Human mobility drives geographical diffusion of airborne infectious diseases at different scales. During the COVID-19 pandemic mobility data was made available and has been widely used, but few studies focus on mobility itself. We used public data from February 14th 2020, to May 9th 2021, in Spain to characterize mobility patterns and study geographical diffusion phenomena using network science methods. With 135 (out of 2.264) connections and the 52 provinces of Spain, a weighted, directed network was built: the Epidemic Diffusion Network (EDN). Centrality measures (degree and strength), community structure and shortest distances were obtained using the EDN. The resulting network was highly clustered (modularity: 63%) with 7 communities. Madrid, Valladolid and Araba/Álaba act as mobility hubs of their communities and the whole network. Shortest distances unveil a geographical wave-like diffusion pattern with occasional distance jumps, a small-world network characteristic, and COVID-19 cumulative incidence curves showed a pattern of proximity within provinces of the same community. These properties remain constant in time despite factors like seasonality or restrictions and could inform public health authorities in preparedness and response plans for diseases and other threads. Further studies are needed to better understand relationship between network measures and epidemiological outcomes in real life.