Graphs and complex networks

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Table of contents

1 Brief history and state of the art 
1.1 Graphs and complex networks
1.2 Network metrology
1.3 Dynamic networks and metrology
1.4 Models of networks
2 Model 
2.1 Motivation: tracetree measurements
2.2 Causes of the observed dynamics
2.3 Model description
2.4 Preliminary results of simulation
2.4.1 Evolution of the number of distinct nodes
2.4.2 Observation number vs. block number
List of definitions
3 Characterising the dynamics 
3.1 Linearity of the evolution
3.1.1 Correlation coefficient
3.1.2 Coefficients of segmented linear regression
3.2 Characterisation of the slope
3.2.1 Random graphs
3.2.2 Power-law graphs
3.3 In search of unified laws
3.3.1 Impact of the size of the shortest path subgraph
3.3.2 Probability of shortest path subgraph modifications
3.4 Real-world measurements
3.4.1 Frequency of measurements
3.4.2 Size of the shortest path tree vs slope
4 Size of shortest path subgraphs 
4.1 Definitions
4.2 Complete and quasi-complete graphs
4.3 Dense random graphs (p is fixed, n Ñ 8)
4.4 Sparse random graphs with unbounded mean degree (p Ñ 0 and np Ñ 8 as n Ñ 8)
4.4.1 Approximated expectation of the size
4.4.2 Classification of sparse graphs according to the distance distribution
5 Real and observed dynamics 
5.1 Impact of the measurement frequency
5.2 Inferring the evolution speed
5.2.1 Poisson process
5.2.2 sps-process
5.2.3 spt-process
5.3 Nonuniform dynamics
Conclusion
A Résumé
Bibliography