  A bridge in a graph is an edge whose removal disconnects the graph and
increases the number of connected components. We calculate the fraction of
bridges in a wide range of real-world networks and their randomized
counterparts. We find that real networks typically have more bridges than their
completely randomized counterparts, but very similar fraction of bridges as
their degree-preserving randomizations. We define a new edge centrality
measure, called bridgeness, to quantify the importance of a bridge in damaging
a network. We find that certain real networks have very large average and
variance of bridgeness compared to their degree-preserving randomizations and
other real networks. Finally, we offer an analytical framework to calculate the
bridge fraction , the average and variance of bridgeness for uncorrelated
random networks with arbitrary degree distributions.
