text: We consider two competing first passage percolation processes started from uniformly chosen subsets of a random regular graph on @math vertices. The processes are allowed to spread with different rates, start from vertex subsets of different sizes or at different times. We obtain tight results regarding the sizes of the vertex sets occupied by each process, showing that in the generic situation one process will occupy @math vertices, for some @math . The value of @math is calculated in terms of the relative rates of the processes, as well as the sizes of the initial vertex sets and the possible time advantage of one process. The motivation for this work comes from the study of viral marketing on social networks. The described processes can be viewed as two competing products spreading through a social network (random regular graph). Considering the processes which grow at different rates (corresponding to different attraction levels of the two products) or starting at different times (the first to market advantage) allows to model aspects of real competition. The results obtained can be interpreted as one of the two products taking the lion share of the market. We compare these results to the same process run on @math dimensional grids where we show that in the generic situation the two products will have a linear fraction of the market each.
  @cite_30: Though word-of-mouth (w-o-m) communications is a pervasive and intriguing phenomenon, little is known on its underlying process of personal communications. Moreover as marketers are getting more interested in harnessing the power of w-o-m, for e-business and other net related activities, the effects of the different communications types on macro level marketing is becoming critical. In particular we are interested in the breakdown of the personal communication between closer and stronger communications that are within an individual's own personal group (strong ties) and weaker and less personal communications that an individual makes with a wide set of other acquaintances and colleagues (weak ties).
  @cite_37: 
  @cite_22: 
  @cite_15: Infinite particle systems on lattices have been extensively studied in recent years. The main questions of interest concern the ergodic and limiting behaviour of these processes, and their relationship with the dimension of the underlying lattice. A comprehensive review is given by Durrett( 6 ). One of the more tractable of these processes is the voter model introduced by Clifford and Sudbury( 3 ) and much studied since, see for example the monograph by Griffeath( 8 ), or the papers by Harris( 11 ), Holley and Liggett( 13 ), Bramson and Griffeath( 1 ) and ( 2 ) or, for a more general approach, Kelly( 16 ). In this paper we consider the case where the underlying spatial structure is finite and examine the transient behaviour of the voter process and also the infection process introduced by Williams and Bjerknes( 21 ).
  @cite_48: The individual spins of the Ising model are assumed to interact with an external agency (e.g., a heat reservoir) which causes them to change their states randomly with time. Coupling between the spins is introduced through the assumption that the transition probabilities for any one spin depend on the values of the neighboring spins. This dependence is determined, in part, by the detailed balancing condition obeyed by the equilibrium state of the model. The Markoff process which describes the spin functions is analyzed in detail for the case of a closed N‐member chain. The expectation values of the individual spins and of the products of pairs of spins, each of the pair evaluated at a different time, are found explicitly. The influence of a uniform, time‐varying magnetic field upon the model is discussed, and the frequency‐dependent magnetic susceptibility is found in the weak‐field limit. Some fluctuation‐dissipation theorems are derived which relate the susceptibility to the Fourier transform of the tim...
  @cite_44: 
  @cite_23: 
  @cite_49: Aggregate level simulation procedures have been used in many areas of marketing. In this paper we show how individual level simulations may be used support marketing theory development. More specifically, we show how a certain type of simulations that is based on complex systems studies (in this case Stochastic Cellular Automata) may be used to generalize diffusion theory one of the fundamental theories of new product marketing. Cellular Automata models are simulations of global consequences, based on local interactions between individual members of a population, that are widely used in complex system analysis across disciplines. In this study we demonstrate how the Cellular Automata approach can help untangle complex marketing research problems. Specifically, we address two major issues facing current theory of innovation diffusion: The first is general lack of data at the individual level, while the second is the resultant inability of marketing researchers to empirically validate the main assumptions used in the aggregate models of innovation diffusion. Using a computer-based Cellular Automata Diffusion Simulation, we demonstrate how such problems can be overcome. More specifically, we show that relaxing the commonly used assumption of homogeneity in the consumers’ communication behavior is not a barrier to aggregate modeling. Thus we show that notwithstanding some exceptions, the well-known Bass model performs well on aggregate data when the assumption that that all adopters have a possible equal effect on all other potential adopters is relaxed. Through Cellular Automata we are better able to understand how individual level assumptions influence aggregate level parameter values, and learn the strengths and limitations of the aggregate level analysis. We believe that this study can serve as a demonstration towards a much wider use of Cellular Automata models for complex marketing research phenomena.
  @cite_5: 
  @cite_47: Two of the simplest interacting particle systems are the coalescing random walks and the voter model. We are interested here in the asymptotic density and growth of these systems ast→∞. More specifically, let (ζ t Zd ) be a system of coalescing random walks with initial stateZd, and (ζ t O ) a voter model with a single individual originating atO. We analyse (p_t = P(0 _t^ Zd ) = .5em @math P ( _t^0 ) ), and show that (p_t 1 t t ) ast→∞ ford=2, andpt∼(γdt)−1 ast→∞ ford≧3 for someγd. As a consequence, conditioned on non-extinction ofζ t O , Pt¦ζ t O ¦ approaches an exponential distribution. Results of a recent paper by Sawyer are applied.
  @cite_11: 
summary: As mentioned earlier, diffusion and growth processes have been studied intensely in the past few years in relation to many areas such as sociology, economics and engineering. Among the models studied are (see, for example @cite_44 , @cite_49 , @cite_30 ), which was first introduced by Clifford and Sudbury in @cite_5 and has been much studied since in, for example, @cite_23 , @cite_47 , @cite_15 , (see, for example, @cite_22 ), (see @cite_48 , @cite_11 ), and (see @cite_37 ).