Uniform Approximations for the M/G/1 Queue with Subexponential Processing Times Abstract: This paper studies the asymptotic behavior of the steady-state waiting time, W∞, of the M/G/1 queue with subexponenential processing times for different combinations of traffic intensities and overflow levels. In particular, we provide insights into the regions of large deviations where the so-called heavy traffic approximation and heavy tail asymptotic hold. For queues whose service time distribution decays slower than e − √ t we identify a third region of asymptotics where neither the heavy traffic nor the heavy tailed approximations are valid. These results are obtained by deriving approximations for P (W∞ > x) that are either uniform in the traffic intensity as the tail value goes to infinity or uniform on the positive axis as the traffic intensity converges to one. Our approach makes clear the connection between the asymptotic behavior of the steady-state waiting time distribution and that of an associated random walk. We study in this paper the asymptotic behavior of the steady-state waiting time distribution of an M/G/1 queue with subexponential service time distribution and first-in-first-out (FIFO) discipline. The goal is to provide expressions that will allow us to identify the different types of asymptotic behavior that the queue experiences depending on different combinations of traffic intensity and overflow levels. We give our results for the special case of an M/G/1 queue with the idea that the insights that we obtain are applicable to more general queues and even to networks of queues.. The special case of an M/G/1 queue with regularly varying processing times was previously analyzed in [17] , where it was shown that the behavior of P (W ∞ > x), the steady-state waiting time distribution, can be fully described by the so-called heavy traffic approximation and heavy tail asymptotic (see Theorems 2.1 and 2.2 in [17] ). As pointed out in that work, the same type of results can be derived for a larger subclass of the subexponential family, in particular, for service time distributions whose tails decay slower than e − √ t . As the main results of this paper show, the behavior of W ∞ for lighter subexponential service time distributions may include a third region where neither the heavy traffic approximation nor the heavy tail asympotic are valid, and where the higher order moments of the service time distribution start playing a role. The exact way in which these higher order moments appear in the distribution of W ∞ is closely related to the large deviations behavior of an associated random walk and its corresponding Cramér series.. The approach that we take to understand the asymptotics of P (W ∞ > x) over the entire line is to provide approximations that hold uniformly across all values of the traffic intensity for large values of the tail, or alternatively, uniformly across all tail values for traffic intensities close to one. From such uniform approximations it is possible to compute the exact thresholds separating the different regions of deviations of W ∞ , which for service time distributions decaying slower than e − √ t are simply the heavy traffic and heavy tail regions, and, for lighter subexponential distributions, include a third region where neither the heavy traffic approximation nor the heavy tail asymptotic hold. Similar uniform approximations have been derived in the literature for the tail distribution of a random walk with subexponential increments in [6] , [7] , and [20] , where the uniformity is on the number of summands for large values of the tail or across all tail values as the number of summands grows to infinity. The results in the paper are in some sense the equivalent for the single-server queue.. To explain the idea behind our main results let us recall that one can approximate the tail distribution of the steady-state waiting time of a single-server queue with subexponential processing times, P (W ∞ > x), via two well known approximations: the heavy traffic approximation and the heavy tail asymptotic exp − 2(Eτ 1. respectively, where V 1 denotes the service time, τ 1 the inter-arrival time, and ρ the traffic intensity of the queue. We refer the reader to Chapter X of [1] and the references therein for more details on the history and the exact formulation of these limit theorems. The heavy traffic approximation is valid for the general GI/GI/1 queue and can be derived by using a functional Central Limit Theorem type of analysis (see, e.g. [11, 12] ). The theorem that justifies this approximation is obtained by taking the limit as the traffic intensity approaches one and is applicable for bounded values of x. The heavy tail asymptotic is valid for the GI/GI/1 FIFO queue with subexponential service time distribution (see, e.g., [10] ), and is obtained by taking the limit as x goes to infinity for a fixed traffic intensity, that is, it is applicable for large values of x. One can then think of combining these two approximations to obtain an expression that is uniformly valid on the entire positive axis.. The approach we take in the derivation of the main theorems is to start with the Pollaczek-Khintchine formula for the distribution of the steady-state waiting time of the M/G/1 queue, which expresses it as a geometric random sum, and use the asymptotics for the tail distribution of the random walk. One of the difficulties in obtaining uniform asymptotics for the distribution of W ∞ lies in the highly complex asymptotic behavior of the random walk. Surprisingly, most of the cumbersome details of the asymptotics for the random walk disappear in the queue, but showing that this is indeed the case requires a considerable amount of work. The qualitative difference between queues with service time distributions with tails decaying slower than e. and their lighter-tailed counterparts comes from the asymptotic behavior of the random walk associated to the geometric random sum. The function e − √ t has been identified as a threshold in the behavior of heavy tailed sums and queues in [6, 16] , and [3, 13, 14] , respectively, to name a few references, and we provide here yet another example.. As mentioned before, the approximations we provide can be used to derive the exact regions where the heavy traffic and heavy tail approximations hold, but we do not provide the details in this paper since our focus is on deriving uniform expressions for P (W ∞ > x) under minimal conditions on the service time distribution. The setting we consider is the same from [3, 14] where the busy period was analyzed. More detailed comments about the third region of asymptotics that arises when the service time distribution is lighter than e − √ t can be found in Remark 2 right after Theorem 3.4. For clarity, we state all our assumptions and notation in the following section, and our main results in Section 3.. Finally, we mention that the expressions given in the main theorems can be of practical use as numerical approximations for P (W ∞ > x), and based on simulation experiments done for service times with a Pareto (α > 3) or Weibull (0 < α < 1/2) distribution, they seem to perform very well (see Section 4 in [17] ). It is worth pointing out that the uniform approximations given here are far superior than the heavy traffic or heavy tail approximations individually even in the regions where these are valid, which is to be expected since they are based on the entire Pollaczek-Khintchine formula; they are also easy to compute given the integrated tail distribution of the processing times and its first few moments (cumulants).

B I O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A Complete Axiomatization for Blocks World Blocks World (BW) is one of the most popular model domains in AI. In this paper, we model a state of BW by a nite collection of nite chains, and call the theory of all these structures BW theory. We present seven simple axioms and prove that their consequences are precisely BW theory. We present a simple decision procedure for the theory which can be implemented in exponential space, and prove that every decision procedure (even if nondeterministic) for the theory must take at least exponential time. We characterize both the de nable predicates in the theory and its nonstandard models. Blocks World (BW) is one of the most popular model domains in arti cial intelligence. This domain consists of a set of blocks of various shapes, sizes and colors sitting on a table. A robot can pick up a block and move it to another position, either onto the table or on top of some other block(s). A simple and well-known version of BW consists of cubic blocks of equal size.. BW is most extensively used in studies of planning. Roughly, the planning task is to nd a sequence of actions which transforms a given initial state into a given goal state. Gupta and Nau 5] showed that optimal BW planning is NP-hard. Slaney and Thi ebaux 10] presented linear time algorithms for near-optimal BW planning within a ratio of 2.. BW is also used in studies of action theories, a subarea of AI concerned with representing and reasoning about actions and their e ects. See Reiter 9] for an action theory for BW in the situation calculus. Liu 7] used BW in veri cation of robot programs written in GOLOG 9] , a situation calculus-based logic programming language for high-level robotic control. This research suggested the need for a formal theory of BW.. Such a theory would address two issues relevant to program veri cation in 7] . First, it would be useful to have a complete axiomatization of the state constraints of BW. By state constraints of BW, we mean those properties which hold in every state of BW. For example, \no block can be above itself" is such a property. Without such an axiomatization, it is even impossible to prove some simple properties of BW programs. Despite the popularity of BW, to the best of our knowledge, there has not been serious work on axiomatizing the state constraints of BW and giving justi cation for its soundness and completeness.. The second issue concerns characterizing the de nable predicates in the BW domain. In proving relative completeness of Hoare Logic, Cook 2] de ned the notion of expressiveness. Roughly, we say that a rst-order language L is expressive relative to a set S of programs, if for any L-formula and any program P in S, there is a L-formula expressing the post relation corresponding to and P. A similar notion of expressiveness can be de ned in the context of robot programs, and this raises the question of whether the BW language is expressive.. In this paper, we address these two issues. We model a state of BW by a nite collection of nite chains, and call the theory of all these structures BW theory.. In Section 2, we introduce the syntax and semantics of BW theory. In Section 3, we give a set of axioms and prove that their consequences are precisely BW theory by a game-theoretic argument. We give a simple decision procedure for BW theory requiring exponential space, and prove that every decision procedure for BW theory requires at least exponential time, even if it is nondeterministic. We also give a characterization of all nonstandard models for the theory. In Section 4, we present an expansion of BW theory and show that it admits elimination of quanti ers. As a result we are able to characterize all de nable predicates in BW theory, and give simple examples of unde nable predicates.

O O O O B I B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Gaussian filtering and variational approximations for Bayesian smoothing in continuous-discrete stochastic dynamic systems The Bayesian smoothing equations are generally intractable for systems described by nonlinear stochastic differential equations and discrete-time measurements. Gaussian approximations are a computationally efficient way to approximate the true smoothing distribution. In this work, we present a comparison between two Gaussian approximation methods. The Gaussian filtering based Gaussian smoother uses a Gaussian approximation for the filtering distribution to form an approximation for the smoothing distribution. The variational Gaussian smoother is based on minimizing the Kullback-Leibler divergence of the approximate smoothing distribution with respect to the true distribution. The results suggest that for highly nonlinear systems, the variational Gaussian smoother can be used to iteratively improve the Gaussian filtering based smoothing solution. We also present linearization and sigma-point methods to approximate the intractable Gaussian expectations in the Variational Gaussian smoothing equations. In addition, we extend the variational Gaussian smoother for certain class of systems with singular diffusion matrix. The continuous-discrete system refers to a system whose process dynamics are governed by a continuous-time stochastic differential equation (SDE) and whose measurements are taken at discrete time instants. Bayesian filtering and smoothing equations give the solution to the problem of estimating the state of the system from the noisy measurements. Computing the filtering and smoothing distributions involves solving the related partial differential equations [1, 2] , and is only tractable for linear-Gaussian systems (and some other special cases [3] ). In this paper, we consider two different approaches for computing a Gaussian approximation for the smoothing distribution.. The Gaussian approximation for the filtering and smoothing equations is well known in the literature. The first Gaussian approximations were based on linearization using the Taylor series based methods [4, 1] . The Taylor series based methods can be seen as a special case of the more general Gaussian filtering and smoothing framework, where different filters and smoothers arise based on the numerical method for computing the Gaussian expectations [5, 6] . A different approach for Gaussian smoothing was considered in [7, 8, 9] , where a Gaussian approximation is sought by approximating the stochastic process giving the smoothed distribution with a linear process. The method is based on the fixedform variational Bayes approximation [10] and minimizes the Kullback-Leibler divergence of the approximate distribution with respect to the true distribution.. The variational Gaussian approximation is considered further in [11, 12, 13, 14] . Shen et al. [11] compared the variational approximation to a Monte Carlo Markov Chain (MCMC) solution for the one dimensional double well system and found that the variational method performed comparatively to the MCMC solution when the uncertainties in the measurements were not so large as to cause the true posterior to be multimodal. In [12] the variational Gaussian smoothing solution is used as a proposal distribution in an MCMC method to improve the efficiency of the algorithm. The variational MCMC method was found to outperform the hybrid Monte Carlo method for sparsely observed diffusion processes. Vrettas et al. [13, 14] considered approximating the variational Gaussian smoothing equations using a radial basis function representation for the variational parameters. By imposing a certain structure for the variational parameter functions, the overall number of parameters to be optimized can be reduced.. The variational Gaussian smoothing equations derived by Archambeau et al. [7, 8, 9 ] require a non-singular effective diffusion matrix. Our first goal is to extend the variational Gaussian approximation to a certain class of singular models by considering an alternative derivation based on Girsanov's theorem.. Another problem in the variational Gaussian algorithm is the need to compute Gaussian expectations over nonlinear functions. Previous works on the variational approximation [7, 8, 11, 13, 14] have not presented details for computing the Gaussian expectations for general nonlinear systems. Our second goal is to extend the variational Gaussian smoothing method for general nonlinear systems by considering numerical approximations for the Gaussian expectations. The treatment is similar to the one in [6] , where Taylor series based linearization, cubature and unscented transform based sigma-point methods and Gauss-Hermite quadrature were used to compute Gaussian expectations in the Gaussian smoothers based on the classical Gaussian filtering framework.. In addition, we provide in this paper a comparison between the variational Gaussian approximation and the Gaussian smoothers presented in [6] . Using a suitable change of variables the Gaussian smoothing equations can be converted to a variational form similar to the variational Gaussian smoothing equations. The computation of the Gaussian filtering based smoothing solution is numerically stable and provides good initial conditions for the variables in the variational Gaussian smoothing algorithm. This might help to overcome the problems reported by Vrettas et al. [13] in the initialization of the variational Gaussian algorithm for high dimensional systems. Also, we study if the variational Gaussian smoother can be used to iteratively improve the results from the Gaussian filtering based smoother.. The organization of this paper is as follows. First we present the variational Gaussian smoother and the Gaussian smoother based on the Gaussian filtering framework for the continuous-discrete system. Here we extend the variational Gaussian smoothing equations for a certain class of singular models. Next we compare theoretically the Gaussian smoothers by presenting a conversion of the Gaussian filtering based smoothing equations to the variational form by a suitable change of variables. This also provides the initial values for the variational Gaussian smoothing algorithm. The problem of numerically computing the Gaussian expectations in the variational Gaussian smoothing equations is treated in the next section. We present Taylor series linearization and sigma-point methods to approximate the Gaussian expectations in the variational Gaussian smoothing equations. The paper concludes with two synthetic-data examples that are used to compare the Gaussian smoothers, and also provides comparison of the different numerical methods for the computation of the Gaussian expectations. Compared to the Gaussian filtering based Gaussian smoother the variational Gaussian smoother is more complex to implement and is computationally heavier, since each iteration requires approximately the same amount of computations as one run of the filtering and smoothing equations. The Gaussian filtering based Gaussian smoother provides good initial conditions for the variational Gaussian smoother and could solve the problems with initialization mentioned in [13] . The variational Gaussian smoother provides Gaussian approximation that is optimal in the sense that it minimizes the Kullback-Leibler divergence of the approximating distribution with respect to the true distribution. However, the examples considered here show that this will not always improve the estimate with respect to other commonly used metrics.. Using statistical linearization with respect to the filtering distribution in the variational Gaussian smoothing equations gives formally the Gaussian filtering based smoother as a special case. This suggests that for highly nonlinear systems, the variational Gaussian smoother could better capture the nonlinearities. This is demonstrated in the numerical experiment for the double well system, where the variational Gaussian smoother clearly improves the Gaussian filtering based smoother estimate for small measurement variances. However, no clear improvement was observed in the second numerical experiment for the reentry problem. The reason could be that the nonlinearities in the reentry system are not high enough to gain benefit from using the variational Gaussian smoother equations. A drawback of the variational Gaussian smoother is that it tends to underestimate the variance compared to the Gaussian filtering based smoother.. For general nonlinear systems the Gaussian expectations can be computed using Taylor seires based linearization or standard sigma-point methods. The Taylor series based variational Gaussian smoother linearizes the drift and measurement function with respect to the current mean estimate and is therefore similar in idea to the iterated Extended Kalman smoother [23] . Using unscented transform and cubature rule based sigma-point methods in the variational Gaussian smoother resulted in some numerical problems. This is caused since the unscented and cubature methods are only accurate up to third order monomials, but even for linear systems some of the expectations are over fourth order polynomials. The numerical problems can be reduced by computing the Jacobians of the drift and measurement functions and using an alternative form for the expectations. Also, higher order unscented transform could be used for these expectations (see [19, 24] ). Third order Gauss-Hermite integration rule worked well for both examples considered in this paper, but the computational cost is high especially for high dimensional systems. For some high dimensional systems the computational cost of the Gauss-Hermite method could be reduced by using Rao-Blackwellisation [25] .. The KL-divergence term in the right side of Equation (A.1) is then given by KL(Q X || P X ) = E QX [− ln Z(t)] = 1 2 t 0 E q {f 2 (s 1 (t), s 2 (t), t) − g 2 (s 1 (t), s 2 (t), t)} T Q −1 (t) {f 2 (s 1 (t), s 2 (t), t) − g 2 (s 1 (t), s 2 (t), t)}] .. Inserting g 2 (s 1 (t), s 2 (t), t) = −A(t)s(t)+b(t) gives then the desired KL-divergence.

O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


SPAIS: A novel Self-checking Pollution Attackers Identification Scheme in network coding-based wireless mesh networks Pollution attacks refer to ones where attackers modify and inject corrupted data packets into the wireless network with network coding to disrupt the decoding process. In the context of network coding, the epidemic effect of pollution attacks can degrade network throughput significantly because of the mixing nature of network coding. To address this issue, a number of schemes to identify malicious nodes have been developed in the past. Nonetheless, these schemes have their limitations and cannot effectively deal with pollution attacks. In this paper, we propose a novel light-weight Self-checking Pollution Attackers Identification Scheme (SPAIS), which can identify the pollution attackers effectively and efficiently. By making full use of the broadcast nature of wireless media and the insight that a well-behaved node can monitor its downstream neighboring nodes locally by cooperating with other nodes, SPAIS hierarchically organizes the network as levels such that the nodes in the same level can monitor their downstream level nodes cooperatively. Through the combination of theoretical analysis and extensive simulations, our experimental data demonstrates that SPAIS can more effectively identify pollution attackers with a lower cost in comparison with the existing candidate schemes. For example, even if the quality of network connections is not in good condition and the malicious nodes send only one corrupted packet, the pollution attackers can be identified with a high probability. Wireless mesh networks (WMNs) have become a promising technology for providing easy wireless Internet access because of highly self-organized and self-configured features of WMNs [1] . Nonetheless, the wireless links between the nodes are lossy due to the temporal and spatial fading of wireless channels, leading to a distributed WMN infrastructure and relatively low performance. To this end, network coding [2, 3] , which is a technique that integrates the coding and routing schemes, has been introduced to improve the performance of WMNs. Generally speaking, in contrast to the traditional "store and forward" method, network coding allows relay nodes to mix the information content in packets before forwarding them, which can improve the network throughput significantly. Network coding has been applied to various applications, including the content distribution [4] an epidemic propagation, the corrupted packets can pollute the whole network in a short time and degrade the network throughput dramatically [7] .. In recent years, a number of authentication schemes have been developed to address pollution attacks against network coding [8] [9] [10] [11] [12] [13] [14] . In addition, a number of identification schemes [15] [16] [17] [18] [19] have been proposed to identify and locate the pollution attackers. In comparison with the authentication schemes, identification schemes are more proactive in defending pollution attacks. Because authentication schemes are all receivers-based, the detection may be too late as the attack may already consume the bandwidth and significantly disrupt network performance. If attackers can be quickly identified and immediately excluded from the network, the attack consequence can be largely mitigated.. Nonetheless, the existing identification schemes have their limitations and cannot be used to effectively deal with pollution attacks in WMNs where network coding is used. For example, Watchdog scheme [15] relies on a large number of extra trusted nodes to serve as watchdogs, which incurs a high cost to large networks. In MIS [16] , a trusted server is responsible for computing the checksum of suspicious packets. The checksum transmission incurs a significant communication overhead. In [18] , Algebraic-Watchdog scheme develops the hypothesis testing method to make the decision of identifying a malicious node. Nonetheless, that scheme needs to collect a large number of detecting samples and its computation cost is high. Similar to MIS [16] , SpaceMac [19] relies on a trusted controller to identify the malicious nodes. In addition, SpaceMac and PAI [17] do not consider the lossy wireless environment, which leads to the limited applicability.. In this paper, we propose a novel light-weight Selfchecking Pollution Attackers Identification Scheme (SPAIS) to effectively and efficiently identify the pollution attackers against network coding based legacy WMNs where MIMO (Multiple-Input and Multiple-Output) systems and directed antennas are not in use. Our proposed scheme has several novelties. First, unlike existing research efforts, SPAIS does not need any extra nodes to serve as watchdogs or any centralized server or controller. Second, it is self-checking and fully distributed. Third, it is light-weight and incurs only a little bandwidth overhead on the system. Moreover, network topology can be dynamic. If the identification is triggered and the topology has changed during data transmission, the source can re-determine the network level and re-distribute the random seeds to the corresponding nodes to ensure the secrecy of the random seeds. Our main contributions are summarized as follows:. • Novel scheme. In our design of SPAIS, we make full use of the broadcast nature of wireless media and adopt the principle that a well-behaved node can monitor its downstream neighbor nodes locally by cooperating with other nodes. To this end, our SPAIS hierarchically organizes the network as levels and the nodes in the same level can cooperatively monitor the nodes in the downstream level.. To further reduce the bandwidth overhead, we develop a homomorphic hash function for network coding.. • Theoretical analysis. We conduct theoretical analysis and prove that our scheme is correct and secure. For example, under typical system settings, our analytical data shows that SPAIS allows only 1 out of 65,536 polluted packets passing the verification of the upstream level of the attacker if the network is lossless.. • Evaluation. We evaluate the effectiveness of SPAIS in terms of false positive probability and false negative probability through simulation experiments. The simulation data shows that even if in a wireless network where the average link quality is only 0.7 and the attacker sends out only one corrupted packet to launch a pollution attack, the attacker can be located with a probability of 95.62%. In addition, even if there are 30% of malicious nodes in the network, the probability that the malicious nodes can disparage a well-behaved node is only 1.45%. We also compare the computation overhead and bandwidth overhead with the existing schemes. The experimental data confirms that our scheme is more efficient than existing schemes.. The remainder of the paper is organized as follows: we introduce the network and threat models in Section 2. We present our proposed scheme in Section 3. We conduct the security analysis of our scheme in Section 4. In Section 5, we show both analytical and experimental results to validate the effectiveness of our proposed scheme in comparison with existing schemes. We discuss the limitation and the potential future work in Section 6. We review the related work in Section 7 and conclude the paper in Section 8, respectively.

O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


ON THE IMPLICATION OF MULTIVALUED DEPENDENCIES IN PARTIAL DATABASE RELATIONS The implication of multivalued dependencies (MVDs) in relational databases has originally and independently been defined in the context of some fixed finite universe by Delobel, Fagin, and Zaniolo. Biskup observed that the original axiomatisation for MVD implication does not reflect the fact that the complementation rule is merely a means to achieve database normalisation. He proposed two alternative ways to overcome this deficiency: i) an axiomatisation that does represent the role of the complementation rule adequately, and ii) a notion of MVD implication in which the underlying universe of attributes is left undetermined together with an axiomatisation of this notion. In this paper we investigate multivalued dependencies with null values (NMVDs) as defined and axiomatised by Lien. We show that Lien's axiomatisation does not adequately reflect the role of the complementation rule, and extend Biskup's findings for MVDs in total database relations to NMVDs in partial database relations. Moreover, a correspondence between (minimal) axiomatisations in fixed universes that do reflect the property of complementation and (minimal) axiomatisations in undetermined universes is shown. Relational databases still form the core of most database management systems, even after more than three decades following their introduction (Codd 1970 ). The relational model organises data into a collection of relations. These structures permit the storage of inconsistent data, inconsistent in the semantic sense. Since this is not acceptable additional assertions, called dependencies, are formulated that every legal database is compelled to obey. There are many different classes of dependencies which can be utilised for improving the representation of the target 692 S. Link database. Excellent surveys on relational dependencies can be found in (Fagin & Vardi 1986 , Thalheim 1991 .. Multivalued dependencies (MVDs) (Delobel 1978 , Fagin 1977 , Zaniolo 1976 ) are an important class of dependencies. A relation exhibits an MVD precisely when it is decomposable into two of its projections without loss of information (Fagin 1977 ). This property is fundamental to relational database design, in particular 4NF (Fagin 1977) , and a lot of research has therefore been devoted to studying the behaviour of these dependencies. Recently, extensions of multivalued dependencies have been found very useful for various design problems in advanced data models such as the nested relational data model (Fischer, Saxton, Thomas & Van Gucht 1985) , the Entity-Relationship model (Thalheim 2003) , data models that support nested lists (Hartmann, Link & Schewe 20066) and XML .. The classical notion of an MVD (Fagin 1977 ) is dependent on the underlying universe R. This dependence is reflected syntactically by the ^-complementation rule which is part of the axiomatisation for MVD implication in fixed universes R, from now on referred to as R-implication (Beeri, Fagin & Howard 1977) . The complementation rule CR is special in the sense that it is the only inference rule which is dependent on R. This observation has led to further research (Mendelzon 1979 , Biskup 1978 , Biskup 1980 , Hartmann & Link 20066, Hartmann et al. 20066, Hartmann, Link & Kohler 2007 , Link 2006a ) on the complementation rule. In particular, Biskup proposed an alternative notion of semantic implication in which the underlying universe is left undetermined (Biskup 1980 ). Biskup shows that this notion can be captured syntactically by a sound and complete set of inference rules, denoted by 05. If 23c results from adding the Rcomplementation rule CR to 03, then 23c is i?-sound and incomplete for the Rimplication of MVDs for all relation schemata R. Moreover, every inference of an MVD by 03c can be turned into an inference of the same MVD in which the Rcomplementation rule CR is applied at most once, and if it is applied, then in the last step of the inference (03c is said to be ^-complementary). This indicates that the ^-complementation rule simply represents a part of the schema normalisation process, and does not necessarily infer semantically meaningful consequences. The ^-complementary system 03c reflects this property indicating that the underlying universe can be fixed in the very last step of an inference. Based on a common set of inference rules (Link 2006 a) identifies all complementary axiomatisations for MVDs in fixed universes as well as all axiomatisations for MVDs in undetermined universes. Moreover, the time-complexity of the associated implication problem in undetermined universes is also studied. Apart from this work research has not been continued in this direction but focused on the original notion of i?-implication. Since research on MVDs seems to experience a recent revival in the context of other data models (Fischer et al. 1985 , Thalheim 2003 , Hartmann & Link 20066, Hartmann et al. 20066, Hartmann et al. 2007 , Link 2006a , Link 20066, Vincent & Liu 2003 , Vin-cent et al. 2003 it seems desirable to further extend the knowledge on relational MVDs. An advancement of such knowledge may also simplify the quest of finding suitable and comprehensible extensions of MVDs to currently popular data models. It is very rare in practice that the information in a database is complete. This observation has led to many extensions of the relational data model (Codd 1979 , Lien 1982 , Atzeni & Morfuni 1986 , Levene & Loizou 1993 , Levene & Loizou 1998 , Johnson & Rosebrugh 2003 ) that can handle incomplete information. In particular, multivalued dependencies in the presence of null values (NMVDs) have been studied (Lien 1982) . The original notion of an NMVD (Lien 1982 ) is again dependent on the underlying set R of attributes. This dependence is reflected syntactically by the .^-complementation rule CR which is part of the axiomatisation of NMVDs (Lien 1982) . To the author's best knowledge no efforts have been made so far to study the role of the ^-complementation rule for the implication of NMVDs. Moreover, the implication of NMVDs has not been investigated in undetermined universes. We have explored the notion of MVD implication in the presence of null values (NMVDs) with meaning "no information". We observed that Lien's original axiomatisation of NMVDs (Lien 1982) does not adequately reflect the role of the Rcomplementation rule as a mere means of database normalisation. This observation is analagous to Biskup's findings for total database relations. We have then proposed sound and complete sets of inference rules for the i?-implication of NMVDs that are indeed adequate. Moreover, Biskup's alternative notion of MVD implication, in which the underlying universe is left undetermined, was extended to the presence of null values. Several sound and complete sets of inference rules for NMVD implication in undetermined universes have been proposed, which were also extended to cover both functional and multivalued dependencies in the presence of null values. The results clarify the role of the ^-complementation rule for NMVDs, and may simplify the quest of finding suitable and comprehensible notions of multivalued dependencies in the context of advanced database models. Moreover, the results clarify the power of several R-incomplete subsets.. Some interesting problems warrant future research. While the i?-implication problem of MVDs has received a considerable amount of interest with the best current time bound proposed in (Galil 1982) , no research has been devoted to the corresponding R-implication problem of NMVDs. In the spirit of our article it seems also interesting to investigate the implication problem of (N)MVDs in undetermined universes, and maybe derive further correspondences between implication and Rimplication. An interesting open problem is to generalise the approach in (Levene & Loizou 1998 ) from functional to multivalued dependencies. The approach uses a possible world semantics exploring all extensions of an incomplete database to a complete database. Weak MVDs must be satisfied by some possible world while strong MVDs are satisfied by all possible worlds.. The probably most important open problem in the context of multivalued dependencies is the absence of a synthesis algorithm that extends the well-known synthesis approach to functional dependencies (Bernstein 1976 , Biskup et al. 1979 . Clearly, the notion of MVD implication in undetermined universes provides a much better basis for the development of such an algorithm than the original notion of MVD implication in fixed universes.

O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O


Tweeting the terror: modelling the social media reaction to the Woolwich terrorist attack Abstract Little is currently known about the factors that promote the propagation of information in online social networks following terrorist events. In this paper we took the case of the terrorist event in Woolwich, London in 2013 and built models to predict information flow size and survival using data derived from the popular social networking site Twitter. We define information flows as the propagation over time of information posted to Twitter via the action of retweeting. Following a comparison with different predictive methods, and due to the distribution exhibited by our dependent size measure, we used the zerotruncated negative binomial (ZTNB) regression method. To model survival, the Cox regression technique was used because it estimates proportional hazard rates for independent measures. Following a principal component analysis to reduce the dimensionality of the data, social, temporal and content factors of the tweet were used as predictors in both models. Given the likely emotive reaction caused by the event, we emphasize the influence of emotive content on propagation in the discussion section. From a sample of Twitter data collected following the event (N = 427,330) we report novel findings that identify that the sentiment expressed in the tweet is statistically significantly predictive of both size and survival of information flows of this nature. Furthermore, the number of offline press reports relating to the event published on the day the tweet was posted was a significant predictor of size, as was the tension expressed in a tweet in relation to survival. Furthermore, time lags between retweets and the cooccurrence of URLS and hashtags also emerged as significant. Open and widely accessible social micro-blogging technologies, such as Twitter, are increasingly being used by citizens on a global scale to publish content in reaction to real-world events. The diffusion of this information following events can manifest itself in a number of ways, ranging from supporting social resilience through calls for assistance or help, and spreading of advice and information (Procter et al. 2013a) ; to the socially disruptive, by providing a platform for the distribution of misinformation, rumour (Procter et al. 2013b ) and antagonist commentary .. The recent terrorist events in Boston, USA and in Woolwich, London, UK sparked widespread reaction and news reporting via social media. In both cases, information pertaining to the event had both positive and negative impacts in the immediate aftermath. In relation to positive impacts, Twitter was used by law enforcement officials and journalists to request information and relay assurances to the public. On the negative side, in Boston the vast amount of information posted by the public on Twitter led to law enforcement becoming overwhelmed with multiple lines of enquiry; in the UK there were a number of arrests made following the event due to alleged religiously offensive comments being posted on Twitter. Given such significant impacts following these kind of events, it is important for those with a remit to ensure community safety to understand the predictive factors for the propagation of information flows (Lotan et al. 2011 ) as a first step towards being able to mitigate their impact. We define information flow propagation as the process of information spreading to a greater number of people over time via Twitter through the action of retweeting. That is, when a Twitter user reads a tweet, they perform an action to 'forward' that tweet to all other Twitter users who follow them, incrementally widening the potential readership of the original tweet. This information can contain textual content, hashtags and URLs, from which a variety of temporal, content and social metrics can be derived for modelling purposes.. This modelling is important to understand how long a piece of antagonistic text might continue to be propagated in the Twittersphere as it may pose a risk to social cohesion. Likewise, it could be important to understand the factors that are likely to lead to information from official sources, such as law enforcement, reaching a large number of people in a short space of time (Procter et al. 2013a) . In this paper we are particularly interested in information flow propagation in the aftermath of a terrorist event. This can be considered as an aspect of the broader study of information diffusion (Guille and Hacid 2012) . Understanding the diffusion of information in social networks has received significant attention from a topological and social influence perspective, but as a recent survey identified, further research is required to better understand the significance of opinion and social factors (Guille et al. 2013) . Therefore, our key research question is--what are the opinion and social factors that predict large (the size) and long lasting (the survival) information flows following a terrorist event?. To model information diffusion we derived two features of an information flow: size and survival. We took the frequency of retweets surrounding the event as our size dependent measure, and the duration between the first and last retweet as our survival dependent measure. These two features are independent as they measure two distinct properties of information flow propagation. In terms of size, the number of retweets is a measure of the level of public interest and endorsement of the information, while survival is a measure of persistence of interest over time. The two features are visualized in Sect. 4 (see Figs. 1 , 2) where we discuss the features of the study dataset. This is consistent with previous work on modelling information diffusion in social networks (e.g. Zaman et al. 2010 , Lin et al. 2013 .. This paper contributes to the existing literature by studying quantitative measures of opinion and emotion (sentiment and tension) expressed in tweets for the purposes of predicting information flow size and survival in the case of a terrorist event. Guille et al. (2013) identified the requirement for additional work examining such features in the study of information diffusion. In addition to this, we also examine other relatively unexplored factors, including the influence of daily newspaper coverage of the event; the co-occurrence of content linking factors (hashtags and URLs) within the tweet; and build on previous work investigating the significance of rapidity of early user interaction intervals in predicting ''thread'' size (Backstrom et al. 2013) . We also employ Kaplan-Meier estimation, which produces a survivor function plot of the declining propagation of information flows over time, allowing us to visually interpret the influence of independent variables on survival. Suh et al. (2011) aimed to better understand the features that are important indicators of whether an individual tweet would be retweeted. They identified three factors relating to (1) author profile--number of followers, followees, and tweets; (2) content features--URLs and hashtags; and (3) retweets and followers--separating those who have been retweeted a lot and have a large number of followers, from those who tweet a lot and favourite a lot. They built a Generalized Linear Model (GLM) that suggested URL, hashtag and age of account to be most useful for retweet prediction, with follower/followee status less significant, but still important. Similarly, Zaman et al. (2010) used the Matchbox algorithm to predict retweet probability for individual tweets, finding that attributes of the tweeter and the retweeter (similar to author profile of Suh et al.) , were most accurate for prediction. However, they identified that using the content of the tweet was detrimental to prediction performance, even when normalized. Tsur and Rappoport (2012) also investigated Twitter content, specifically hashtags, in the context of the spread of ideas and memes. They identified that the emotive aspects of hashtags were not predictive of the spread of information. This is interesting as Berger and Milkman (2012) studied the likelihood of people propagating news stories based on the emotion invoked by the content and found that, contrary to common perception, people were more likely to share stories that invoke positive emotion as opposed to negative. Bandari et al. (2012) undertook the task of predicting the popularity of news stories on Twitter prior to their release. Using classification and regression techniques with features relating to content subjectivity, source and topic, they were able to achieve a reasonable accuracy in predicting a range of propagation likelihood scores, but were less efficient in predicting information flow size. Guille and Hacid (2012) developed a model to predict the diffusion of information in online social networks, focusing on social, temporal and content factors. They identified a Bayesian logistic regression as a favoured predictive model. While the model performed well for diffusion, it was less effective at predicting size, indicating that predictive features of information diffusion and information flow size are independent. Zaman et al. (2013) also used Bayesian models for predicting the number of retweets using a time-series model, predicting at certain points in time as opposed to projecting the final size in the early stages of tweet lifetime. Neither of these models included latent subjectivity and emotion/opinion within the tweet as a feature. Backstrom et al. (2013) identified that temporal factors including the rapidity of comments posted in response to a Facebook status update were predictive of overall thread length. Twitter interaction is slightly different to that of Facebook as retweets are propagated within and between networks as opposed to being visualized in structured ''conversations'' between ''friends'' or groups, but it could be possible to use temporal retweet factors in a similar predictive manner, i.e. to predict total number of retweets using the rapidity of occurrence of initial retweets. Macskassy and Michelson (2011) built information propagation behaviour models for Twitter using temporal features such as the time lapsed since the original tweet was published and the recency of communication between a tweeter and other users. Both papers suggest that time is an important factor in modelling information propagation and that more investigation is required to use rapidity of retweeting to predict size and survival. Yang and Counts (2010) constructed a topic-based diffusion model based on user mentions, where a mention constitutes the propagation of information from one user to another. They aimed to predict speed (time taken to reach first mention), scale (the number of first-order mentions of the user), and reach (the number of hops the mention produced). They employed the Cox proportional hazards regression model to quantify the degree to which features of the tweet or the original tweeting user were useful in predicting speed, scale and reach. They found that predictive features varied across topics for speed and scale with the amount a user was mentioned in the past emerging as most predictive. In this paper we built models that predicted information flow size and survival on Twitter following a terrorist event. We defined information flows as the propagation over time of information posted to Twitter via the action of retweeting. We used zero-truncated negative binomial and Cox proportional hazards regression techniques to investigate the predictive value of social, temporal and content factors of the tweet. The novel findings were that time lags between retweets, the co-occurrence of URLS and hashtags, and the sentiment expressed in the tweet, were statistically significantly predictive of both size and survival of information flows of this nature. Furthermore, the number of offline press reports relating to the event published on the day the tweet was made were significant predictors of size, and the tension expressed in a tweet was predictive of survival.. The sub-model analysis for size suggested that social factors explained the largest amount of variance, followed by content factors and temporal factors. Interestingly, despite all but one of the content features being highly significant, it appeared that the number of followers and previous tweets alone accounted for more of the variance in the dependent of size for an event of this nature. Thus, to create a large information flow following a terrorist event, the most important factor is the social features of the tweeter. In contrast, for survival, content factors explained the largest amount of variance, followed by temporal factors and social factors. This suggests that to create a longlasting information flow following such an event, the tweeter should focus on content features, such as including hashtags, URLs, positive sentiment and low tension.. This study has provided evidence that the opinion/ emotional factors of tweets are statistically significant predictors of both information flow size and survival following a terrorist event. Contrary to popular perception, while negative and tense content did emerge on Twitter following the event, it failed to propagate via retweeting. In short, negative and tense content remained small in number and lasted for short periods, dying out quickly. This indicates that on the whole, members of the public who used Twitter in this case chose to propagate positive and supporting content. Following on from this work we intend to test the predictive efficacy of our models on other cases that exhibit similar characteristics (e.g. the Boston terrorist event and the 2011 riots in England). The outcome of this ongoing research will be a general predictive model for Twitter information flow size and survival following major socially disruptive events.. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. Aside from the work of Lin et al. (2013) on the growth, survival and context of hashtags, our study of the existing literature at the time of writing failed to identify any work in the field related to information flow content and its relationship to survival over time, beyond hashtags alone.

O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B I O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B I O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A threshold for the Maker-Breaker clique game We study the Maker-Breaker k-clique game played on the edge set of the random graph G(n, p). In this game, two players, Maker and Breaker, alternately claim unclaimed edges of G(n, p), until all the edges are claimed. Maker wins if he claims all the edges of a k-clique; Breaker wins otherwise. We determine that the threshold for the graph property that Maker can win this game is at n − 2 k+1 , for all k > 3, thus proving a conjecture from [20] . More precisely, we conclude that there exist constants c, C > 0 such that when p > Cn − 2 k+1 the game is Maker's win a.a.s., and when p < cn − 2 k+1 it is Breaker's win a.a.s. For the triangle game, when k = 3, we give a more precise result, describing the hitting time of Maker's win in the random graph process. We show that, with high probability, Maker can win the triangle game exactly at the time when a copy of K 5 with one edge removed appears in the random graph process. As a consequence, we are able to give an expression for the limiting probability of Maker's win in the triangle game played on the edge set of G(n, p). Let X be a finite set and let F ⊆ 2 X be a family of subsets of X. In the positional game (X, F), two players take turns in claiming one previously unclaimed element of X. The set X is called the "board", and the members of F are referred to as the "winning sets". In a Maker-Breaker positional game, the two players are called Maker and Breaker. Maker wins the game if he occupies all elements of some winning set; Breaker wins otherwise. We will assume that Maker starts the game. A game (X, F) is said to be a Maker's win if Maker has a strategy that ensures his win against any strategy of Breaker; otherwise it is a Breaker's win. Note that F alone determines whether the game is Maker's win or Breaker's win.. A well-studied class of positional games are the games on graphs, where the board is the set of edges of a graph. The winning sets in this case are usually representatives of some graph theoretic structure. Research in this area was initiated by Lehman [18] , who studied the connectivity game, a generalization of the well-known Shannon switching game, where Maker's goal is to claim a spanning connected graph by the end of the game. We denote the game by (E(K n ), T ). Another important game is the Hamilton cycle game (E(K n ), H), where H = H n consists of the edge sets of all Hamilton cycles of K n .. In the clique game the winning sets are the edge sets of all k-cliques, for a fixed integer k ≥ 3. We denote this game with (E(K n ), K k ). Note that the size of the winning sets is fixed and does not depend on n, which distinguishes it from the connectivity game and the Hamilton cycle game. A simple Ramsey argument coupled with the strategy stealing argument (see [3] for details) ensures Maker's win if n is large. Beck [2] determined the largest clique Maker can claim on E(K n ) with remarkable precision.. All three games that we introduced are straightforward Maker's wins when n is large enough. This is however not the end of the story. We present two general approaches to even out the odds, giving Breaker more power -biased games and random games. Biased games. Biased games are a widely studied generalization of positional games, introduced by Chvátal and Erdős in [9] . Given two positive integers a and b and a positional game (X, F), in the biased (a : b) game Maker claims a previously unclaimed elements of the board in one move, while Breaker claims b previously unclaimed elements. The rules determining the outcome remain the same. The games we introduced initially are (1 : 1) games, also referred to as unbiased games. Now, if an unbiased game (X, F) is a Maker's win, we choose to play the same game with (1 : b) bias, increasing b until Breaker starts winning. Formally, we want to answer the following question: What is the largest integer b F for which Maker can win the biased (1 : b F ) game? This value is called the threshold bias of F.. For the connectivity game, it was shown by Chvátal and Erdős [9] and Gebauer and Szabó [11] that the threshold bias is b T = (1 + o(1)) n log n . The result of Krivelevich [16] gives the leading term of the threshold bias for the Hamilton cycle game, b H = (1 + o(1)) n log n . In the k-clique game, Bednarska and Luczak [4] found the order of the threshold bias, b K k = Θ(n 2 k+1 ). Determining the leading constant inside the Θ(.) remains an open problem that appears to be very challenging. Some lower and upper bounds can be derived from the calculations in [4] , and for k = 3 better bounds are obtained in [1] . Random games. An alternative way to give Breaker more power in a positional game, introduced by the second author and Szabó in [20] , is to randomly thin out the board before the game starts, thus eliminating some of the winning sets.. Ben-Shimon et al. [5] closed this question by giving a very precise description of the low order terms of the limiting probability.. Moving to the clique game, it was shown in [20] that for every k ≥ 4 and every ε > 0 we have n. Moreover, it was proved that there exist a constant C > 0 such that for p ≥ Cn. Maker wins the k-clique game on G(n, p) a.a.s. The threshold for the triangle game was determined to be p K 3 = n − 5 9 , showing that the behavior of the triangle game is different from the k-clique game for k ≥ 4, as 9 5 < 3+1 2 = 2. Our main result is the following theorem. It gives a lower bound on the threshold for the k-clique game, when k ≥ 4, which matches the upper bound from [20] up to the leading constant. Breaker wins the Maker-Breaker k-clique game played on the edge set of G(n, p) a.a.s.. The threshold probability for the k-clique game for k ≥ 4 was conjectured to be [20] . The previous theorem resolves this conjecture in the affirmative. Summing up the results of Theorem 1.1 and Theorem 19 from [20] , we now have the following. Corollary 1.2 Let k ≥ 4 and consider the Maker-Breaker k-clique game on the edge set of G(n, p). There exist constants c, C > 0 such that the following hold: A result of this type is sometimes called a "semi-sharp threshold" in the random graphs literature. Hitting time of Maker's win. We look at the same collection of positional games on graphs, now played in a slightly different random setting. Let V be a set of cardinality n, and let π be a permutation of the set V 2 . If by G i we denote the graph on the vertex set V whose edges are the first i edges in the permutation π, G i = (V, π −1 ([i])), then we say thatG = {G i } ( n 2 ) i=0 is a graph process. Given a monotone increasing graph property P and a graph processG, we define the hitting time of P with τ (G; P) = min{t : G t ∈ P}. If π is chosen uniformly at random from the set of all permutations of the set V 2 , we say thatG is a random graph process. Such processes are closely related to the model of random graph we described above, see, e.g., [14] .. Given a positional game, our general goal is to describe the hitting time of the graph property "Maker's win" in a typical graph process. For a game F, by M F we denote the graph property "Maker wins F". It was shown in [20] that in the connectivity game (with the technical assumption that Breaker is the first to play), for a random graph process G, we have τ (G; M T ) = τ (G; δ 2 ), where δ is the graph property "minimum degree at least ". Recently, Ben-Shimon et al. [5] resolved the same question for the Hamilton cycle game, obtaining τ (G; M H ) = τ (G; δ 4 ). Note that inequality in one direction for both of these equalities holds trivially.. Moving on to the clique game, we denote the property "the graph contains K 5 − e as a subgraph" with G K − 5. . We are able to show the following hitting time result for Maker's win in the triangle game. Theorem 1.3 For a random graph processG, the hitting time for Maker's win in the triangle game is asymptotiaclly almost surely the same as the hitting time for appearance of K 5 − e, i.e., τ (G;. By considering the number of copies of K 5 − e, Theorem 1.3 together with the result proved independently by Bollobás [6] and Karoński and Ruciński [15] (also occurring as Theorem 3.19 in [14] ) we are able to give a precise expression for the probability that Maker wins the triangle game on G(n, p). Corollary 1.4 Let p = p(n) be an arbitrary sequence of numbers ∈ [0, 1] and let us write

O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A supervised manifold learning method Abstract. The Locally Linear Embedding (LLE) algorithm is an unsupervised nonlinear dimensionality-reduction method, which reports a low recognition rate in classification because it gives no consideration to the label information of sample distribution. In this paper, a classification method of supervised LLE (SLLE) based on Linear Discriminant Analysis (LDA) is proposed. First, samples are classified according to their label values, and low dimensional features of intraclass data are expressed through LLE manifold learning. Then, the base vectors in Fisher subspace of the low dimensional features are generated through LDA learning. This method increases inter-class variation, and decreases the intra-class variation when samples are projected to the Fisher subspace. Hence, the samples of different labels can be recognized, and the recognition rate and robustness of the LLE learning are improved. Experiments on handwritten digit recognition show that the proposed method is featuring high recognition rate. To learn from the mechanism people perceive the world and improve machine intelligence has long been a great challenge facing artificial intelligence. Literature in 1985 [1] showed that the key for human beings to recognize objects is to uncover the inherent law of high-dimensionality data and form our own cognition. In other words, low-dimensionality manifolds of objects formed in human brains enable us to quickly recognize the objects. Another article [2] in Science, 2000, by H.S. Seung, et al. discussed the manifold learning based on cognition and put forward the idea that human beings have the ability to instantaneously perceive the inherent low-dimensionality structure of objects and can thus recognize the corresponding objects under complex surroundings. It proposed the assumption that human perception acts in the form of manifolds. Therefore, computer-based recognition can be inspired from this assumption, that is, the low dimensional data of objects acquired from LLE manifold learning can reveal their features sufficiently. The essence of manifold learning is that when the sampling data are in a low-dimensionality manifold space, their dimensionalities can be reduced to show the inherent geometrical distribution rule of the low-dimensionality manifold. In this way, the manifold learning approach can better reveal the substantive characteristics of objects. This non-linear unsupervised learning approach is simple, intuitive and effective. At present, its mainstream methods include: Locally Linear Embedding (LLE) [3] , Isotacticity Mapping (ISOMAP) [4] , Laplace Characteristic Mapping [5] , Local Preserving Projection (LPP) [6] , Stochastic Neighbor Embedding (SNE) [7] , Charting a Manifold [8] , Locally Linear Smoothing [9] , etc.. As for classification, the purpose of dimensionality reduction is not only to reduce the dimensionalities of known sample points, but more importantly to discover the inherent manifold of unknown sample points and augment, to the greatest extent, the margin between sample points and classification plane so as to improve its robustness. Consequently, the label information plays an important role in classification. Recently, the supervised LLE manifold learning approach has also experienced some progress [10] [15] and shown favorable results in pattern recognition. For supervised feature extraction, sufficient consideration should be given to the generalization of dimensionality reduction learning, ensuring precise reflection of manifold structure in the embedded space and largest variation between sub-manifolds in the embedded space after dimensionality reduction, so as to increase the margin between different classes and reduce the inner-class data variation as much as possible. The Fisher subspace with LDA algorithm is undoubtedly the best learning approach with the largest intra-class and smallest inner-class variation, and improved separability of data.. This paper aims to improve the LLE manifold learning method with the idea of LDA learning. In the experiment of handwritten digit classification, this method shows a better recognition result compared with unsupervised LLE.. The following section will review the traditional LLE and the main contribution of this paper will be set out in section 3, where the structure and process of the proposed SLLE method is described. Section 4 will display experiments of its application to handwritten digit recognition.

O O B I O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Distribution of event complexity in the British Columbia court system an analysis based on the CourBC analytical system This paper reports an exploratory research on the distribution of event complexity in the British Columbia court system. Analysis of event distribution shows that the frequency of events sharply decreases with the increase in the number of persons and counts. The most frequently observed type of event is the event that has one person involved with one count. The number of events observed sharply declines when we query for events with a larger number of people involved or more counts charged. It is found that the number of events observed exponentially decreases when more complex events comprising more counts are analyzed. The same exponential decrease is observed for events with two or more people. This means that, in general, the least complex events are the most frequently observed ones. The events with more than one person involved have a mode of two counts. A first approximation model for the distribution of the load on the system based on different levels of complexity is proposed. The proposed model can be used for and be evaluated by predicting the load distribution in the BC criminal court system. Common law criminal justice systems have experienced a series of major problems in recent years including declining rates of case clearances and prosecutions, rising rates of remand in custody, increasing delays between charge and trial dates, and increasing rates of case collapse at trial [1] [2] [3] [4] . Sources of these problems may reside at many points in the system with important operational feedbacks between decisions and events occurring in the investigative, prosecutorial, judicial and correctional elements of the system [5] . Case complexity is thought to be a major contributor to justice system problems, but little systemic science addresses the issue at any agency level [6] [4] .. Interoperable justice agency databases could be used to identify systemic trouble points relate them to case complexity and perhaps develop improvements. For a variety of good reasons ranging from issues of privacy to the requirements of fair trials judicial data systems are rarely made available to researchers interested in understanding system problems.. The Institute for Canadian Urban Research Studies (ICURS) has collected information about all crimes handled in the British Columbia criminal court system during a 3-year period encompassing 2008 through 2010 using publically available court data published by the courts from a judicial Records Management System named JUSTIN. These published data have been reverse engineered into a research data warehouse called CourBC. A research tool called the CourBC Analytical System has been developed to facilitate repetitive queries and data mining in this database.. A file, in a British Columbian criminal court, represents all linked courtroom actions. This would involve crimes committed by one or more individuals, that is, all crimes associated with co-accused. The crimes may be divided into informations or indictments when the crimes are linked by occurrence time or place, called folders in CourBC. A file is processed in a specific court and has a folder number, one or more associated documents, one or more persons involved, one or more crime codes involved, and one or more counts arising from those crime codes [7] . Figure 1 is an Entity Relationship Diagram showing the structure of the relationship between entities in the court system. A unique folder in a court can contain many documents, a document in a court-folder can be associated to many persons and a person can be associated to many documents in a folder in a court. In this paper, we refer to a unique folder in a court as an event. A document associated to a person can be related to many crime codes. Moreover, one or more counts can be assigned to the same crime related to a person and a document.. One of the capabilities of the CourBC Analytical System is to find the number of unique court events based on the attributes of the file in court system.

B O B I O O O O O O O O O O O O O O O O O O O O O O B O B I O O O O O O O O B B O O O O O O O O O O O O O O O B O B O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Entropy, Similarity Measure, Inclusion Measure of Intuitionistic Fuzzy Sets and Their Relationships In the present paper, we introduce a new axiomatic definition of the inclusion measure for intuitionistic fuzzy sets (IFSs, for short). The close relationships among entropy, similarity measure, and inclusion measure of IFSs are then discussed in detail. Also, we obtain some important theorems by which the entropy, similarity measure and inclusion measure of IFSs can be transformed into each other based on their axiomatic definitions. Moreover, some formulae for calculating the entropy, similarity measure and inclusion measure of IFSs are put forward. Finally, we compare the proposed new entropy, similarity and distance measures with the existing ones. To alleviate some drawbacks of Zadeh's fuzzy set (Rf.30), the concept of intuitionistic fuzzy set (IFS) was first introduced by Atanassov in 1986 (Rf.1) as an useful generalization of ordinary fuzzy set. It has been found to be more flexible and practical to deal with vagueness and uncertainty due to that intuitionistic fuzzy set can incorporate a hesitation in the membership degree, that is, assign to each element u of U a membership, a nonmembership and a hesitation degree. After the pioneering work of Atanassov, the IFS has received much attention from many researchers and has been used in more wide range of application fields. For example, Gerstenkorn (Rf.10) systematically investigated the correlation coefficients of IFSs. Coker (Rf.6) discussed the intuitionistic fuzzy topologi-cal characteristic. Xu (Rf.25) studied the clustering technique of IFS. Li (Rf.15) proposed some multiattribute decision-making methods using IFSs. In addition, De (Rf.9) applied intuitionistic fuzzy set to medical diagnosis. Ciftcibasi (Rf.4) introduced Two-sided (intuitionistic) fuzzy reasoning approach. Cornelis (Rf.5) studied the implication in intuitionistic fuzzy theory.. As was well known, the entropy and similarity measure, and inclusion measure are three important issues in fuzzy set theory, which have been widely applied to pattern recognition, cluster analysis, image processing and decision making. With respect to intuitionistic fuzzy setting, the three concepts still play significant role in many deployed systems and application problems involving intuitionistic fuzzy sets.. The similarity measure of IFSs indicates the similar degree of two IFSs and plays an important role in pattern recognition, approximate reasoning and decision making. For instance, in (Rf.17) Szmidt presented a similarity measure of IFSs for supporting medical diagnostic reasoning. Later, Xu (Rf. 26) proposed some similarity measures of IFs for multiple attribute decision making. Hung (Rf. 13) also presented a similarity measure of IFSs based on Hausdorff metric and applied it to pattern recognition.. The inclusion measure of fuzzy sets represents the degree to which a fuzzy set is contained in another fuzzy set. Sinha (Rf. 19 ) first introduced an axiomatic definition of the inclusion measure of fuzzy sets. Later on, Cornelis (Rf.7) revised sinha's axiom, and proposed a inclusion for intuitionistic fuzzy set. Bustince (Rf.2) investigated the inclusion grade for interval-valued fuzzy sets. Kehagias (Rf.14) presented the concept of L-fuzzy valued inclusion measure , L-fuzzy valued similarity measure, and Lfuzzy distance.. Entropy, as a measure of fuzziness often used in the literature, describe the fuzziness degree or uncertain information of a fuzzy set and is first mentioned by Zadeh. It has received great attention recently. In 1972 (Rf.8) Deluca and Termini presented some axioms to describe the fuzziness degree of fuzzy set, with which a fuzzy entropy based on Shannon's function was proposed. After that, many other researchers have studied the fuzzy entropy in different ways. Especially, Shang in (Rf.20) introduced another new fuzziness measure. For the information measure of intuitionistic fuzzy set, Szmidt (Rf.21) proposed an entropy for intuitionistic fuzzy set by employing a geometric interpretation of IFS. And Vlachos (Rf.22) studied the intuitionistic fuzzy information entropy and its application to pattern recognition. In 2010 Ye (Rf.29) also proposed two effective measures of intuitionistic fuzzy entropy. Moreover, Burillo (Rf.3) presented an entropy on interval-valued fuzzy sets and on intuitionistic fuzzy sets. And Vlachos (Rf.23) studied the subsethood and entropy of interval-valued fuzzy sets(IvFSs). Further Zeng (Rf.31) investigated the relationship between entropy and similarity measure of IvFSs from a different point of view. Additionally, we presented some new entropy formulae of vague set and interval-valued intuitionistic fuzzy sets, and their relations were then discussed in detail (Rf. 32, 33, 34) .. However, by now, although some entropy formulae, similarity measures and inclusion measures of IFSs have been introduced, and applied to many real life problems, there is little investigation on their relationships among them, which may be widely applied to many fields such as pattern recognition, cluster analysis, image segment. Therefore, it is worthwhile to focus on discovering the close relationships among entropy, similarity measure and inclusion measure for IFS. In section 2, we first summarize the axiomatic definitions of entropy, similarity measure and inclusion measure for IFS. Further, in section 3, some theorems reflecting the relationship among the entropy, the similarity measure, the inclusion measure are obtained, based on which some new formulae to evaluate entropy and similarity, inclusion measure of IFS are then put forward. The conclusion is given in the last section. 0) are the smallest and greatest units of this lattice(Rf.5). In this paper, we summarize many entropy measures, similarity measures and inclusion measures, which have been applied to many real-world fields such as pattern recognition, clustering analysis and image processing. The close relationships among entropy, similarity measure, inclusion measure of IFSs are also investigated. Some new formulae to calculate entropy, similarity measure, and inclusion measure of IFS have been put forward. Also, we compare these measures with the existing ones and obtain some important theorems by which entropy, similarity measure, and inclusion measure of IFSs can be transformed into each other.

B B I B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B B I O B I O O O O O O O O O O O O O O O O B B I O B I O O O O O O O O O O O O O O O O O O O B B I O B I O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O B O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O B I O O O B I O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B I I O O O O B O B I I O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O B I O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B I O B I O O O O O O O O O O O O B B I O B I O O O O O O O O O O O O O B O B I O B I O O O O O O O O O O B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B B I B I O O O O O O O O O O B B I O B I O O O O O O O O O O O O O O O O O O O O O O B B I O B I O O O O O O O O