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Approximation of the Diagonal of a Laplacian's Pseudoinverse for Complex Network Analysis [article]

Eugenio Angriman, Maria Predari, Alexander van der Grinten, Henning Meyerhenke
2021 arXiv   pre-print
From a computational point of view, the three measures have in common that their crucial component is the diagonal of the graph Laplacian's pseudoinverse, L^†.  ...  We are motivated here by three electrical measures for the analysis of large small-world graphs G = (V, E) – i.e., graphs with diameter in O(log |V|), which are abundant in complex network analysis.  ...  Matrix Computations for Mining Large Dynamic Complex Networks).  ... 
arXiv:2006.13679v2 fatcat:e6c57yt5nnepbitwrqljuozeqe

Approximation of the Diagonal of a Laplacian's Pseudoinverse for Complex Network Analysis

Eugenio Angriman, Maria Predari, Alexander van der Grinten, Henning Meyerhenke, Peter Sanders, Fabrizio Grandoni, Grzegorz Herman
2020 European Symposium on Algorithms  
From a computational point of view, the three measures have in common that their crucial component is the diagonal of the graph Laplacian's pseudoinverse, L^+.  ...  We are motivated here by three electrical measures for the analysis of large small-world graphs G = (V, E) - i. e., graphs with diameter in O(log |V|), which are abundant in complex network analysis.  ...  Acknowledgements We thank our colleague Fabian Brandt-Tumescheit for his technical support for the experiments.  ... 
doi:10.4230/lipics.esa.2020.6 dblp:conf/esa/AngrimanPGM20 fatcat:ucyir4c7bbaala4zhfsxp6ocmq

InfiniteWalk: Deep Network Embeddings as Laplacian Embeddings with a Nonlinearity [article]

Sudhanshu Chanpuriya, Cameron Musco
2020 arXiv   pre-print
Further, we show that by a applying a simple nonlinear entrywise transformation to this pseudoinverse, we recover a good approximation of the finite-T objective and embeddings that are competitive with  ...  Recent work of Qiu et al. (2018) provides a closed-form expression for the DeepWalk objective, obviating the need for sampling for small datasets and improving accuracy.  ...  [33] shows that much of the complexity of GCNs comes from components inspired by other forms of deep learning that have limited utility for network data.  ... 
arXiv:2006.00094v2 fatcat:bqwymzoy4bc77pap3nizlwuqhi

Frequency Response and Gap Tuning for Nonlinear Electrical Oscillator Networks

Harish S. Bhat, Garnet J. Vaz, Jesus Gomez-Gardenes
2013 PLoS ONE  
For such networks, we develop two algorithms to compute the steady-state response when a subset of nodes are driven at the same fixed frequency.  ...  We develop a Newton-type method that solves for the network inductances such that the graph Laplacian achieves a desired set of eigenvalues; this method enables one to move the eigenvalues while keeping  ...  Performed the experiments: GJV HSB. Analyzed the data: HSB GJV. Wrote the paper: HSB GJV. Designed the software used in analysis: GJV HSB.  ... 
doi:10.1371/journal.pone.0078009 pmid:24223751 pmcid:PMC3817173 fatcat:dkqg77cll5fthkzc6j4rbiabci

Higher-order interactions can better optimize network synchronization

Per Sebastian Skardal, Lluís Arola-Fernández, Dane Taylor, Alex Arenas
2021 Physical Review Research  
Here, we study the optimization of collective behavior in networks with higher-order interactions encoded in clique complexes.  ...  We find that this phenomenon stems from the broadening of a composite Laplacian's eigenvalue spectrum, which improves the optimal collective behavior and widens the range of possible behaviors.  ...  INTRODUCTION Complex networks provide the structural architecture for dynamical processes from a wide array of disciplines, and therefore their study constitutes an important fundamental area of research  ... 
doi:10.1103/physrevresearch.3.043193 fatcat:jp6ozrt4ajgk7i4ohncpu22xui

Higher-order interactions improve optimal collective dynamics on networks [article]

Per Sebastian Skardal, Lluís Arola-Fernández, Dane Taylor, Alex Arenas
2021 arXiv   pre-print
Here, we study the optimization of collective behavior in networks with higher-order interactions encoded in clique complexes.  ...  We find that this phenomenon stems from the broadening of a composite Laplacian's eigenvalue spectrum, which improves the optimal collective behavior and widens the range of possible behaviors.  ...  INTRODUCTION Complex networks provide the structural architecture for dynamical processes from a wide array of disciplines, and therefore their study constitutes an important fundamental area of research  ... 
arXiv:2108.08190v2 fatcat:nf4fjwzmabdgpeoqx54aenvhxe

Kirchhoff Index as a Measure of Edge Centrality in Weighted Networks: Nearly Linear Time Algorithms [chapter]

Huan Li, Zhongzhi Zhang
2018 Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms  
The Kirchhoff index of a network is defined as the sum of effective resistances over all vertex pairs.  ...  The centrality of an edge e is reflected in the increase of Kirchhoff index of the network when the edge e is partially deactivated, characterized by a parameter θ.  ...  Kirchhoff Index Into Quadratic Forms of L † By Fact 2.4, the Kirchhoff Index of a graph equals n times the trace the Laplacian's pseudoinverse.  ... 
doi:10.1137/1.9781611975031.153 dblp:conf/soda/LiZ18 fatcat:hzpwu4oh7fb7veav53wq6mh6pu

Machine learning and geodesy: A survey

Jemil Butt, Andreas Wieser, Zan Gojcic, Caifa Zhou
2021 Journal of Applied Geodesy  
Machine learning methods are more flexible with respect to assumed regularity of the input and the form of the desired outputs and allow for nonparametric stochastic models at the cost of substituting  ...  This article aims at examining common grounds of geodetic data analysis and machine learning and showcases applications of algorithms for supervised and unsupervised learning to tasks concerned with optimal  ...  Acknowledgment: The authors acknowledge the work of the two anonymous reviewers who contributed to this paper by providing suggestions on content and formatting of the paper thereby improving its readability  ... 
doi:10.1515/jag-2020-0043 fatcat:uavqma3z6nh7vaxqani5oh3u7q

Graph Structured Normal Means Inference

James Sharpnack
2018
We will also show how one can form a decomposition of the graph from a spanning tree which will lead to a test for activity in the graph.  ...  We will develop asymptotic guarantees of the performance of statistical estimators and tests, by stating conditions for consistency by properties of the graph (e.g. graph spectra).  ...  Because G −B is disconnected ∆ −B is block diagonal. Hence, U is block diagonal with blocks being the eigen- vectors of each component of A.  ... 
doi:10.1184/r1/6718457.v1 fatcat:t7t3owpj7bbe7j76v7iyfynb6m