Most real-world systems are networked, composed of nodes connected to each other via edges, whose overall behaviour can depend on the underlying connectivity structure in non-trivial ways. They are also of a large scale, consisting of numerous nodes and edges not all of which can be viably observed, making the study of key structural properties computationally intractable. This thesis responds to the challenge by leveraging sparsity of edges to determine (1) structural statistics analytically, and (2) the relevance of structure in describing behaviour, based on (3) an inferred statistical model of the structure of sparse and partially observed large-scale networks. First, we determine higher-order structural statistics of networks: in particular, the length of a shortest path connecting any two nodes in the network. Assuming a statistical model for the network structure, we establish an analytic distribution of shortest path lengths, whose approximate closed-form has a natural interpretation of traversing independent walks. This formalism yields new results for both network- and node-level properties like percolation phenomena and distance-based centralities of closeness and betweenness, for a large family of statistical network models. Then, we contextualise behaviour in a network to health behaviours in a social network: in particular, we consider how misinformation influences vaccination outcomes. Through a randomised experiment we find that exposure to misinformation about COVID-19 vaccines negatively impacts individuals' intentions to vaccinate against the disease. To address one possible psycho-social mechanism, we use a psychometric test to show that the ability to detect misinformation positively predicts regional vaccine uptake. Assuming a statistical model for the network structure, we observe that the ability is mediated by the social network. Finally, we infer a statistical model for social networks from a source of partial observations. We use publicly available data from spatially aggregated friendship counts to infer a Bayesian socio-physical connectivity model. The approach is grounded in a microscopic process of friendship formation describing how people in a society connect to each other jointly in social and physical space, that explains the empirically observed decay of connection probabilities in physical space. It provides a multidimensional social inequality measure quantifying the surprisal of social connections that we term the "social Gini" index.