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Table of contents
1 Models and frameworks
1.1 The role of models
1.2 Forward problems, inverse problems
1.2.1 Same phenomena, dierent questions
1.2.2 A more precise denition
1.3 A physics modelling framework: the Ising model
1.3.1 Motivation
1.3.2 The Ising model
1.3.3 Universality of critical phenomena
1.3.4 The generalisations
1.3.5 Inverse problems in disordered systems
1.4 A computer science modelling framework: the constraint satisfaction problems
1.4.1 Denitions and examples
1.4.2 Infeasibility of the straightforward solutions
1.4.3 The worst case scenario
1.4.4 The typical case scenario
1.5 Spin glasses, rCSP and large deviations
1.5.1 Spin glasses and CSP: two languages, same problems
1.5.2 Looking for atypical events
1.5.3 The large deviations in a simple case
1.6 Statistical mechanics for non-physics problems
1.6.1 A common framework, many elds of interest
1.6.2 Social systems
1.6.3 Economics and nance
1.6.4 Neural networks
1.6.5 Other biological systems
1.6.6 Signal processing
1.6.7 Computer science
2 The network theory
2.1 Motivations and denitions
2.1.1 Ubiquity of networks
2.1.2 The birth of graph theory
2.1.3 The main features of a graph
2.1.4 Enriching the network metaphor
2.2 A path towards more realistic networks
2.2.1 The comparison between observed and reproduced features
2.2.2 Reproducing the observed average distance and clustering coe cient
2.2.3 Reproducing the observed degree distribution
2.3 Studying graphs as a branch of probability theory
2.3.1 A mathematical description
2.3.2 Combinatorial problems on graphs
2.4 Phase transitions, critical phenomena and statistical physics approach
2.4.1 Dynamics on networks, dynamics of networks
2.4.2 Statistical physics and network theory
2.4.3 Critical phenomena on networks
2.5 Examples of other problems
2.5.1 The detection of communities
2.5.2 Networked versions of other problems
3 Contagion dynamics on graphs
3.1 General framework
3.1.1 The problem and its range of applicability
3.1.2 Epidemic processes and compartmental models
3.1.3 The choice of the dynamics
3.1.4 The characteristics of the graph
3.1.5 One spreading phenomenon, many possible questions
3.1.6 Dierent approaches
3.2 Minimal contagious sets in random regular graphs
3.2.1 Denition of the problem
3.2.2 The energy function
3.2.3 Mapping to other standard problems in graph theory
3.3 The cavity method treatment
3.3.1 The replica symmetric RS formalism
3.3.2 The breaking down of the RS assumption
3.3.3 1RSB and energetic 1RSB formalism
3.3.4 The energetic 1RSB formalism
3.4 Main results
3.4.1 The solutions of the problem
3.4.2 Analytical results
3.4.3 Numerical results
3.5 Future perspectives
4 Exploring networks
4.1 Graph exploration and random walks
4.1.1 Exploring a graph, a very general problem
4.1.2 Interest and applications in dierent elds
4.1.3 Random walks on graphs
4.1.4 Joining the two: graph exploration through random walks
4.2 Rare event statistics on random walks on networks
4.2.1 The problem
4.2.2 The model
4.2.3 The large deviations
4.3 Main results
4.3.1 A degree-based approximation
4.3.2 Localisation transition
4.3.3 Mode-switching transition
4.4 Summary and future perspectives
4.4.1 Summary
4.4.2 Better understanding of the critical behaviour
4.4.3 Dierent functions
4.4.4 Functions taking value on the links
4.4.5 Dierent topologies
4.4.6 Improvement of clustering techniques
5 Inferring the graph 99
5.1 Motivation, applications and connection to other problems
5.1.1 Motivation
5.1.2 The matrix completion problem
5.1.3 The collaborative ltering
5.2 The inference of a network of interactions from a partial knowledge of the correlations
5.2.1 Denition of the problem
5.2.2 Motivation, applications and previous work
5.3 The model
5.3.1 Statistical context of the project
5.3.2 Prior, posterior and optimal estimate of the model
5.4 Choosing the couplings and evaluating the performances
5.4.1 Articial models
5.4.2 The geometry of J
5.4.3 Generating synthetic J
5.4.4 The observables
5.5 Main analytical results
5.5.1 Reasons justifying the study of the Hessian
5.5.2 Derivation of the Hessian
5.5.3 Derivation of the Hessian in the eigenbasis of C
5.5.4 Perturbative approximation of the Hessian
5.6 The heuristics
5.6.1 Choices based on inference
5.6.2 Heuristics making use of the Hessian
5.6.3 Intractability of the complete Hessian
5.6.4 Diagonal elements of the Hessian
5.6.5 Validity of the perturbative approximation
5.7 Initial knowledge and eects of measurements
5.7.1 Choice of the initial condition
5.7.2 Update of C after a measurement
5.8 Results
5.8.1 Dependency on the initial condition
5.8.2 The exploration-exploitation trade-o
5.8.3 Dierent heuristics on a xed geometry
5.8.4 Dierent graphs, dierent features
5.9 Future perspectives
5.9.1 Biological applications
5.9.2 Theoretical challenges
5.9.3 More complicated heuristics
5.10 Appendices
5.10.1 Caveat on the procedure for creating synthetic J
5.10.2 The value of the couplings
5.10.3 Regularisation and noise
5.10.4 Uniqueness of the minimum
5.10.5 Completion of the initial condition
6 Conclusions and perspectives
6.1 The network theory in the big data era
6.1.1 The end of theory?
6.1.2 An example of rich data analysis: Quora
6.1.3 New solutions to old questions
6.1.4 The future of network theory
6.2 More data, same problems
6.2.1 Optimisation of spreading on networks
6.2.2 Extreme events in network exploration
6.2.3 Inferring networks with partial information
References
A Minimal contagious sets in random regular graphs
B Rare events statistics of random walks on networks: localization and
other dynamical phase transitions
