  Quantum networks are of great interest of late which apply quantum mechanics
to transfer information securely. One of the key properties which are exploited
is entanglement to transfer information from one network node to another.
Applications like quantum teleportation rely on the entanglement between the
concerned nodes. Thus, efficient entanglement distribution among network nodes
is of utmost importance. Several entanglement distribution methods have been
proposed in the literature which primarily rely on attributes, such as,
fidelities, link layer network topologies, proactive distribution, etc. This
paper studies the centralities of the network when the link layer topology of
entanglements (referred to as entangled graph) is driven by usage patterns of
peer-to-peer connections between remote nodes (referred to as connection graph)
with different characteristics. Three different distributions (uniform,
gaussian, and power law) are considered for the connection graph where the two
nodes are selected from the same distribution. For the entangled graph, both
reactive and proactive entanglements are employed to form a random graph.
Results show that the edge centralities (measured as usage frequencies of
individual edges during entanglement distribution) of the entangled graph
follow power law distributions whereas the growth in entanglements with
connections and node centralities (degrees of nodes) are monomolecularly
distributed for most of the scenarios. These findings will help in quantum
resource management, e.g., quantum technology with high reliability and lower
decoherence time may be allocated to edges with high centralities.
