In two studies, we examined the influence of behavioral inhibition system (BIS) and need for closure (NFC) on information processing in decision making. We expected that BIS would regulate behavior in a decisional context and that this relationship would be mediated by epistemic motivation expressed by NFC. In addition, drawing on contradictory findings in the literature on anxiety, NFC, and information processing, we investigated the moderating role of decision rules. The results supported our predictions. BIS was strongly and positively related to NFC, and through NFC it was related to decision-making style. Moreover, decision task characteristics moderated the relationship between NFC and decision making. When a task did not offer a confident decision rule, high NFC participants prolonged the information search more than low NFC individuals. However, when a reliable strategy was suggested, high NFC participants behaved in line with it. These results are discussed within an uncertainty management framework. Decision making involves a considerable amount of uncertainty. When faced with a choice, a person may not know all of the available options, may be unable to recognize the quality of each option, or may select an ineffective decision making strategy, all of which create stressful and psychologically demanding conditions. One variable that is responsible for the regulation of behavior in such situations is the behavioral inhibition system . In contrast, when a specific decision rule was offered, we expected that an anchoring effect would be found such that participants high in NFC would behave more in line with the suggested rule than participants low in NFC (Study 2).The aim of the Study 1 was to explore the relationship between self-reported BIS, NFC and behavior in an abstract and unfamiliar decision making task that offered no prior information about the best strategy by which to make decisions. We used the Information Sampling Task . In exchange for their participation, each participant could win one of four tickets in a lottery (each worth approximately $15).The study was conducted on-line. Participants were 115 students from a large Polish university recruited via a student mailing list  round, participants could win 100 points if they made a correct decision, irrespective of the number of opened boxes, and they lost 100 points if they were wrong. In the Decreasing Wins (DW) round, there were costs of opening boxes such that the maximum amount to win (250 points) decreased by ten points for every box opened. Again, after an incorrect decision, participants lost 100 points, regardless of the number of boxes opened. A test trial took place before each round. Although we expected that effects would occur in the FW round when there were no costs of gathering information, we included the DW condition to explore the effects of NFC under conditions of trade-off between uncertainty and loss. Decision behavior was measured by the average number of boxes opened, the decision time, and the number of correct judgments made in each condition. The total points won indicated overall task performance. In the present article, we will focus on information search as indicated by the number of boxes opened and decision time. Decision time was measured from the moment participants were first presented with the matrix to the moment they made a decision. It was log transformed.To verify whether BIS and NFC were related to information search and decision time we analyzed the pattern of correlations between those variables. In the second step we tested the mediation model for each of the dependent variables. Both analyses were conducted separately for the FW condition and the DW condition. Table b = 0.50, SE = 0.08, β = 0.51, p < 0.001). NFC had a positive effect on the number of opened boxes . Neither the direct effect of BIS on the number of opened boxes , nor the total effect  were significant. The entire model was marginally significant . More importantly, the indirect effect of BIS on the number of opened boxes estimated with 20,000 bootstrapped samples was significant , which indicates that BIS was related to information search through its impact on NFC. The mediation model is presented in Fig. In both FW and DW rounds bivariate correlations between BIS and the number of opened boxes and NFC and the number of opened boxes were not significant. To test the simultaneous effects of BIS and NFC on information search, we used the PROCESS program Hayes . We testr = 0.07, p = 0.451; in the DW round: r = 0.09, p = 0.363) but correlations between NFC and decision time were significant in both rounds .Correlations between BIS and decision time were not significant . NFC had a positive effect on decision time . Neither the direct effect of BIS on decision time  nor the total effect  were significant. The entire model was marginally significant . Although the confidence intervals for the indirect effect estimated with 20,000 bootstrapped samples included zero , the Sobel test for this indirect effect was marginally significant . The mediation model for the DW round was not significant .In the mediational analysis for the FW condition BIS had a significant and positive effect on NFC  to parallel Study 1.Mage = 23.30, SD = 5.59). In exchange for their participation, each participant could win one of eight tickets in a lottery (each worth approximately $15).The study was conducted on-line. Two hundred twenty-three participants were recruited via invitation placed on a popular free ads web portal  = 103.11, p < 0.001, η2 = 0.49; in the DW round: F = 83.80, p < 0.001, η2 = 0.43]. Participants in the low anchor condition remembered that other people opened fewer boxes  than participants in high anchor condition . In the control condition, participants thought that others opened 13.58 boxes (SD = 6.73) in the FW round and 7.69 (SD = 4.55) in the DW round.A one-way ANOVA revealed a significant effect of condition on the manipulation check items . In line with our expectations, we found that BIS had a positive effect on NFC . We also found a significant interaction between NFC and the experimental manipulation  = 4.60, p = 0.011). The effect of NFC on the number of opened boxes was positive and significant in the control condition . The effect was positive but not significant in the high anchor condition  and was negative but not significant in the low anchor condition . These results are presented in Fig. We used the PROCESS program to test the interactional effect of NFC and the manipulation, and the indirect effect of BIS on the information search through NFC . However, the indirect effect estimated with 20,000 bootstrapped samples was significant in the control condition . This effect was positive but not significant in the high anchor condition , and it was negative but not significant in the low anchor condition . We also performed a test of moderated mediation , which means that participants opened more boxes in the FW round when this round was first. There were no significant effects of NFC and BIS on number of opened boxes in the DW condition.The effect of order of rounds was significant  was negative and not significant. However, the correlations between BIS and decision time in both rounds were not significant.F = 7.10, p < 0.001, R2 = 0.19]. BIS had a positive effect on NFC . We found a significant interaction between NFC and the experimental manipulation  = 5.41, p = 0.005). Further analyses showed that the effect of NFC on decision time (in FW) was positive and significant in the control condition  and in the high anchor condition , but it was negative and not significant in the low anchor condition . The visualization of this interaction is presented in Fig. The mediation model was significant in the FW round  and in the high anchor condition ; however, this effect was not significant in the low anchor condition . The difference in the indirect effects between the control condition and the high anchor condition was not significant [95 % CI ], but it was significant between the control condition and the low anchor condition [95 % CI ].Although the direct effect of BIS on decision time (FW) was not significant , which means that decision time in the FW round was shorter when this round was second. There were no effects of NFC and BIS on decision time in the DW condition.Finally, the effect of order of rounds was significant , the effects of NFC with regard to decision time and number of opened boxes were not significant. These results are consistent with our expectations that high NFC individuals would behave in line with accessible and reliable rules. Our experimental manipulation induced differences among high NFC individuals but individuals low in NFC were not affected by it. It is an interesting result, because in the high anchor condition the suggested style of processing—longer decision making process and extended information search—was opposite to the one usually associated with high NFC. We think that this anchoring effect occurred because such rules decrease uncertainty, which is important for high NFC individuals. This result corroborates past research showing how psychological states similar to NFC  resulted in stronger judgmental anchoring and insufficient adjustment (Epley and Gilovich In Study 2, we replicated the strong and positive relationship between NFC and BIS found in Study 1. More importantly, in the FW round we also replicated the indirect effect of BIS on decision making in the control group. This pattern of results offers support for our initial assumption that the effects of sensitivity to uncertainty on decision making result from avoidance of uncertainty and a stronger motivation to reach closure among highly uncertain individuals. Significant moderation of this indirect effect by situational conditions indicates that when the available information suggested that a prolonged search was the best strategy, high BIS individuals, due to their higher level of NFC, gathered more information to reduce uncertainty than low BIS individuals. However, the rule available in the low anchor condition, which advised a limited information search, diminished the indirect relationship between BIS and a prolonged decision making strategy. We think that in this condition, reduction of uncertainty to an acceptable level was achieved with a shorter decision making process. A stronger manipulation may be needed to significantly reverse this effect.Although no difference was found between the control group and the high anchor group, we think that there still may be some psychological differences between those two conditions. In the high anchor condition, the decision rule was externally given to participants, while in the control condition and in the Study 1 it was not available from the outside. Therefore, it is possible that the control condition and high-anchor condition may differ with regard to other aspects of the decision making process such as confidence in the decision or satisfaction with a decision, even though on a behavioral level the groups were similar. For example, it could be expected that in the high anchor condition, the uncertainty regarding the chosen strategy was lower than in the control condition. It should be noted that the abovementioned effects were restricted to FW condition only. We will discuss this limitation in the general discussion.We began this article with the observation that decision making can be a demanding activity, but at the same time not everyone experiences the process in the same way. In two studies, we investigated the effects of individual differences in BIS and NFC on the extent of information gathering involved in a decision making process. In both studies, BIS was strongly and positively related to NFC. This result confirms the relationship between BIS and NFC, which was found in one recent study (Corr et al. Moreover, BIS was related to decision making behavior through NFC. This indirect effect is in line with the interpretation of BIS as a system that controls reactions to new and ambiguous situations (Amodio et al. Our interpretation of the relationship between BIS activity and NFC focuses on the functionality of NFC, which we assume is one of the mechanisms applied by highly uncertain individuals to cope with a stressful environment. However, further research is needed to determine whether behavior elicited by high NFC indeed reduces feelings of uncertainty and anxiety. In addition, although the causal direction assumed in the present study seems convincing in terms of the biological underpinnings of BIS, future research should verify the causal relationship between uncertainty and NFC with an experimental design. To fully understand the nature of the relationship between BIS and decision making, further studies are definitely needed.With regard to the relationship between NFC and decision making process, in both of our studies, when a decision task did not offer a clear and confident cue, high NFC participants sought more information and spent more time making a decision than low NFC individuals. In contrast, when a strategy was suggested, high NFC participants behaved in line with the suggested strategy. This result is important, because it confirms that NFC is not always related to limited information processing; rather, under certain circumstances, NFC may lead to a more complex decision making style (e.g., Kruglanski et al. One important limitation to the generalizability of our findings stems from the fact that we found differences between the FW and DW rounds. The pattern of results for the FW rounds was consistent across two studies but not for the DW rounds, which may suggest that the effects of BIS and NFC are different with regard to situations that involve losses and ambiguity as it was the case in the DW round. The lack of effect of NFC in this condition could be a result of two competing tendencies—to reduce uncertainty and to reduce costs—both of which may play a role for highly uncertain individuals. In such a case we suspect that the differences between conditions would be stronger if the gains and losses were more meaningful to participants than in our study. However, the lack of expected NFC effects might also be due to the lower variance of dependent variables in the DW conditions because on average, participants in the DW conditions opened fewer boxes to decrease the costs of making a decision and spent less time on the task.The results of our study also contribute to existing knowledge on the sources of NFC. The strong and positive relationship between BIS and NFC may suggest that limited information processing observed under high NFC, which on the surface seems to be simple and effortless, may instead result from higher emotional costs experienced by high BIS individuals in uncertain situations. Stress and cognitive effort resulting from sensitivity to uncertainty may be responsible for the typical effects of NFC on judgment formation and decision making, which involves a shorter and less extensive search. Simultaneous inclusion of measures of cognitive resources and motivational factors as predictors of NFC would help to clarify the origins of NFC in future studies.One of the limitations of our study is that there were more female than male participants. Gender may be an important factor determining the relationships explored in our study as previous studies have demonstrated higher scores of BIS among women than among men (e.g., Carver and White In summary, we believe that our research, despite its limitations, offers the first attempt to integrate the basic personality variable represented by BIS and a specific epistemic motivation in a coherent model and to verify the simultaneous effect of those factors on decision-making style.


Model checking has been recently introduced as an integrated framework for extracting information of the phylogenetic trees using temporal logics as a querying language, an extension of modal logics that imposes restrictions of a boolean formula along a path of events. The phylogenetic tree is considered a transition system modeling the evolution as a sequence of genomic mutations (we understand mutation as different ways that DNA can be changed), while this kind of logics are suitable for traversing it in a strict and exhaustive way. Given a biological property that we desire to inspect over the phylogeny, the verifier returns true if the specification is satisfied or a counterexample that falsifies it. However, this approach has been only considered over qualitative aspects of the phylogeny.In this paper, we repair the limitations of the previous framework for including and handling quantitative information such as explicit time or probability. To this end, we apply current probabilistic continuous-time extensions of model checking to phylogenetics. We reinterpret a catalog of qualitative properties in a numerical way, and we also present new properties that couldn’t be analyzed before. For instance, we obtain the likelihood of a tree topology according to a mutation model. As case of study, we analyze several phylogenies in order to obtain the maximum likelihood with the model checking tool PRISM. In addition, we have adapted the software for optimizing the computation of maximum likelihoods.We have shown that probabilistic model checking is a competitive framework for describing and analyzing quantitative properties over phylogenetic trees. This formalism adds soundness and readability to the definition of models and specifications. Besides, the existence of model checking tools hides the underlying technology, omitting the extension, upgrade, debugging and maintenance of a software tool to the biologists. A set of benchmarks justify the feasibility of our approach. A phylogenetic tree is a description of the evolution process which is discovered via molecular sequencing data and morphological data matrices . ComputeModel checking is a paradigm stemming from computer science based on temporal logics which has been successfully applied in industry for system modeling and verification . The basComputational Tree Logic  alreadygic  or contTA ). Much mTA   it is possible to verify if a property is true or false according to some temporal requirements and the probabilistic behavior of the model and b) it is also possible to obtain the minimum probability that makes the specification true within an interval of time. In order to show the feasibility of our approach, we apply our framework to a specific problem: the probability of obtaining a phylogeny with a certain topology. Specially, we focus on the computation of maximum likelihood estimations (MLE) for a phylogenetic tree using models of DNA evolution, that simulate the changes in the genome through the time.The paper is arranged as follows. The next Section “The process of model checking is divided in three phases. The first part consists of a correct modeling of the system. Next, a set of properties are written using a specification logic. Finally, a model checking tool takes the description of the model and the hypotheses, and analyze them in an automatic and exhaustive way. The quality of the analysis is influenced by the identification of the main characteristics of the system during modeling. To this end, the next paragraphs present the principal keystones that we try to capture of the evolution model and how they are translated into a computational model  suitable for the study.Topological models. Birth-death Markovian models describe the macroevolution process of speciation =si, while π(t) returns the state si of the path in which the system is found after s0 =s0). For any set of infinite paths Π starting in the initial state s0, the subset Π(πn) selects the paths π∈Π whose prefix equals to the finite sequence πn=s0s1s2…sn of length n+1 states.The computation of probabilities depends on the paths starting in the initial state. Given an infinite path of successive states sn is determined by the probability measure Pr(Π(πn))=Pr(Π(s0s1s2…sn)) over all the paths sharing the prefix πn with πn(0)=s0. For the trivial case over Π(s0), Pr(Π(s0))=1. In general for Π(πn), Pr(Π(s0s1s2…sn))=Pt. Starting in s, the probability of arriving to a state s′ at a particular time instant t is represented by Pt=Pr{π∈Π | π(t)=s′}, with Pt an exponential matrix indicating the transient probabilities for time t and Q the infinitesimal generator matrix of M: The likelihood of reaching a state Q is used for the calculation of transient states and it is defined as: The infinitesimal generator matrix Pt is presented in  is a real number denoting a probability, and ∼ in {<,≤,=,>,≥} is a comparison operator: A temporal logic formula M considering all paths π emerging from a state s0. Notice that s0 satisfies ϕ. The semantics of well-formed formulas is as follows (let π=s0s1s2…): The formulas are checked against a structure Prob requires the identification of the set of infinite paths π satisfying the path formula t∈I, t′<t with I an interval of reals.The calculation of the probability Π(πn) by  is computed as the summation of probabilities in all possible prefixes πn by  by , or at least exists one computation path (E). These two must be immediately qualified by one of five temporal operators, of which three express the satisfaction of a property eventually in a future state (F), generally at all states (G), or in the next state (X); and two are conditional constructs in which a precedent is verified until a consequent comes into force (U), or until and including the moment when it does, if it does (R). A complete grammar and semantics of CTL formulas can be defined from a minimal subset of logical operators =σ (s[i]≠σ) of a state as σi  means that σ∈Σ appears (not) in position i in a state sequence . Then, at some point in the subtree hanging from that intermediate the symbol from the root reoccurs.F operators: a node satisfies a property which a) eventually some other descendant does not satisfy, but b) it is fulfilled once again at some point in the future. The check is repeated for every symbol that may occur in the node. The formula must model this condition by nesting detectBM iterates the check over the positions of the alignment and extends it to all tree nodes. Finally, the global formula In these two formulas we present a non-trivial modeling example of a cladistic property with a heavy use of sequence data. The goal is to detect if the tree is free of back mutations . In the case of finite domains, such as the set of DNA sequences, the evaluation of logical quantifiers ∀ and ∃ can be substituted by multiple instances of boolean formulas connected by the Identifying if a phylogenetic tree reflects the correct structure of the evolution is one of the most important topological questions. Given a phylogenetic tree inferred from an alignment, it should be compared to the speciation model that we use as reference, for instance, a DNA mutation model. The computation of maximum likelihoods quantifies the fitness or probability that the topology obtained by the alignment follows the same trace imposed by a mutation model .true evolution process generating the tree of life, the estimation of maximum likelihoods works as a metric for comparing the topology of two or more trees built over the same data set . The selection of other inferred phylogenies, alignments or alternative DNA models modifies the probabilities of the arrangements and returns different scores  with I= with t≤t′;I= with The computation of the probability for the until operator depends on the value of the interval I = , the probability Prob is equal to: (i) 1, if s∈Sat(ϕ)(ii) 0, if s∈Sat(¬ψ∧¬ϕ) or t=0(iii)For the case of I= , then Prob=Prob(emb(M),s,ψUϕ). It is calculated indefinitely as follows until a stop condition is reached (i-ii): When The embedded DTMC must be free of loops (infinite paths) of intermediate states so as to have a finite and solvable numerical equation system.I= , the until operator must consider a) the time t spent in states satisfying ψ plus b) up to time t′−t required for reaching ϕ. The first part can be compared to Ft] is similar to the previous I=  but changing ψUϕ due to the infinity. As the until operator is unbounded, it is evaluated in the embedded DTMC: The last case TA formula ϕ against a CTMC is linear in the number of logical connectives and temporal operators of the formula (|ϕ|) and polynomial in the size of S. More generally, the complexity is: tmax is the maximal bound of a path subformula ψ1UIψ2 of ϕ, with tmax=1 if it doesn’t contain any U subformula. The parameter q is equal to q=maxs∈S|Q|.The temporal complexity of verifying a CSLA model checking tool or verifier requires two input files for the verification process: a first file with the description of the model, and a second file with the specification of the properties. Traditionally, it returns if the specification is satisfied in the model or not, and a counterexample that falsifies it if required. In probabilistic model checking, extra information shall be provided such as the minimum probability for which the property succeed.The modularity and independence of model and specifications allow the evaluation of several properties over the same model and, vice versa, the test of the same phylogenetic requirements over different computational models. The encapsulation of the underlying technological implementation details  is one of the most important features of this kind of tools. The main advantages of the model checking framework are the abstract description of model and properties using logics and mathematical formalisms, and the portability of these files between computers that have installed the same model checking tool.There exists a considerable diversity of verifiers with different performances, qualities and designs . Among aIn our case, we have selected PRISM for the experimentation for several reasons. PRISM is a generic state-based model checking tool capable of handling probabilistic and timed specifications over Markov chains. PRISM offers Java portability, a powerful syntax for handling time and probabilities in models and specifications, and a good scientific community support. Besides, it is open source, which allows the modification and optimization of its code. The real performance depends on the particular structure of the model and specifications. This fact is shown in the example of case of study.Formal methods help for describing mutation models and evaluating properties over phylogenetic trees. Among all the potential applications, we focus on the problem of analyzing the statistical probability of a phylogeny. Maximum likelihood is one of the most common methods for phylogenetic inference. Given a mutation model, it scores the quality of the topology for a inferred tree. However, the incorporation of new models into current software tools requires an expertise in programming languages and a deep knowledge of the source code. This fact limits the extension, upgrade, debugging and maintenance of a software tool with new models defined by the biologists.TA and CTMCs. A phylogenetic tree is formulated as a succession of nucleotide mutations that are checked over a mutation model. The verification process returns the likelihood of obtaining the tree from a particular model. We have analyzed several phylogenies using the PRISM model checking tool. In addition, we have adapted the software for optimizing the computation of maximum likelihoods.To this end, we show how to define a tree and a mutation model using the description language of CSLworker for computing the maximum likelihood during the evaluation of the tree topology in current inference algorithms and tools The model checking tool requires as input a) a description of the DNA mutation model, and b) the specification of the tree in terms of likelihood equations. Figure nu equals to 1. It expresses the number of ticks needed for the activation of the transition rather than an explicit probability.The computation of the probability is model-dependent. Some of the historical mutation models are Jukes-Cator (JC), Kimura, Felsenstein or the Generalized time-reversible (GTR) . For simspecify the structure of phylogenetic tree with respect to a DNA mutation model: they depict the trace of mutations from the root to the leaves.The translation of the mathematical equations to the syntax of the stochastic logic supported by PRISM is exemplified in Fig. TA syntax, but ? is the threshold calculated by the model checking tool instead of an explicit probability.Besides asking for a particular bound, PRISM includes iterative methods for calculating maximal  probabilities of a path of the tree. For example, the operator P=?[F<=5 x1 = A] returns the probability of reaching a nucleotide A at position x1 of the alignment within 5 time steps in the future. It is equivalent to PRISM allows the specification of the initial state. For example, filter returns the minimum probability taking the sequence x1 = C as the initial state. It ensures that there exists a path starting in x1 = C that eventually reaches x1 = A. The properties are annotated with names for defining the partial likelihoods Pij(·) in Eq. However, the current release of PRISM does not store the partial results Figure All tests have been run on a Intel Core 2 Duo E6750 @ 2.66 GHz with 8 GB RAM and Linux. Memory is not a limiting factor because the experiments have been successfully executed with only 4GB assigned to the Java virtual machine. The factorization of the equations and the initialization cost in PRISM are delays that must be considered for the judgment of the results.Pij(d). In this way, PRISM unfolds the DNA mutation model and generates all the possible traces between the bases i and j, applying implicitly an exhaustive searching method. PRISM looks for all the paths of n nodes with distance Σk=1…ndk=d that maximize Pij(d), being i (j) the initial  state and dk the distance between each pair of intermediate states. The combinatorial exploration of the paths satisfying the previous restrictions of length and probability leads to the exponential cost for calculating Pij(d).The exponential trend observed in the graphic is explained by the exploration method selected by PRISM for calculating the probability function We propose an alternative configuration of the equations in order to take advantage of the peculiar procedure in PRISM for managing probabilities. Figure Figure This framework becomes competitive for obtaining an upper-bound approximation for MLE efficiently. Future optimizations should focus on the implementation of more efficient libraries and dynamic programming techniques that solve, or at least ease, the aforementioned limitations and bottlenecks in PRISM.Model checking is a generic framework that has been applied for qualitative analysis over phylogenies. The main advantages of this approach are the legibility and soundness of formal methods based on temporal logics, the independence of the model from the specifications and the availability of powerful verifiers that automatically analyze the specifications and find counterexamples. Besides, the encapsulation of the implementation in a model checking tool hides and simplifies the access to the underlying technology.Nevertheless, some phylogenetic properties are beyond the expressiveness of first-order temporal logics used in that environment. In this paper, we have completed the previous framework including explicit time and probabilistic information. To this end, we have applied stochastic extensions of the model checking framework for querying quantitative properties over phylogenetic trees and mutation models. The inclusion of explicit time and probabilities matches naturally with the speciation models. In this sense, we have introduced a stochastic logic, data structure and methods adapted for manipulating and computing probabilities over continuous-time systems. Several quantitative properties have been specified.In order to prove the feasibility of our approach, we focused on the problem of analyzing the statistical probability of a phylogeny. We have studied the likelihood of obtaining a phylogenetic tree through the evaluation of maximum likelihood estimations over DNA mutation models. A phylogenetic tree is formulated as a set of paths using temporal logics . The verification process returns the likelihood of obtaining that tree from a mutation model.We have shown how to translate the model and specifications to the particular notation of this framework. We have analyzed several synthetic (random) phylogenies using the PRISM model checking tool. A linear-time heuristic has been proposed for the calculation of an upper bound of the likelihood score. We have customized the tool in order to slightly improve the verification costs.This work opens the door for the review of bigger systems with quantitative properties similar to those defined in this paper. The modularity of our framework allows the evaluation of hypotheses and the comparison of results for a set of phylogenetic trees by only changing the tree file (the specification of the property remains constant). Finally, the search for the valuations that verify a certain specification leads to an intensive exploration of the formula space or the solution of linear systems. The introduction of parametric model checking for the automatic discovery and mining of phylogenetic information outlines our future work.TA, continuous stochastic logic with timed automata; CTL, computational tree logic; CTMC, continuous-time Markov chain; DTMC, discrete-time Markov chain; GTR, generalized time-reversible; JC, Jukes-Cator; MLE, maximum likelihood estimation; MRMC, Markov reward model checker; PCTL, probabilistic CTL; PTS, probabilistic transition systemAP, atomic proposition; BM, back mutation; CADP, construction and analysis of distributed processes; CSL


A large proportion of mental health costs is inpatient care but little is known about their variation between patients. The aim of this study was to measure and identify the predictors of costs of staff contacts and activities on inpatient wards.Inpatients from psychiatric hospital wards in south London were interviewed in 2008 and 2009 and staff contacts and use of activities recorded over a week and costs calculated. Regression analyses identified predictors.Of 334 participants, 78 % used activities and 90 % had staff contacts. However, 41 % reported no nurse contact. Mean staff contact and activity costs were £197 and £30 per week, respectively. Staff contact costs were inversely related to age, and activity costs were higher for patients with higher levels of education. Patient satisfaction was positively associated with both costs.The costs of self-reported staff contacts and use of activities account for a small amount of total inpatient costs. Patients with higher costs appeared to have higher levels of satisfaction. An increased focus on providing care in community settings has taken place in developed countries in recent decades. Despite this trend, psychiatric hospital inpatient services remain an important element of the mental health care system. In 2010/11 in England, investment in inpatient care for working age adults was estimated to be £2 billion, representing 38 % of all direct investment for this population [The care and interventions offered on inpatient wards need to be evaluated, just as any other health care service, in terms of costs and outcomes. Such evaluations require data on the use of services and related costs incurred by those using these services, and for inpatient care this cost has been usually obtained by multiplying the number of days spent in hospital by the unit cost per day. While in some circumstances this may be appropriate, it does not take into account the very likely variation between patients. For evaluations of inpatient interventions it would be more helpful to identify and cost all the care inputs received whilst on a ward. This distinction is important because, even if the length of stay of two patients is the same, the use of resources might be quite different depending on activities attended and the care received. Data on the amount of care received by inpatients in terms of staff contacts and activities attended would be informative for establishing if these wards have the therapeutic ethos that is required to fulfil the purpose of this level of care. Such care will inevitably vary between patients and identifying reasons for such variation is also important for service planning.Previous literature suggests that only a limited amount of time is spent by patients in contact with health care professionals or engaged in therapeutic activities . Most ecThis paper asks the following questions: (1) what amount of ‘meaningful’ care is received by patients on psychiatric wards in a large inner-city hospital in south London in 2008/9? (2) What is the cost of this care? and (3) What demographic and clinic characteristics can predict these costs?This study is part of the “patient involvement in improving the evidence base on inpatient care” (PERCEIVE) research programme. PERCEIVE focussed on the therapeutic environment on inpatient psychiatric wards emphasising service user views and also taking feedback from staff. The programme took place in 17 acute mental health wards of a large service (the South London and Maudsley NHS Foundation Trust) providing mental health care in South London to a population of around 1.1 million people. Inpatients were eligible to be participants in PERCEIVE if they had been on a ward for a minimum of 7 days (with the exception of a triage ward where length of stay would be for up to 3 days), could communicate in English and had the ability to consent. Data for the analyses in this paper were collected between September 2008 and October 2009.Bexley and Greenwich Research Ethics Committee granted approval for this study (Ref: 07/H0809/49).(1) Socio-demographic data: sex, age, ethnicity, level of education, marital status, and living and employment situation, 2) clinical information: diagnosis, whether the current admission to hospital was under the Mental Health Act (i.e. involuntary) and at what age the service user had their first contact with mental health services and time since their first psychiatric admission. 3) Functioning: global assessment of functioning (GAF) [ clinical FunctionData on use of services were collected using a service use schedule, the CITRINE (client services receipt inventory—inpatient), developed as part of PERCEIVE. This tool is completed in a 5–10 min interview between a service user and a researcher or to be self-completed by service users. Its main objective is to obtain the number and location of therapeutic activities and number and duration of one-to-one staff contacts for each service user in the week prior to the interview that they consider to be meaningful.  It is adapted for each ward to reflect the differences in the range of activities provided .The service use data obtained with the questionnaire were combined with appropriate unit costs for 2007/8 obtained from national sources for care professionals . SimilarMultiple regression models were constructed to identify factors that explained variations in costs. Dependent variables were the cost of one-to-one care contacts and costs of therapeutic activities. A simple regression model for each potential independent variable was run and those variables with statistical significance for any of the two dependent variables were included in the final model. All selected variables were entered in a single block in a fixed effects regression to take account of data being obtained from service users on 17 different wards. Cost data are often positively skewed and this may result in regression residuals that are similarly skewed, which is a violation of the assumptions underlying the linear regression model. Consequently non-parametric bootstrapping methods were used . Bootstrp < 0.01).Of those service users who fulfilled the inclusion criteria, 62 % agreed to participate in the study. Service use data were available for 402 service users. However, some values for socio-demographic and clinical variables were missing for some participants or their length of stay was less than 7 days. It was decided to include in the analysis only participants with complete data, and the final sample size was 334. It was found that those with incomplete data had lower GAF functioning scores (35.6) than those included (39.8) (The characteristics of the sample are shown in Table Table More than 40 % of participants reported no contacts with a member of nursing staff in the past week and for those who did have contact, their frequency was slightly higher than one contact every 2 days (Table The service use data were combined with the unit costs for 2008 (see “Appendix Table Table The regression analysis of the variation in costs of one-to-one contacts Table  showed tThis is the first study to present a detailed analysis of the cost of care contacts and activities on adult psychiatric inpatient wards. The results show that service users report low levels of contacts with staff members. It was striking that 40 % did not report contact with nursing staff. The results for therapeutic activities attended do not indicate that the low level of contacts with staff was compensated by more attendances. We have estimated that the cost of care contacts and therapeutic activities is around £30 per day, which is just slightly higher than 10 % of the average cost per bed day £288 for 2008/9 . The latHigh scores on the VOICE questionnaire, representing low satisfaction, were related to low costs of staff contacts and activities. However, the direction of the association is not clear; it is possible that attending fewer activities and having fewer contacts with staff is a result of a worse perception of care received or low levels of care received may lead to poor perceptions of care. In relation to socio-demographic characteristics of the patients, age was inversely related to staff contact costs while service users with more than 12 years of education had activity costs 23 % higher than those with fewer years of education. Differing results have been found on the impact of socio-demographic characteristics on mental health care costs . NeverthThese findings need to be considered taking account of the limitations of the study. First, a relevant proportion of inpatients, almost 40 %, refused to participate in the study. It was not possible to collect information on demographic or clinical characteristics for this group of patients, therefore preventing a comparison with those who agreed to participate. This analysis would have allowed us to establish any significant difference in these variables that would indicate a possible different pattern in their use of services. Second, limitations of the instrument used to collect the data, the CITRINE questionnaire, need to be considered. These have been already discussed , specifiThe low costs found reflect low levels of reported interaction between care professionals and inpatients. This is concerning because it has been established that in inpatient care outcomes are associated to the level of attention patients received from staff . The rel


Indirect reciprocity is often claimed as one of the key mechanisms of human cooperation. It works only if there is a reputational score keeping and each individual can inform with high probability which other individuals were good or bad in the previous round. Gossip is often proposed as a mechanism that can maintain such coherence of reputations in the face of errors of transmission. Random errors, however, are not the only source of uncertainty in such situations. The possibility of deceptive communication, where the signallers aim to misinform the receiver cannot be excluded. While there is plenty of evidence for deceptive communication in humans the possibility of deception is not yet incorporated into models of indirect reciprocity. Here we show that when deceptive strategies are allowed in the population it will cause the collapse of the coherence of reputations and thus in turn it results the collapse of cooperation. This collapse is independent of the norms and the cost and benefit values. It is due to the fact that there is no selection for honest communication in the framework of indirect reciprocity. It follows that indirect reciprocity can be only proposed plausibly as a mechanism of human cooperation if additional mechanisms are specified in the model that maintains honesty. Indirect reciprocity (IR), i.e. repeated encounters between donors and beneficiaries where the donor receives a payback not from the beneficiary itself but later from another individual in the population, is identified to be one of the key mechanisms of human cooperation ,2,3. It c for the donor and a benefit b for the beneficiary (where b > c) or to be defective where there is no donation (no benefit and no cost). Further there are only two states of reputation: bad and good [q) is larger than a given threshold; namely q>c/b [μ), where: q = 1–2 μ [Indirect reciprocity can only work if it is possible to discriminate between individuals with different reputation ,2. In thand good . It was and good  that indly q>c/b . If thisc to the donor and increases the fitness of the recipient with a benefit b. Defection has no cost and gives no benefit. Punishment has a cost α to the punisher (donor) and it incurs a cost β to the receiver (where α < β) [To investigate the problem we use the model of indirect reciprocity described by Ohtsuki et al. . The mode α < β) . Individe α < β) . Ohtsukie α < β)  investigOhtsuki et al.  show than = 100, c = 1, α = 1, β = 4, the value of b changed from 1 to 5 with steps of 0.5). Note that neither of our simulations reproduce the original figures of the Ohtsuki et al. [q) was a parameter controlled by the authors, whereas here q depends on the proportion of the cheaters which is allowed to change in our model and which depends on the success of various signalling strategies. To investigate the effect of the composition of the initial strategies we also repeated these runs starting with 90% cooperators and honest behavioural strategies respectively. Further, to investigate the effect of punishment we repeated the simulations with β = 6, for those norms that utilise punishment (i.e. scoring and shunning). All in all we had three main scenarios: (i) the original one, where the frequency of the cooperative behavioral strategy and the honest signalling strategy is 80% (MST80); (ii) the next one where these frequencies were set to 90% (MST90), and finally where these frequencies were 80% again but we set β = 6 (BETA6). Note, that both of these changes should favour cooperators: in the first case their initial frequency is much higher; in the second case punishment is more efficient. Collapse of cooperation was defined when the cooperative behavioural rule (CD or CP) that was the major strategy at the start in the population (i.e. at 80% or 90%) completely disappeared from the population and the average level of cooperativeness in the population fall below the threshold 0,05. When both conditions were satisfied the run was terminated and the frequencies of behavioural and signalling strategies were recorded alongside with the time of collapse.We repeated the simulations with the same initial conditions as specified in Ohtsuki et al.  allowingi et al.  article:b for the 3 main scenarios respectively  that assign reputation after an action to receivers in bad reputation based on the action of the donor itself collapse fast and they are not sensitive to the changes in b (i.e. collapse as fast regardless of high benefit). This is due to the fact that these later norms  are more sensitive to the information uncertainty created by the cheaters than the former ones. Also, there is a difference between norms in the sequence of events that leads to the disappearance of cooperation: in case of norms using punishment (shunning and scoring) the original behavioural strategy  is very often replaced by other cooperative strategies: ,  or , and in turn these strategies are invaded by selfish ones. On the other hand, in case of norms without punishment  the original cooperative strategy  is usually directly replaced by selfish ones, mostly by . We think that this difference is due to the fact that punishment is costly for both donors and receivers; thus in the face of information uncertainty donors are better off not punishing at all. This selects for switching to other cooperative strategies, even to indiscriminate cooperation , especially if the frequency of cooperators is high (which it is at the beginning of the simulations). Of course, these cooperative strategies are also vulnerable to the information uncertainty, and indiscriminate cooperators are highly vulnerable to selfish behavioural strategies so cooperation disappears even after the switch.Indirect reciprocity crucially depends on the coherence of reputational scores in the population. We have shown that the introduction of deceptive strategies destroys this coherence, thus destroys cooperation itself. This effect is highly robust against the variation of the norms and parameters used in the simulation, though some norms and parameter combination can delay this effect more than some others. Norms treating donors that interact with receivers in bad reputation non-discriminately  favours cooperation, thus the collapse of cooperation can be delayed, but the system is not evolutionarily stable, it is at the mercy of random drift.What is the reason behind this collapse of cooperation? Assuming a finite group of cooperators playing an indirect reciprocity (IR) game, for cooperation to collapse two conditions have to be met: (i) the chance of assigning incorrect reputation has to be over the threshold, (ii) non-cooperative strategies have to be present. The first condition depends on the error of communication and on the frequency of dishonest signallers in the population. Ohtsuki et al.  have shoFew existing studies that investigate the role of deception or cost in the formation of reputational systems support our conclusions. Nakamaru and Kagawa  investigSuzuki & Kimura  investigA related problem is that there is a growing number of models using reputational scores as a tool to generate and maintain cooperative behaviour in social dilemma games without even considering the possibility that the communication of these reputational scores need not be honest , 23, 24.Recent experiments on the role of gossip might seem to contradict our conclusion as all of them show that gossip serves as a reliable tool to increase the cooperative effort in social dilemma games ,28,29,30at most once with each other’ [sensu Nowak & Sigmund [within the IR framework.Last but not least, one may argue that a more realistic implementation of communication could favour cooperative behavioural rules . There ah other’ , (ii) lah other’  and  that anWe have shown that allowing cheating signalling strategies to evolve in games of IR undermines the coherence of reputational scores maintained by gossip, which in turn results in the collapse of cooperation. We have further shown that this is a direct consequence of the general assumption of the IR framework. Thus to maintain the coherence of the IR framework the honesty of communication requires special explanation. In turn, the current model predicts that cooperation maintained by indirect reciprocity should be rare or absent in those societies which lack those means that can maintain the coherence of reputational scores in the face of deceptive communicational strategies.i number of individuals giving incorrect and N-i individuals transmitting correct information about the reputation of a third individual in the population. The information exchange algorithm defined by Ohtsuki et al. [We have i et al.  :ti=i = Nμ/2 thusIn our case μ<<1 then previous equationIf N = 100 and μ = 0.02 in [T = N chances of asking in communication round within two cycle of IR games [Since The information exchange algorithm is the same as in Appendix 1, however if a dishonest individual accepts the opinion of another individual it will transmit just the opposite information later when it becomes an information donor.  We will show that in the above defined information system the opinion followed by the minority about any given individual increases—regardless whether it is correct or not-, thus fixation of an opinion is highly improbable (more precisely fixation needs a very long time ).h and h or G→B is the probability that a G changes to a B Click here for additional data file.S2 File(PDF)Click here for additional data file.S3 File(PDF)Click here for additional data file.S4 File(PDF)Click here for additional data file.S1 Table(PDF)Click here for additional data file.S2 Table(PDF)Click here for additional data file.S3 Table(PDF)Click here for additional data file.


In their recent study (Yaung  See also: SJ Yaung et al (March 2015)et al  that are depleted from the insertion sequence library. An advantage of this method, termed insertion sequencing (INSeq), is that it probes gene function in the context of an organism's native genome rather than by transplanting genes into a foreign host organism, with perhaps markedly different functional potential. These INSeq libraries can provide information about the depletion of tens to hundreds of genes, whereas overexpression-based functional metagenomics methods like TFUMseq might be more biased toward identifying a smaller number of genetic regions that improve fitness. This is due to the fact that large increases in fitness can quickly bottleneck a library into a few dominant loci that outcompete the remainder of the library and limit the breadth of information gathered from any individual experiment. Importantly, Goodman et al found numerous genetic regions that were important in the context of a diverse community of microbes but not in the context of mono-colonization. Moving forward, it is essential to apply TFUMseq to microbial communities of increasing complexity.Similar functional metagenomic approaches have been used l, et al  used a ret al for understanding the functional capacity of gut microbes drives to the heart of the most fundamental requirement of any organism in the microbiome, namely its ability to stably colonize a host in the context of the rich genetic diversity of competing organisms and host factors. Notably, the microbiota in an adult human gut is stable, with the majority of organisms likely remaining there for decades (Faith et al, et al, et al, The framework presented by Yaung


Dark Matter (DM) detection prospects at future  The search for dark matter is of prime importance for our understanding of the universe. This goal is pursued using a wide variety of approaches, given the very large spectrum of interpretations predicting particles with a mass range between On recent times several hints for detection have come from Indirect Dark Matter Detection (ID) searches, i.e. lines at 3.5 KeV , 2 and 1c center .Similarly previous hints in Direct Detection (DD) seem contradicted by recent results by SuperCDMS , XENON10Collider searches are therefore the necessary complement for a safe conclusion on this essential investigation. Here we will focus on the prospects offered by future see also –17 for sZ width from LEP1 and the invisible H width from LHC. We also remark that direct production at linear colliders of the mediators allows for the observation of distinctive signatures, in the form of resonances, and to profit of the optimal capability of mass reconstruction of these kind of colliders.For our investigation we will consider simplified scenarios .A heavy ussed in . If the In scenario 1 the width of the It should be underlined that this invisible From Fig. Z where RL is a luminosity ratio which takes care of the difference of mass between Z contribution would be A quasi-invisible see e.g. –52), as ent data , 54 takeFigures If LHC observes only an excess at large In conclusion, for the invisible In scenario 2, the DM being a Majorana fermion, the decay width of the An interaction between a SM singlet fermionic DM and the SM Higgs can be achieved, contrary the case of a scalar DM (see e.g.  for a reetection –60; for It is anyway worth investigating whether As the DM mass increases, the coupling to the DM should rapidly vanish in order to comply with the correct DM relic density. For }% level .If orted in . Here itWe now consider the case that the main interactions of the DM are with two Higgs singlets, As we notice the contribution from pure pseudoscalar mediation is not velocity suppressed. It is then possible to fit the GC signal in case this is the dominant contribution. This requirement is actually rather easy to fulfill since the couplings of the pseudoscalar are substantially irrelevant for DM direct detection  while, oM setups –67. TherM setups , 38 of t section , which c as well .The GC excess could be fitted, alternatively, in agreement with the DM relic density, in the case the annihilation channel  120 GeV . As alreIn the setup depicted the a/s particles can be produced through top loops and detected through decay into At LHC the At a In the presence of A larger variety of experimental signatures can be provided in a 2 Higgs-doublet scenario. Particularly interesting would be a type II two Higgs-doublet scenario  (SUSY moThe DM annihilation cross section and the decay widths of the  leptons . This kiFigures A events  with \doFor what concerns H, which will serve to tag the presence of an invisible A, the standard decay mode is dominated by hh Figure In the case of ± decays –74 whichAn alternative detection strategy for an invisible ested in . The relested in  indicateWe have considered some simple DM scenarios, possibly providing an interpretation of the recently reported Z width is the most sensitive SM observable to monitor this scenario. With the accuracy given by LEP1, one can already disfavor In the case of In the case of scalar mediators the SM Higgs is ruled out by limits from DM Direct Detection and invisible decay width of the Higgs itself.Interactions mediated by a pseudoscalar Higgs single state are a rather simple and economic way to account for the GC excess. This state can be lighter than the SM Higgs boson and then observed in its decays. Associate production of this particle with a A larger variety of collider signatures in the case the pseudoscalar state is achievable in a two-doublet Higgs extension of the SM. Moreover, in this kind of scenarios it is possible to naturally achieve a DM mostly annihilating into bottom pairs. In this kind of scenario the DM is assumed to couple exclusively with the pseudoscalar component of the Higgs spectrum which decays mostly invisibly. It can be observed at a TeV The interpretation of the GC excess seems to disfavor SM particles as mediators of the DM interaction. If this excess would be confirmed it would provide an indication of the existence of extra particles, besides the DM, beyond the SM. Although several scenarios can already tested at LHC, a TeV Note added: Soon after the completion of this work new analyses [analyses , 78 haveented in .


Investigation of divisibility properties of natural numbers is one of the most important themes in the theory of numbers. Various tools have been developed over the centuries to discover and study the various patterns in the sequence of natural numbers in the context of divisibility. In the present paper, we study the divisibility of natural numbers using the framework of a growing complex network. In particular, using tools from the field of statistical inference, we show that the network is scale-free but has a non-stationary degree distribution. Along with this, we report a new kind of similarity pattern for the local clustering, which we call “stretching similarity”, in this network. We also show that the various characteristics like average degree, global clustering coefficient and assortativity coefficient of the network vary smoothly with the size of the network. Using analytical arguments we estimate the asymptotic behavior of global clustering and average degree which is validated using numerical analysis. The study of complex networks has become a very important part of many disciplines like information5688et al.et al.In the present paper, we report an analysis for a particular deterministic network that resembles real networks in many aspects. This network consists of natural numbers 1, 2, 3, … as nodes and if a given number divides another, then their corresponding nodes are connected by an undirected link. The network thus constructed, though deterministic, can be studied on an equal footing with the other random networks because of the irregular distribution of primes which makes divisibility relations themselves irregular. It is helpful to view this network as a growing network where nodes are added one at a time. A similar network with nodes as composite numbers has already been studiedIn the present work we consider a more general set up where we put all the natural numbers on a complex network with their divisibility relations as the underlying deterministic rule of connections. Here the network is undirected with links to both divisors and multiples. Using tools from statistical inference, we confirm that this network is scale-free and show that average degree, global clustering coefficient and assortativity coefficient vary smoothly with the size of the network. This is surprising in view of the fact that distribution of primes is quite irregular in the sequence of natural numbers. We provide analytical results for the asymptotic behavior of average degree and global clustering coefficient for this network. In particular, we show that the global clustering coefficient of this network decays to zero whereas average degree increases logarithmically. We also report an interesting and novel similarity exhibited by local clustering coefficients of nodes in this network which we call “stretching similarity”.The remaining paper is organized as follows: In the next section we describe the construction of the network and show that the network is scale-free. We then describe the existence of stretching similarity in this network. Finally we show the behavior of average degree, global clustering coefficient and assortativity coefficient as a function of size of the network and analytically obtain the asymptotic trends for average degree and clustering.t = 1 network starts with a single node n = 1 and at every time t, a node with the number n = t is added to the network.At time This node connects to all the existing nodes whose numbers divide it.The nodes of the present network are natural numbers 1, 2, 3, … and there is a link between two nodes if either divides the other. We avoid self-links and all the links are undirected. Since the sequence of natural numbers has natural ordering, it is helpful to view this network as a growing network with the addition of a new node at each discrete time as follows:t = 16 and t = 32 which would correspond to networks of size N = 16 and N = 32 respectively. To find the distribution of degrees of this network, we grow the network till the size reaches N = 225 = 3, 35, 54, 432. The resulting distribution shown in α ~ 2. We establish the existence of power-law in the distribution  using the approach described in Clauset et al.i with degree ki this can be written asThe network thus constructed is shown in Ei is the actual number of edges among the neighbors of node i.where In ci of node i varies seemingly irregularly. However, when ci is plotted against i, a global pattern is seen. In N some discontinuous vertical steps occur approximately at values N/3 and N/2 and these numbers correspond to prime numbers and their powers in that range. This can be seen by the following argument. Consider any prime number p in the interval . On the lower side, it is connected only to 1 while on the upper side, it would be connected only to its multiples. However, all the numbers in this range would have only one multiple 2p up to N. Thus, three numbers 1, p, 2p form a triangle and hence clustering coefficient of number p must be 1. A similar argument for prime powers in this range tells that they also have clustering coefficient 1. There is another band of numbers with clustering coefficient exactly 0 between N/2 and N which are also prime numbers. This is because all the primes in this range are connected only to 1 making their clustering 0.We now discuss an interesting behavior that sets network of natural numbers apart from other complex networks. In the network presented here, each node has an identity which is the number attached to it and this defines a natural order on the nodes. This means that we can study various properties of nodes as a function of their labels. This is not possible for other networks because no such unique labeling exists for the nodes. Here we specifically consider local clustering coefficient of nodes and study its behavior as a function of node index. We find that the clustering coefficient N/2 and N. For a vertex n, the only neighbors are the proper divisors of n i.e. m such that 1 ≤ m < n and n is divisible by m.Now we discuss the local clustering coefficient for the composite numbers between n as the product of (distinct) prime powers. The fundamental theorem of arithmetic states that such a factorization is unique up to a reordering of the primes pi’s. It can be observed that every divisor m of n is of the form i ≤ k.Let m < m′ are adjacent to each other if and only if i.Any two nodes n in the network of size N is given by,Thus the clustering coefficient of where cn depends only on the number of distinct prime factors of n and the powers ji’s which appear in the prime factorization of n; but not on the actual primes which appear there. Thus for any given j1, j2, … jk, the value cn is constant for every n in the range N/2 < n ≤ N such that k distinct primes p1, p2, … pk. This explains the occurrence of horizontal dotted lines in the plot for local clustering coefficients.From the above expression it follows that value of n can be computed and it can be observed that they depend on the powers and the number of distinct prime factors of n as well as the range in which n belongs that is r such that n in the range 1, 2, …, N. This leads to possibly different values of clustering coefficients. This explains the occurrence of demarked regions like N/2 to N, N/3 to N/2, N/4 to N/3 etc in the plot for local clustering coefficients. For any N there will be sufficient number of primes in the range  and choices for ji such that the pattern of horizontal lines between N/2 to N remains the same. Also, the demarked regions have similar structures. This provides a possible explanation for the observed stretching similarity in the clustering coefficients as N is changed . With increasing size of the network, this pattern also shows stretching similarity.We also observe an interesting pattern when we plot the difference  In the present section, we discuss how three of the most important quantities average degree, global clustering and assortativity coefficient vary with the size of the network.Here we derive an approximate expression for the average degree of the network as a function of its size. By definition, the average degree of the network is given by:m is the total number of edges in the network and n is the size of the network. The value of m is also equal to the sum of the elements in lower (or upper) triangular part of the adjacency matrix. To find this sum, we interpret the second index of element Aij of adjacency matrix to be the divisor of first index if Aij = 1. In other words, let Aij = 1 if and only if i > j and j|i. Then the sum of the elements in the lower triangular part of the matrix is equal to the number of integers of the form kj with k ≥ 2 and kj ≤ n. However, whenever jth column of the lower triangular part of A are zero. Let x. Then m is given by:where It is well known that the first term on the right satisfies an estimate as followsγ is Euler-Mascheroni constant. Also we observe that:where From γ ≈ 0.5772, in the limit of large n, we get,Since N = 225 and the results obtained, shown by solid dots in This means that the average degree of the network increases logarithmically with the size and this variation is plotted in The global clustering coefficient of the network quantifies the density of closed triplets in the network. A connected triplet in the network is the set of 3 nodes connected to each other with at least 2 links. A closed triplet is the set of 3 nodes connected to each other with exactly 3 links. A triangle in the network counts as three closed triplets (one centered at each node of the triangle). The global clustering coefficient of the network is then defined as:Tn in the network using the following strategy. Let us fix a vertex i and calculate the number of triangles in which i is the smallest vertex. The number i has n]. Each of them is of the form ki where Tn is given by:We estimate the number of triangles Using the integral approximation for the above:The above is bounded by,A and B are constants. Hence we see that:Here In particular,U(n) be the the number of connected triplets in the network after nth stage. Then U(n) is given by:Let kα− with α ~ 2, we see that the proportion p(k) of nodes with degree k is ~k−2.Since we have  observedk2 satisfies:Thus, the expectation of the variable Hence we see thatCWS of the network th element of the adjacency matrix, m is the total number of edges in the network and δij is the Kronecker delta.where r on the size of the network and in spite of irregularity in the divisibility pattern, r has a smooth behavior with n. It can be seen that r always remains negative though asymptotically it seems to reach the value 0 implying that the network is dissortative. The dissortative nature of the network of natural numbers is understandable from the following argument. For any link in this network, the one end of the link is divisor (node A) and other is multiple (node B). Hence node A is also connected to all the nodes which are multiples of B but the reverse is not true. This means that the degree of node A always tends to be very high as compared to degree of node B for a given size of the network giving the negative value for the overall correlation coefficient.In We also find that all the important statistical properties of the network like stretching similarity, degree distribution, clustering-degree correlation etc. are very robust to the removal of even the biggest hubs like numbers 1, 2, 3, …. This shows that the global divisibility pattern of natural numbers does not depend only on the few nodes but instead is built by contributions from all the nodes. (See Methods)et al.CWS saturates to a high value. All these results are validated by extensive numerical calculations for network up to size 225.The network of natural numbers constructed using divisibility relations looks like real networks in many characteristics like degree distribution, clustering and degree correlations. We show how insights into the divisibility patterns of natural numbers can be obtained using the framework of complex networks, where we consider both composite and prime numbers in a single undirected network with links generated using both multiples and divisors. Some of the interesting results that we get are the scale-free nature of the network with a non-stationary distribution and the existence of stretching similarity. We validate the existence of power-law in the distribution and estimate the corresponding power-law index using rigorous techniques from statistical inference advocated by Clauset We also find that there exists a pattern in the local clustering coefficients that reflects universality in the organization of natural numbers in terms of their prime constituents. We observe that this pattern has a stretching similarity which is a reflection of the nature of prime factorization of natural numbers. Also, the behavior of characteristics like average degree, global clustering and assortativity coefficients for this network vary quite smoothly and hence may help us to understand better the divisibility relations between natural numbers. In conclusion, the work presented here describes an interesting perspective on the divisibility relations of natural numbers and has potential to become an important tool in the investigation of the properties of natural numbers.k ≥ kmin). However a visual inspection to find kmin and least square fit and related methods to find the exponent α of the power law are known to produce very bad estimatesα of the power-law distributionkα− for all k after k ≥ kmin. To find this kmin, we use the approach proposed by Clauset et al.k as kmin which makes the probability distribution of the data and best-fit power-law model as similar as possible above kmin where we use Kolmogorov-Smirnov statistic as the distance between two distributionskmin using this method, the best estimation for scaling exponent α is given by:The shape of the degree distribution of the network in ki, i = 1, …, N are values of k such that ki ≥ kmin. For the network of size 215, the value α is obtained here as ~2.where et al.p–value is defined as the fraction of the synthetic-empirical distances that are larger than the empirical distance. A large p–value is indicative of existence of power law in the data. In the present work we calculate the p–values for three different sizes of the network: N = 256, 512, 1024. For this, we generate 2500 synthetic data sets which gives p–values accurate up to two decimal places as 0.62, 0.95 and 0.98 respectively. The existence of power-law degree distribution for this network is thus confirmed by the fact that p–values rapidly converge to 1 as the network size increases.To validate the existence of power law, we use the approach described in Clauset c(k) on degree k in The distribution in c around the horizontal axis Δc = 0  of a particular pixel  as the ratio of the number of points present in the pixel to the maximum number that can be there which is equal to 2N−7 . For each x we calculate the absolute difference between the corresponding pixels on each side of the line Δc = 0. If the pattern is symmetric then these absolute differences are expected to be small. We calculate the average of such differences as:To establish the global symmetry of difference in local clustering values Δs Δc = 0  for any ϕ(x) as a function of x and as is clear from the figure, all ϕ values are very close to 0 confirming that the pattern is indeed symmetric.In N/2 and N become isolated and these remain as the only isolated nodes. This means that in this case the network consists of a giant component along with many isolated nodes. We find that such a removal does not affect the degree distribution and clustering-degree correlation too much and qualitatively the network remains scale-free with the same power-law index as for the original network. The other properties like average degree, clustering coefficients and assortativity do change to some extent by this removal but qualitatively remain the same. The plot of degree distributions after removing hubs is shown in To test the robustness of the various statistical properties of the network against the removal of hubs from the network, we simulated the network of natural numbers removing numbers 1 to 4 step by step. When number 1 is removed from the network, all the prime numbers between How to cite this article: Shekatkar, S. M., Bhagwat, C. and Ambika, G. Divisibility patterns of natural numbers on a complex network. Sci. Rep .5, 14280; doi: 10.1038/srep14280 (2015).


A method to evaluate chemical element concentrations in samples by generating an effective polychromatic beam using as initial input real monochromatic beam data is presented. There is a great diversity of research being conducted at synchrotron facilities around the world and a diverse set of beamlines to accommodate this research. Time is a precious commodity at synchrotron facilities; therefore, methods that can maximize the time spent collecting data are of value. At the same time the incident radiation spectrum, necessary for some research, may not be known on a given beamline. A preliminary presentation of a method applicable to X-ray fluorescence spectrocopic analyses that overcomes the lack of information about the incident beam spectrum that addresses both of these concerns is given here. The method is equally applicable for other X-ray sources so long as local conditions are considered. It relies on replacing the polychromatic spectrum in a standard fundamental parameters analysis with a set of effective monochromatic photon beams. A beam is associated with each element and can be described by an analytical function allowing extension to elements not included in the necessary calibration measurement(s). Placing a detector at the sample position itself may be complicated due to a high count rate. In any case, all of this would imply the use of valuable beam time, especially if new electronics are to be introduced or the beamline set-up significantly changed in order to allow the measurement of the incident beam spectrum. On the other hand, the method proposed here requires nothing more than a brief (∼1 min) XRF measurement on the VESPERS beamline to assess the incident spectrum without changing any aspect of the beamline set-up.A collaboration between the University of Regina and the Royal Saskatchewan Museum  is conducting research to probe fossils at microscopic level using the synchrotron radiation facilities available at the Canadian Light Source in Saskatoon, Saskatchewan. This research is focused on the qualitative and the quantitative analysis of fossil chemistry in the pursuit of palaeo-ecological information. An important part of this work is the use of hard X-rays for fluorescence (XRF) analysis. There are pre-existing packages for the analysis of X-ray fluorescence spectra . The incident and detection angles are ϕ and ψ, respectively. The variables i, which lead to the emission of the line of interest. ith element. X at photon energy Y. The matrix is denoted as M and single elements denoted with i or j. When N beams is known, the concentration of the ith element can be determined through an iterative method.One way to develop an effective polychromatic beam is by modifying the standard fundamental parameters equation, as presented by Beckhoff  al. 2006, for pri3.−1. The standard is a 3 mm-thick circular wafer with a diameter of ∼13 mm.To properly determine the effective beam, a standard is needed. The standard must have known concentrations of several elements. Here the standard used is SRM 610 from the National Institute of Standards and Technology (NIST)  of the certified and reference values. A systematic error based on this assumption is included, taken to be one half of the difference between the results computed assuming an uncertainty of 1σsd and 3σsd for the information values.The element concentrations (mass fractions) in the standard are guaranteed to varying degrees by the manufacturer. There are 15 certified elements, whose concentrations are given with a high degree of confidence. Two reference values are given. These are not certified, but are best estimates which do not meet the certification criteria. Information values are given for seven elements. These are values considered of interest, but with uncertainties that were not able to be determined  ≤10 µm. Some deviations from the nominal concentrations can be expected at this small scale.The elements composing the glass matrix are not included in this classification. The matrix is specified as having mass fractions: 72% SiO4.4.1.E0 is set to 18995 eV as only one of the beam energy or the incident intensity needs to be varied in this equation in order to construct the effective beam. This value was chosen such that the cross sections  of the various elements will be similar and at the same time the approximation et al., 1993N monochromatic beams for the elements detected in the sample. Fig. 1To calibrate the effective beam, the element concentrations as given by NIST are treated as known constants in equation (1)Kα emission lines in the detectable range. The method, at this level, could just as well be applied to other shells.The elements in Fig. 1Differences between samples are expected in that the spectrum ‘seen’ by the sample depends on its composition. The effective beam is compensating for this spectrum and must vary to some extent with sample composition. For this reason, the best results are expected when the calibration standard and analytes are similar in their absorption/enhancement properties. Preliminary measurements with other references suggest that the general spectrum shape varies little when derived from a variety of natural minerals such as apatite, clinopyroxene and biotite. However, the errors associated with those measurements are large due to the low statistics collected, and the results are not conclusive enough to be used in the construction of an effective beam. For this reason we have opted to not include them in this paper. On the other hand, given the fact that the effective beam constructed with data collected with SRM 610 successfully determined the concentrations of elements in natural minerals, as discussed in §4.54.2.K-shell emissions are considered. A functional form can be obtained by considering the detected intensity in the first approximation (primary excitation only). It is further assumed that the incident spectrum can be described by a normalized distribution i is theni is related to the detected intensity, N to be the concentration of absorbers in the material, where m is the mass of an electron and k is the photon energy, this has the formE is used to denote the energies of the incident photon spectrum, and k is the energies of other photons. For the case at hand As a general trend appears in the effective beam spectrum, it is reasonable to fit it with a functional form for application to all detectable elements. Only tler 1954 and are Ec is the critical energy of the synchrotron. This result gives the standard form for synchrotron radiation shown in Fig. 2It is expected that the form of kson 1999. The relkson 1999, where  4.3.et al.  al. 1993. The choi the value of et al. tler 1954. It is a al. 1993, can be et al., 2007The corresponding parameter values are given in Table 14.4.It must also be asked whether the concentrations for the SRM 610 sample are reproduced if the effective beam is used for an iterative concentration determination with equation (1)There are two variations of the calculation shown in the figure. One compensates for the undetected elements in the sample by including them in the mass absorption coefficient of the matrix and the normalization of the concentrations. A reference element is chosen and all undetected elements are included in their nominal concentrations relative to the reference element. The other uses only detected elements in the calculation. The latter case is relevant, as in general the presence of elements is only known if they are detected.Looking at the absolute concentrations in Fig. 5It is critical that there is agreement between the non-compensated and nominal concentrations as the non-compensated scenario is the one that matches the general experimental conditions. If agreement is reached here, then it is anticipated that the relative concentration values will be reliable for other samples where some fraction of the elements in the matrix are unknown.It can be seen in Fig. 54.5.2O6] and biotite [K3(AlSi3O10)(OH)2]. Concentrations referred to will be atomic fractions, whether discussing nominal or determined values. Because these are natural samples, they contain many trace elements in addition to the nominal ones. Deviations from nominal concentrations are also possible. Only the nominal elements will be considered, because, to test the determination of trace elements, measurements through another method must be performed. The nominal ratio CFe/CK = 3 is expected for biotite. The nominal ratio CFe/CCa = 1 is expected for clinopyroxene. X-ray fluorescence measurements on the biotite reference give non-compensated concentrations CK = 0.28 ± 0.08 and CFe = 0.62 ± 0.10 and a ratio of CFe/CK = 2.2 ± 0.8. Similarly for the clinopyroxene sample it was determined that CCa = 0.51 ± 0.12 and CFe = 0.41 ± 0.07, yielding a ratio CFe/CCa = 0.80 ± 0.2. Both of these results agree with expectations from the nominal mineral compositions. As with the SRM 610 sample, the relative concentrations agree with the nominal values without considering undetected elements. It should also be noted that these samples all represent different matrices, suggesting validity of the method for different sample types. The matrices of SRM 610 and these minerals are similar in that a considerable portion of each material is low-Z elements, which is typical of many minerals. In the event that all elements can be detected, the absolute concentrations should be valid estimates of the real concentrations and accuracy of results should improve. The precision of the concentrations presented here is largely constrained by the uncertainty in fitting the peaks in the collected XRF spectra. Reduction in the uncertainty can be achieved through improved statistics and fitting methods SiIt is also interesting to note that the clinopyroxene data and biotite data were taken during different runs on the VEPSERS beamline. The biotite and SRM 610 data were taken during the same run. Further, the two runs used different XRF detectors. It can be suggested that the effective beam results can be applied across runs, assuming that no major change in beamline conditions has occurred that would greatly effect the beam spectrum.5.Evidence has been given supporting that the effective beam method, in the form given herein, provides an accurate means to assess the concentration of elements in XRF spectra, taken from polyatomic samples, using a polychromatic incident spectrum. Though presented for synchrotron data, it should be useful for other sources. The semi-physical function used here could be remodelled for other sources, or a simple polynomial fit used.Further definition and testing of the effective beam spectrum given here can be performed through the use of additional standards. Improvements in the accuracy and precision of the results are expected with the use of well defined references on the scale of measurement and higher statistics data, respectively.