Graph Complexity Measures Research Articles

Functionality of Box Intersection Graphs

Functionality is a graph complexity measure that extends a variety of parameters, such as vertex degree, degeneracy, clique-width, or twin-width. In the present paper, we show that functionality is bounded for box intersection graphs in R1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^1$$\\end{document}, i.e. for interval graphs, and unbounded for box intersection graphs in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}. We also study a parameter known as symmetric difference, which is intermediate between twin-width and functionality, and show that this parameter is unbounded both for interval graphs and for unit box intersection graphs in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^2$$\\end{document}.

Structure of Poincaré plots revealed by their graph analysis and low pass filtering of the RRI time series

ObjectivesIn order to reveal their structure, Poincaré plots (PP) of electrocardiogram (ECG) RR intervals (RRI) were studied as linear edge planar directed graphs, obtained by connecting all their sequential points. We were also aimed at studying their graph complexity properties. MethodsRRI signals were subjected to a series of different window length (WL) Moving Average Low Pass (MALP) filters. For each filtered graph, four standard PP descriptors: Pearson’s coefficient, SD1, SD2, and SD2/SD1 were calculated, as well as four new graph complexity measures: mean angle between adjacent graph edges; mean number of edge crossings; directional complexity and directional entropy. This approach was applied to signals of twenty young healthy subjects, recorded in four experimental conditions – combination of two body postures (supine and standing) and two breathing regimes (spontaneous and slow 0.1 Hz). ResultsWe found that PP graphs consist of two superimposed components: one originating from Respiratory Sinus Arrhythmia (RSA) oscillations, the other from slow variations (SV) of the RRI time series. This result was further corroborated by observing the transformation of a PP cloud shape occurring in filtered graphs. When applied to subjects, the outcome was that three measures significantly differentiated the two breathing regimes in the RSA region of the WL domain, while four other measures were able to differentiate two body postures in the SV WL region. DiscussionAfter obtaining these results in healthy, we expect to successfully apply this approach to patients suffering from different pathological conditions.

Network analysis of quantitative encephalography

This is a technical video on hands-on format introducing the concepts of neural connectivity and complexity measures with applications to neurostimulation assessment, based on the brain network analysis. The scientific relevance of this submission is the introduction of the small-world parameter, a graph complexity measure considered as a biomarker of brain disordes and used to validate neurostimulation therapies. The small-world is a technical network feature but addressed in this presentation as a brain network efficiency measure, in a concise and assertive way, not common in the literature. The narrative is didactic and accessible to researchers from several fields, suitable for professional and educational applications, allowing the generalization of the addressed techniques.

On the Similarity Between von Neumann Graph Entropy and Structural Information: Interpretation, Computation, and Applications

The von Neumann graph entropy is a measure of graph complexity based on the Laplacian spectrum. It has recently found applications in various learning tasks driven by the networked data. However, it is computationally demanding and hard to interpret using simple structural patterns. Due to the close relation between the Laplacian spectrum and the degree sequence, we conjecture that the structural information, defined as the Shannon entropy of the normalized degree sequence, might be a good approximation of the von Neumann graph entropy that is both scalable and interpretable. In this work, we thereby study the difference between the structural information and the von Neumann graph entropy named as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">entropy gap</i> . Based on the knowledge that the degree sequence is majorized by the Laplacian spectrum, we for the first time prove that the entropy gap is between 0 and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\log _{2} e$ </tex-math></inline-formula> in any undirected unweighted graphs. Consequently we certify that the structural information is a good approximation of the von Neumann graph entropy that achieves provable accuracy, scalability, and interpretability simultaneously. This approximation is further applied to two entropy-related tasks: network design and graph similarity measure, where a novel graph similarity measure and the corresponding fast algorithms are proposed. Meanwhile, we show empirically and theoretically that maximizing the von Neumann graph entropy can effectively hide the community structure, and then propose an alternative metric called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">spectral polarization</i> to guide the community obfuscation. Our experimental results on graphs of various scales and types show that the very small entropy gap readily applies to a wide range of simple/weighted graphs. As an approximation of the von Neumann graph entropy, the structural information is the only one that achieves both high efficiency and high accuracy among the prominent methods. It is at least two orders of magnitude faster than SLaQ (Tsitsulin <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> , 2020) with comparable accuracy. Our structural information based methods also exhibit superior performance in downstream tasks such as entropy-driven network design, graph comparison, and community obfuscation.

A graph complexity measure based on the spectral analysis of the Laplace operator

In this work we introduce a concept of complexity for undirected graphs in terms of the spectral analysis of the Laplacian operator defined by the incidence matrix of the graph. Precisely, we compute the norm of the vector of eigenvalues of both the graph and its complement and take their product. Doing so, we obtain a quantity that satisfies two basic properties that are the expected for a measure of complexity. First, complexity of fully connected and fully disconnected graphs vanish. Second, complexity of complementary graphs coincide. This notion of complexity allows us to distinguish different kinds of graphs by placing them in a “croissant-shaped” region of the plane link density - complexity, highlighting some features like connectivity, concentration, uniformity or regularity and existence of clique-like clusters. Indeed, considering graphs with a fixed number of nodes, by plotting the link density versus the complexity we find that graphs generated by different methods take place at different regions of the plane. We consider some of the paradigmatic randomly generated graphs, in particular the Erdös-Rényi, the Watts-Strogatz and the Barabási-Albert models. Also, we place some particular, let us say deterministic, well known hand-crafted graphs, to wit, lattices, stars, hyper-concentrated and cliques-containing graphs. It is worthy noticing that these deterministic classical models of graphs depict the boundary of the croissant-shaped region. Finally, as an application to graphs generated by real measurements, we consider the brain connectivity graphs from two epileptic patients obtained from magnetoencephalography (MEG) recording, both in a baseline period and in ictal periods (epileptic seizures). In this case, our definition of complexity could be used as a tool for discerning between states, by the analysis of differences at distinct frequencies of the MEG recording.

Fast computation of von Neumann entropy for large-scale graphs via quadratic approximations

The von Neumann graph entropy (VNGE) can be used as a measure of graph complexity, which can be the measure of information divergence and distance between graphs. However, computing VNGE is extensively demanding for a large-scale graph. We propose novel quadratic approximations for fast computing VNGE. Various inequalities for error between the quadratic approximations and the exact VNGE are found. Our methods reduce the cubic complexity of VNGE to linear complexity. Computational simulations on random graph models and various real network datasets demonstrate superior performance.

Deep depth-based representations of graphs through deep learning networks

Graph-based representations are powerful tools in structural pattern recognition and machine learning. In this paper, we propose a framework of computing the deep depth-based representations for graph structures. Our work links the ideas of graph complexity measures and deep learning networks. Specifically, for a set of graphs, we commence by computing depth-based representations rooted at each vertex as vertex points. In order to identify an informative depth-based representation subset, we employ the well-known k-means method to identify M dominant centroids of the depth-based representation vectors as prototype representations. To overcome the burdensome computation of using depth-based representations for all graphs, we propose to use the prototype representations to train a deep autoencoder network, that is optimized using Stochastic Gradient Descent together with the Deep Belief Network for pretraining. By inputting the depth-based representations of vertices over all graphs to the trained deep network, we compute the deep representation for each vertex. The resulting deep depth-based representation of a graph is computed by averaging the deep representations of its complete set of vertices. We theoretically demonstrate that the deep depth-based representations of graphs not only reflect both the local and global characteristics of graphs through the depth-based representations, but also capture the main structural relationship and information content over all graphs under investigations. Experimental evaluations demonstrate the effectiveness of the proposed method.

Quantification of Graph Complexity Based on the Edge Weight Distribution Balance: Application to Brain Networks

The aim of this study was to introduce a novel global measure of graph complexity: Shannon graph complexity (SGC). This measure was specifically developed for weighted graphs, but it can also be applied to binary graphs. The proposed complexity measure was designed to capture the interplay between two properties of a system: the 'information' (calculated by means of Shannon entropy) and the 'order' of the system (estimated by means of a disequilibrium measure). SGC is based on the concept that complex graphs should maintain an equilibrium between the aforementioned two properties, which can be measured by means of the edge weight distribution. In this study, SGC was assessed using four synthetic graph datasets and a real dataset, formed by electroencephalographic (EEG) recordings from controls and schizophrenia patients. SGC was compared with graph density (GD), a classical measure used to evaluate graph complexity. Our results showed that SGC is invariant with respect to GD and independent of node degree distribution. Furthermore, its variation with graph size [Formula: see text] is close to zero for [Formula: see text]. Results from the real dataset showed an increment in the weight distribution balance during the cognitive processing for both controls and schizophrenia patients, although these changes are more relevant for controls. Our findings revealed that SGC does not need a comparison with null-hypothesis networks constructed by a surrogate process. In addition, SGC results on the real dataset suggest that schizophrenia is associated with a deficit in the brain dynamic reorganization related to secondary pathways of the brain network.

From tree-decompositions to clique-width terms

Tree-width and clique-width are two important graph complexity measures that serve as parameters in many fixed-parameter tractable algorithms. We give two algorithms that transform tree-decompositions represented by normal trees into clique-width terms (a rooted tree is normal for a graph if its nodes are the vertices of the graph and every two adjacent vertices are on a path of the tree starting at the root). As a consequence, we obtain that, for certain classes of sparse graphs, clique-width is polynomially bounded in terms of tree-width. It is even linearly bounded for planar graphs and incidence graphs. These results are useful in the construction of model-checking algorithms for problems described by monadic second-order formulae, including those allowing edge set quantifications.

Novel measure of the weigh distribution balance on the brain network: graph complexity applied to schizophrenia.

The aim of this study was to assess brain complexity dynamics in schizophrenia (SCH) patients during an auditory oddball task. For this task, we applied a novel graph measure based on the balance of the node weights distribution. Previous studies applied complexity parameters that were strongly dependent on network topology. This could bias the results, as well as making correction techniques, such as surrogating process, necessary. In the present study, we applied a novel graph complexity measure derived from the information theory: Shannon Graph Complexity (SGC). Complexity patterns from electroencephalographic recordings of 20 healthy controls and 20 SCH patients during an auditory oddball task were analyzed. Results showed a significantly more pronounced decrease of SGC for controls than for SCH patients during the cognitive task. These findings suggest an important change in the brain configuration towards a more balanced network, mainly in the connections related to long-range interactions. Since these changes are significantly more pronounced in controls, a deficit in the neural network reorganization can be associated with SCH. In addition, an accuracy of 72.5% was obtained using a receiver operating characteristic curve with a leave-one-out cross-validation procedure. The independence of network topology has been demonstrated by the novel complexity measure proposed in this study, therefore, it complements traditional graph measures as a means to characterize brain networks.

The Hosoya Entropy of a Graph

This paper demonstrates properties of Hosoya entropy, a quantitative measure of graph complexity based on a decomposition of the vertices linked to partial Hosoya polynomials. Connections between the information content of a graph and Hosoya entropy are established, and the special case of Hosoya entropy of trees is investigated.

Graph characterizations from von Neumann entropy

In this paper we explore how the von Neumann entropy can be used as a measure of graph complexity. We also develop a simplified form for the von Neumann entropy of a graph that can be computed in terms of node degree statistics. We compare the resulting complexity with Estrada’s heterogeneity index which measures the heterogeneity of the node degree across a graph and reveal a new link between Estrada’s index and the commute time on a graph. Finally, we explore how the von Neumann entropy can be used in conjunction with thermodynamic depth. This measure has been shown to overcome problems associated with iso-spectrality encountered when using complexity measures based on spectral graph theory. Our experimental evaluation of the simplified von Neumann entropy explores (a) the accuracy of the underlying approximation, (b) a comparison with alternative graph characterizations, and (c) the application of the entropy-based thermodynamic depth to characterize protein–protein interaction networks.

Heat diffusion: Thermodynamic depth complexity of networks

In this paper we use the Birkhoff-von Neumann decomposition of the diffusion kernel to compute a polytopal measure of graph complexity. We decompose the diffusion kernel into a series of weighted Birkhoff combinations and compute the entropy associated with the weighting proportions (polytopal complexity). The maximum entropy Birkhoff combination can be expressed in terms of matrix permanents. This allows us to introduce a phase-transition principle that links our definition of polytopal complexity to the heat flowing through the network at a given diffusion time. The result is an efficiently computed complexity measure, which we refer to as flow complexity. Moreover, the flow complexity measure allows us to analyze graphs and networks in terms of the thermodynamic depth. We compare our method with three alternative methods described in the literature (Estrada's heterogeneity index, the Laplacian energy, and the von Neumann entropy). Our study is based on 217 protein-protein interaction (PPI) networks including histidine kinases from several species of bacteria. We find a correlation between structural complexity and phylogeny (more evolved species have statistically more complex PPIs). Although our methods outperform the alternatives, we find similarities with Estrada's heterogeneity index in terms of network size independence and predictive power.

On Minimal Tree Realizations of Linear Codes

A tree decomposition of the coordinates of a code is a mapping from the coordinate set to the set of vertices of a tree. A tree decomposition can be extended to a tree realization, i.e., a cycle-free realization of the code on the underlying tree, by specifying a state space at each edge of the tree, and a local constraint code at each vertex of the tree. The constraint complexity of a tree realization is the maximum dimension of any of its local constraint codes. A measure of the complexity of maximum-likelihood (ML) decoding for a code is its treewidth, which is the least constraint complexity of any of its tree realizations.It is known that among all tree realizations of a linear code that extends a given tree decomposition, there exists a unique minimal realization that minimizes the state-space dimension at each vertex of the underlying tree. In this paper, we give two new constructions of these minimal realizations. As a by-product of the first construction, a generalization of the state-merging procedure for trellis realizations, we obtain the fact that the minimal tree realization also minimizes the local constraint code dimension at each vertex of the underlying tree. The second construction relies on certain code decomposition techniques that we develop. We further observe that the treewidth of a code is related to a measure of graph complexity, also called treewidth. We exploit this connection to resolve a conjecture of Forney's regarding the gap between the minimum trellis constraint complexity and the treewidth of a code. We present a family of codes for which this gap can be arbitrarily large.

Graph operations characterizing rank-width

Graph complexity measures such as tree-width, clique-width and rank-width are important because they yield Fixed Parameter Tractable algorithms. These algorithms are based on hierarchical decompositions of the considered graphs, and on boundedness conditions on the graph operations that, more or less explicitly, recombine the components of decompositions into larger graphs. Rank-width is defined in a combinatorial way, based on a tree, and not in terms of graph operations. We define operations on graphs that characterize rank-width and help for the construction of Fixed Parameter Tractable algorithms, especially for problems specified in monadic second-order logic.

What is a complex graph?

Many papers published in recent years show that real-world graphs G ( n , m ) ( n nodes, m edges) are more or less “complex” in the sense that different topological features deviate from random graphs. Here we narrow the definition of graph complexity and argue that a complex graph contains many different subgraphs. We present different measures that quantify this complexity, for instance C 1 e , the relative number of non-isomorphic one-edge-deleted subgraphs (i.e. DECK size). However, because these different subgraph measures are computationally demanding, we also study simpler complexity measures focussing on slightly different aspects of graph complexity. We consider heuristically defined “product measures”, the products of two quantities which are zero in the extreme cases of a path and clique, and “entropy measures” quantifying the diversity of different topological features. The previously defined network/graph complexity measures Medium Articulation and Offdiagonal complexity ( OdC) belong to these two classes. We study OdC measures in some detail and compare it with our new measures. For all measures, the most complex graph G C max has a medium number of edges, between the edge numbers of the minimum and the maximum connected graph n − 1 < m ( G C max ) < n ( n − 1 ) / 2 . Interestingly, for some measures C ̃ this number scales exactly with the geometric mean of the extremes: m ( G C ̃ max ) = n / 2 ( n − 1 ) ∼ n 1.5 . All graph complexity measures are characterized with the help of different example graphs. For all measures the corresponding time complexity is given. Finally, we discuss the complexity of 33 real-world graphs of different biological, social and economic systems with the six computationally most simple measures (including OdC). The complexities of the real graphs are compared with average complexities of two different random graph versions: complete random graphs (just fixed n , m ) and rewired graphs with fixed node degrees.

The monadic second-order logic of graphs XIV: uniformly sparse graphs and edge set quantifications

We consider the class USk of uniformly k-sparse simple graphs, i.e., the class of finite or countable simple graphs, every finite subgraph of which has a number of edges bounded by k times the number of vertices. We prove that for each k, every monadic second-order formula (intended to express a graph property) that uses variables denoting sets of edges can be effectively translated into a monadic second-order formula where all set variables denote sets of vertices and that expresses the same property of the graphs in USk. This result extends to the class of uniformly k-sparse simple hypergraphs of rank at most m (for any k and m).It follows that every subclass of USk consisting of finite graphs of bounded clique-width has bounded tree-width. Clique-width is a graph complexity measure similar to tree-width and relevant to the construction of polynomial algorithms for NP-complete problems on special classes of graphs.

An axiomatic approach to network complexity

Despite the recent wave of interest in the social and physical sciences regarding “complexity,” relatively title attention has been given to the logical foundation of complexity measurement. With this in mind, a number of fairly simple, “reasonable” axioms for the measurement of network complexity are here presented, and some of the implications of these axioms are considered. It is shown that the only family of graph complexity measures satisfying the “reasonable” axioms is of limited theoretical utility, and hence that those seeking more interesting measures of complexity must be willing to sacrifice at least one intuitively reasonable constraint. Several existing complexity measures are also described, and are differentiated from one another on an axiomatic basis. Finally, some suggestions are offered regarding future efforts at measuring graph complexity.