Quantifying Complexity in Networks: The von Neumann Entropy

The authors introduce a novel entropic notion with the purpose of quantifying disorder/uncertainty in networks. This is based on the Laplacian and it is exactly the von Neumann entropy of certain quantum mechanical states. It is remarkable that the von Neumann entropy depends on spectral properties and it can be computed efficiently. The analytical results described here and the numerical computations lead us to conclude that the von Neumann entropy increases under edge addition, increases with the regularity properties of the network and with the number of its connected components. The notion opens the perspective of a wide interface between quantum information theory and the study of complex networks at the statistical level. DOI: 10.4018/jats.2009071005 IGI PUBLISHING This paper appears in the publication, International Journal of Agent Technologies and Systems, Volume 1, Issue 4 edited by Goran Trajkovski © 2009, IGI Global 701 E. Chocolate v ue, Hershey PA 17033-1240, USA Tel: 717/533-8845; Fax 717/533-8661; URL-http://www.igi-global.com ITJ 5307 International Journal of Agent Technologies and Systems, 1(4), 58-67, October-December 2009 59 Copyright © 2009, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. ful mapping between discrete Laplacians and quantum states, firstly introduced by Braunstein, Ghosh, and Severini (2006); see also Hildebrand, Mancini, and Severini (2008). We interpret the set of eigenvalues of an appropriately normalized discrete Laplacian as a distribution and we compute its Shannon entropy. Let us recall that the Shannon entropy measures the amount of uncertainty of a random variable, or the amount of information obtained when its value is revealed. The topic is extensively covered by, e.g., Cover and Tomas (1991). It is not simple to give a combinatorial interpretation to the von Neumann entropy. Superficially, we give evidence that this can be seen as a measure of regularity, i.e., regular graphs have in general higher entropy when the number of edges has been fixed. This is not the end of the story. Quantum entropy seems to depend on the number of connected components, long paths, and nontrivial symmetries (in terms of the automorphism group of the graph). Fixed the number of edges, entropy is smaller for graphs with large cliques and short paths, i.e., graphs in which the vertices form an highly connected cluster. The remainder of the article is organized as follows. In the next section we introduce the required definitions and focus on first properties. By adding edges one by one to the empty graph (that is, the graph with zero edges), we attept to construct graphs with minimum and maximum entropy, respectively. We then explore the influence of the graph structure on the entropy. We consider different classes of graphs: regular graphs, random graphs, and the star as an extremal case of scale-free graph (i.e., graphs for which the degree distribution follows a power law). We have chosen these classes because these are well-studied and considered in many different contexts. The asymptotic behavior for large number of vertices shows that regular graphs tend to have maximum entropy. We study numerically how the entropy increases when adding edges with different prescriptions. Once fixed the number of edges, the entropy is minimized by graphs with large cliques. In the concluding section, we will indicate a number of directions for future research. ThE voN NEumaNN ENTropy The state of a quantum mechanical system with a Hilbert space of finite dimension n is described by a density matrix. Each density matrix ρ is a positive semidefinite matrix with Tr(ρ) = 1. Here we consider a matrix representation based on the combinatorial Laplacian to associate graphs to specific density matrices. Let G = (V, E) be a simple undirec ted graph wi th se t o f ver t ices V(G) = {1, 2, ...,n} and set of edges E G V G V G v v v V G ( ) ( ) ( ) {{ , } : ( )} Í ́ Î . The adjacency matrix of G is denoted by A(G) and defined by [A(G)]u,v = 1 if {u, v} ∈ E(G) and [A(G)]u,v = 0, otherwise. The degree of a vertex v ∈ V(G), denoted by d(v), is the number of edges adjacent to v. A graph G is d-regular if d(v) = d for all v ∈ V(G). Let dG be the degree-sum of the graph, i.e. dG = ∑v∈V(G)d(v). The average degree of G is defined by