Complex network zeta function

Abstract: In this short note we point out that a recent article in this journal incorrectly attributes some properties to Ihara zeta functions. The properties actually are attributable to another class of complex network zeta functions, which are used in computer science and mathematical physics applications. In an interesting study [ AGarrido2009] have presented many aspects of Ihara zeta functions of graphs. While the bulk of the analysis is valid, we note that the properties attributed to the Ihara zeta functions on pages 261 and 262, and the discussion of monotonicity, stability and Lipschitz Invariance, are not correct. These properties seem to be taken from a class of functions dierent from Ihara zeta functions, and the other class of functions are reported in [ OShanker2007]. The class of functions for which these properties actually hold are now called complex network zeta functions [ OShanker2008], to avoid confusion with Ihara graph zeta functions. They have been studied in the mathematical physics and computer science [OShanker2009, OShanker2010] literature. To help clear up the confusion, we briefly present here the definition of the complex network zeta function, with particular reference to the properites presented in [ AGarrido2009] which actually hold for the complex network zeta functions and not for Ihara graph zeta functions. Let us denote by rij the distance from node i to node j of a complex network (the length of the shortest path connecting the node i to node j). rij is 1 if there is no path from node i to node j. This definition of distance satisfies the triangle inequality, and hence the nodes of the complex network form a metric space.