On the Von Neumann Entropy of Graphs

The von Neumann entropy of a graph is a spectral complexity measure that has recently found applications in complex networks analysis and pattern recognition. Two variants of the von Neumann entropy exist based on the graph Laplacian and normalized graph Laplacian, respectively. Due to its computational complexity, previous works have proposed to approximate the von Neumann entropy, effectively reducing it to the computation of simple node degree statistics. Unfortunately, a number of issues surrounding the von Neumann entropy remain unsolved to date, including the interpretation of this spectral measure in terms of structural patterns, understanding the relation between its two variants, and evaluating the quality of the corresponding approximations. In this paper we aim to answer these questions by first analysing and comparing the quadratic approximations of the two variants and then performing an extensive set of experiments on both synthetic and real-world graphs. We find that 1) the two entropies lead to the emergence of similar structures, but with some significant differences; 2) the correlation between them ranges from weakly positive to strongly negative, depending on the topology of the underlying graph; 3) the quadratic approximations fail to capture the presence of non-trivial structural patterns that seem to influence the value of the exact entropies; 4) the quality of the approximations, as well as which variant of the von Neumann entropy is better approximated, depends on the topology of the underlying graph

Demystification of Graph and Information Entropy

Shannon entropy is an information-theoretic measure of unpredictability in probabilistic models. Recently, it has been used to form a tool, called the von Neumann entropy, to study quantum mechanics and network flows by appealing to algebraic properties of graph matrices. But still, little is known about what the von Neumann entropy says about the combinatorial structure of the graphs themselves. This paper gives a new formulation of the von Neumann entropy that describes it as a rate at which random movement settles down in a graph. At the same time, this new perspective gives rise to a generalization of von Neumann entropy to directed graphs, thus opening a new branch of research. Finally, it is conjectured that a directed cycle maximizes von Neumann entropy for directed graphs on a fixed number of vertices

Scaling of the von Neumann entropy across a finite temperature phase transition

The spectrum of the reduced density matrix and the temperature dependence of the von Neumann entropy (VNE) are analytically obtained for a system of hard core bosons on a complete graph which exhibits a phase transition to a Bose-Einstein condensate at $T=T_c$. It is demonstrated that the VNE undergoes a crossover from purely logarithmic at T=0 to purely linear in block size $n$ behaviour for $T\geq T_{c}$. For intermediate temperatures, VNE is a sum of two contributions which are identified as the classical (Gibbs) and the quantum (due to entanglement) parts of the von Neumann entropy.Comment: 4 pages, 2 figure

Tournament Entropy

Across disciplines, the term \u27entropy\u27 is used to quantify the complexity of a system. In particular, Rényi and Von Neumann entropies measure the structural complexity of discrete binary structures (undirected graphs). Here, the spectrum of the graph\u27s normalized Laplacian matrix is treated as a discrete probability distribution by which the entropy functionals operate. This works well for undirected graphs because the normalized Laplacian matrices are positive semidefinite, so each eigenvalue is a real number between 0 and 1, and because of the normalization, the eigenvalues add to 1. In the case of directed graphs, little research has been done because eigenvalues are often complex. We find that by extending the entropy definitions to complex numbers, Rényi and Von Neumann entropies still give insight into structure of the graph. We focus our attention on a certain class of directed graphs called tournaments, which consist of pairwise comparisons on a set. We find that Rényi and Von Neumann entropies can be analyzed combinatorially and that tournaments with high transitivity tend to have lower entropy, while tournaments with more chaotic structures tend to have higher entropy

Edge centrality via the Holevo quantity

In the study of complex networks, vertex centrality measures are used to identify the most important vertices within a graph. A related problem is that of measuring the centrality of an edge. In this paper, we propose a novel edge centrality index rooted in quantum information. More specifically, we measure the importance of an edge in terms of the contribution that it gives to the Von Neumann entropy of the graph. We show that this can be computed in terms of the Holevo quantity, a well known quantum information theoretical measure. While computing the Von Neumann entropy and hence the Holevo quantity requires computing the spectrum of the graph Laplacian, we show how to obtain a simplified measure through a quadratic approximation of the Shannon entropy. This in turns shows that the proposed centrality measure is strongly correlated with the negative degree centrality on the line graph. We evaluate our centrality measure through an extensive set of experiments on real-world as well as synthetic networks, and we compare it against commonly used alternative measures

Towards an approximate graph entropy measure for identifying incidents in network event data

A key objective of monitoring networks is to identify potential service threatening outages from events within the network before service is interrupted. Identifying causal events, Root Cause Analysis (RCA), is an active area of research, but current approaches are vulnerable to scaling issues with high event rates. Elimination of noisy events that are not causal is key to ensuring the scalability of RCA. In this paper, we introduce vertex-level measures inspired by Graph Entropy and propose their suitability as a categorization metric to identify nodes that are a priori of more interest as a source of events. We consider a class of measures based on Structural, Chromatic and Von Neumann Entropy. These measures require NP-Hard calculations over the whole graph, an approach which obviously does not scale for large dynamic graphs that characterise modern networks. In this work we identify and justify a local measure of vertex graph entropy, which behaves in a similar fashion to global measures of entropy when summed across the whole graph. We show that such measures are correlated with nodes that generate incidents across a network from a real data set

Quantum entanglement in states generated by bilocal group algebras

Given a finite group G with a bilocal representation, we investigate the bipartite entanglement in the state constructed from the group algebra of G acting on a separable reference state. We find an upper bound for the von Neumann entropy for a bipartition (A,B) of a quantum system and conditions to saturate it. We show that these states can be interpreted as ground states of generic Hamiltonians or as the physical states in a quantum gauge theory and that under specific conditions their geometric entropy satisfies the entropic area law. If G is a group of spin flips acting on a set of qubits, these states are locally equivalent to 2-colorable (i.e., bipartite) graph states and they include GHZ, cluster states etc. Examples include an application to qudits and a calculation of the n-tangle for 2-colorable graph states.Comment: 9 pages, no figs; updated to the published versio