Bipartite quantum states and random complex networks

We introduce a mapping between graphs and pure quantum bipartite states and show that the associated entanglement entropy conveys non-trivial information about the structure of the graph. Our primary goal is to investigate the family of random graphs known as complex networks. In the case of classical random graphs we derive an analytic expression for the averaged entanglement entropy $\bar S$ while for general complex networks we rely on numerics. For large number of nodes $n$ we find a scaling $\bar{S} \sim c \log n +g_e$ where both the prefactor $c$ and the sub-leading O(1) term $g_e$ are a characteristic of the different classes of complex networks. In particular, $g_e$ encodes topological features of the graphs and is named network topological entropy. Our results suggest that quantum entanglement may provide a powerful tool in the analysis of large complex networks with non-trivial topological properties.

Introduction. Complex networks are models of graphs that appear able to capture the phenomenology of a plethora of systems, from biology to the World Wide Web [1]. The departure from regular lattices, the most common background geometry in solid state physics, allows for the rich static and dynamic behavior of complex networks. This is due to the simultaneous presence of a global compact structure and a sophisticated architecture of interactions. By compact structures we refer to the typical small distance (with respect to a regular lattice) between nodes in the network. The complexity in the network architecture is manifest in the entangled pattern of links and paths that such objects display, see Fig.1. This architecture encodes a type of strong disorder that requires for its analysis some of the techniques developed in Statistical Mechanics [2]. There exist a plethora of phenomenological quantities that provide information on the architecture of a network: degree distribution, clustering coefficient, community structure measures and many others [1]. In particular a few classical entropic measures have been introduced to describe the structure of complex networks [3].
In this letter we address the problem of the entropic analysis and discrimination of networks using quantum information tools, notably entanglement entropy. Entanglement, a purely quantum measure of correlation, is one of the fundamental concepts in quantum information [4]. We provide a recipe, in a way the simplest possible one, to construct a pure bipartite quantum state for any given graph. This allows us to study entanglement properties of quantum states that are related to the topological features of the original graphs, and that are able to distinguish between different complex network topologies. Although at the first sight it may seem a bit artificial to look for a graph-entropy measure in a quantum context the synergy between quantum information and complex network tools is not new. For example, in [5] and [6] the authors have discussed different interesting ways to asso- ciate graphs to quantum states and investigated in which sense complex networks may play a role in the quantum domain. All these constructions are then similar in spirit but substantially different from the present approach.
The paper is organized as follows. We first describe the construction of the quantum bipartite network states. Then we introduce the families of complex networks that we consider in this work. Subsequently we define the notion of topological network entropy and apply it to study the structure of different complex network topologies. Finally we briefly discuss relation with former works and state our conclusions.
Network quantum states. Any graph G A with n nodes is completely specified by its adjacency matrix A: a 2dimensional array of size n, where each entry a i,j characterizes the connection between nodes i and j. The domain of a i,j determines the kind of graph one is considering: directed, undirected, weighted or unweighted.
In this work we focus on undirected unweighted graphs, so called simple graphs, for which a i,j ∈ {0, 1} (the same analysis can be naturally extended to directed and weighted graphs). In particular if a i,j = 0 it means there is no edge connecting the two nodes i and j, otherwise a i,j = 1. Sometimes it will be useful to refer also to non simple graphs with loops. Given the graph G A (A = 0) we define the following bipartite quantum state where A F := √ TrA † A denotes the Frobenius norm of the matrix A and H 1 ∼ = H 2 ∼ = C n . In the fixed local basis {|i : i = 1, . . . , n} we refer to |A as a pure network state. It corresponds to the state of two n-level systems, or analogously to the state of two n-qubit systems where each subsystem of n qubits is constrained to the oneexcitation manifold. The isomorphism class of a graph corresponds to the orbit of the permutation group on the adjacency matrix: P AP t , where P ∈ S n are permutation matrices. This implies that the adjacency matrix of isomorphic graphs is unique up to permutations of rows and columns, and the same holds true for the bipartite states in Eq.1. The reduced density matrix of the subsystem whose Hilbert space is H 1 is given (in the given basis) by We refer to this reduced density matrix as the mixed network state. Notice that both definitions (1) and (2) do not rely on A being symmetric and therefore extends immediately to oriented (and weighted) graphs. We are interested in the correlation properties between H 1 and H 2 , quantified by the entanglement entropy [7]. Since this quantity depends only on the spectrum of the adjacency matrix, it is a property of the isomorphism class of the graph, i.e. isomorphic graphs will have the same entanglement entropy. Indeed, if P is an (orthogonal) permutation matrix: |P AP t =P ⊗ 2 |A , wherê P = n i,j=1 P i,j |i j|. Namely isomorphic graphs give rise to locally equivalent network states.
Before considering complex topologies, it is instructive to play with the simplest possible examples and try to characterize maximally and minimally entangled network states. Notice that the unnormalized bipartite state can be written as A ⊗ I|I , where |I = n i |i, i is an unnormalized maximally entangled state. The network corresponding to the state |I consists of n nodes with loops, and by construction its adjacency matrix is the identity. Entanglement does not change under local unitary transformations [7], so in order to construct other maximally entangled network states we need to characterize all the adjacency matrices that correspond to unitary operators. It is easy to prove that the set of unitary adjacency matrices coincides with the set of permutation involutions, i.e. the permutation matrices that square to the identity. This is also consistent with the fact that the square of the adjacency matrix is the unnormalized totally mixed network state. Unitary adjacency matrices correspond to networks made of only loops or disconnected linked pairs of nodes. On the other hand factorized states (i.e. unentangled) correspond to complete graphs with loops.
In the following we study properties of ensembles of random network states. The probability measure in the space of network states is the one induced by the measure on the space of random networks, according to the construction in Eq.1 and Eq.2.
Complex networks. In order to make the paper selfcontained, let us briefly introduce three network structures that we will use in the following. The seminal paper of Erdős and Rényi in 1959 defined what is now the standard example of a random network [8]. The ER random graph model, denoted by G ER n,m , is an ensemble of graphs where each element has n nodes and m edges. ER graphs are also related to so-called Gilbert models [9], denoted by G ER n,p , where an edge between each pair of n nodes is present with a probability p. The Gilbert model is better suited for analytical investigations, while G ER n,m graphs are numerically easier to study. In the thermodynamic limit, fixing the average degree q of a node, one can constrain the two models to be related by q = 2m/n = pn. If, for n → ∞, q/n → 0 the network is said to be sparse. The other example of complex network that we consider is known as the Barabási-Albert model [10], denoted by G BA , based on a growth process and a preferential attachment mechanism. The rationale is that nodes with higher degree acquire new nodes at higher rates than other lower-degree nodes. Nodes are added successively, and for each node a number d of edges are generated, with bias towards connections with higher-degree nodes. The distribution for the number of links emanating from a node is not Poissonian, like for ER graphs, but rather follows a power-law. Another way to model stochasticity in the connectivity pattern of a graph is by randomly destroying the periodicity of a regular lattice. This is the idea behind small-world networks, denoted by G SW p,k , as proposed by Watts and Strogatz [11]. They can be created by randomly adding bonds to a regular one dimensional ring, this way building a superposition between regular lattices and classical random graphs. The probability p according to which new bonds are added at random is a parameter characterizing the ensemble, and it allows to interpolate from regular graphs (p = 0) to ER random graphs (p = 1). The other parameter for this kind of networks is denoted with k, and it quantifies the number of next-nearest-neighbor links present in the original regular graph. For each of the above complex network ensembles we shall construct the associated ensemble of random network states denoted by ψ ER , ψ BA and ψ SW , and we will consider scaling properties of the average entanglement entropies.
Topological network entanglement. We start by evaluating analytically the averaged Renyi entropy of network states in ψ ER , the states associated to the ensemble of ER random graphs. The α-Renyi entropy of a state is defined by R α (ρ) ≡ (1 − α) −1 log 2 Trρ α . Using the definition of ρ A given in Eq.2 we have We are interested in the scaling in n of the average Renyi entropy R α . In order to provide an explicit expression for R α we use the fact that for each α there exists a constant c 2α such that lim n→∞ n −1 TrA 2α = c 2α , i.e. for sparse ER graphs the thermodynamic limit of the moments of the graph spectrum exist and are finite [12]. Furthermore one can check numerically that the difference between the quenched average (1 − α) −1 log 2 ρ α and the annealed average (1 − α) −1 log 2 ρ α scales like n −1 . Putting this together we can write where g(α) := log 2 c 2 + (1 − α) −1 (log 2 c 2α − log 2 c 2 ) is a sub-leading O(1) term. This equation tells us that the Renyi entropy is almost maximal for any α. Notice that, even though the logarithmic scaling for these network states is consistent with the one of general (Haar distributed) random states [14], one could not predict a priori this behavior for the particular family of random states we introduced. Remarckably the sub-leading term contains information about the topology of the graphs. In fact, the term c 2α is directly related to the average number of closed paths of length 2α in the graph. In Fig.2a we provide a numerical check of Eq.4, which supports in particular the approximation of the quenched with the annealed average. The figure shows a perfect agreement between Eq.4 and the results of the simulation. It is tempting to extrapolate our analysis from the Renyi entropy to the von Neumann entanglement entropy, which is defined by S ≡ lim α→1 R α . By construction it follows immediately the logarithmic scaling of the von Neumann entropy, while for the sub-leading term we have g e := lim α→1 g(α) = log 2 c 2 − d log c2x dx | x=1 and then, using the definition of c 2α , one finds It is interesting to observe that the second term in Eq.5 can be regarded as a sort of topological susceptibility of the given family of networks. In fact this term is equal to − log c2x dx | x=1 , and it tells us how the logarithm of the rescaled averaged number of loops of length 2α changes as the length is changed continuously around α = 1. For this reason, and in view of its conceptual similarity with the topological entanglement entropy introduced in [13] we call the O(1) quantity in Eq.5 topological network entanglement. In Fig.2b it is shown a comparison between the analytical expression for the von Neumann entropy, obtained using Eq.5, and the empirical average of the entanglement entropy over different realizations. As can be seen from the figure there is a perfect overlap between the two. The ensemble G ER n,m is parameterized by the number of edges m. One can wonder about the behavior of g e as m varies in some particular interval, but in such a way that the graph is always sparse. Fig. 2c shows the value of g e as a function of a parameter β ∈ (0, 1] which is related to the number of edges by m = βn log 2 n. The figure shows that for ER-graphs the greater the number of links the greater the entanglement. We now evaluate numerically g e for small worlds and preferential attachment networks. Considering first networks in G SW p,k we checked numerically that the scaling is logarithmic and that the prefactor is always 1. In Fig.2d we see the dependence of g e on p, the probability of rewiring edges. The figure shows that g e increases monotonically from regular to more random graphs. This is consistent with the intuition that adding randomness to a graph increases its entropy, as measured by g e . From these results it is clear that the properties of the entanglement entropy provide information on the complex network structure, supporting its interpretation as a graph entropic quantity. Considering the ability to discriminate between different network topologies, in Fig.3 we compare the scaling of the average von Neumann entropy of ψ ER n,n , ψ SW 0.1,2 , and ψ BA random network states. The simulations show that, unlike for ER and small-world networks, the loga-rithmic prefactor for ψ BA states is slightly smaller than 1. From the figure it is clear that the von Neumann entropy distinguishes different complex network ensembles. For sufficiently big networks the fluctuations due to disorder are strongly suppressed. On the one hand this is an indication of the robustnes of this graph-entropic measure; on the other hand it can also be useful from a computational point of view. In fact one has a very good estimate of the entanglement entropy already from few realizations. Hence, if the network is big enough the scaling analysis could in principle be done on one single realization, for example evaluating the entanglement entropy on sub-graphs of increasing size. Furthermore whenever computational efficiency is an issue we point out that instead of the Von Neumann entropy one could evaluate the so called single-copy entanglement (lim α→∞ R α ) [15], for which efficient numerical techniques can be used [16].
Discussion and conclusions. In this letter we have exploited a natural mapping from graphs to quantum bipartite states and we have defined the entropy of a graph as the entanglement entropy of the associated quantum state. We have then used this quantum measure of correlations to study the structure of complex networks. The scaling of the entanglement entropy is logarithmic in the system dimension, and both the prefactor and the subleading O(1) term (topological network entropy) can be used to characterize the network family and to distinguish between different network topologies. In particular, we showed that the Barabási-Albert model has a scaling behavior that differs significantly from the one of smallworld and ER graphs. While these last two have a similar scaling, but still distinguishable comparing graphs with the same number of edges. This is consistent with the fact that small-world networks are mixture of regular lattices and ER graphs. Furthermore, we provided an analytic expression, exact in the thermodynamic limit, for the averaged Renyi and von Neumann entropy associated to ER random graphs. It is desirable to achieve a clear and general understanding of the relations between the quantum entropic measures we introduced and the standard graph-theoretic observables analyzed in the complex-network community. One would like also to gain a deeper insight into the measure concentration (large size convergence) properties of the various probabilistic objects we discussed for the different families of complex networks. While the primary goal of this paper has been to show how to use quantum tools to investigate complex networks it should be clear that also the converse task i.e., using properties of complex newtwork to study the novel class of random quantum states we introduced, is of interest on its own right. Moreover on the quantum side it is a challenge to find a consistent inverse mapping that allows one to associate to a general bipartite quantum state a specific network. Finally, one would like to devise efficient and physically feasible preparation schemes for the network quantum states we proposed. We thank N. Toby Jacobson for a careful reading of the manuscript. P.Z. acknowledges support from NSF grants PHY-803304, DMR-0804914