This paper proposes and investigates a Boolean gossip model as a simplified but non-trivial probabilistic Boolean network. With positive node interactions, in view of standard theories from Markov chains, we prove that the node states asymptotically converge to an agreement at a binary random variable, whose distribution is characterized for large-scale networks by mean-field approximation. Using combinatorial analysis, we also successfully count the number of communication classes of the positive Boolean network explicitly in terms of the topology of the underlying interaction graph, where remarkably minor variation in local structures can drastically change the number of network communication classes. With general Boolean interaction rules, emergence of absorbing network Boolean dynamics is shown to be determined by the network structure with necessary and sufficient conditions established regarding when the Boolean gossip process defines absorbing Markov chains. Particularly, it is shown that for the majority of the Boolean interaction rules, except for nine out of the total @math possible nonempty sets of binary Boolean functions, whether the induced chain is absorbing has nothing to do with the topology of the underlying interaction graph, as long as connectivity is assumed. These results illustrate possibilities of relating dynamical properties of Boolean networks to graphical properties of the underlying interactions.
In this paper, we study the impact of edge weights on distances in sparse random graphs. We interpret these weights as delays and take them as independent and identically distributed exponential random variables. We analyze the weighted flooding time defined as the minimum time needed to reach all nodes from one uniformly chosen node and the weighted diameter corresponding to the largest distance between any pair of vertices. Under some standard regularity conditions on the degree sequence of the random graph, we show that these quantities grow as the logarithm of @math when the size of the graph @math tends to infinity. We also derive the exact value for the prefactor. These results allow us to analyze an asynchronous randomized broadcast algorithm for random regular graphs. Our results show that the asynchronous version of the algorithm performs better than its synchronized version: in the large size limit of the graph, it will reach the whole network faster even if the local dynamics are similar on average. Context-sensitive probabilistic Boolean networks (PBNs) have been recently introduced as a paradigm for modeling genetic regulatory networks and have served as the main model for the application of intervention methods, including optimal control strategies, to favorably effect system dynamics. Since it is believed that the steady-state behavior of a context-sensitive PBN is indicative of the phenotype, it is important to study the alternation in the steady-state probability distribution due to any variations in the formulations of the context-sensitive PBNs. Furthermore, the huge computational complexity of the context-sensitive PBN model necessitates generation of size-reduction techniques and approximate methods for calculation of the steady-state probability distribution of context-sensitive PBNs. The goal of this paper is threefold: i) to study the effects of the various definitions of context-sensitive PBNs on the steady-state probability distributions and the downstream control policy design; ii) to propose a reduction technique that maintains the steady-state probability distribution; and iii) to provide an approximation method for calculating the steady-state probability distribution of a context-sensitive PBN. Fine-scale models based on stochastic master equations can provide the most detailed description of the dynamics of gene expression and imbed, in principle, all the information about the biochemical reactions involved in gene interactions. However, there is limited time-series experimental data available for inference of such fine-scale models. Furthermore, the computational complexity involved in the design of optimal intervention strategies to favorably effect system dynamics for such detailed models is enormous. Thus, there is a need to design mappings from fine-scale models to coarse-scale models while maintaining sufficient structure for the problem at hand and to study the effect of intervention policies designed using coarse-scale models when applied to fine-scale models. In this paper, we propose a mapping from a fine-scale model represented by a stochastic master equation to a coarse-scale model represented by a probabilistic Boolean network that maintains the collapsed steady state probability distribution of the detailed model. We also derive bounds on the performance of the intervention strategy designed using the coarse-scale model when applied to the fine-scale model. A voter sits on each vertex of an infinite tree of degree @math , and has to decide between two alternative opinions. At each time step, each voter switches to the opinion of the majority of her neighbors. We analyze this majority process when opinions are initialized to independent and identically distributed random variables. In particular, we bound the threshold value of the initial bias such that the process converges to consensus. In order to prove an upper bound, we characterize the process of a single node in the large @math -limit. This approach is inspired by the theory of mean field spin-glass and can potentially be generalized to a wider class of models. We also derive a lower bound that is nontrivial for small, odd values of @math . Boolean networks form a class of disordered dynamical systems that have been studied in physics owing to their relationships with disordered systems in statistical mechanics and in biology as models of genetic regulatory networks. Recently they have been generalized to probabilistic Boolean networks (PBNs) to facilitate the incorporation of uncertainty in the model and to represent cellular context changes in biological modeling. In essence, a PBN is composed of a family of Boolean networks between which the PBN switches in a stochastic fashion. In whatever framework Boolean networks are studied, their most important attribute is their attractors. Left to run, a Boolean network will settle into one of a collection of state cycles called attractors. The set of states from which the network will transition into a specific attractor forms the basin of the attractor. The attractors represent the essential long-run behavior of the network. In a classical Boolean network, the network remains in an attractor once there; in a Boolean network with perturbation, the states form an ergodic Markov chain and the network can escape an attractor, but it will return to it or a different attractor unless interrupted by another perturbation; in a probabilistic Boolean network, so long as the PBN remains in one of its constituent Boolean networks it will behave as a Boolean network with perturbation, but upon a switch it will move to an attractor of the new constituent Boolean network. Given the ergodic nature of the model, the steady-state probabilities of the attractors are critical to network understanding. Heretofore they have been found by simulation; in this paper we derive analytic expressions for these probabilities, first for Boolean networks with perturbation and then for PBNs.