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A Boolean network (BN) is called observable if any initial state can be uniquely determined from the output sequence. In the existing literature on observability of BNs, there is almost no research on the relationship between the number of observation nodes and the observability of BNs, which is an important and practical issue. In this paper, we mainly focus on three types of BNs with $n$ nodes (i.e., $K$-AND-OR-BNs, $K$-XOR-BNs, and $K$-NC-BNs, where $K$ is the number of input nodes for each node and NC means nested canalyzing) and study the upper and lower bounds of the number of observation nodes for these BNs. First, we develop a novel technique using information entropy to derive a general lower bound of the number of observation nodes, and conclude that the number of observation nodes cannot be smaller than $\left[(1-K)+\frac{2^{K}-1}{2^{K}}\log_{2}(2^{K}-1)\right]n$ to ensure that any $K$-AND-OR-BN is observable, and similarly, some lower bound is also obtained for $K$-NC-BNs. Then for any type of BN, we also develop two new techniques to infer the general lower bounds, using counting identical states at time 1 and counting the number of fixed points, respectively. On the other hand, we derive nontrivial upper bounds of the number of observation nodes by combinatorial analysis of several types of BNs. Specifically, we indicate that $\left(\frac{2^{K}-K-1}{2^{K}-1}\right)n,~1$, and $\lceil \frac{n}{K}\rceil$ are the best case upper bounds for $K$-AND-OR-BNs, $K$-XOR-BNs, and $K$-NC-BN, respectively., Comment: 31 pages, 2 figures, 15 tables

Boolean Network (BN) and its extension Probabilistic Boolean Network (PBN) are popular mathematical models for studying genetic regulatory networks. BNs and PBNs are also applied to model manufacturing systems, financial risk and healthcare service systems. In this paper, we propose a novel Greedy Entry Removal (GER) algorithm for constructing sparse PBNs. We derive theoretical upper bounds for both existing algorithms and the GER algorithm. Furthermore, we are the first to study the lower bound problem of the construction of sparse PBNs, and to derive a series of related theoretical results. In our numerical experiments based on both synthetic and practical data, GER gives the best performance among state-of-the-art sparse PBN construction algorithms and outputs sparsest possible decompositions on most of the transition probability matrices being tested., Comment: 74 pages, 0 figures

Nonnegative matrix factorization arises widely in machine learning and data analysis. In this paper, for a given factorization of rank r, we consider the sparse stochastic matrix factorization (SSMF) of decomposing a prescribed m-by-n stochastic matrix V into a product of an m-by-r stochastic matrix W and an r-by-n stochastic matrix H, where both W and H are required to be sparse. With the prescribed sparsity level, we reformulate the SSMF as an unconstrained nonconvex-nonsmooth minimization problem and introduce a column-wise update algorithm for solving the minimization problem. We show that our algorithm converges globally. The main advantage of our algorithm is that the generated sequence converges to a special critical point of the cost function, which is nearly a global minimizer over each column vector of the W-factor and is a global minimizer over the H-factor as a whole if there is no sparsity requirement on H. Numerical experiments on both synthetic and real data sets are given to demonstrate the effectiveness of our proposed algorithm., Comment: 28 pages,8 figures

Observing the internal state of the whole system using a small number of sensor nodes is important in analysis of complex networks. Here, we study the problem of determining the minimum number of sensor nodes to discriminate attractors under the assumption that each attractor has at most K noisy nodes. We present exact and approximation algorithms for this minimization problem. The effectiveness of the algorithms is also demonstrated by computational experiments using both synthetic data and realistic biological data.

Probabilistic Boolean Networks play a remarkable role in the modelling and control of gene regulatory networks. In this paper, we consider the inverse problem of constructing a sparse probabilistic Boolean network from the prescribed transition probability matrix. We propose a modified orthogonal matching pursuit for solving the inverse problem. We provide some conditions under which the proposed algorithm can recover a sparse probabilistic Boolean network. We also report some numerical results to illustrate the effectiveness of the proposed algorithm., Comment: 18 pages, 14 figures

An autoencoder is a layered neural network whose structure can be viewed as consisting of an encoder, which compresses an input vector of dimension $D$ to a vector of low dimension $d$, and a decoder which transforms the low-dimensional vector back to the original input vector (or one that is very similar). In this paper we explore the compressive power of autoencoders that are Boolean threshold networks by studying the numbers of nodes and layers that are required to ensure that the numbers of nodes and layers that are required to ensure that each vector in a given set of distinct input binary vectors is transformed back to its original. We show that for any set of $n$ distinct vectors there exists a seven-layer autoencoder with the smallest possible middle layer, (i.e., its size is logarithmic in $n$), but that there is a set of $n$ vectors for which there is no three-layer autoencoder with a middle layer of the same size. In addition we present a kind of trade-off: if a considerably larger middle layer is permissible then a five-layer autoencoder does exist. We also study encoding by itself. The results we obtain suggest that it is the decoding that constitutes the bottleneck of autoencoding. For example, there always is a three-layer Boolean threshold encoder that compresses $n$ vectors into a dimension that is reduced to twice the logarithm of $n$., Comment: 13 pages, 3 figures, 1 table

In this paper, we study optimal liquidation problems in a randomly-terminated horizon. We consider the liquidation of a large single-asset portfolio with the aim of minimizing a combination of volatility risk and transaction costs arising from permanent and temporary market impact. Three different scenarios are analyzed under Almgren-Chriss's market impact model to explore the relation between optimal liquidation strategies and potential inventory risk arising from the uncertainty of the liquidation horizon. For cases where no closed-form solutions can be obtained, we verify comparison principles for viscosity solutions and characterize the value function as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equation.

In this paper, we generalize the Almgren-Chriss's market impact model to a more realistic and flexible framework and employ it to derive and analyze some aspects of optimal liquidation problem in a security market. We illustrate how a trader's liquidation strategy alters when multiple venues and extra information are brought into the security market and detected by the trader. This study gives some new insights into the relationship between liquidation strategy and market liquidity, and provides a multi-scale approach to the optimal liquidation problem with randomly varying volatility.

In this paper we consider a reduced-form intensity-based credit risk model with a hidden Markov state process. A filtering method is proposed for extracting the underlying state given the observation processes. The method may be applied to a wide range of problems. Based on this model, we derive the joint distribution of multiple default times without imposing stringent assumptions on the form of default intensities. Closed-form formulas for the distribution of default times are obtained which are then applied to solve a number of practical problems such as hedging and pricing credit derivatives. The method and numerical algorithms presented may be applicable to various forms of default intensities.

In this paper, we employ the Heston stochastic volatility model to describe the stock's volatility and apply the model to derive and analyze the optimal trading strategies for dealers in a security market. We also extend our study to option market making for options written on stocks in the presence of stochastic volatility. Mathematically, the problem is formulated as a stochastic optimal control problem and the controlled state process is the dealer's mark-to-market wealth. Dealers in the security market can optimally determine their ask and bid quotes on the underlying stocks or options continuously over time. Their objective is to maximize an expected profit from transactions with a penalty proportional to the variance of cumulative inventory cost.

In this paper, the optimal pricing strategy in Avellande-Stoikov's for a monopolistic dealer is extended to a general situation where multiple dealers are present in a competitive market. The dealers' trading intensities, their optimal bid and ask prices and therefore their spreads are derived when the dealers are informed the severity of the competition. The effects of various parameters on the bid-ask quotes and profits of the dealers in a competitive market are also discussed. This study gives some insights on the average spread, profit of the dealers in a competitive trading environment.

In this paper, we study a continuous time structural asset value model for two correlated firms using a two-dimensional Brownian motion. We consider the situation of incomplete information, where the information set available to the market participants includes the default time of each firm and the periodic asset value reports. In this situation, the default time of each firm becomes a totally inaccessible stopping time to the market participants. The original structural model is first transformed to a reduced-form model. Then the conditional distribution of the default time together with the asset value of each name are derived. We prove the existence of the intensity processes of default times and also give the explicit form of the intensity processes. Numerical studies on the intensities of the two correlated names are conducted for some special cases. We also indicate the possible future research extension into three names case by considering a special correlation structure., Comment: 24 pages, 6 figures

In the aftermath of the global financial crisis, much attention has been paid to investigating the appropriateness of the current practice of default risk modeling in banking, finance and insurance industries. A recent empirical study by Guo et al.(2008) shows that the time difference between the economic and recorded default dates has a significant impact on recovery rate estimates. Guo et al.(2011) develop a theoretical structural firm asset value model for a firm default process that embeds the distinction of these two default times. To be more consistent with the practice, in this paper, we assume the market participants cannot observe the firm asset value directly and developed a reduced-form model to characterize the economic and recorded default times. We derive the probability distribution of these two default times. The numerical study on the difference between these two shows that our proposed model can both capture the features and fit the empirical data., Comment: arXiv admin note: text overlap with arXiv:1012.0843 by other authors

In this paper, we propose a two-sector Markovian infectious model, which is an extension of Greenwood's model. The central idea of this model is that the causality of defaults of two sectors is in both direction, which enrich dependence dynamics. The Bayesian Information Criterion is adopted to compare the proposed model with the two-sector model in credit literature using the real data. We find that the newly proposed model is statistically better than the model in past literature. We also introduce two measures: CRES and CRVaR to give risk evaluation of our model.

Corporate defaults may be triggered by some major market news or events such as financial crises or collapses of major banks or financial institutions. With a view to develop a more realistic model for credit risk analysis, we introduce a new type of reduced-form intensity-based model that can incorporate the impacts of both observable "trigger" events and economic environment on corporate defaults. The key idea of the model is to augment a Cox process with trigger events. Both single-default and multiple-default cases are considered in this paper. In the former case, a simple expression for the distribution of the default time is obtained. Applications of the proposed model to price defaultable bonds and multi-name Credit Default Swaps (CDSs) are provided.

In this paper we propose a simple and efficient method to compute the ordered default time distributions in both the homogeneous case and the two-group heterogeneous case under the interacting intensity default contagion model. We give the analytical expressions for the ordered default time distributions with recursive formulas for the coefficients, which makes the calculation fast and efficient in finding rates of basket CDSs. In the homogeneous case, we explore the ordered default time in limiting case and further include the exponential decay and the multistate stochastic intensity process. The numerical study indicates that, in the valuation of the swap rates and their sensitivities with respect to underlying parameters, our proposed model outperforms the Monte Carlo method.

Theoretical frameworks to estimate the tolerance of metabolic networks to various failures are important to evaluate the robustness of biological complex systems in systems biology. In this paper, we focus on a measure for robustness in metabolic networks, namely, the impact degree, and propose an approximation method to predict the probability distribution of impact degrees from metabolic network structures using the theory of branching process. We demonstrate the relevance of this method by testing it on real-world metabolic networks. Although the approximation method possesses a few limitations, it may be a powerful tool for evaluating metabolic robustness., Comment: 17 pages, 4 figures, 4 tables