Concept: Library networks were established in India to promote resource sharing, catalogue sharing, inter-library cooperation and access to bibliographic information.
Step 1: Identify the earlier network.
CALIBNET is one of the earlier library networks among the given options.
\[
B = \text{CALIBNET}
\]
Step 2: Place INFLIBNET next.
INFLIBNET was established for academic library networking and information sharing.
\[
E = \text{INFLIBNET}
\]
Step 3: Place PUNENET after INFLIBNET.
PUNENET was developed as a Pune-based library network.
\[
D = \text{PUNENET}
\]
Step 4: Place ADINET and BONET later.
ADINET and BONET came later in the sequence.
\[
A = \text{ADINET}, \quad C = \text{BONET}
\]
Therefore, the ascending order is:
\[
B,\ E,\ D,\ A,\ C
\]