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Showing papers on "Spectral graph theory published in 2013"


Proceedings ArticleDOI
26 May 2013
TL;DR: A novel algorithm to interpolate data defined on graphs, using signal processing concepts, and imposes a `bilateral' weighting scheme on the links between known samples, which improves accuracy and complexity.
Abstract: In this paper, we propose a novel algorithm to interpolate data defined on graphs, using signal processing concepts. The interpolation of missing values from known samples appears in various applications, such as matrix/vector completion, sampling of high-dimensional data, semi-supervised learning etc. In this paper, we formulate the data interpolation problem as a signal reconstruction problem on a graph, where a graph signal is defined as the information attached to each node (scalar or vector values mapped to the set of vertices/edges of the graph). We use recent results for sampling in graphs to find classes of bandlimited (BL) graph signals that can be reconstructed from their partially observed samples. The interpolated signal is obtained by projecting the input signal into the appropriate BL graph signal space. Additionally, we impose a `bilateral' weighting scheme on the links between known samples, which further improves accuracy. We use our proposed method for collaborative filtering in recommendation systems. Preliminary results show a very favorable trade-off between accuracy and complexity, compared to state of the art algorithms.

254 citations


Journal ArticleDOI
TL;DR: A spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed, which provides a fundamental tradeoff between a signal's localization on a graph and in its spectral domain.
Abstract: The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed. Just as the classical result provides a tradeoff between signal localization in time and frequency, this result provides a fundamental tradeoff between a signal's localization on a graph and in its spectral domain. Using the eigenvectors of the graph Laplacian as a surrogate Fourier basis, quantitative definitions of graph and spectral “spreads” are given, and a complete characterization of the feasibility region of these two quantities is developed. In particular, the lower boundary of the region, referred to as the uncertainty curve, is shown to be achieved by eigenvectors associated with the smallest eigenvalues of an affine family of matrices. The convexity of the uncertainty curve allows it to be found to within e by a fast approximation algorithm requiring O(e-1/2) typically sparse eigenvalue evaluations. Closed-form expressions for the uncertainty curves for some special classes of graphs are derived, and an accurate analytical approximation for the expected uncertainty curve of Erd-s-Renyi random graphs is developed. These theoretical results are validated by numerical experiments, which also reveal an intriguing connection between diffusion processes on graphs and the uncertainty bounds.

185 citations


Proceedings ArticleDOI
23 Jun 2013
TL;DR: A graph-Laplacian PCA (gLPCA) to learn a low dimensional representation of X that incorporates graph structures encoded in W that is capable to remove corruptions and shows promising results on image reconstruction and significant improvement on clustering and classification.
Abstract: Principal Component Analysis (PCA) is a widely used to learn a low-dimensional representation. In many applications, both vector data X and graph data W are available. Laplacian embedding is widely used for embedding graph data. We propose a graph-Laplacian PCA (gLPCA) to learn a low dimensional representation of X that incorporates graph structures encoded in W. This model has several advantages: (1) It is a data representation model. (2) It has a compact closed-form solution and can be efficiently computed. (3) It is capable to remove corruptions. Extensive experiments on 8 datasets show promising results on image reconstruction and significant improvement on clustering and classification.

150 citations


Journal ArticleDOI
TL;DR: In this article, the authors formulate and prove a Cheeger-type inequality that relates a measure of how well it is possible to solve the synchronization problem with the spectra of an operator, the graph connection Laplacian.
Abstract: The $O(d)$ synchronization problem consists of estimating a set of $n$ unknown orthogonal $d\times d$ matrices $O_1,\ldots,O_n$ from noisy measurements of a subset of the pairwise ratios $O_iO_j^{-1}$. We formulate and prove a Cheeger-type inequality that relates a measure of how well it is possible to solve the $O(d)$ synchronization problem with the spectra of an operator, the graph connection Laplacian. We also show how this inequality provides a worst-case performance guarantee for a spectral method to solve this problem.

140 citations


Journal ArticleDOI
TL;DR: This framework generalizes the recently proposed spectral graph wavelet transform (SGWT) and proposes a design for multislice graphs that is based on the higher-order singular value decomposition (HOSVD), a powerful tool from multilinear algebra.
Abstract: We present a framework for the design of wavelet transforms tailored to data defined on multislice graphs (i.e., multiplex or dynamic graphs). Graphs with multiple types of interactions are ubiquitous in real life, motivating the extension of wavelets to these complex domains. Our framework generalizes the recently proposed spectral graph wavelet transform (SGWT) [D. Hammond, P. Vandergheynst, and R. Gribonval, “Wavelets on Graphs via Spectral Graph Theory,” Appl. Comput. Harmon. Anal., vol. 30, pp. 129-150, Mar. 2011], which is designed in the spectral (frequency) domain of an arbitrary finite weighted graph. We extend the SGWT to form a tight frame, which conserves energy in the wavelet domain, and define the relationship between conventional and spectral graph wavelets. We then propose a design for multislice graphs that is based on the higher-order singular value decomposition (HOSVD), a powerful tool from multilinear algebra. In particular, the multiple adjacency matrices are stacked to form a tensor and the HOSVD decomposition provides information about its third-order structure, analogous to that provided by matrix factorizations. We obtain a set of “eigennetworks” and from these graph wavelets, which exploit the variability across the graphs. We demonstrate the feasibility of our method 1) by capturing different orientations of a gray-scale image and 2) by decomposing brain signals from functional magnetic resonance imaging. We show its effectiveness to identify variability across graph edges and provide meaningful decompositions.

136 citations


Proceedings ArticleDOI
01 Jun 2013
TL;DR: In this paper, the spectral partitioning algorithm is shown to be a constant factor approximation algorithm for finding a sparse cut if lk is a constant for some constant k. This bound is improved to O(k) l2/√lk by Cheeger's inequality.
Abstract: Let φ(G) be the minimum conductance of an undirected graph G, and let 0=λ1 ≤ λ2 ≤ ... ≤ λn ≤ 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k ≥ 2, [φ(G) = O(k) l2/√lk,] and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheeger's inequality, and the bound is optimal up to a constant factor for any $k$. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if lk is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to spectral algorithms for other graph partitioning problems, including multi-way partition, balanced separator, and maximum cut.

98 citations


Posted Content
TL;DR: A large amount of robustness measures on simple, undirected and unweighted graphs are surveyed in order to offer a tool for network administrators to evaluate and improve the robustness of their network.
Abstract: Network robustness research aims at finding a measure to quantify network robustness. Once such a measure has been established, we will be able to compare networks, to improve existing networks and to design new networks that are able to continue to perform well when it is subject to failures or attacks. In this paper we survey a large amount of robustness measures on simple, undirected and unweighted graphs, in order to offer a tool for network administrators to evaluate and improve the robustness of their network. The measures discussed in this paper are based on the concepts of connectivity (including reliability polynomials), distance, betweenness and clustering. Some other measures are notions from spectral graph theory, more precisely, they are functions of the Laplacian eigenvalues. In addition to surveying these graph measures, the paper also contains a discussion of their functionality as a measure for topological network robustness.

97 citations


Book ChapterDOI
28 Jun 2013
TL;DR: This work proposes a very fast alternative based on an application of spectral graph theory on a novel association graph that can generate a diffeomorphic correspondence map within a few minutes on high-resolution meshes while avoiding the sign and multiplicity ambiguities of conventional spectral matching methods.
Abstract: Accurate matching of cortical surfaces is necessary in many neuroscience applications. In this context diffeomorphisms are often sought, because they facilitate further statistical analysis and atlas building. Present methods for computing diffeomorphisms are based on optimizing flows or on inflating surfaces to a common template, but they are often computationally expensive. It typically takes several hours on a conventional desktop computer to match a single pair of cortical surfaces having a few hundred thousand vertices. We propose a very fast alternative based on an application of spectral graph theory on a novel association graph. Our symmetric approach can generate a diffeomorphic correspondence map within a few minutes on high-resolution meshes while avoiding the sign and multiplicity ambiguities of conventional spectral matching methods. The eigenfunctions are shared between surfaces and provide a smooth parameterization of surfaces. These properties are exploited to compute differentials on highly folded cortical surfaces. Diffeomorphisms can thus be verified and invalid surface folding detected. Our method is demonstrated to attain a vertex accuracy that is at least as good as that of FreeSurfer and Spherical Demons but in only a fraction of their processing time. As a practical experiment, we construct an unbiased atlas of cortical surfaces with a speed several orders of magnitude faster than current methods.

77 citations


Journal ArticleDOI
TL;DR: This paper proposes a general framework for the trapping problem on a weighted network with a perfect trap fixed at an arbitrary node, and deduces an explicit expression for average trapping time (ATT) in terms of the eigenvalues and eigenvectors of the Laplacian matrix associated with the weighted graph.
Abstract: Trapping processes constitute a primary problem of random walks, which characterize various other dynamical processes taking place on networks. Most previous works focused on the case of binary networks, while there is much less related research about weighted networks. In this paper, we propose a general framework for the trapping problem on a weighted network with a perfect trap fixed at an arbitrary node. By utilizing the spectral graph theory, we provide an exact formula for mean first-passage time (MFPT) from one node to another, based on which we deduce an explicit expression for average trapping time (ATT) in terms of the eigenvalues and eigenvectors of the Laplacian matrix associated with the weighted graph, where ATT is the average of MFPTs to the trap over all source nodes. We then further derive a sharp lower bound for the ATT in terms of only the local information of the trap node, which can be obtained in some graphs. Moreover, we deduce the ATT when the trap is distributed uniformly in the whole network. Our results show that network weights play a significant role in the trapping process. To apply our framework, we use the obtained formulas to study random walks on two specific networks: trapping in weighted uncorrelated networks with a deep trap, the weights of which are characterized by a parameter, and L\'evy random walks in a connected binary network with a trap distributed uniformly, which can be looked on as random walks on a weighted network. For weighted uncorrelated networks we show that the ATT to any target node depends on the weight parameter, that is, the ATT to any node can change drastically by modifying the parameter, a phenomenon that is in contrast to that for trapping in binary networks. For L\'evy random walks in any connected network, by using their equivalence to random walks on a weighted complete network, we obtain the optimal exponent characterizing L\'evy random walks, which have the minimal average of ATTs taken over all target nodes.

73 citations


Proceedings ArticleDOI
11 Mar 2013
TL;DR: In this paper, the bilateral filter is viewed as a spectral domain transform defined on a weighted graph and the graph spectrum is defined in terms of the eigenvalues and eigenvectors of the graph Laplacian matrix.
Abstract: In this paper we study the bilateral filter proposed by Tomasi and Manduchi and show that it can be viewed as a spectral domain transform defined on a weighted graph. The nodes of this graph represent the pixels in the image and a graph signal defined on the nodes represents the intensity values. Edge weights in the graph correspond to the bilateral filter coefficients and hence are data adaptive. The graph spectrum is defined in terms of the eigenvalues and eigenvectors of the graph Laplacian matrix. We use this spectral interpretation to generalize the bilateral filter and propose new spectral designs of “bilateral-like” filters. We show that these spectral filters can be implemented with k-iterative bilateral filtering operations and do not require expensive diagonalization of the Laplacian matrix.

68 citations


Journal ArticleDOI
TL;DR: This paper considers ad hoc networks where the nodes can process and exchange data in a synchronous fashion, and proposes a distributed algorithm for in-network estimation of the Fiedler vector and the algebraic connectivity of the corresponding network graph.

Proceedings ArticleDOI
01 Dec 2013
TL;DR: A parametric dictionary learning algorithm is proposed to design structured dictionaries that sparsely represent graph signals that incorporate the graph structure by forcing the learned dictionaries to be concatenations of subdictionaries that are polynomials of the graph Laplacian matrix.
Abstract: We propose a parametric dictionary learning algorithm to design structured dictionaries that sparsely represent graph signals. We incorporate the graph structure by forcing the learned dictionaries to be concatenations of subdictionaries that are polynomials of the graph Laplacian matrix. The resulting atoms capture the main spatial and spectral components of the graph signals of interest, leading to adaptive representations with efficient implementations. Experimental results demonstrate the effectiveness of the proposed algorithm for the sparse approximation of graph signals.

Proceedings Article
16 Jun 2013
TL;DR: The bigraphical lasso is introduced, an estimator for precision matrices of matrix-normals based on the Cartesian product of graph theory, a prominent product in spectral graph theory that has appealing properties for regression, enhanced sparsity and interpretability.
Abstract: The i.i.d. assumption in machine learning is endemic, but often flawed. Complex data sets exhibit partial correlations between both instances and features. A model specifying both types of correlation can have a number of parameters that scales quadratically with the number of features and data points. We introduce the bigraphical lasso, an estimator for precision matrices of matrix-normals based on the Cartesian product of graphs. A prominent product in spectral graph theory, this structure has appealing properties for regression, enhanced sparsity and interpretability. To deal with the parameter explosion we introduce l1 penalties and fit the model through a flip-flop algorithm that results in a linear number of lasso regressions. We demonstrate the performance of our approach with simulations and an example from the COIL image data set.

Proceedings ArticleDOI
17 Jun 2013
TL;DR: Using bifurcation theory and spectral graph theory, the epidemic threshold of one network is found as a function of the infection strength of the other coupled network and adjacency matrices of each graph and their interconnection, and a quantitative measure to distinguish weak and strong interconnection topology is provided.
Abstract: In epidemic spreading models, if the infection strength is higher than a certain critical value - which we define as the epidemic threshold - then the epidemic spreads through the population. For a single arbitrary graph representing the contact network of the population under consideration, the epidemic threshold turns out to be equal to the inverse of the spectral radius of the contact graph. However, in a real world scenario, it is not possible to isolate a population completely: there is always some interconnection with another network, which partially overlaps with the contact network. In this paper, we study the spreading process of a susceptible-infected-susceptible (SIS) epidemic model in an interconnected network of two generic graphs with generic interconnection and different epidemic-related parameters. Using bifurcation theory and spectral graph theory, we find the epidemic threshold of one network as a function of the infection strength of the other coupled network and adjacency matrices of each graph and their interconnection, and provide a quantitative measure to distinguish weak and strong interconnection topology. These results have implications for the broad field of epidemic modeling and control.

Journal ArticleDOI
TL;DR: In this paper, the spectral conditions for a graph to be Hamilton-connected in terms of the spectral radius of the adjacency matrix or signless Laplacian of the graph or its complement are established.
Abstract: Some spectral conditions for a graph to be Hamilton-connected in terms of the spectral radius of the adjacency matrix or signless Laplacian of the graph or its complement are established, and then the condition on the signless Laplacian spectral radius of a graph for the existence of Hamiltonian paths or cycles is given.

Journal ArticleDOI
TL;DR: A unified theoretical framework for estimating various transmission costs in wireless networks is developed by generalizing the random walk theory that has been primarily developed for undirected graphs to digraphs and demonstrating that the proposed digraph-based analytical model can achieve more accurate transmission cost estimation over existing methods.
Abstract: Various applications in wireless networks, such as routing and query processing, can be formulated as random walks on graphs. Many results have been obtained for such applications by utilizing the theory of random walks (or spectral graph theory), which is mostly developed for undirected graphs. However, this formalism neglects the fact that the underlying (wireless) networks in practice contain asymmetric links, which are best characterized by directed graphs (digraphs). Therefore, random walk on digraphs is a more appropriate model to consider for such networks. In this paper, by generalizing the random walk theory (or spectral graph theory) that has been primarily developed for undirected graphs to digraphs, we show how various transmission costs in wireless networks can be formulated in terms of hitting times and cover times of random walks on digraphs. Using these results, we develop a unified theoretical framework for estimating various transmission costs in wireless networks. Our framework can be applied to random walk query processing strategy and the three routing paradigms--best path routing, opportunistic routing, and stateless routing--to which nearly all existing routing protocols belong. Extensive simulations demonstrate that the proposed digraph-based analytical model can achieve more accurate transmission cost estimation over existing methods.

Journal ArticleDOI
TL;DR: This paper uses algebraic graph theory and convex optimization to study how structural properties influence the spectrum of eigenvalues of the network, and can compute, with low computational overhead, global spectral properties of a network from its local structural properties.
Abstract: The eigenvalues of matrices representing the structure of large-scale complex networks present a wide range of applications, from the analysis of dynamical processes taking place in the network to spectral techniques aiming to rank the importance of nodes in the network. A common approach to study the relationship between the structure of a network and its eigenvalues is to use synthetic random networks in which structural properties of interest, such as degree distributions, are prescribed. Although very common, synthetic models present two major flaws: 1) These models are only suitable to study a very limited range of structural properties; and 2) they implicitly induce structural properties that are not directly controlled and can deceivingly influence the network eigenvalue spectrum. In this paper, we propose an alternative approach to overcome these limitations. Our approach is not based on synthetic models. Instead, we use algebraic graph theory and convex optimization to study how structural properties influence the spectrum of eigenvalues of the network. Using our approach, we can compute, with low computational overhead, global spectral properties of a network from its local structural properties. We illustrate our approach by studying how structural properties of online social networks influence their eigenvalue spectra.

01 Jan 2013
TL;DR: This thesis develops highly efficient and parallelizable algorithms for solving linear systems involving graph Laplacian matrices and gives two solvers that take diametrically opposite approaches, the first highly efficient solver for Laplachian linear systems that parallelizes almost completely.
Abstract: Spectral graph theory is the interplay between linear algebra and combinatorial graph theory. Laplace’s equation and its discrete form, the Laplacian matrix, appear ubiquitously in mathematical physics. Due to the recent discovery of very fast solvers for these equations, they are also becoming increasingly useful in combinatorial optimization, computer vision, computer graphics, and machine learning. In this thesis, we develop highly efficient and parallelizable algorithms for solving linear systems involving graph Laplacian matrices. These solvers can also be extended to symmetric diagonally dominant matrices and M -matrices, both of which are closely related to graph Laplacians. Our algorithms build upon two decades of progress on combinatorial preconditioning, which connects numerical and combinatorial algorithms through spectral graph theory. They in turn rely on tools from numerical analysis, metric embeddings, and random matrix theory. We give two solver algorithms that take diametrically opposite approaches. The first is motivated by combinatorial algorithms, and aims to gradually break the problem into several smaller ones. It represents major simplifications over previous solver constructions, and has theoretical running time comparable to sorting. The second is motivated by numerical analysis, and aims to rapidly improve the algebraic connectivity of the graph. It is the first highly efficient solver for Laplacian linear systems that parallelizes almost completely. Our results improve the performances of applications of fast linear system solvers ranging from scientific computing to algorithmic graph theory. We also show that these solvers can be used to address broad classes of image processing tasks, and give some preliminary experimental results.

Journal ArticleDOI
TL;DR: The purpose of this article is to improve existing lower bounds on the chromatic number chi by using a new technique of converting the adjacency matrix into the zero matrix by conjugating with unitary matrices and use majorization of spectra of self-adjoint matrices.
Abstract: The purpose of this article is to improve existing lower bounds on the chromatic number $\chi$. Let $\mu_1,\ldots,\mu_n$ be the eigenvalues of the adjacency matrix sorted in non-increasing order. First, we prove the lower bound $\chi \ge 1 + \max_m\{\sum_{i=1}^m \mu_i / -\sum_{i=1}^m \mu_{n - i +1}\}$ for $m=1,\ldots,n-1$. This generalizes the Hoffman lower bound which only involves the maximum and minimum eigenvalues, i.e., the case $m=1$. We provide several examples for which the new bound exceeds the Hoffman lower bound. Second, we conjecture the lower bound $\chi \ge 1 + s^+ / s^-$, where $s^+$ and $s^-$ are the sums of the squares of positive and negative eigenvalues, respectively. To corroborate this conjecture, we prove the bound $\chi \ge s^+/s^-$. We show that the conjectured lower bound is true for several families of graphs. We also performed various searches for a counter-example, but none was found. Our proofs rely on a new technique of considering a family of conjugates of the adjacency matrix, which add to the zero matrix, and use majorization of spectra of self-adjoint matrices. We also show that the above bounds are actually lower bounds on the normalized orthogonal rank of a graph, which is always less than or equal to the chromatic number. The normalized orthogonal rank is the minimum dimension making it possible to assign vectors with entries of modulus one to the vertices such that two such vectors are orthogonal if the corresponding vertices are connected. All these bounds are also valid when we replace the adjacency matrix $A$ by $W * A$ where $W$ is an arbitrary self-adjoint matrix and $*$ denotes the Schur product, that is, entrywise product of $W$ and $A$.

Journal ArticleDOI
TL;DR: In this article, the phase transition phenomenon of the graph Laplacian eigenvectors constructed on a certain type of unweighted trees, which was previously observed through numerical experiments, was described.

Journal ArticleDOI
TL;DR: The notion of soft nodes is introduced, which gives sufficient conditions for their existence in general graphs and may be of critical importance for complex physical networks and engineering networks like power grids.
Abstract: To describe the flow of a miscible quantity on a network, we consider the graph wave equation where the standard continuous Laplacian is replaced by the graph Laplacian. The structure of the graph influences strongly the dynamics. Assuming the graph is forced and damped at specific nodes, we derive the amplitude equations using a basis of eigenvectors of the graph Laplacian. These lead us to introduce the notion of soft nodes. We give sufficient conditions for their existence in general graphs. They can cause several effects as we show on small graphs, for example the ineffectiveness of damping applied to them. Soft nodes may be of critical importance for complex physical networks and engineering networks like power grids.

Journal ArticleDOI
TL;DR: Some new and improved sharp upper bounds on the spectral radius q"1(G) of the signless Laplacian matrix of a graph G are obtained.

Proceedings ArticleDOI
12 Dec 2013
TL;DR: Two new spectral quantities, single- and multi-weighted Cheeger constants and corresponding eigenvalue variants, are constructed to direct motions of the defender and the attacker in this dynamic adaptive competition.
Abstract: In this work, the mobility of network nodes is explored as a new promising approach for jamming defense. To fulfill it, properly designed node motion that can intelligently adapt to the jammer's action is crucial. In our study, anti-jamming mobility control is investigated in the context of the single and multiple commodity flow problems, in the presence of one intelligent mobile jammer which can respond to the evasion of legitimate nodes as well. Based on spectral graph theory, two new spectral quantities, single- and multi-weighted Cheeger constants and corresponding eigenvalue variants, are constructed to direct motions of the defender and the attacker in this dynamic adaptive competition. Both analytical and simulation results are presented to justify the effectiveness of the proposed approach.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the branching chain of rings as a quantum graph and used the transfer matrix method to obtain the spectral equation of the system and proved the existence of bound states.

Journal ArticleDOI
TL;DR: In this paper, a generalization of some results concerning the spectral properties of block matrices is presented, and some of its implications on nonnegative matrices and doubly stochastic matrices as well as on graph spectra and graph energy.
Abstract: In this note, we present a generalization of some results concerning the spectral properties of a certain class of block matrices. As applications, we study some of its implications on nonnegative matrices and doubly stochastic matrices as well as on graph spectra and graph energy.

Journal ArticleDOI
TL;DR: This paper proposes a spectral-multiplicity-tolerant graph matching approach that is more robust to noise and structural corruption and has a comparable complexity.

Posted Content
TL;DR: In this article, the spectral properties of the adjacency matrix and random Schrodinger operators on a random regular graph of fixed degree with n vertices were studied and convergence of the integrated density of states on the graph converges to the integrated densities on the infinite regular tree.
Abstract: Consider a random regular graph of fixed degree $d$ with $n$ vertices. We study spectral properties of the adjacency matrix and of random Schrodinger operators on such a graph as $n$ tends to infinity. We prove that the integrated density of states on the graph converges to the integrated density of states on the infinite regular tree and we give uniform bounds on the rate of convergence. This allows to estimate the number of eigenvalues in intervals of size comparable to $\log_{d-1}^{-1}(n)$. Based on related estimates for the Green function we derive results about delocalization of eigenvectors.

Journal ArticleDOI
TL;DR: This paper proposes a nodal domain theorem for the eigenvectors of $M$; points out several relations occurring between the graph's communities and nonnegative eigenvalues of £M; and derives a Cheeger-type inequality for the graph modularity.
Abstract: One of the most relevant tasks in network analysis is the detection of community structures, or clustering. Most popular techniques for community detection are based on the maximization of a quality function called modularity, which in turn is based upon particular quadratic forms associated to a real symmetric modularity matrix $M$, defined in terms of the adjacency matrix and a rank one null model matrix. That matrix could be posed inside the set of relevant matrices involved in graph theory, alongside adjacency, incidence and Laplacian matrices. This is the reason we propose a graph analysis based on the algebraic and spectral properties of such matrix. In particular, we propose a nodal domain theorem for the eigenvectors of $M$; we point out several relations occurring between graph's communities and nonnegative eigenvalues of $M$; and we derive a Cheeger-type inequality for the graph optimal modularity.

Journal ArticleDOI
Enrico Bozzo1
TL;DR: In this article, the Moore-Penrose inverse of the Laplacian matrix of an undirected graph is shown to be symmetric and is strictly related to its connectivity properties.

Journal ArticleDOI
TL;DR: The spectral evolution model, which characterizes the growth of large networks in terms of the eigenvalue decomposition of their adjacency matrices, is studied and two link prediction algorithms based on the learning of the changes to a network’s spectrum are introduced.
Abstract: We introduce and study the spectral evolution model, which characterizes the growth of large networks in terms of the eigenvalue decomposition of their adjacency matrices: In large networks, changes over time result in a change of a graph’s spectrum, leaving the eigenvectors unchanged. We validate this hypothesis for several large social, collaboration, rating, citation, and communication networks. Following these observations, we introduce two link prediction algorithms based on the learning of the changes to a network’s spectrum. These new link prediction methods generalize several common graph kernels that can be expressed as spectral transformations. The first method is based on reducing the link prediction problem to a one-dimensional curve-fitting problem which can be solved efficiently. The second algorithm extrapolates a network’s spectrum to predict links. Both algorithms are evaluated on fifteen network datasets for which edge creation times are known.