special critical point at which this second transition first appears. In networks with degree distributions whose second moment diverges (but whose first moment does not), we find a qualitatively different behavior. In this case the giant active component appears for any f>0 and p>0, and the discontinuous transition is absent. This means that the giant active component is robust to damage, and also is very easily activated. We also formulate a generalized bootstrap process in which each vertex can have an arbitrary threshold.

Fixation and consensus times on a network: a unified approach.

We investigate a set of stochastic models of biodiversity, population genetics, language evolution, and opinion dynamics on a network within a common framework. Each node has a state 0

Exact solution of the multi-allelic diffusion model.

We give an exact solution to the Kolmogorov equation describing genetic drift for an arbitrary number of alleles at a given locus. This is achieved by finding a change of variable which makes the equation separable, and therefore reduces the problem with an arbitrary number of alleles to the solution of a set of equations that are essentially no more complicated than that found in the two-allele case. The same change of variable also renders the Kolmogorov equation with the effect of mutations added separable, as long as the mutation matrix has equal entries in each row. Thus, this case can also be solved exactly for an arbitrary number of alleles. The general solution, which is in the form of a probability distribution, is in agreement with the previously known results. Results are also given for a wide range of other quantities of interest, such as the probabilities of extinction of various numbers of alleles, mean times to these extinctions, and the means and variances of the allele frequencies. To aid dissemination, these results are presented in two stages: first of all they are given without derivations and too much mathematical detail, and then subsequently derivations and a more technical discussion are provided.

Utterance selection model of language change.

We present a mathematical formulation of a theory of language change. The theory is evolutionary in nature and has close analogies with theories of population genetics. The mathematical structure we construct similarly has correspondences with the Fisher-Wright model of population genetics, but there are significant differences. The continuous time formulation of the model is expressed in terms of a Fokker-Planck equation. This equation is exactly soluble in the case of a single speaker and can be investigated analytically in the case of multiple speakers who communicate equally with all other speakers and give their utterances equal weight. Whilst the stationary properties of this system have much in common with the single-speaker case, time-dependent properties are richer. In the particular case where linguistic forms can become extinct, we find that the presence of many speakers causes a two-stage relaxation, the first being a common marginal distribution that persists for a long time as a consequence of ultimate extinction being due to rare fluctuations.

Urinary excretion of salicyluric and salicylic acids by non-vegetarians, vegetarians, and patients taking low dose aspirin.

AIM: To compare amounts of salicyluric acid (SU) and salicylic acid (SA) excreted daily in the urine of non-vegetarians and vegetarians not taking salicylate drugs, and patients taking 75 or 150 mg aspirin/day. METHODS: Urine excreted over 24 hours was collected from volunteers in the four groups. The volumes were recorded and the concentrations of SU and SA were determined electrochemically after separation by high performance liquid chromatography. RESULTS: Significantly more SU was excreted daily by vegetarians (median, 11.01; range, 4.98-26.60 micro mol/24 hours) than by non-vegetarians (median, 3.91; range, 0.87-12.23 micro mol/24 hours), although amounts were significantly lower than those excreted by patients taking aspirin. Median amounts of SU excreted by patients taking 75 and 150 mg/day of low dose aspirin were 170.69 (range, 13.15-377.18) micro mol/24 hours and 165.17 (range, 5.61-429.12) micro mol/24 hours, respectively. The amount of SU excreted by patients taking either 75 or 150 mg of aspirin/day was not significantly different. Significantly more SA was excreted by vegetarians (median, 1.19; range, 0.02-3.55 micro mol/24 hours) than by non-vegetarians (median, 0.31; range, 0.01-2.01 micro mol/24 hours). The median amounts of SA excreted by vegetarians and the patients taking aspirin were not significantly different. CONCLUSIONS: More SU and SA is excreted in the urine of vegetarians than in non-vegetarians, consistent with the observation that fruits and vegetables are important sources of dietary salicylates. However, significantly less SU was excreted by vegetarians than patients taking aspirin, indicating that the daily intake of bioavailable salicylates by vegetarians is considerably lower than that supplied by a single 75 or 150 mg dose of aspirin.

Identification and determination of salicylic acid and salicyluric acid in urine of people not taking salicylate drugs.

BACKGROUND: Salicylic acid (SA) is present in the serum of people who have not taken salicylate drugs. Now we have examined the urine of these subjects and found that it contains SA and salicyluric acid (SU). We have established the identities of these phenolic acids and determined their concentrations. METHODS AND RESULTS: The acidic hydrophobic compounds of urine were separated using high-performance liquid chromatography (HPLC) and were detected and quantified electrochemically. Two approaches were used to establish the identity of SA and SU. First, the retention times (Rt) of the substances extracted and those of SA and SU were compared under two sets of chromatographic conditions; the Rt of the compounds suspected to be SA and SU and those of the authentic substances were very similar under both sets of conditions. Second, the unknown substances, isolated by HPLC, were treated with acetyl chloride in methanol and compared with the methyl esters of SA and SU by using gas chromatography-mass spectrometry; the unknown compounds after esterification had very similar mass spectra and gas chromatographic R, to those of methyl salicylate and methyl salicylurate. The median (n = 10) urinary concentration of SA was 0.56 micromol/L (range 0.07-0.89 micromol/L) and that of SU was 3.20 micromol/L (range 1.32-6.54 micromol/L). SA and its major urinary metabolite, SU, were found in the urine of all of the 10 people examined.

Salicylic acid in soups prepared from organically and non-organically grown vegetables.

BACKGROUND: Salicylic acid is a chemical signal in plants infected by pathogens and it is responsible for the anti-inflammatory action of aspirin. Patients who take aspirin have a reduced risk of developing atherosclerosis and colorectal cancer, both of these pathologies having an inflammatory component. Dietary salicylic acid may help to prevent these conditions. We wondered if foods made from organically-reared plants might have a higher content of salicylic acid than those made from non-organic plants, since the latter are more likely to be protected from infection by the application of pesticides. OBJECTIVE: To determine if organic vegetable soups have a higher salicylic acid content than non-organic vegetable soups. METHODS: The contents of salicylic acid in organic and non-organic vegetable soups purchased from supermarkets were determined. Salicylic acid was identified by varying the chromatographic conditions and comparing the retention times of the unknown substance in the extracts with salicylic acid; by treating extracts of the soups with salicylate hydroxylase; and by using GCMS. Salicylic acid was determined by using HPLC with electrochemical detection. RESULTS: Salicylic acid was present in all of the organic and most of the non-organic vegetable soups. The median contents of salicylic acid in the organic and non-organic vegetable soups were 117 (range, 8-1040) ng x g(-1) and 20 (range, 0-248) ng x g(-1) respectively. The organic soups had a significantly higher content of salicylic acid (p=0.0032 Mann Whitney U test), with a median difference of 59 ng g(-1) (95 % confidence interval, 18-117ng x g(-1)). CONCLUSIONS: Organic vegetable soups contained more salicylic acid than non-organic ones, suggesting that the vegetables and plants used to prepare them contained greater amounts of the phenolic acid than the corresponding non-organic ingredients. Consumption of organic foods may result in a greater intake of salicylic acid.

Glutamate receptor subunits associated with rat sympathetic preganglionic neurons.

The purpose of this study was to determine what subunits of the glutamate (alpha-amino-3-hydroxy-5-methyl-4-isoxazole propionic acid (AMPA)) receptor are expressed by sympathetic preganglionic neurons in the spinal cord of adult rats. Preganglionic neurons were retrogradely labelled with Fluorogold, double-labelled with choline acetyltransferase immunofluorescence, and examined with confocal microscopy for evidence of immunoreactivity for GluR1, GluR2, GluR2/3 and GluR4 subunits. Quantitative analysis revealed that 92, 63 and 85% of preganglionic cells in the T8 segment express GluR1, GluR2 and GluR2/3 subunits, respectively. Cells were not immunoreactive for the GluR4 subunit. This evidence is consistent with the idea that most sympathetic preganglionic neurons form heteromeric AMPA receptors. Cells with GluR2 subunits will assemble receptors which are impermeable to calcium ions and may be resistant to excitotoxic cell death.

Changes in neuropeptide immunoreactivity in cultured adult mouse sensory neurons following methylmercury chloride treatments.

Changes in the neuropeptide expression of sensory neurons, related to functional modulation, have been widely reported both following physical injury in vivo, and after toxic insult in vitro and in vivo. The current immunocytochemical study aimed to monitor the neuropeptide status of neuronal cultures prepared from adult mouse dorsal root ganglia (DRG), and to ascertain whether changes occurred following treatments with 0.1-1 microM methylmercury (MeHg). Proportions of both substance P (SP) and calcitonin gene related peptide (CGRP) containing neurons increased significantly, and were maintained throughout the 24 h exposure period. In contrast the numbers of somatostatin (SOM)-ir neurons decreased. Substance P- and CGRP-ir neuron increases may be related to nociceptive responses, whereas the decreases in SOM containing neurons could reflect a differential loss in this subset of sensory neurons.