  The connectivity of a network contains information about the relationships
between nodes, which can denote interactions, associations, or dependencies. We
show that this information can be analyzed by measuring the uncertainty (and
certainty) contained in paths along nodes and links in a network. Specifically,
we derive from first principles a measure known as effective information and
describe its behavior in common network models. Networks with higher effective
information contain more information in the relationships between nodes. We
show how subgraphs of nodes can be grouped into macro-nodes, reducing the size
of a network while increasing its effective information (a phenomenon known as
causal emergence). We find that informative higher scales are common in
simulated and real networks across biological, social, informational, and
technological domains. These results show that the emergence of higher scales
in networks can be directly assessed and that these higher scales offer a way
to create certainty out of uncertainty.
