Diffusion and Superposition Distances for Signals Supported on Networks

We introduce the diffusion and superposition distances as two metrics to compare signals supported in the nodes of a network. Both metrics consider the given vectors as initial temperature distributions and diffuse heat through the edges of the graph. The similarity between the given vectors is determined by the similarity of the respective diffusion profiles. The superposition distance computes the instantaneous difference between the diffused signals and integrates the difference over time. The diffusion distance determines a distance between the integrals of the diffused signals. We prove that both distances define valid metrics and that they are stable to perturbations in the underlying network. We utilize numerical experiments to illustrate their utility in classifying signals in a synthetic network as well as in classifying ovarian cancer histologies using gene mutation profiles of different patients. We also utilize diffusion as part of a label propagation method in semi-supervised learning to classify handwritten digits.

[1]  D. Bu,et al.  Topological structure analysis of the protein-protein interaction network in budding yeast. , 2003, Nucleic acids research.

[2]  Yann LeCun,et al.  The mnist database of handwritten digits , 2005 .

[3]  Santiago Segarra,et al.  Hierarchical clustering and consensus in trust networks , 2013, 2013 5th IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

[4]  T. Fujimoto,et al.  TWO CHARACTERIZATIONS OF INVERSE-POSITIVE MATRICES: THE HAWKINS-SIMON CONDITION AND THE LE CHATELIER-BRAUN PRINCIPLE ∗ , 2004 .

[5]  D. Long.  Networks of the Brain , 2011 .

[6]  J. Alison Noble,et al.  Ultrasound image segmentation: a survey , 2006, IEEE Transactions on Medical Imaging.

[7]  R. Mittler,et al.  Reactive oxygen gene network of plants. , 2004, Trends in plant science.

[8]  A. Lykov,et al.  Analytical heat diffusion theory , 1968 .

[9]  C. Loan.  The Sensitivity of the Matrix Exponential , 1977 .

[10]  David Kempe,et al.  A decentralized algorithm for spectral analysis , 2004, STOC '04.

[11]  Martin A. Nowak,et al.  Evolutionary dynamics on graphs , 2005, Nature.

[12]  Sunil K. Narang,et al.  Downsampling graphs using spectral theory , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[13]  Wei Ren,et al.  Consensus based formation control strategies for multi-vehicle systems , 2006, 2006 American Control Conference.

[14]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[15]  Alexander J. Smola,et al.  Kernels and Regularization on Graphs , 2003, COLT.

[16]  Mark Freidlin,et al.  Diffusion Processes on Graphs and the Averaging Principle , 1993 .

[17]  M. Newman,et al.  Finding community structure in networks using the eigenvectors of matrices. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[19]  Gene H. Golub,et al.  Matrix computations , 1983 .

[20]  John D. Lafferty,et al.  Diffusion Kernels on Graphs and Other Discrete Input Spaces , 2002, ICML.

[21]  Saiful Islam,et al.  Mahalanobis Distance , 2009, Encyclopedia of Biometrics.

[22]  George J. Pappas,et al.  Stable flocking of mobile agents, part I: fixed topology , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[23]  Pascal Frossard,et al.  The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains , 2012, IEEE Signal Processing Magazine.

[24]  Andrew M. Gross,et al.  Network-based stratification of tumor mutations , 2013, Nature Methods.

[25]  Gary D. Bader,et al.  Pathway Commons, a web resource for biological pathway data , 2010, Nucleic Acids Res..

[26]  M. Degroot.  Reaching a Consensus , 1974 .

[27]  Ronald R. Coifman,et al.  Regularization on Graphs with Function-adapted Diffusion Processes , 2008, J. Mach. Learn. Res..

[28]  S. Wasserman,et al.  Models and Methods in Social Network Analysis , 2005 .

[29]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

[30]  D. Burago,et al.  A Course in Metric Geometry , 2001 .

[31]  E. Eckert,et al.  Analysis of heat and mass transfer , 1971 .

[32]  Seif Haridi,et al.  Distributed Algorithms , 1992, Lecture Notes in Computer Science.

[33]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[34]  E.M. Atkins,et al.  A survey of consensus problems in multi-agent coordination , 2005, Proceedings of the 2005, American Control Conference, 2005..

[35]  Janke Dittmer.  Consensus formation under bounded confidence , 2001 .

[36]  Patrick J. Wolfe,et al.  Toward signal processing theory for graphs and non-Euclidean data , 2010, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing.

[37]  Bernhard Schölkopf,et al.  Comparing support vector machines with Gaussian kernels to radial basis function classifiers , 1997, IEEE Trans. Signal Process..

[38]  Pascal Frossard,et al.  Cooperative Special Issue on Graph Signal Processing in the IEEE Journal of Selected Topics in Signal Processing and the IEEE Transactions on Signal and Information Processing Over Networks , 2017, IEEE Trans. Signal Inf. Process. over Networks.

[39]  Louis A. Hageman,et al.  Iterative Solution of Large Linear Systems. , 1971 .

[40]  Michael C. Hout,et al.  Multidimensional Scaling , 2003, Encyclopedic Dictionary of Archaeology.

[41]  José M. F. Moura,et al.  Discrete Signal Processing on Graphs , 2012, IEEE Transactions on Signal Processing.