Spectra of "real-world" graphs: beyond the semicircle law

Beyond nearest-neighbor universality of spectral fluctuations in quantum chaotic and complex many-body systems

Discerning chaos in quantum systems is an important problem as the usual route of Lyapunov exponents in classical systems is not straightforward in quantum systems. A standard route is the comparison of statistics derived from model physical systems to those from random matrix theory (RMT) ensembles, of which the most popular is the nearest-neighbor-spacing distribution, which almost always shows good agreement with chaotic quantum systems. However, even in these cases, the long-range statistics (like number variance and spectral rigidity), which are also more difficult to calculate, often show disagreements with RMT. As such, a more stringent test for chaos in quantum systems, via an analysis of intermediate-range statistics, is needed, which will additionally assess the extent of agreement with RMT universality. In this paper, we deduce the effective level-repulsion parameters and the corresponding Wigner-surmise-like results of the next-nearest-neighbor-spacing distribution (nNNSD) for integrable systems (semi-Poissonian statistics) as well as the three classical quantum chaotic Wigner-Dyson regimes, by stringent comparisons to numerical RMT models and benchmarking against our exact analytical results for 3×3 Gaussian matrix models, along with a semi-analytical form for the nNNSD in the orthogonal-to-unitary symmetry crossover. To illustrate the robustness of these RMT based results, we test these predictions against the nNNSD obtained from quantum chaotic models as well as disordered lattice spin models. This reinforces the Bohigas-Giannoni-Schmit and the Berry-Tabor conjectures, extending the associated universality to longer-range statistics. In passing, we also highlight the equivalence of nNNSD in the apparently distinct orthogonal-to-unitary and diluted-symplectic-to-unitary crossovers.

Effects of clustering heterogeneity on the spectral density of sparse networks

We derive exact equations for the spectral density of sparse networks with an arbitrary distribution of the number of single edges and triangles per node. These equations enable a systematic investigation of the effects of clustering on the spectral properties of the network adjacency matrix. In the case of heterogeneous networks, we demonstrate that the spectral density becomes more symmetric as the fluctuations in the triangle-degree sequence increase. This phenomenon is explained by the small clustering coefficient of networks with a large variance of the triangle-degree distribution. In the homogeneous case of regular clustered networks, we find that both perturbative and nonperturbative approximations fail to predict the spectral density in the high-connectivity limit. This suggests that traditional large-degree approximations may be ineffective in studying the spectral properties of networks with more complex motifs. Our theoretical results are fully confirmed by numerical diagonalizations of finite adjacency matrices.

Topological and spectral properties of random digraphs

We investigate some topological and spectral properties of Erdős-Rényi (ER) random digraphs of size n and connection probability p, D(n,p). In terms of topological properties, our primary focus lies in analyzing the number of nonisolated vertices V_{x}(D) as well as two vertex-degree-based topological indices: the Randić index R(D) and sum-connectivity index χ(D). First, by performing a scaling analysis, we show that the average degree 〈k〉 serves as a scaling parameter for the average values of V_{x}(D), R(D), and χ(D). Then, we also state expressions relating the number of arcs, largest eigenvalue, and closed walks of length 2 to (n,p), the parameters of ER random digraphs. Concerning spectral properties, we observe that the eigenvalue distribution converges to a circle of radius sqrt[np(1-p)]. Subsequently, we compute six different invariants related to the eigenvalues of D(n,p) and observe that these quantities also scale with sqrt[np(1-p)]. Additionally, we reformulate a set of bounds previously reported in the literature for these invariants as a function (n,p). Finally, we phenomenologically state relations between invariants that allow us to extend previously known bounds.

Structural network characteristics affect epidemic severity and prediction in social contact networks

Understanding and mitigating epidemic spread in complex networks requires the measurement of structural network properties associated with epidemic risk. Classic measures of epidemic thresholds like the basic reproduction number (R0) have been adapted to account for the structure of social contact networks but still may be unable to capture epidemic potential relative to more recent measures based on spectral graph properties. Here, we explore the ability of R0 and the spectral radius of the social contact network to estimate epidemic susceptibility. To do so, we simulate epidemics on a series of constructed (small world, scale-free, and random networks) and a collection of over 700 empirical biological social contact networks. Further, we explore how other network properties are related to these two epidemic estimators (R0 and spectral radius) and mean infection prevalence in simulated epidemics. Overall, we find that network properties strongly influence epidemic dynamics and the subsequent utility of R0 and spectral radius as indicators of epidemic risk.

Does the brain behave like a (complex) network? I. Dynamics

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.

Spectral density-based clustering algorithms for complex networks

Introduction

Clustering is usually the first exploratory analysis step in empirical data. When the data set comprises graphs, the most common approaches focus on clustering its vertices. In this work, we are interested in grouping networks with similar connectivity structures together instead of grouping vertices of the graph. We could apply this approach to functional brain networks (FBNs) for identifying subgroups of people presenting similar functional connectivity, such as studying a mental disorder. The main problem is that real-world networks present natural fluctuations, which we should consider.

Methods

In this context, spectral density is an exciting feature because graphs generated by different models present distinct spectral densities, thus presenting different connectivity structures. We introduce two clustering methods: k-means for graphs of the same size and gCEM, a model-based approach for graphs of different sizes. We evaluated their performance in toy models. Finally, we applied them to FBNs of monkeys under anesthesia and a dataset of chemical compounds.

Results

We show that our methods work well in both toy models and real-world data. They present good results for clustering graphs presenting different connectivity structures even when they present the same number of edges, vertices, and degree of centrality.

Discussion

We recommend using k-means-based clustering for graphs when graphs present the same number of vertices and the gCEM method when graphs present a different number of vertices.

Density of states for fast embedding node-attributed graphs

Given a node-attributed graph, how can we efficiently represent it with few numerical features that expressively reflect its topology and attribute information? We propose A-DOGE, for attributed DOS-based graph embedding, based on density of states (DOS, a.k.a. spectral density) to tackle this problem. A-DOGE is designed to fulfill a long desiderata of desirable characteristics. Most notably, it capitalizes on efficient approximation algorithms for DOS, that we extend to blend in node labels and attributes for the first time, making it fast and scalable for large attributed graphs and graph databases. Being based on the entire eigenspectrum of a graph, A-DOGE can capture structural and attribute properties at multiple (“glocal”) scales. Moreover, it is unsupervised (i.e., agnostic to any specific objective) and lends itself to various interpretations, which makes it suitable for exploratory graph mining tasks. Finally, it processes each graph independent of others, making it amenable for streaming settings as well as parallelization. Through extensive experiments, we show the efficacy and efficiency of A-DOGE on exploratory graph analysis and graph classification tasks, where it significantly outperforms unsupervised baselines and achieves competitive performance with modern supervised GNNs, while achieving the best trade-off between accuracy and runtime.

Evolution of Cohesion between USA Financial Sector Companies before, during, and Post-Economic Crisis: Complex Networks Approach

Various mathematical frameworks play an essential role in understanding the economic systems and the emergence of crises in them. Understanding the relation between the structure of connections between the system’s constituents and the emergence of a crisis is of great importance. In this paper, we propose a novel method for the inference of economic systems’ structures based on complex networks theory utilizing the time series of prices. Our network is obtained from the correlation matrix between the time series of companies’ prices by imposing a threshold on the values of the correlation coefficients. The optimal value of the threshold is determined by comparing the spectral properties of the threshold network and the correlation matrix. We analyze the community structure of the obtained networks and the relation between communities’ inter and intra-connectivity as indicators of systemic risk. Our results show how an economic system’s behavior is related to its structure and how the crisis is reflected in changes in the structure. We show how regulation and deregulation affect the structure of the system. We demonstrate that our method can identify high systemic risks and measure the impact of the actions taken to increase the system’s stability.

Maximum Entropy Approach to Massive Graph Spectrum Learning with Applications

We propose an alternative maximum entropy approach to learning the spectra of massive graphs. In contrast to state-of-the-art Lanczos algorithm for spectral density estimation and applications thereof, our approach does not require kernel smoothing. As the choice of kernel function and associated bandwidth heavily affect the resulting output, our approach mitigates these issues. Furthermore, we prove that kernel smoothing biases the moments of the spectral density. Our approach can be seen as an information-theoretically optimal approach to learning a smooth graph spectral density, which fully respects moment information. The proposed method has a computational cost linear in the number of edges, and hence can be applied even to large networks with millions of nodes. We showcase the approach on problems of graph similarity learning and counting cluster number in the graph, where the proposed method outperforms existing iterative spectral approaches on both synthetic and real-world graphs.

Eigenvalue-based entropy and spectrum of bipartite digraph

Graph entropy is an important measure of the evolution and complexity of networks. Bipartite graph is a special network and an important mathematical model for system resource allocation and management. In reality, a network system usually has obvious directionality. The direction of the network, or the movement trend of the network, can be described with spectrum index. However, little research has been done on the eigenvalue-based entropy of directed bipartite network. In this study, based on the adjacency matrix, the in-degree Laplacian matrix and the in-degree signless Laplacian matrix of directed bipartite graph, we defined the eigenvalue-based entropy for the directed bipartite network. Using the eigenvalue-based entropy, we described the evolution law of the directed bipartite network structure. Aiming at the direction and bipartite feature of the directed bipartite network, we improved the generation algorithm of the undirected network. We then constructed the directed bipartite nearest-neighbor coupling network, directed bipartite small-world network, directed bipartite scale-free network, and directed bipartite random network. In the proposed model, spectrum of those directed bipartite network is used to describe the directionality and bipartite property. Moreover, eigenvalue-based entropy is empirically studied on a real-world directed movie recommendation network, in which the law of eigenvalue-base entropy is observed. That is, if eigenvalue-based entropy value of the recommendation system is large, the evolution of movie recommendation network becomes orderless. While if eigenvalue-based entropy value is small, the structural evolution of the movie recommendation network tends to be regular. The simulation experiment shows that eigenvalue-based entropy value in the real directed bipartite network is between the values of a directed bipartite small world and a scale-free network. It shows that the real directed bipartite network has the structural property of the two typical directed bipartite networks. The coexistence of the small-world phenomena and the scale-free phenomena in the real network is consistent with the evolution law of typical network models. The experimental results show that the validity and rationality of the definition of eigenvalue-based entropy, which serves as a tool in the analysis of directed bipartite networks.

On the Normalized Laplacian Spectra of Random Geometric Graphs

In this work, we study the spectrum of the normalized Laplacian and its regularized version for random geometric graphs (RGGs) in various scaling regimes. Two scaling regimes are of special interest, the connectivity and the thermodynamic regime. In the connectivity regime, the average vertex degree grows logarithmically in the graph size or faster. In the thermodynamic regime, the average vertex degree is a constant. We introduce a deterministic geometric graph (DGG) with nodes in a grid and provide an upper bound to the probability that the Hilbert–Schmidt norm of the difference between the normalized Laplacian matrices of the RGG and DGG is greater than a certain threshold in both the connectivity and thermodynamic regime. Using this result, we show that the RGG and DGG normalized Laplacian matrices are asymptotically equivalent with high probability (w.h.p.) in the full range of the connectivity regime. The equivalence is even stronger and holds almost surely when the average vertex degree $$a_n$$ a n satisfies the inequality $$a_n > 24 \log (n).$$ a n > 24 log ( n ) . Therefore, we use the regular structure of the DGG to show that the limiting eigenvalue distribution of the RGG normalized Laplacian matrix converges to a distribution with a Dirac atomic measure at zero. In the thermodynamic regime, we approximate the eigenvalues of the regularized normalized Laplacian matrix of the RGG by the eigenvalues of the DGG regularized normalized Laplacian and we provide an error bound which is valid w.h.p. and depends upon the average vertex degree.

Physics in nonfixed spatial dimensions via random networks

We study the quantum statistical electronic properties of random networks which inherently lack a fixed spatial dimension. We use tools like the density of states (DOS) and the inverse participation ratio to uncover various phenomena, such as unconventional properties of the energy spectrum and persistent localized states (PLS) at various energies, corresponding to quantum phases with zero-dimensional (0D) and one-dimensional (1D) order. For small ratio of edges over vertices in the network R we find properties resembling graphene(honeycomb) lattices, like a similar DOS containing a linear dispersion relation at the band center at energy E=0. In addition, we find PLS at various energies including E=-1,0,1, and others, for example, related to the golden ratio. At E=0 the PLS lie at vertices that are not directly connected with an edge, due to partial bipartite symmetries of the random networks (0D order). At E=-1,1 the PLS lie mostly at pairs of vertices (bonds), while the rest of the PLS at other energies, like the ones related to the golden ratio, lie at lines of vertices of fixed length (1D order), at the spatial boundary of the network, resembling the edge states in confined graphene systems with zigzag edges. As the ratio R is increased the DOS of the network approaches the Wigner semicircle, corresponding to random symmetric matrices(Hamiltonians) and the PLS are reduced and gradually disappear as the connectivity in the network increases. Finally, we calculate the spatial dimension D of the network and its fluctuations. We obtain both integer and noninteger D and a logarithmic dependence on R. In addition, we examine the relation of D and its fluctuations to the electronic properties derived. Our results imply that universal physics can manifest in physical systems irrespectively of their spatial dimension. Relations to emergent spacetime in quantum and emergent gravity approaches are also discussed.

Autopoietic Influence Hierarchies in Pancreatic β Cells

β cells are biologically essential for humans and other vertebrates. Because their functionality arises from cell-cell interactions, they are also a model system for collective organization among cells. There are currently two contradictory pictures of this organization: the hub-cell idea pointing at leaders who coordinate the others, and the electrophysiological theory describing all cells as equal. We use new data and computational modeling to reconcile these pictures. We find via a network representation of interacting β cells that leaders emerge naturally (confirming the hub-cell idea), yet all cells can take the hub role following a perturbation (in line with electrophysiology).

Eigenvalue-based entropy in directed complex networks

Entropy is an important index for describing the structure, function, and evolution of network. The existing research on entropy is primarily applied to undirected networks. Compared with an undirected network, a directed network involves a special asymmetric transfer. The research on the entropy of directed networks is very significant to effectively quantify the structural information of the whole network. Typical complex network models include nearest-neighbour coupling network, small-world network, scale-free network, and random network. These network models are abstracted as undirected graphs without considering the direction of node connection. For complex networks, modeling through the direction of network nodes is extremely challenging. In this paper, based on these typical models of complex network, a directed network model considering node connection in-direction is proposed, and the eigenvalue entropies of three matrices in the directed network is defined and studied, where the three matrices are adjacency matrix, in-degree Laplacian matrix and in-degree signless Laplacian matrix. The eigenvalue-based entropies of three matrices are calculated in directed nearest-neighbor coupling, directed small world, directed scale-free and directed random networks. Through the simulation experiment on the real directed network, the result shows that the eigenvalue entropy of the real directed network is between the eigenvalue entropy of directed scale-free network and directed small-world network.

Spectral analysis of transient amplifiers for death-birth updating constructed from regular graphs

A central question of evolutionary dynamics on graphs is whether or not a mutation introduced in a population of residents survives and eventually even spreads to the whole population, or becomes extinct. The outcome naturally depends on the fitness of the mutant and the rules by which mutants and residents may propagate on the network, but arguably the most determining factor is the network structure. Some structured networks are transient amplifiers. They increase for a certain fitness range the fixation probability of beneficial mutations as compared to a well-mixed population. We study a perturbation method for identifying transient amplifiers for death–birth updating. The method involves calculating the coalescence times of random walks on graphs and finding the vertex with the largest remeeting time. If the graph is perturbed by removing an edge from this vertex, there is a certain likelihood that the resulting perturbed graph is a transient amplifier. We test all pairwise nonisomorphic regular graphs up to a certain order and thus cover the whole structural range expressible by these graphs. For cubic and quartic regular graphs we find a sufficiently large number of transient amplifiers. For these networks we carry out a spectral analysis and show that the graphs from which transient amplifiers can be constructed share certain structural properties. Identifying spectral and structural properties may promote finding and designing such networks.

Potential energy of complex networks: a quantum mechanical perspective

We propose a characterization of complex networks, based on the potential of an associated Schrödinger equation. The potential is designed so that the energy spectrum of the Schrödinger equation coincides with the graph spectrum of the normalized Laplacian. Crucial information is retained in the reconstructed potential, which provides a compact representation of the properties of the network structure. The median potential over several random network realizations, which we call ensemble potential, is fitted via a Landau-like function, and its length scale is found to diverge as the critical connection probability is approached from above. The ruggedness of the ensemble potential profile is quantified by using the Higuchi fractal dimension, which displays a maximum at the critical connection probability. This demonstrates that this technique can be successfully employed in the study of random networks, as an alternative indicator of the percolation phase transition. We apply the proposed approach to the investigation of real-world networks describing infrastructures (US power grid). Curiously, although no notion of phase transition can be given for such networks, the fractality of the ensemble potential displays signatures of criticality. We also show that standard techniques (such as the scaling features of the largest connected component) do not detect any signature or remnant of criticality.

Self-isolation or borders closing: What prevents the spread of the epidemic better?

Pandemic propagation of COVID-19 motivated us to discuss the impact of the human network clustering on epidemic spreading. Today, there are two clustering mechanisms which prevent of uncontrolled disease propagation in a connected network: an "internal" clustering, which mimics self-isolation (SI) in local naturally arranged communities, and an "external" clustering, which looks like a sharp frontiers closing (FC) between cities and countries, and which does not care about the natural connections of network agents. SI networks are "evolutionarily grown" under the condition of maximization of small cliques in the entire network, while FC networks are instantly created. Running the standard SIR model on clustered SI and FC networks, we demonstrate that the evolutionary grown clustered network prevents the spread of an epidemic better than the instantly clustered network with similar parameters. We find that SI networks have the scale-free property for the degree distribution P(k)∼k^{η}, with a small critical exponent -2<η<-1. We argue that the scale-free behavior emerges as a result of the randomness in the initial degree distributions.

Heatmap centrality: A new measure to identify super-spreader nodes in scale-free networks

The identification of potential super-spreader nodes within a network is a critical part of the study and analysis of real-world networks. Motivated by a new interpretation of the "shortest path" between two nodes, this paper explores the properties of the heatmap centrality by comparing the farness of a node with the average sum of farness of its adjacent nodes in order to identify influential nodes within the network. As many real-world networks are often claimed to be scale-free, numerical experiments based upon both simulated and real-world undirected and unweighted scale-free networks are used to illustrate the effectiveness of the proposed "shortest path" based measure with regards to its CPU run time and ranking of influential nodes.

Stochastic and mixed flower graphs

Stochasticity is introduced to a well studied class of recursively grown graphs: (u,v)-flower nets, which have power-law degree distributions as well as small-world properties (when u=1). The stochastic variant interpolates between different (deterministic) flower graphs thus adding flexibility to the model. The random multiplicative growth process involved, however, leads to a spread ensemble of networks with finite variance for the number of links, nodes, and loops. Nevertheless, the degree exponent and loopiness exponent attain unique values in the thermodynamic limit of infinitely large graphs. We also study a class of mixed flower networks, closely related to the stochastic flowers, but which are grown recursively in a deterministic way. The deterministic growth of mixed flower-nets eliminates ensemble spreads, and their recursive growth allows for exact analysis of their (uniquely defined) mixed properties.

Metrics for graph comparison: A practitioner's guide

Comparison of graph structure is a ubiquitous task in data analysis and machine learning, with diverse applications in fields such as neuroscience, cyber security, social network analysis, and bioinformatics, among others. Discovery and comparison of structures such as modular communities, rich clubs, hubs, and trees yield insight into the generative mechanisms and functional properties of the graph. Often, two graphs are compared via a pairwise distance measure, with a small distance indicating structural similarity and vice versa. Common choices include spectral distances and distances based on node affinities. However, there has of yet been no comparative study of the efficacy of these distance measures in discerning between common graph topologies at different structural scales. In this work, we compare commonly used graph metrics and distance measures, and demonstrate their ability to discern between common topological features found in both random graph models and real world networks. We put forward a multi-scale picture of graph structure wherein we study the effect of global and local structures on changes in distance measures. We make recommendations on the applicability of different distance measures to the analysis of empirical graph data based on this multi-scale view. Finally, we introduce the Python library NetComp that implements the graph distances used in this work.

May's instability in large economies

Will a large economy be stable? Building on Robert May's original argument for large ecosystems, we conjecture that evolutionary and behavioural forces conspire to drive the economy towards marginal stability. We study networks of firms in which inputs for production are not easily substitutable, as in several real-world supply chains. Relying on results from random matrix theory, we argue that such networks generically become dysfunctional when their size increases, when the heterogeneity between firms becomes too strong, or when substitutability of their production inputs is reduced. At marginal stability and for large heterogeneities, we find that the distribution of firm sizes develops a power-law tail, as observed empirically. Crises can be triggered by small idiosyncratic shocks, which lead to "avalanches" of defaults characterized by a power-law distribution of total output losses. This scenario would naturally explain the well-known "small shocks, large business cycles" puzzle, as anticipated long ago by Bak, Chen, Scheinkman, and Woodford.

Random temporal connections promote network synchronization

We report a phenomenon of collective dynamics on discrete-time complex networks: a random temporal interaction matrix even of zero or/and small average is able to significantly enhance synchronization with probability one. According to current knowledge, there is no verifiably sufficient criterion for the phenomenon. We use the standard method of synchronization analytics and the theory of stochastic processes to establish a criterion, by which we rigorously and accurately depict how synchronization occurring with probability one is affected by the statistical characteristics of the random temporal connections such as the strength and topology of the connections as well as their probability distributions. We also illustrate the enhancement phenomenon using physical and biological complex dynamical networks.

Multifractality in random networks with power-law decaying bond strengths

In this paper we demonstrate numerically that random networks whose adjacency matrices A are represented by a diluted version of the power-law banded random matrix (PBRM) model have multifractal eigenfunctions. The PBRM model describes one-dimensional samples with random long-range bonds. The bond strengths of the model, which decay as a power-law, are tuned by the parameter μ as A_{mn}∝|m-n|^{-μ}; while the sparsity is driven by the average network connectivity α: for α=0 the vertices in the network are isolated and for α=1 the network is fully connected and the PBRM model is recovered. Though it is known that the PBRM model has multifractal eigenfunctions at the critical value μ=μ_{c}=1, we clearly show [from the scaling of the relative fluctuation of the participation number I_{2} as well as the scaling of the probability distribution functions P(lnI_{2})] the existence of the critical value μ_{c}≡μ_{c}(α) for α<1. Moreover, we characterize the multifractality of the eigenfunctions of our random network model by the use of the corresponding multifractal dimensions D_{q}, that we compute from the finite network-size scaling of the typical eigenfunction participation numbers exp〈lnI_{q}〉.

Extreme value analysis of gut microbial alterations in colorectal cancer

Gut microbes play a key role in colorectal carcinogenesis, yet reaching a consensus on microbial signatures remains a challenge. This is in part due to a reliance on mean value estimates. We present an extreme value analysis for overcoming these limitations. By characterizing a power-law fit to the relative abundances of microbes, we capture the same microbial signatures as more complex meta-analyses. Importantly, we show that our method is robust to the variations inherent in microbial community profiling and point to future directions for developing sensitive, reliable analytical methods.

Spectra of random networks with arbitrary degrees

We derive a message-passing method for computing the spectra of locally treelike networks and an approximation to it that allows us to compute closed-form expressions or fast numerical approximates for the spectral density of random graphs with arbitrary node degrees-the so-called configuration model. We find that the latter approximation works well for all but the sparsest of networks. We also derive bounds on the position of the band edges of the spectrum, which are important for identifying structural phase transitions in networks.

Network spectra for drug-target identification in complex diseases: new guns against old foes

The fundamental understanding of altered complex molecular interactions in a diseased condition is the key to its cure. The overall functioning of these molecules is kind of jugglers play in the cell orchestra and to anticipate these relationships among the molecules is one of the greatest challenges in modern biology and medicine. Network science turned out to be providing a successful and simple platform to understand complex interactions among healthy and diseased tissues. Furthermore, much information about the structure and dynamics of a network is concealed in the eigenvalues of its adjacency matrix. In this review, we illustrate rapid advancements in the field of network science in combination with spectral graph theory that enables us to uncover the complexities of various diseases. Interpretations laid by network science approach have solicited insights into molecular relationships and have reported novel drug targets and biomarkers in various complex diseases.

Spectral properties of complex networks

This review presents an account of the major works done on spectra of adjacency matrices drawn on networks and the basic understanding attained so far. We have divided the review under three sections: (a) extremal eigenvalues, (b) bulk part of the spectrum, and (c) degenerate eigenvalues, based on the intrinsic properties of eigenvalues and the phenomena they capture. We have reviewed the works done for spectra of various popular model networks, such as the Erdős-Rényi random networks, scale-free networks, 1-d lattice, small-world networks, and various different real-world networks. Additionally, potential applications of spectral properties for natural processes have been reviewed.

Random Matrix Analysis for Gene Interaction Networks in Cancer Cells

Investigations of topological uniqueness of gene interaction networks in cancer cells are essential for understanding the disease. Although cancer is considered to originate from the topological alteration of a huge molecular interaction network in cellular systems, the theoretical study to investigate such complex networks is still insufficient. It is necessary to predict the behavior of a huge complex interaction network from the behavior of a finite size network. Based on the random matrix theory, we study the distribution of the nearest neighbor level spacings P(s) of interaction matrices of gene networks in human cancer cells. The interaction matrices are computed using the Cancer Network Galaxy (TCNG) database which is a repository of gene interactions inferred by a Bayesian network model. 256 NCBI GEO entries regarding gene expressions in human cancer cells have been used for the inference. We observe the Wigner distribution of P(s) when the gene networks are dense networks that have more than ~38,000 edges. In the opposite case, when the networks have smaller numbers of edges, the distribution P(s) becomes the Poisson distribution. We investigate relevance of P(s) both to the sparseness of the networks and to edge frequency factor which is the reliance (likelihood) of the inferred gene interactions.

Comment on "Spectral analysis of deformed random networks"

We comment on the findings presented in [Phys. Rev. E 80, 046101 (2009)10.1103/PhysRevE.80.046101] concerning the spacing distribution of the spectrum of sparsely connected random networks. We point out that for clustered networks without any connection among them the spacing distribution is not described by Poisson but rather by a superposition of multiple Gaussian orthogonal ensemble (GOE) statistics. Therefore, the spacing distribution for a network having two dense subnetworks with few connections between the two subnetworks follows a transition from two-GOE to a GOE statistics and not as the article suggests a transition from Poisson statistics to the GOE.

On-the-Fly Computation of Frontal Orbitals in Density Matrix Expansions

We propose a method for computation of frontal (homo and lumo) orbitals in recursive polynomial expansion algorithms for the density matrix. Such algorithms give a computational cost that increases only linearly with system size for sufficiently sparse systems, but a drawback compared to the traditional diagonalization approach is that molecular orbitals are not readily available. Our method is based on the idea to use the polynomial of the density matrix expansion as an eigenvalue filter giving large separation between eigenvalues around homo and lumo [ Rubensson et al. J. Chem. Phys. 2008 , 128 , 176101 ]. This filter is combined with a shift-and-square (folded spectrum) method to move the desired eigenvalue to the end of the spectrum. In this work we propose a transparent way to select recursive expansion iteration and shift for the eigenvector computation that results in a sharp eigenvalue filter. The filter is obtained as a byproduct of the density matrix expansion, and there is no significant additional cost associated either with its construction or with its application. This gives a clear-cut and efficient eigenvalue solver that can be used to compute homo and lumo orbitals with sufficient accuracy in a small fraction of the total recursive expansion time. Our algorithms make use of recent homo and lumo eigenvalue estimates that can be obtained at negligible cost [ Rubensson et al. SIAM J. Sci. Comput . 2014 , 36 , B147 ]. We illustrate our method by performing self-consistent field calculations for large scale systems.

Emergent spectral properties of river network topology: an optimal channel network approach

Characterization of river drainage networks has been a subject of research for many years. However, most previous studies have been limited to quantities which are loosely connected to the topological properties of these networks. In this work, through a graph-theoretic formulation of drainage river networks, we investigate the eigenvalue spectra of their adjacency matrix. First, we introduce a graph theory model for river networks and explore the properties of the network through its adjacency matrix. Next, we show that the eigenvalue spectra of such complex networks follow distinct patterns and exhibit striking features including a spectral gap in which no eigenvalue exists as well as a finite number of zero eigenvalues. We show that such spectral features are closely related to the branching topology of the associated river networks. In this regard, we find an empirical relation for the spectral gap and nullity in terms of the energy dissipation exponent of the drainage networks. In addition, the eigenvalue distribution is found to follow a finite-width probability density function with certain skewness which is related to the drainage pattern. Our results are based on optimal channel network simulations and validated through examples obtained from physical experiments on landscape evolution. These results suggest the potential of the spectral graph techniques in characterizing and modeling river networks.

Spectral partitioning in equitable graphs

Graph partitioning problems emerge in a wide variety of complex systems, ranging from biology to finance, but can be rigorously analyzed and solved only for a few graph ensembles. Here, an ensemble of equitable graphs, i.e., random graphs with a block-regular structure, is studied, for which analytical results can be obtained. In particular, the spectral density of this ensemble is computed exactly for a modular and bipartite structure. Kesten-McKay's law for random regular graphs is found analytically to apply also for modular and bipartite structures when blocks are homogeneous. An exact solution to graph partitioning for two equal-sized communities is proposed and verified numerically, and a conjecture on the absence of an efficient recovery detectability transition in equitable graphs is suggested. A final discussion summarizes results and outlines their relevance for the solution of graph partitioning problems in other graph ensembles, in particular for the study of detectability thresholds and resolution limits in stochastic block models.

How the site degree influences quantum probability on inhomogeneous substrates

We investigate the effect of the node degree and energy E on the electronic wave function for regular and irregular structures, namely, regular lattices, disordered percolation clusters, and complex networks. We evaluate the dependency of the quantum probability for each site on its degree. For a class of biregular structures formed by two disjoint subsets of sites sharing the same degree, the probability P_{k}(E) of finding the electron on any site with k neighbors is independent of E≠0, a consequence of an exact analytical result that we prove for any bipartite lattice. For more general nonbipartite structures, P_{k}(E) may depend on E as illustrated by an exact evaluation of a one-dimensional semiregular lattice: P_{k}(E) is large for small values of E when k is also small, and its maximum values shift towards large values of |E| with increasing k. Numerical evaluations of P_{k}(E) for two different types of percolation clusters and the Apollonian network suggest that this observed feature might be generally valid.

Localization in random bipartite graphs: Numerical and empirical study

We investigate adjacency matrices of bipartite graphs with a power-law degree distribution. Motivation for this study is twofold: first, vibrational states in granular matter and jammed sphere packings; second, graphs encoding social interaction, especially electronic commerce. We establish the position of the mobility edge and show that it strongly depends on the power in the degree distribution and on the ratio of the sizes of the two parts of the bipartite graph. At the jamming threshold, where the two parts have the same size, localization vanishes. We found that the multifractal spectrum is nontrivial in the delocalized phase, but still near the mobility edge. We also study an empirical bipartite graph, namely, the Amazon reviewer-item network. We found that in this specific graph the mobility edge disappears, and we draw a conclusion from this fact regarding earlier empirical studies of the Amazon network.

EIGENVECTOR-BASED CENTRALITY MEASURES FOR TEMPORAL NETWORKS. MULTISCALE MODELING & SIMULATION : A SIAM INTERDISCIPLINARY JOURNAL 2017;15:537-574. [PMID: 29046619 PMCID: PMC5643020 DOI: 10.1137/16m1066142]

Numerous centrality measures have been developed to quantify the importances of nodes in time-independent networks, and many of them can be expressed as the leading eigenvector of some matrix. With the increasing availability of network data that changes in time, it is important to extend such eigenvector-based centrality measures to time-dependent networks. In this paper, we introduce a principled generalization of network centrality measures that is valid for any eigenvector-based centrality. We consider a temporal network with N nodes as a sequence of T layers that describe the network during different time windows, and we couple centrality matrices for the layers into a supra-centrality matrix of size NT × NT whose dominant eigenvector gives the centrality of each node i at each time t. We refer to this eigenvector and its components as a joint centrality, as it reflects the importances of both the node i and the time layer t. We also introduce the concepts of marginal and conditional centralities, which facilitate the study of centrality trajectories over time. We find that the strength of coupling between layers is important for determining multiscale properties of centrality, such as localization phenomena and the time scale of centrality changes. In the strong-coupling regime, we derive expressions for time-averaged centralities, which are given by the zeroth-order terms of a singular perturbation expansion. We also study first-order terms to obtain first-order-mover scores, which concisely describe the magnitude of nodes' centrality changes over time. As examples, we apply our method to three empirical temporal networks: the United States Ph.D. exchange in mathematics, costarring relationships among top-billed actors during the Golden Age of Hollywood, and citations of decisions from the United States Supreme Court.

Exploring the "Middle Earth" of network spectra via a Gaussian matrix function

We study a Gaussian matrix function of the adjacency matrix of artificial and real-world networks. We motivate the use of this function on the basis of a dynamical process modeled by the time-dependent Schrödinger equation with a squared Hamiltonian. In particular, we study the Gaussian Estrada index-an index characterizing the importance of eigenvalues close to zero. This index accounts for the information contained in the eigenvalues close to zero in the spectra of networks. Such a method is a generalization of the so-called "Folded Spectrum Method" used in quantum molecular sciences. Here, we obtain bounds for this index in simple graphs, proving that it reaches its maximum for star graphs followed by complete bipartite graphs. We also obtain formulas for the Estrada Gaussian index of Erdős-Rényi random graphs and for the Barabási-Albert graphs. We also show that in real-world networks, this index is related to the existence of important structural patterns, such as complete bipartite subgraphs (bicliques). Such bicliques appear naturally in many real-world networks as a consequence of the evolutionary processes giving rise to them. In general, the Gaussian matrix function of the adjacency matrix of networks characterizes important structural information not described in previously used matrix functions of graphs.

Eigenvalue tunneling and decay of quenched random network

We consider the canonical ensemble of N-vertex Erdős-Rényi (ER) random topological graphs with quenched vertex degree, and with fugacity μ for each closed triple of bonds. We claim complete defragmentation of large-N graphs into the collection of [p^{-1}] almost full subgraphs (cliques) above critical fugacity, μ_{c}, where p is the ER bond formation probability. Evolution of the spectral density, ρ(λ), of the adjacency matrix with increasing μ leads to the formation of a multizonal support for μ>μ_{c}. Eigenvalue tunneling from the central zone to the side one means formation of a new clique in the defragmentation process. The adjacency matrix of the network ground state has a block-diagonal form, where the number of vertices in blocks fluctuates around the mean value Np. The spectral density of the whole network in this regime has triangular shape. We interpret the phenomena from the viewpoint of the conventional random matrix model and speculate about possible physical applications.

Low-dimensional dynamics of structured random networks

Using a generalized random recurrent neural network model, and by extending our recently developed mean-field approach [J. Aljadeff, M. Stern, and T. Sharpee, Phys. Rev. Lett. 114, 088101 (2015)], we study the relationship between the network connectivity structure and its low-dimensional dynamics. Each connection in the network is a random number with mean 0 and variance that depends on pre- and postsynaptic neurons through a sufficiently smooth function g of their identities. We find that these networks undergo a phase transition from a silent to a chaotic state at a critical point we derive as a function of g. Above the critical point, although unit activation levels are chaotic, their autocorrelation functions are restricted to a low-dimensional subspace. This provides a direct link between the network's structure and some of its functional characteristics. We discuss example applications of the general results to neuroscience where we derive the support of the spectrum of connectivity matrices with heterogeneous and possibly correlated degree distributions, and to ecology where we study the stability of the cascade model for food web structure.

Distinct types of eigenvector localization in networks

The spectral properties of the adjacency matrix provide a trove of information about the structure and function of complex networks. In particular, the largest eigenvalue and its associated principal eigenvector are crucial in the understanding of nodes’ centrality and the unfolding of dynamical processes. Here we show that two distinct types of localization of the principal eigenvector may occur in heterogeneous networks. For synthetic networks with degree distribution P(q) ~ q^−γ, localization occurs on the largest hub if γ > 5/2; for γ < 5/2 a new type of localization arises on a mesoscopic subgraph associated with the shell with the largest index in the K-core decomposition. Similar evidence for the existence of distinct localization modes is found in the analysis of real-world networks. Our results open a new perspective on dynamical processes on networks and on a recently proposed alternative measure of node centrality based on the non-backtracking matrix.

The Mixing Time of the Newman-Watts Small-World Model

‘Small worlds’ are large systems in which any given node has only a few connections to other points, but possessing the property that all pairs of points are connected by a short path, typically logarithmic in the number of nodes. The use of random walks for sampling a uniform element from a large state space is by now a classical technique; to prove that such a technique works for a given network, a bound on the mixing time is required. However, little detailed information is known about the behaviour of random walks on small-world networks, though many predictions can be found in the physics literature. The principal contribution of this paper is to show that for a famous small-world random graph model known as the Newman-Watts small-world model, the mixing time is of order log2 n. This confirms a prediction of Richard Durrett [5, page 22], who proved a lower bound of order log2 n and an upper bound of order log3 n.

The Mixing Time of the Newman-Watts Small-World Model

‘Small worlds’ are large systems in which any given node has only a few connections to other points, but possessing the property that all pairs of points are connected by a short path, typically logarithmic in the number of nodes. The use of random walks for sampling a uniform element from a large state space is by now a classical technique; to prove that such a technique works for a given network, a bound on the mixing time is required. However, little detailed information is known about the behaviour of random walks on small-world networks, though many predictions can be found in the physics literature. The principal contribution of this paper is to show that for a famous small-world random graph model known as the Newman-Watts small-world model, the mixing time is of order log2n. This confirms a prediction of Richard Durrett [5, page 22], who proved a lower bound of order log2n and an upper bound of order log3n.

Neural Network Spectral Robustness under Perturbations of the Underlying Graph

Recent studies have been using graph-theoretical approaches to model complex networks (such as social, infrastructural, or biological networks) and how their hardwired circuitry relates to their dynamic evolution in time. Understanding how configuration reflects on the coupled behavior in a system of dynamic nodes can be of great importance, for example, in the context of how the brain connectome is affecting brain function. However, the effect of connectivity patterns on network dynamics is far from being fully understood. We study the connections between edge configuration and dynamics in a simple oriented network composed of two interconnected cliques (representative of brain feedback regulatory circuitry). In this article our main goal is to study the spectra of the graph adjacency and Laplacian matrices, with a focus on three aspects in particular: (1) the sensitivity and robustness of the spectrum in response to varying the intra- and intermodular edge density, (2) the effects on the spectrum of perturbing the edge configuration while keeping the densities fixed, and (3) the effects of increasing the network size. We study some tractable aspects analytically, then simulate more general results numerically, thus aiming to motivate and explain our further work on the effect of these patterns on the network temporal dynamics and phase transitions. We discuss the implications of such results to modeling brain connectomics. We suggest potential applications to understanding synaptic restructuring in learning networks and the effects of network configuration on function of regulatory neural circuits.

Eigenvalues for the transition matrix of a small-world scale-free network: Explicit expressions and applications

The eigenvalues of the transition matrix for random walks on a network play a significant role in the structural and dynamical aspects of the network. Nevertheless, it is still not well understood how the eigenvalues behave in small-world and scale-free networks, which describe a large variety of real systems. In this paper, we study the eigenvalues for the transition matrix of a network that is simultaneously scale-free, small-world, and clustered. We derive explicit simple expressions for all eigenvalues and their multiplicities, with the spectral density exhibiting a power-law form. We then apply the obtained eigenvalues to determine the mixing time and random target access time for random walks, both of which exhibit unusual behaviors compared with those for other networks, signaling discernible effects of topologies on spectral features. Finally, we use the eigenvalues to count spanning trees in the network.

Ground-state phase-space structures of two-dimensional ±J spin glasses: A network approach

We illustrate a complex-network approach to study the phase spaces of spin glasses. By mapping the whole ground-state phase spaces of two-dimensional Edwards-Anderson bimodal (±J) spin glasses exactly into networks for analysis, we discovered various phase-space properties. The Gaussian connectivity distribution of the phase-space networks demonstrates that both the number of free spins and the visiting frequency of all microstates follow the Gaussian distribution. The spectra of phase-space networks are Gaussian, which is proven to be exact when the system is infinitely large. The phase-space networks exhibit community structures. By coarse graining to the community level, we constructed a network representing the entropy landscape of the ground state and discovered its scale-free property. The phase-space networks exhibit fractal structures, as a result of the rugged entropy landscape. Moreover, we show that the connectivity distribution, community structures, and fractal structures change drastically at the ferromagnetic-to-glass phase transition. These quantitative measurements of the ground states provide new insight into the study of spin glasses. The phase-space networks of spin glasses share a number of common features with those of lattice gases and geometrically frustrated spin systems and form a new class of complex networks with unique topology.

Collective relaxation dynamics of small-world networks

Complex networks exhibit a wide range of collective dynamic phenomena, including synchronization, diffusion, relaxation, and coordination processes. Their asymptotic dynamics is generically characterized by the local Jacobian, graph Laplacian, or a similar linear operator. The structure of networks with regular, small-world, and random connectivities are reasonably well understood, but their collective dynamical properties remain largely unknown. Here we present a two-stage mean-field theory to derive analytic expressions for network spectra. A single formula covers the spectrum from regular via small-world to strongly randomized topologies in Watts-Strogatz networks, explaining the simultaneous dependencies on network size N, average degree k, and topological randomness q. We present simplified analytic predictions for the second-largest and smallest eigenvalue, and numerical checks confirm our theoretical predictions for zero, small, and moderate topological randomness q, including the entire small-world regime. For large q of the order of one, we apply standard random matrix theory, thereby overarching the full range from regular to randomized network topologies. These results may contribute to our analytic and mechanistic understanding of collective relaxation phenomena of network dynamical systems.