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. * Electronic address: shinancao@ruc.edu.cn † Electronic address: wenxuwang@bnu.edu.cn 1 Introduction.-Many large-scale dynamical processes taking place on complex networks can be triggered from a small number of nodes. Prototypical examples include epidemic spreading on a global scale, rumor propagation through micro-blogs on the Internet, wide-ranging blackouts across North America and financial crises accompanied by the bankruptcy of a large number of financial institutions. The self-organization theory introduced by Bak and his collaborators [1] has provided a theoretical explanation: when a complex system enters a self-organized criticality state, small perturbations to even single individuals are able to initiate a big event, such as the avalanche of collapses in the sandpile model [2] . Moreover, the development of modern technology considerably facilitates the spreading of disease and information via public traffic systems and the Internet, which enables propagation across a large area from a source, such as the worldwide H1N1 pandemic in 2009 [3, 4] and the irrational and panicked acquisition of salt in southeast Asian countries caused by a rumor relevant to the nuclear leak in Japan. These phenomena raise a challenging question: how to locate the source in a huge network relying on relatively limited accessibility to nodal states, answers to which are of paramount importance for many aspects of nature and society, such as disease control, anti-terrorism, and economic health. Despite some pioneering approaches attempting to locate sources [5] [6] [7] [8] [9] [10] [11] and superspreaders [12, 13], we still lack a comprehensive understanding of our ability to precisely identify the original source of spreading in a large complex network. The difficulty stems from the lack of a general locatability condition to predict if the source at any possible locations is fully locatable in terms of a given set of observers. We develop a general locatability framework based on the time reversible characteristic of diffusion-like processes. This allows us to perform a time-reversal backward spreading to accurately locate the source, and offer a locatability condition that guarantees that a source will be fully locatable at any position. The algorithm and locatability condition are applicable in both directed and undirected networks with inherently limited knowledge of nodes and a time delay along links. We validate the tools by using a variety of complex networks in combination with two typical diffusion-like dynamical processes, i.e., epidemic spreading [14] [15] [16] and consensus dynamics [17, 18] . We have also applied our method to real networked systems by employing empirical data from the 2009 H1N1 pandemic in China, focusing on the Chinese airline and train networks as the epidemic spreading network. The four sources predicted by our tools are in good agreement with empirical findings. Our framework has further potential applications in locating, for example, a spammer who abuses email systems and pollution sources in river networks. Time-reversal backward spreading.-Our goal is to locate the source that initiates a diffusionlike process taking place on an already-known undirected or directed complex network using only the limited time information pertaining to the diffusion observed from a fraction of nodes. This limited information could be the time period during which a person is being invaded by a virus, or the appearance of an abnormal signal at a node. To better mimic a real-world scenario, we assume that we are unable to detect communications between the observable nodes and their neighbors. For example, hospital records tell us when a patient became ill, but do not tell us who passed the disease to the patient. Even knowing all of the sick persons with whom the patient has had recent
In REF , a time-reversal backward spreading (TRBS) algorithm was proposed to infer a single source in a weighted network.
Locating the source of diffusion in complex networks by time-reversal backward spreading.
Physical review. E
Physical review. E
Physics Medicine