  Graph-based methods pervade the inference toolkits of numerous disciplines
including sociology, biology, neuroscience, physics, chemistry, and
engineering. A challenging problem encountered in this context pertains to
determining the attributes of a set of vertices given those of another subset
at possibly different time instants. Leveraging spatiotemporal dynamics can
drastically reduce the number of observed vertices, and hence the cost of
sampling. Alleviating the limited flexibility of existing approaches, the
present paper broadens the existing kernel-based graph function reconstruction
framework to accommodate time-evolving functions over possibly time-evolving
topologies. This approach inherits the versatility and generality of
kernel-based methods, for which no knowledge on distributions or second-order
statistics is required. Systematic guidelines are provided to construct two
families of space-time kernels with complementary strengths. The first
facilitates judicious control of regularization on a space-time frequency
plane, whereas the second can afford time-varying topologies. Batch and online
estimators are also put forth, and a novel kernel Kalman filter is developed to
obtain these estimates at affordable computational cost. Numerical tests with
real data sets corroborate the merits of the proposed methods relative to
competing alternatives.
