We investigate the impact of community structure on information spreading with the linear threshold model. Contrary to the common belief that communities hinder information diffusion, we show that strong communities can facilitate global cascades by enhancing local, intra-community spreading. Using both analytical approaches and numerical simulations, we demonstrate the existence of optimal clustering, where global cascades require the minimal number of early adopters. The study of information diffusion -fads, diffusion of innovations, collective action, viral memes, etc. -is relevant to a number of disciplines, including mathematical, physical and social sciences, communication, marketing and economics [1] [2] [3] [4] [5] [6] . The most common approach is focusing on the resemblances between information contagion and infectious diseases [7, 8] ; a piece of information can travel from one individual to another through social contacts and the 'infected' individuals can, in turn, propagate the information to others, possibly generating a large-scale cascade, like an epidemic outbreak [9, 10] . In addition to classical epidemic models, two main types of information diffusion models have been proposed: the independent cascade model, which was initially adopted to study the dynamics of viral marketing [11] [12] [13] [14] [15] , considers information contagion as a simple branching process; the threshold model, which was originally proposed to study collective social behaviors [2, [16] [17] [18] , incorporates the idea of 'social reinforcement', assuming that each adoption requires a certain number of exposures. Since information spreads through social contacts, the structure of the underlying social network is a crucial ingredient in studying information diffusion. In particular, the role of hubs and degree distribution have been studied extensively, mainly because of their critical role in epidemic spreading [19] [20] [21] . However, it is not yet fully understood how information diffusion differs from epidemic spreading and how the impact of network structure is changed by the differences. One notable difference is the social reinforcement: unlike epidemic spreading where each exposure acts independently, each additional exposure to the information seems to markedly increase the probability of adoption [22] [23] [24] . As a consequence, it has been suggested that clustering plays a more important role in information spreading than in epidemics [6, 22, 24] . These results suggest that clustering is a crucial element in understanding information diffusion, and that social reinforcement is a key ingredient in modeling information diffusion. Some studies have investigated the role of communities in information diffusion [25] [26] [27] [28] [29] [30] , but most of them either overlook the effect of social reinforcement or do not fully explore the implications of community structure. The most common notion about the role of communities is that they simply hinder the spreading of information, keeping it confined in the community of origin, as one would expect for epidemics [25, 31] . However, recent empirical work has suggested that clustering may, counterintuitively, facilitate information diffusion [24] and therefore may provide an important clue to understanding fundamental differences between epidemics and information contagion [6] . In this letter we use the linear threshold modelwhich incorporates the simplest form of social reinforcement -to study how the global information cascades are affected by community structures. We expose two roles played by community structure: enhancing local spreading and hindering global spreading. Strong communities facilitate social reinforcement and thereby enhance local spreading [6, 24] ; weak community structure makes global spreading easier, because it provides more bridges between communities. The more cohesive communities are, the better information can spread locally and the worse it can spread between communities. We show that there exists an optimal balance between these two effects, where community structure counterintuitively enhances -rather than hinders -global cascades of information. This is analogous to the small-world phenomenon where a small number of shortcuts greatly reduces the average path length of the network while maintaining high clustering [32] ; a small number of bridges between communities allow inter-community spreading while maintaining intra-community spreading. We adopt the linear threshold model to account for recent observations and experiments that have demonstrated the relevance of social reinforcement in the diffusion of information [6, [22] [23] [24] . Let us formally define the linear threshold model first. Consider a set of N nodes (agents) connected by M undirected edges. The state of an agent i at time t is described by a binary variable s i (t) = {0, 1}, where 1 represents 'active' state and 0 indicates 'inactive'. At time t = 0 a fraction ρ 0 of randomly selected agents, 'seeds', is initialized in the active state. At each time step, agents' states are updated
Nematzadeh et al. REF demonstrated that strong communities with high modularity can facilitate global diffusion by enhancing local, intra-community spreading.
Optimal network modularity for information diffusion
Phys. Rev. Lett. 113, 088701 (2014)
Physics Computer Science Medicine