Structural Analysis of Laplacian Spectral Properties of Large-Scale Networks

Preciado, Víctor M.; Jadbabaie, Ali; Verghese, George C.

doi:10.1109/tac.2013.2261187

Cited by 33 publications

(14 citation statements)

References 10 publications

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“…Concretely, we use the eigenvalues to compare graphs because the spectral moments of the non-backtracking matrix describe certain aspects of the length spectrum (see "Operationalizing the length spectrum" section). The spectral moments of other matrices (e.g., adjacency and Laplacian matrices) also describe structural features of networks (Estrada 1996;Preciado et al 2013). Many distance methods have been proposed recently (Soundarajan et al 2014;Koutra et al 2016;Bagrow and Bollt 2018;Bento and Ioannidis 2018;Onnela et al 2012;Schieber et al 2017;Chowdhury and Mémoli 2017;2018;Berlingerio et al 2013;Yaveroglu et al 2014).…”

Section: Related Workmentioning

confidence: 99%

Non-backtracking cycles: length spectrum theory and graph mining applications

2019

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Graph distance and graph embedding are two fundamental tasks in graph mining. For graph distance, determining the structural dissimilarity between networks is an ill-defined problem, as there is no canonical way to compare two networks. Indeed, many of the existing approaches for network comparison differ in their heuristics, efficiency, interpretability, and theoretical soundness. Thus, having a notion of distance that is built on theoretically robust first principles and that is interpretable with respect to features ubiquitous in complex networks would allow for a meaningful comparison between different networks. For graph embedding, many of the popular methods are stochastic and depend on black-box models such as deep networks. Regardless of their high performance, this makes their results difficult to analyze which hinders their usefulness in the development of a coherent theory of complex networks. Here we rely on the theory of the length spectrum function from algebraic topology, and its relationship to the non-backtracking cycles of a graph, in order to introduce two new techniques: Non-Backtracking Spectral Distance (NBD) for measuring the distance between undirected, unweighted graphs, and Non-Backtracking Embedding Dimensions (NBED) for finding a graph embedding in low-dimensional space. Both techniques are interpretable in terms of features of complex networks such as presence of hubs, triangles, and communities. We showcase the ability of NBD to discriminate between networks in both real and synthetic data sets, as well as the potential of NBED to perform anomaly detection. By taking a topological interpretation of non-backtracking cycles, this work presents a novel application of topological data analysis to the study of complex networks.

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Section: Related Workmentioning

confidence: 99%

Non-backtracking cycles: length spectrum theory and graph mining applications

2019

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“…We make the standing assumption that the local dynamics (19) admit a stable fixed point x * and that the units synchronize, so that the solutions X(t) of (19)- (20) converge to the (stable) fixed point X * = [x * . .…”

Section: Nonlinear Systems With Identical Unitsmentioning

confidence: 99%

“…Koopman operator Let X(t) be a trajectory solution of (19)- (20) associated with the initial condition X 0 . We suppose that p ≪ n measurements of X(t) are obtained through a possibly nonlinear observation function f : R mn → R p , which depends on a few local states in our case.…”

Section: Propositionmentioning

confidence: 99%

Spectral Identification of Networks Using Sparse Measurements

Mauroy¹,

Hendrickx²

2017

SIAM J. Appl. Dyn. Syst.

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We propose a new method to recover global information about a network of interconnected dynamical systems based on observations made at a small number (possibly one) of its nodes. In contrast to classical identification of full graph topology, we focus on the identification of the spectral graph-theoretic properties of the network, a framework that we call spectral network identification.The main theoretical results connect the spectral properties of the network to the spectral properties of the dynamics, which are well-defined in the context of the so-called Koopman operator and can be extracted from data through the Dynamic Mode Decomposition algorithm. These results are obtained for networks of diffusively-coupled units that admit a stable equilibrium state. For large networks, a statistical approach is considered, which focuses on spectral moments of the network and is well-suited to the case of heterogeneous populations.Our framework provides efficient numerical methods to infer global information on the network from sparse local measurements at a few nodes. Numerical simulations show for instance the possibility of detecting the mean number of connections or the addition of a new vertex using measurements made at one single node, that need not be representative of the other nodes' properties.Network identification methods developed in the framework of dynamical systems theory are usually not well-suited to the analysis of real networks such as biological networks, social networks, etc. Most of them are invasive, requiring the modification of the network connectivity or dynamics. In addition, some of them cannot be used "offline" for data analysis, since they require to interact dynamically with the network. More importantly, all the methods proposed so far for full network reconstruction require measurements at all the nodes of the network. Partial measurements have been considered in [9] in the context of linear timeinvariant systems for a partial reconstruction of the network between the measured states, and yet the authors showed that the problem cannot be solved without additional information on the system. It can actually be shown that measuring all nodes is necessary for a full network reconstruction, and this is usually out of reach in large real networks. Indeed, the number of sensors is limited and typically (much) smaller that the number of nodes. Some nodes of real networks might also not be accessible, or the only available information might be the averaged activity of a group of nodes lying in a given region of the network (e.g. electrical activity in a region of the brain). All these limitations motivate the network identification framework developed in this paper, which overcomes them.In this work, we take the view that identifying the exact complete topology of large networks is not only practically impossible, as mentioned above, but also often unnecessary. The presence or absence of an edge between two specific nodes is for instance often only marginally relevant when analyzing the global ...

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“…Let us consider a discrete random variable X whose probability density function is µ LH . The moments of this random variable satisfy the following [19]:…”

Section: Moment-based Spectral Boundsmentioning

confidence: 99%

“…Applying Theorem 4.1 to the spectral density µ LH of a given graph H with spectral moments (m 0 , m 1 , ..., m 2r+1 ), we can find a lower bound on its largest eigenvalue, λ 1 (L H ), as follows [19]:…”

Section: Moment-based Spectral Boundsmentioning

confidence: 99%

Laplacian Spectral Properties of Graphs from Random Local Samples

Wu¹,

Preciado

2014

Proceedings of the 2014 SIAM International Conference on Data Mining

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The Laplacian eigenvalues of a network play an important role in the analysis of many structural and dynamical network problems. In this paper, we study the relationship between the eigenvalue spectrum of the normalized Laplacian matrix and the structure of 'local' subgraphs of the network. We call a subgraph local when it is induced by the set of nodes obtained from a breath-first search (BFS) of radius r around a node. In this paper, we propose techniques to estimate spectral properties of the normalized Laplacian matrix from a random collection of induced local subgraphs. In particular, we provide an algorithm to estimate the spectral moments of the normalized Laplacian matrix (the power-sums of its eigenvalues). Moreover, we propose a technique, based on convex optimization, to compute upper and lower bounds on the spectral radius of the normalized Laplacian matrix from local subgraphs. We illustrate our results studying the normalized Laplacian spectrum of a large-scale e-mail network. IntroductionUnderstanding the relationship between the structure of a network and its eigenvalues is of great relevance in the field of network science (see [3], [16] and references therein). The growing availability of massive databases, computing facilities, and reliable data analysis tools has provided a powerful framework to explore this relationship for many real-world networks. On the other hand, in many cases of practical interest, one cannot efficiently retrieve and/or store the exact full topology of a large-scale network. Alternatively, it is usually easy to retrieve local samples of the network structure. In this paper, we focus our attention on local sample of the network structure given in the form of a subgraph induced by the set of nodes obtained from a breath-first search (BFS) of small radius r around a particular node.We study the relationship between the normalized Laplacian spectrum of a graph and a random collection of local subgraphs. We show how local structural information contained in these localized subgraphs can be efficiently aggregated to infer global properties of the

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Structural Analysis of Laplacian Spectral Properties of Large-Scale Networks

Cited by 33 publications

References 10 publications

Non-backtracking cycles: length spectrum theory and graph mining applications

Non-backtracking cycles: length spectrum theory and graph mining applications

Spectral Identification of Networks Using Sparse Measurements

Laplacian Spectral Properties of Graphs from Random Local Samples

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