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Based on the concepts of ''word-of-mouth'' effect and viral marketing, the diffusion of an innovation may be triggered starting from a set of initial users. Estimating the influence spread is a preliminary step to determine a suitable or even optimal set of initial users to reach a given goal. In this paper, we focus on a stochastic model called the independent cascade model and compare a few approaches to compute activation probabilities of nodes in a social network, i.e., the probability that a user adopts the innovation. First, we propose the path method that computes the exact value of the activation probabilities but has high complexity. Second, an approximated method, called SSS-Noself, is obtained by the modification of the existing SteadyStateSpread algorithm, based on fixed-point computation, to achieve better accuracy. Finally, an efficient approach, also based on fixed-point computation, is proposed to compute the probability that a node is activated through a path of minimal length from the seed set. This algorithm, called SSS-Bounded-Path algorithm, can provide a lower bound for the computation of activation probabilities. Furthermore, these proposed approaches are applied to the influence maximization problem combined with the SelectTopK algorithm, the RankedReplace algorithm, and the greedy algorithm.INDEX TERMS Independent cascade model, influence maximization, social networks.

We give an overview of GRIP, a symmetry reduction tool for the probabilistic model checker PRISM, together with experimental results for a selection of example specifications. An Overview of GRIPGRIP (generic representatives in PRISM), introduced in [1], is a symmetry reduction tool for the PRISM model checker [6]. GRIP is based on the generic representatives approach of [2], which aims to overcome the inherent problem of combining symmetry reduction with symbolic state-space representation. We present an overview of GRIP version 2.0 (referred to henceforth as GRIP), an improved version of the original tool, and compare GRIP to PRISM-symm, an alternative symmetry reduction tool for PRISM [5]. GRIP, together with the PRISM examples used for experiments in Section 3 can be downloaded from our website [4].The top panel of Figure 1 shows a simple leader election protocol in PRISM, adapted from [1]. The underlying model here is a Markov decision process (MDP). GRIP works by translating this specification into a reduced form, as shown in the bottom-left panel of the figure. The reduced specification abstracts away from specific modules, instead using a single generic module comprised of variables which count the number of modules in each potential local state. Symmetric temporal properties can also be translated into reduced form. PRISM can then be used, unchanged, to check reduced properties of a reduced specification. New Features of GRIPThe original version of GRIP required specifications to consist of multiple instantiations of a single symmetric module type, specified using a single local state variable. This model of computation is in keeping with the presentation of the generic representatives approach for non-probabilistic model checking [2]. While a wide class of symmetric systems can, in theory, be specified in this way, accurately modelling complex protocols via a single state variable quickly becomes impractical. GRIP now supports: multiple local state variables; a wide range of arithmetic and boolean expressions over these variables; communication via shared global variables, and multiple asymmetric modules in parallel with a single family of symmetric modules. In addition, GRIP handles models with continuous time Markov chain (CTMC) semantics.Multiple local variables can result in a large number of local states, which translates to many counters in the specification output by GRIP. This in turn can lead to large MTBDDs (the symbolic data structure used by PRISM). To combat this, we have implemented an optimisation suggested in [3]: we use PRISM for local reachability analysis during the translation process, to reduce the number of counters in the output specification. In addition, since the sum of counter variables should always equal N (the number of symmetric modules), the last counter variable can be eliminated and replaced with the formulaThis second optimisation offers a modest reduction in MTBDD size. The bottom-right panel of Figure 1 shows the effect of these optimisations: local reachabi...

Based on the concepts of word-of-mouth effect and viral marketing, the diffusion of an innovation may be triggered starting from a set of initial users. Estimating the influence spread is a preliminary step to determine a suitable or even optimal set of initial users to reach a given goal. In this paper, we focus on a stochastic model called the Independent Cascade model, and compare a few approaches to compute activation probabilities of nodes in a social network, i.e., the probability that a user adopts the innovation. In the paper, first we propose the Path Method which computes the exact value of the activation probabilities but it has high complexity. Second an approximated method, called SSS-Noself, is obtained by modification of the existing SteadyStateSpread algorithm, based on fixed-point computation, to achieve a better accuracy. Finally an efficient approach, also based on fixed-point computation, is proposed to compute the probability that a node is activated though a path of minimal length from the seed set. This algorithm, called SSS-Bound-t algorithm, can provide a lowerbound for the computation of activation probabilities.

International audienceStochastic automata networks (Sans) are high-level formalisms for modeling very large and complex Markov chains in a compact and structured manner. To date, the exponential distribution has been the only distribution used to model the passage of time in the evolution of the different San components. In this paper we show how phase-type distributions may be incorporated into Sans thereby providing the wherewithal by which arbitrary distributions can be used which in turn leads to an improved ability for more accurately modeling numerous real phenomena. The approach we develop is to take a San model containing phase-type distributions and to translate it into another, stochastically equivalent, San model having only exponential distributions. In the San formalism, it is the events that are responsible for firing transitions and our procedure is to associate a stochastic automaton with each event having a phase-type distribution. This automaton models the distribution of time until the event occurs. In this way, the size of the elementary matrices remain small, because the size of the automata are small: the automata are either those of the original San, or are those associated with the phase-type events and are of size k, the number of phases in the representation of the distribution

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