Locating the source that triggers a dynamical process is a fundamental but challenging problem in complex networks, ranging from epidemic spreading in society and on the Internet to cancer metastasis in the human body. An accurate localization of the source is inherently limited by our ability to simultaneously access the information of all nodes in a large-scale complex network. This thus raises two critical questions: how do we locate the source from incomplete information and can we achieve full localization of sources at any possible location from a given set of observable nodes. Here we develop a time-reversal backward spreading algorithm to locate the source of a diffusion-like process efficiently and propose a general locatability condition. We test the algorithm by employing epidemic spreading and consensus dynamics as typical dynamical processes and apply it to the H1N1 pandemic in China. We find that the sources can be precisely located in arbitrary networks insofar as the locatability condition is assured. Our tools greatly improve our ability to locate the source of diffusion in complex networks based on limited accessibility of nodal information. Moreover, they have implications for controlling a variety of dynamical processes taking place on complex networks, such as inhibiting epidemics, slowing the spread of rumors, pollution control, and environmental protection.

Data based source localization in complex networks has a broad range of applications. Despite recent progress, locating multiple diffusion sources in time varying networks remains to be an outstanding problem. Bridging structural observability and sparse signal reconstruction theories, we develop a general framework to locate diffusion sources in time varying networks based solely on sparse data from a small set of messenger nodes. A general finding is that large degree nodes produce more valuable information than small degree nodes, a result that contrasts that for static networks. Choosing large degree nodes as the messengers, we find that sparse observations from a few such nodes are often sufficient for any number of diffusion sources to be located for a variety of model and empirical networks. Counterintuitively, sources in more rapidly varying networks can be identified more readily with fewer required messenger nodes.

SignificanceUnderstanding how communities emerge is a fundamental problem in social and economic systems. Here, we experimentally explore the emergence of communities in social networks, using the ultimatum game as a paradigm for capturing individual interactions. We find the emergence of diverse communities in static networks is the result of the local interaction between responders with inherent heterogeneity and rational proposers in which the former act as community leaders. In contrast, communities do not arise in populations with random interactions, suggesting that a static structure stabilizes local communities and social diversity. Our experimental findings deepen our understanding of self-organized communities and of the establishment of social norms associated with game dynamics in social networks.

Volatility clustering, Fat tail, Prospect theory, Risk appetite, Market efficiency,

Experiments on the Ultimatum Game (UG) repeatedly show that people's behaviour is far from rational. In UG experiments, a subject proposes how to divide a pot and the other can accept or reject the proposal, in which case both lose everything. While rational people would offer and accept the minimum possible amount, in experiments low offers are often rejected and offers are typically larger than the minimum, and even fair. Several theoretical works have proposed that these results may arise evolutionarily when subjects act in both roles and there is a fixed interaction structure in the population specifying who plays with whom. We report the first experiments on structured UG with subjects playing simultaneously both roles. We observe that acceptance levels of responders approach rationality and proposers accommodate their offers to their environment. More precisely, subjects keep low acceptance levels all the time, but as proposers they follow a best-response-like approach to choose their offers. We thus find that status equality promotes rational sharing while the influence of structure leads to fairer offers compared to well-mixed populations. Our results are far from what is observed in single-role UG experiments and largely different from available predictions based on evolutionary game theory.The Ultimatum Game (UG) was proposed more than three decades ago 1,2 as a simple and clear way to measure social preferences 3 . In UG experiments, experimenters work with two subjects and give one of them (the "proposer") an amount of money. The proposer makes an offer as to how to split the money to the other player (the "responder"). The responder can only accept the proposal as is or reject it outright, and in case of rejection none of the players receives any money. Clearly, rational people, where the term "rational" is used in the sense of self-interest, will both offer and accept the minimum possible amount, as responders have no incentives to reject any positive amount of money. However, all available experiments provide strong evidence that low offers are often rejected, with low meaning lower than 20-30% of the pot. Correspondingly, it appears that proposers anticipate this behaviour and offer amounts larger than the minimum, with fair splits being frequent. It is worth stressing that in the last three decades literally thousands of experiments have been carried out 4-9 giving the same qualitative results.A common variant of the standard ultimatum game is that the responder precommits a minimum acceptable offer (MAO) that he/she will accept (and any lower offer will be rejected) rather than simply decides whether to accept a specific offer. This MAO variant is more informative about responders' preferences. Experiments found that the minimum acceptable offers for most subjects are around 30% 6,[10][11][12] . This is consistent with the observations of the standard UG that offers lower than 30% are often rejected 8,9 . In particular, the MAO UG has some relation with the coordination game in the sense that w...