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.
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 ).
@cite_30 @cite_37 @cite_22 @cite_15 @cite_48 @cite_44 @cite_23 @cite_49 @cite_5 @cite_47 @cite_11
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Models of networked diffusion that are motivated by analogy with the spread of infectious disease have been applied to a wide range of social and economic adoption processes, including those related to new products, ideas, norms and behaviors. However, it is unknown how accurately these models account for the empirical structure of diffusion over networks. Here we describe the diffusion patterns arising from seven online domains, ranging from communications platforms to networked games to microblogging services, each involving distinct types of content and modes of sharing. We find strikingly similar patterns across all domains. In particular, the vast majority of cascades are small, and are described by a handful of simple tree structures that terminate within one degree of an initial adopting "seed." In addition we find that structures other than these account for only a tiny fraction of total adoptions; that is, adoptions resulting from chains of referrals are extremely rare. Finally, even for the largest cascades that we observe, we find that the bulk of adoptions often takes place within one degree of a few dominant individuals. Together, these observations suggest new directions for modeling of online adoption processes. Topological properties of "scale-free" networks are investigated by determining their spectral dimensions d(S), which reflect a diffusion process in the corresponding graphs. Data bases for citation networks and metabolic networks together with simulation results from the growing network model [A.-L. Barabasi and R. Albert, Science 286, 509 (1999)] are probed. For completeness and comparisons lattice, random and small-world models are also investigated. We find that d(S) is around 3 for citation and metabolic networks, which is significantly different from the growing network model, for which d(S) is approximately 7.5. This signals a substantial difference in network topology despite the observed similarities in vertex-order distributions. In addition, the diffusion analysis indicates that the citation networks are treelike in structure, whereas the metabolic networks contain many loops. (Less) Platforms such as Twitter have provided researchers with ample opportunities to analytically study social phenomena. There are however, significant computational challenges due to the enormous rate of production of new information: researchers are therefore, often forced to analyze a judiciously selected “sample” of the data. Like other social media phenomena, information diffusion is a social process–it is affected by user context, and topic, in addition to the graph topology. This paper studies the impact of different attribute and topology based sampling strategies on the discovery of an important social media phenomena–information diffusion. We examine several widely-adopted sampling methods that select nodes based on attribute (random, location, and activity) and topology (forest fire) as well as study the impact of attribute based seed selection on topology based sampling. Then we develop a series of metrics for evaluating the quality of the sample, based on user activity (e.g. volume, number of seeds), topological (e.g. reach, spread) and temporal characteristics (e.g. rate). We additionally correlate the diffusion volume metric with two external variables–search and news trends. Our experiments reveal that for small sample sizes (30 ), a sample that incorporates both topology and user context (e.g. location, activity) can improve on naive methods by a significant margin of 15-20 . In [N. Alon, M. Feldman, A.D. Procaccia, M. Tennenholtz, A note on competitive diffusion through social networks, Inform. Process. Lett. 110 (2010) 221-225], the authors introduced a game-theoretic model of diffusion process through a network. They showed a relation between the diameter of a given network and existence of pure Nash equilibria in the game. Theorem 1 of their paper says that a pure Nash equilibrium exists if the diameter is at most two. However, we have an example which does not admit a pure Nash equilibrium even if the diameter is two. Hence we correct the statement of Theorem 1 of their paper.