## Random Walks and Diffusions on Graphs and Databases: An IntroductionMost networks and databases that humans have to deal with contain large, albeit finite number of units. Their structure, for maintaining functional consistency of the components, is essentially not random and calls for a precise quantitative description of relations between nodes (or data units) and all network components. This book is an introduction, for both graduate students and newcomers to the field, to the theory of graphs and random walks on such graphs. The methods based on random walks and diffusions for exploring the structure of finite connected graphs and databases are reviewed (Markov chain analysis). This provides the necessary basis for consistently discussing a number of applications such diverse as electric resistance networks, estimation of land prices, urban planning, linguistic databases, music, and gene expression regulatory networks. |

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### Contents

1 | |

Worth Another Binary Relation Graphs | 18 |

Permutations Sieved Through Adjacency Graph Automorphisms | 43 |

Exploring Undirected Graphs by Random Walks | 55 |

Embedding of Graphs in Probabilistic Euclidean Space | 73 |

Random Walks and Electric Resistance Networks | 85 |

Random Walks and Diffusions on Directed Graphs and Interacting Networks | 92 |

Structural Analysis of Networks and Databases | 107 |

When Feedbacks Matter Epidemics Synchronization and Selfregulation in Complex Networks | 171 |

Critical Phenomena on Large Graphs with Regular Subgraphs | 219 |

236 | |

Glossary of Graph Theory | 253 |

259 | |

### Other editions - View all

Random Walks and Diffusions on Graphs and Databases: An Introduction Philippe Blanchard,Dimitri Volchenkov No preview available - 2011 |

Random Walks and Diffusions on Graphs and Databases: An Introduction Philippe Blanchard,Dimitri Volchenkov No preview available - 2011 |

Random Walks and Diffusions on Graphs and Databases: An Introduction Philippe Blanchard,Dimitri Volchenkov No preview available - 2013 |

### Common terms and phrases

ˇ ˇ ˇ adjacency matrix analysis asymptotic Aut.G automorphism behavior Blanchard characterized complex networks components connected graph consider corresponding coupled map lattice coupled maps cycles degree denote diagonal diagram diffusion dimension directed graph dynamical eigenvalues eigenvectors elements entropy equals equation Euclidean expected number feedback circuits finite first-passage function gene geometric representation glottochronology graph G graph G.V;E graph theory infected interactions inverse J.S. Bach language family Laplace operator lattice lexical distances linear Lov´asz Maanyan Markov chain mean musical compositions nodes nonlinear normalized number of edges orthogonal orthonormal partition permutation matrix pitches polynomial probability properties random graph random walks random walks defined randomly regulatory networks relation scale free self-adjoint operators shortest path space spectral stationary distribution statistical stochastic structure subgraph supernodes Swadesh switching parameters symmetric tonal transition matrix tree turbulent fraction undirected vector vertex vertices Volchenkov