Spatial evolution game has traditionally assumed that players interact with direct neighbors on a single network, which is isolated and not influenced by other systems. However, this is not fully consistent with recent research identification that interactions between networks play a crucial rule for the outcome of evolutionary games taking place on them. In this work, we introduce the simple game model into the interdependent networks composed of two networks. By means of imitation dynamics, we display that when the interdependent factor α is smaller than a threshold value αC, the symmetry of cooperation can be guaranteed. Interestingly, as interdependent factor exceeds αC, spontaneous symmetry breaking of fraction of cooperators presents itself between different networks. With respect to the breakage of symmetry, it is induced by asynchronous expansion between heterogeneous strategy couples of both networks, which further enriches the content of spatial reciprocity. Moreover, our results can be well predicted by the strategy-couple pair approximation method.

Diffusion processes are central to human interactions. One common prediction of the current modeling frameworks is that initial spreading dynamics follow exponential growth.Here, we find that, ranging from mobile handsets to automobiles, from smart-phone apps to scientific fields, early growth patterns follow a power law with non-integer exponents. We test the hypothesis that mechanisms specific to substitution dynamics may play a role, by analyzing a unique data tracing 3.6M individuals substituting for different mobile handsets. We uncover three generic ingredients governing substitutions, allowing us to develop a minimal substitution model, which not only explains the power-law growth, but also collapses diverse growth trajectories of individual constituents into a single curve. These results offer a mechanistic understanding of power-law early growth patterns emerging from various domains and demonstrate that substitution dynamics are governed by robust self-organizing principles that go beyond the particulars of individual systems.2 Diffusion processes impact broad aspects of human society 1-5 , ranging from the spread of biological viruses 3, 6-8 to the adoption of innovations 4,[9][10][11][12][13][14] and knowledge 15, 16 and to the spread of information [17][18][19] , cultural norms and social behavior [20][21][22][23] . Despite numerous studies that span multiple disciplines, our knowledge is mainly limited to spreading processes in non-substitutive systems. Yet, a considerable number of ideas, products and behaviors spread by substitution-to adopt a new one, agents often need to give up an existing one. For example, the development of science hinges on scientists' relentlessness in abandoning a scientific framework once one that offers a better description of reality emerges 24 . The same is true for adopting a new healthy habit or other durable items, like mobile phones, cars or homes.While substitutions play a key role from science to economy, our limited understanding of such processes stems from the lack of empirical data tracing their characteristics. To study the dynamics of substitutions, we explore growth patterns in four different substitutive systems where detailed dynamical patterns are captured with fine temporal resolution (See Supplementary Note 1 for detailed data descriptions). Our first dataset captures, with daily resolution, 3.6 Million individuals choosing among different types of mobile handsets, recorded by a Northern European telecommunication company from January 2006 to November 2014. Since an individual is unlikely to keep more than one mobile phone at a time, his or her adoption of a new handset is typically associated with discontinuance of the old one. Here, we focus on handsets that have been released for at least 6 months and used by at least 50 users in total (885 different handset models). Our second dataset captures monthly transaction records of 126 automobiles sold in the North America between 2010 and 2016. These automobiles have been released for at least four mont...