Zhesi Shen

Reconstructing propagation networks with natural diversity and identifying hidden sources

Our ability to uncover complex network structure and dynamics from data is fundamental to understanding and controlling collective dynamics in complex systems. Despite recent progress in this area, reconstructing networks with stochastic dynamical processes from limited time series remains to be an outstanding problem. Here we develop a framework based on compressed sensing to reconstruct complex networks on which stochastic spreading dynamics take place. We apply the methodology to a large number of model and real networks, finding that a full reconstruction of inhomogeneous interactions can be achieved from small amounts of polarized (binary) data, a virtue of compressed sensing. Further, we demonstrate that a hidden source that triggers the spreading process but is externally inaccessible can be ascertained and located with high confidence in the absence of direct routes of propagation from it. Our approach thus establishes a paradigm for tracing and controlling epidemic invasion and information diffusion in complex networked systems. Supplementary information The online version of this article (doi:10.1038/ncomms5323) contains supplementary material, which is available to authorized users.

Robust reconstruction of complex networks from sparse data.

Reconstructing complex networks from measurable data is a fundamental problem for understanding and controlling collective dynamics of complex networked systems. However, a significant challenge arises when we attempt to decode structural information hidden in limited amounts of data accompanied by noise and in the presence of inaccessible nodes. Here, we develop a general framework for robust reconstruction of complex networks from sparse and noisy data. Specifically, we decompose the task of reconstructing the whole network into recovering local structures centered at each node. Thus, the natural sparsity of complex networks ensures a conversion from the local structure reconstruction into a sparse signal reconstruction problem that can be addressed by using the lasso, a convex optimization method. We apply our method to evolutionary games, transportation, and communication processes taking place in a variety of model and real complex networks, finding that universal high reconstruction accuracy can be achieved from sparse data in spite of noise in time series and missing data of partial nodes. Our approach opens new routes to the network reconstruction problem and has potential applications in a wide range of fields.

Locating the source of diffusion in complex networks by time-reversal backward spreading

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.Locating the sources that trigger 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, such as the time at which each individual is infected in a large population. 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 observers. Here we develop an efficient algorithm to locate the source of a diffusion-like process 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, and eliminating cancer seed cells in the human body.

Emergence of communities and diversity in social networks

Significance Understanding 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. Communities are common in complex networks and play a significant role in the functioning of social, biological, economic, and technological systems. Despite widespread interest in detecting community structures in complex networks and exploring the effect of communities on collective dynamics, a deep understanding of the emergence and prevalence of communities in social networks is still lacking. Addressing this fundamental problem is of paramount importance in understanding, predicting, and controlling a variety of collective behaviors in society. An elusive question is how communities with common internal properties arise in social networks with great individual diversity. Here, we answer this question using the ultimatum game, which has been a paradigm for characterizing altruism and fairness. We experimentally show that stable local communities with different internal agreements emerge spontaneously and induce social diversity into networks, which is in sharp contrast to populations with random interactions. Diverse communities and social norms come from the interaction between responders with inherent heterogeneous demands and rational proposers via local connections, where the former eventually become the community leaders. This result indicates that networks are significant in the emergence and stabilization of communities and social diversity. Our experimental results also provide valuable information about strategies for developing network models and theories of evolutionary games and social dynamics.

Reconstructing direct and indirect interactions in networked public goods game

Network reconstruction is a fundamental problem for understanding many complex systems with unknown interaction structures. In many complex systems, there are indirect interactions between two individuals without immediate connection but with common neighbors. Despite recent advances in network reconstruction, we continue to lack an approach for reconstructing complex networks with indirect interactions. Here we introduce a two-step strategy to resolve the reconstruction problem, where in the first step, we recover both direct and indirect interactions by employing the Lasso to solve a sparse signal reconstruction problem, and in the second step, we use matrix transformation and optimization to distinguish between direct and indirect interactions. The network structure corresponding to direct interactions can be fully uncovered. We exploit the public goods game occurring on complex networks as a paradigm for characterizing indirect interactions and test our reconstruction approach. We find that high reconstruction accuracy can be achieved for both homogeneous and heterogeneous networks, and a number of empirical networks in spite of insufficient data measurement contaminated by noise. Although a general framework for reconstructing complex networks with arbitrary types of indirect interactions is yet lacking, our approach opens new routes to separate direct and indirect interactions in a representative complex system.

Interrelations among scientific fields and their relative influences revealed by an input–output analysis

In this paper, we try to answer two questions about any given scientific discipline: First, how important is each subfield and second, how does a specific subfield influence other subfields? We modify the well-known open-system Leontief Input-Output Analysis in economics into a closed-system analysis focusing on eigenvalues and eigenvectors and the effects of removing one subfield. We apply this method to the subfields of physics. This analysis has yielded some promising results for identifying important subfields (for example the field of statistical physics has large influence while it is not among the largest subfields) and describing their influences on each other (for example the subfield of mechanical control of atoms is not among the largest subfields cited by quantum mechanics, but our analysis suggests that these fields are strongly connected). This method is potentially applicable to more general systems that have input-output relations among their elements.

Universal data-based method for reconstructing complex networks with binary-state dynamics

To understand, predict, and control complex networked systems, a prerequisite is to reconstruct the network structure from observable data. Despite recent progress in network reconstruction, binary-state dynamics that are ubiquitous in nature, technology, and society still present an outstanding challenge in this field. Here we offer a framework for reconstructing complex networks with binary-state dynamics by developing a universal data-based linearization approach that is applicable to systems with linear, nonlinear, discontinuous, or stochastic dynamics governed by monotonic functions. The linearization procedure enables us to convert the network reconstruction into a sparse signal reconstruction problem that can be resolved through convex optimization. We demonstrate generally high reconstruction accuracy for a number of complex networks associated with distinct binary-state dynamics from using binary data contaminated by noise and missing data. Our framework is completely data driven, efficient, and robust, and does not require any a priori knowledge about the detailed dynamical process on the network. The framework represents a general paradigm for reconstructing, understanding, and exploiting complex networked systems with binary-state dynamics.

Multi-source localization on complex networks with limited observers

Source localization is a significant task in the contagion process. In this paper, we study the problem of locating multiple sources in complex networks with limited observations. We propose a backward diffusion-based source localization method and apply it on several networks, finding that multiple sources can be located with high accuracy even when the fraction of observers is small and the time delay along the links are not known exactly. By comparing different observer placement strategies, we find that choosing small-degree nodes as observers is better than the other strategies.

Efficient Reconstruction of Heterogeneous Networks from Time Series via Compressed Sensing.

Recent years have witnessed a rapid development of network reconstruction approaches, especially for a series of methods based on compressed sensing. Although compressed-sensing based methods require much less data than conventional approaches, the compressed sensing for reconstructing heterogeneous networks has not been fully exploited because of hubs. Hub neighbors require much more data to be inferred than small-degree nodes, inducing a cask effect for the reconstruction of heterogeneous networks. Here, a conflict-based method is proposed to overcome the cast effect to considerably reduce data amounts for achieving accurate reconstruction. Moreover, an element elimination method is presented to use the partially available structural information to reduce data requirements. The integration of both methods can further improve the reconstruction performance than separately using each technique. These methods are validated by exploring two evolutionary games taking place in scale-free networks, where individual information is accessible and an attempt to decode the network structure from measurable data is made. The results demonstrate that for all of the cases, much data are saved compared to that in the absence of these two methods. Due to the prevalence of heterogeneous networks in nature and society and the high cost of data acquisition in large-scale networks, these approaches have wide applications in many fields and are valuable for understanding and controlling the collective dynamics of a variety of heterogeneous networked systems.

Locating multiple diffusion sources in time varying networks from sparse observations

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.