Suppose we have a virus or one competing idea product that propagates over a multiple profile (e.g., social) network. Can we predict what proportion of the network will actually get "infected" (e.g., spread the idea or buy the competing product), when the nodes of the network appear to have different sensitivity based on their profile? For example, if there are two profiles @math and @math in a network and the nodes of profile @math and profile @math are susceptible to a highly spreading virus with probabilities @math and @math respectively, what percentage of both profiles will actually get infected from the virus at the end? To reverse the question, what are the necessary conditions so that a predefined percentage of the network is infected? We assume that nodes of different profiles can infect one another and we prove that under realistic conditions, apart from the weak profile (great sensitivity), the stronger profile (low sensitivity) will get infected as well. First, we focus on cliques with the goal to provide exact theoretical results as well as to get some intuition as to how a virus affects such a multiple profile network. Then, we move to the theoretical analysis of arbitrary networks. We provide bounds on certain properties of the network based on the probabilities of infection of each node in it when it reaches the steady state. Finally, we provide extensive experimental results that verify our theoretical results and at the same time provide more insight on the problem.
Ecology, is another interesting application field for such propagation models. The principle of states that two species that compete for the exact same resources cannot stably coexist which resembles the aforementioned phenomenon of "winner takes all". This principle has been studied intensively using propagation models such as SIS, SIRS, Lotka-Volterra etc. but there has been no analytical solution presented so far @cite_2 , @cite_22 , @cite_21 .
@cite_21 @cite_22 @cite_2
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The ecological principle of competitive exclusion asserts that two species cannot indefinitely occupy the same niche [Sj. Attempts have been made to state this principle as a mathematical theorem. A standard example is due to Volterra [14]. (See Example 2.1 below.) He constructed a model of two species competing for a single resource and showed that one of the species must go extinct. Although the concept of niche is rather vague, it is generally agreed that Volterra’s model is an example of two species competing for the same niche and is therefore an illustration of the principle of competitive exclusion. The generalizations of Volterra’s model have dealt largely with the introduction of more realistic assumptions about the interactions between species. These assumptions have led to some interesting mathematical problems, many of them unsolved. The purpose of this paper is to state these problems precisely, to give proofs of the known results, and to indicate the unanswered questions. The basic object of study is an idealized ecological community consisting of a certain number of species living together in an isolated geographical area. We shall assume that the community interactions are independent of both space and time and that each species is distributed uniformly over the region. The basic variables are the population densities of each of the species. The For the study of information propagation, one fundamental problem is uncovering universal laws governing the dynamics of information propagation. This problem, from the microscopic perspective, is formulated as estimating the propagation probability that a piece of information propagates from one individual to another. Such a propagation probability generally depends on two major classes of factors: the intrinsic attractiveness of information and the interactions between individuals. Despite the fact that the temporal effect of attractiveness is widely studied, temporal laws underlying individual interactions remain unclear, causing inaccurate prediction of information propagation on evolving social networks. In this report, we empirically study the dynamics of information propagation, using the dataset from a population-scale social media website. We discover a temporal scaling in information propagation: the probability a message propagates between two individuals decays with the length of time latency since their latest interaction, obeying a power-law rule. Leveraging the scaling law, we further propose a temporal model to estimate future propagation probabilities between individuals, reducing the error rate of information propagation prediction from 6.7 to 2.6 and improving viral marketing with 9.7 incremental customers. Cyclic dominance of species has been identified as a potential mechanism to maintain biodiversity, see, e.g., B. Kerr, M. A. Riley, M. W. Feldman and B. J. M. Bohannan Nature 418, 171 (2002)] and B. Kirkup and M. A. Riley Nature 428, 412 (2004)]. Through analytical methods supported by numerical simulations, we address this issue by studying the properties of a paradigmatic non-spatial three-species stochastic system, namely, the "rock-paper-scissors" or cyclic Lotka-Volterra model. While the deterministic approach (rate equations) predicts the coexistence of the species resulting in regular (yet neutrally stable) oscillations of the population densities, we demonstrate that fluctuations arising in the system with a finite number of agents drastically alter this picture and are responsible for extinction: After long enough time, two of the three species die out. As main findings we provide analytic estimates and numerical computation of the extinction probability at a given time. We also discuss the implications of our results for a broad class of competing population systems. Rock is wrapped by paper, paper is cut by scissors and scissors are crushed by rock. This simple game is popular among children and adults to decide on trivial disputes that have no obvious winner, but cyclic dominance is also at the heart of predator–prey interactions, the mating strategy of side-blotched lizards, the overgrowth of marine sessile organisms and competition in microbial populations. Cyclical interactions also emerge spontaneously in evolutionary games entailing volunteering, reward, punishment, and in fact are common when the competing strategies are three or more, regardless of the particularities of the game. Here, we review recent advances on the rock–paper–scissors (RPS) and related evolutionary games, focusing, in particular, on pattern formation, the impact of mobility and the spontaneous emergence of cyclic dominance. We also review mean-field and zero-dimensional RPS models and the application of the complex Ginzburg–Landau equation, and we highlight the importance and usefulness of statistical physics for the successful study of large-scale ecological systems. Directions for future research, related, for example, to dynamical effects of coevolutionary rules and invasion reversals owing to multi-point interactions, are also outlined.