## 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 *G*_{pq} 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 ...