4.5 Communicability in Complex Networks

The communicability between a pair of nodes in a network is usually considered as taking place through the shortest path connecting both nodes. However, it is known that communication between a pair of nodes in a network does not always take place through the shortest paths but it can follow other nonoptimal walks [51–53]. Then, we can consider a communicability measure that accounts for a weighted sum of all walks connecting two nodes in the network. We can design our measure in such a way that the shortest path connecting these two nodes always receives the largest weight. Then, if P(s)pq is the number of shortest paths between nodes p and q having length s and W(k)pq is the number of walks connecting p and q of length k > s, we propose to consider the quantity [54]


In fact, Equation (4.31) can be written as the sum of the p, q entry of the different powers of the adjacency matrix:


which converges to [54]


We call Gpq the communicability between nodes p and q in the network. The communicability should be minimum between the end nodes of a chain, where it vanishes as the length of the chain is increased. On the other hand, the communicability ...

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