Matrix functions in network analysis

Abstract

We review the recent use of functions of matrices in the analysis of graphs and networks, with special focus on centrality and communicability measures and diffusion processes. Both undirected and directed networks are considered, as well as dynamic (temporal) networks. Computational issues are also addressed.

A proposal for ranking through selective computation of centrality measures

“…Over the years, a wide variety of measures have been proposed to quantify the centrality of a node, with each focusing on different concepts; see, e.g., [ 3 ] for more details and [ 15 ] for recent examples. Some centrality measures, which will be recalled in the next section, are defined using functions of the adjacency matrix such as the exponential and the resolvent function; see, e.g., [ 6 ].…”

“…In terms of the adjacency matrix A , d i can be written as: where 1 is the vector of all ones in . Here we concentrate mostly on total communicability introduced in [ 22 ] and on subgraph centrality introduced in [ 23 ], but the discussion can be extended to other walk based indices that can be computed as functions of the adjacency matrix such as those in, e.g., [ 6 ]. These measures are interesting as they can detect a node global influence by looking at the walks (with a suitable weight) in which the node is involved, balancing long and short range connections by giving greater emphasis on short walks rather than long ones.…”

“…Consequently, the demand of a reliable metric that quantify the importance of a node has given rise to a highly popular area of research, resulting in the introduction of various definitions of node centrality [ 3 ] and other analogous tools; see [ 4 , 5 ]. Among the possible centrality measures , we will concentrate on those based on matrix functions of the adjacency matrix because of their ability to take into account certain communicability characteristics of the network; see, e.g., [ 6 ].…”