This study aims at developing a non-(semi-)parametric method to extract the hidden network structure from the -valued distribution flow data with missing observations on the links between nodes. Such an input data type widely exists in the studies of information propagation process, such as the rumor spreading through social media. In that case, a social network does exist as the media of the spreading process, but its link structure is completely unobservable; therefore, it is important to make inference of the structure (links) of the hidden network. Unlike the previous studies on this topic which only consider abstract networks, we believe that apart from the link structure, different social-economic features and different geographic locations of nodes can also play critical roles in shaping the spreading process, which has to be taken into account. To uncover the hidden link structure and its dependence on the external social-economic features of the node set, a multidimensional spatial social network model is constructed in this study with the spatial dimension large enough to account for all influential social-economic factors. Based on the spatial network, we propose a nonparametric mean-field equation to govern the rumor spreading process and apply the likelihood estimator to make inference of the unknown link structure from the observed rumor distribution flows. Our method turns out easily extendible to cover the class of block networks that are useful in most real applications. The method is tested through simulated data and demonstrated on a data set of rumor spreading on Twitter.