The word "network" is an over-loaded term, and "network analysis" means different things to different people. There are many specific forms of network analysis used in diverse studies, such as epidemiology, metabolic pathways, phylogenetics, the internet, transportation systems, electrical circuits, project plans, etc. Here, I am going to ignore all of these quantitative ideas.
In his book A Dictionary of the English Language (1755), Samuel Johnson defined a network as:
"Nétwork. n.s. [net and work.] Any thing reticulated or decussated, at equal distances, with interstices between the intersections."
Biologists, mathematicians and computer scientists have all found this definition to be less than helpful. Still, it was a start.
Oddly, in biology the word "network" has been used to refer to an unrooted tree. This usage arose in the early days of cladistics, from the idea that an unrooted tree represents a set of rooted trees (one potential root per edge in the tree). This usage is usually credited to James S. Farris (1970, Methods for computing Wagner trees. Systematic Zoology 19: 83-92):
"Trees are directed entities in which the root is presumed to represent a point chronologically prior to any descendent point ... If the root is not specified, we have an "undirected tree," or a network. A network with a certain set of nodes may correspond to a wide class of trees with the same nodes, each tree differing from the others in the class only in the position of its root."
Other people have also used the word network with various meanings. The Online Etymology Dictionary (2010) has this to say about the history of the various uses of the word:
"net-like arrangement of threads, wires, etc.," 1560, from net (n.) + work (n.). Extended sense of "any complex, interlocking system" is from 1839 (originally in reference to transport by rivers, canals, and railways). Meaning "broadcasting system of multiple transmitters" is from 1914; sense of "interconnected group of people" is from 1947. The verb, in reference to computers, is from 1972; in reference to persons, it is attested from 1980s.
The idea of trees as line graphs is usually credited to Arthur Cayley (1857, On the theory of the analytical forms called trees. Philosophical Magazine 13: 172-176). Cayley defined his terms with reference to a set of illustrations:
"The inspection of these figures will at once show what is meant by ... the terms root, branches (which may be either main branches, intermediate branches, or free branches), and knots (which may be either the root itself, or proper knots, or the extremities of the free branches)."
It is not clear to me at what point "knots" became "nodes" in mathematical usage, or "branches" became "edges", but biologically "nodes" is more accurate than "knots" although "branches" is more accurate than "edges". Cayley continued to use the term "knots" in his subsequent four papers on trees (1859, 1875, 1881, 1889).