Lectures

Background 
Network theory has recently emerged as a major tool relevant for very different disciplines ranging from physics, applied mathematics, to biology, computer sciences and almost all areas of social sciences such as geography, urbanism and sociology. This lecture will address the characterization of large networks, the most important models and the effect of the network structure on various dynamical processes. This course does not require specific knowledge and is designed to be accessible to anyone with a minimal background in statistical physics, probability, and graph theory.

Outline of the course 
Note that this outline is mainly indicative and I will try to keep it updated
1. The structure of large networks 1.1. Examples in different fields 1.2. Elements of graph theory 1.3. Characterization of large networks: a. Degree distribution b. Shortest paths and diameter, assortativity, clustering, rich club coefficient c. Weighted networks d. Communities, modularity and resolution limit e. Graph Laplacian 1.4. Empirical results and classes of networks 1.5. Characterization of nodes: centralities 1.6. Advanced topics a. Spatial and planar networks: centrality, simplicity and classifications b. Temporal networks: timerespecting paths c. Multilayer networks: interdependency, synchronization efficiency. Information perspective 2. Modeling networks 2.1. The archetype: ErdosRenyi, the hiddenvariable model, and the configuration model 2.2. Smallworlds a. The WattsStrogatz model: a scaling ansatz b. The Kleinberg model. Navigation and message passing 2.3. Preferential attachment and variants a. The BarabasiAlbert model; the fitness model; rank effect b. Effect of aging, bounded knowledge, and space c. The copy model 2.4. Advanced topics a. Modeling weighted networks: coupling weights and topology b. Modeling spatial networks: costbenefit analysis and HOT models, optimal networks c. Modeling temporal networks; core and branches template d. Modeling growing multilayer networks 3. Dynamical processes on networks 3.1. Random walk, diffusion on networks 3.2. Percolation and resilience: ER case, MolloyReed criterium, Random failure 3.3. Traffic: Generalities and ring effect 3.4. Epidemics a. Compartment models; SIS and SIR: Heterogeneous meanfield approach; probabilistic approach b. Immunization strategies c. Advanced topics (i) Rumor spreading (ii) The metapopulation model 3.5. Navigation and routing in complex networks: Kleinberg’s result 
References
Reviews:
Books:
Papers:
Characterization of networks
Models of networks
 Albert, Réka, and AlbertLászló Barabási. "Statistical mechanics of complex networks." Reviews of modern physics 74.1 (2002): 47.
 Dorogovtsev, Sergey N., and Jose FF Mendes. "Evolution of networks." Advances in physics 51.4 (2002): 10791187.
 Newman, Mark EJ. "The structure and function of complex networks." SIAM review 45.2 (2003): 167256.
 Barthelemy, Marc. "Spatial networks." Physics Reports 499.1 (2011): 1101.
Books:
 Newman, Mark. "Networks: an introduction. 2010." Oxford University Press Inc., New York: 12.
 Cohen, Reuven, and Shlomo Havlin. Complex networks: structure, robustness and function. Cambridge University Press, 2010.
 Dorogovtsev, Sergei N., and José FF Mendes. Evolution of networks: From biological nets to the Internet and WWW. OUP Oxford, 2013.
 Barrat, Alain, Marc Barthelemy, and Alessandro Vespignani. Dynamical processes on complex networks. Cambridge university press, 2008.
Papers:
Characterization of networks
 Example of calculation of the BC: Lion, Benjamin, and Marc Barthelemy. "Betweenness centrality patterns in random planar graphs." arXiv preprint arXiv:1611.03232 (2016).
 Effect of the curvature on the max of the BC: Narayan, Onuttom, and Iraj Saniee. "Largescale curvature of networks." Physical Review E 84.6 (2011): 066108 [Note: Hyperbolic case: max(g) scales as N^2; lattice case max(g) scales as N^(1+1/d)].
 Other discussion on the effect of the curvature of networks: Jonckheere, Edmond, et al. "Euclidean versus hyperbolic congestion in idealized versus experimental networks." Internet Mathematics 7.1 (2011): 127
 Evolution of Paris
 Mapping planar graphtree: in botanics, in combinatorics
 A typology of planar graphs
Models of networks
 Calculations on the WattsStrogatz model: Barrat & Weigt paper
 BarabasiAlbert original paper
 Fitness model
 Review article by MEJ Newman
 Weighted network model: preferential attachment and traffic update
 Hidden variable for weighted network
 Book on random geometric graphs by M. Penrose
 Article by Dall & Christensen on random geometric graphs (with calculation of the clustering coeff.)
Slides
Soon