Complex networks: community detection and virus propagation
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brief presentation done for a phd course I attended last semester (fall of 2015) at NTNU about Advanced Information Retrieval
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Complex networks: community detection and virus propagation
- 1. Complex networks: community detection, virus propagation and immunization Eliezer de Souza da Silva eliezer.souza.silva@idi.ntnu.no Friday 27th November, 2015
- 2. Community detection and virus propagation E.S. Silva Community Detection Clustering coefficient Detecting communities Virus propagation and immunization Terminology and basic models In social network References Summary 1 Community Detection Clustering coefficient Detecting communities 2 Virus propagation and immunization Terminology and basic models In social network 3 References
- 3. Community detection and virus propagation E.S. Silva Community Detection Clustering coefficient Detecting communities Virus propagation and immunization Terminology and basic models In social network References Introduction basic intuition A community is a set of nodes with more connection within the set than outside. Elements of a community may: Share common properties; Play similar roles; Compress/summarize collective information;
- 4. Community detection and virus propagation E.S. Silva Community Detection Clustering coefficient Detecting communities Virus propagation and immunization Terminology and basic models In social network References Introduction
- 5. Community detection and virus propagation E.S. Silva Community Detection Clustering coefficient Detecting communities Virus propagation and immunization Terminology and basic models In social network References Introduction main question How to formalize this basic intuition? intra-cluster density and extra-cluster density; clustering coefficient; connectedness, centrality and edge-betweenness; graph partitioning; Authorities and hubs; ...
- 6. Community detection and virus propagation E.S. Silva Community Detection Clustering coefficient Detecting communities Virus propagation and immunization Terminology and basic models In social network References Clustering coefficient Global: measure of how elements of the graph tend to form cluster Local: measure of “transitivity” of the neighbourhood of one vertex. The probability of “friends of friends” being connected.
- 7. Community detection and virus propagation E.S. Silva Community Detection Clustering coefficient Detecting communities Virus propagation and immunization Terminology and basic models In social network References Local Clustering coefficient Given a vertex vi, with ki neighbours with ni edges between the set of neighbours we define the local clustering coefficient [1] for direct (Eq 1) and undirected graphs (Eq 2) for ki > 1 as: Ci = 2ni ki(ki − 1) (1) Ci = ni ki(ki − 1) (2) If ki = 0, 1 then Ci = 0
- 8. Community detection and virus propagation E.S. Silva Community Detection Clustering coefficient Detecting communities Virus propagation and immunization Terminology and basic models In social network References Global Clustering coefficient For a graph with N vertices: Watts and Strogatz (Eq 3) Counting triangles and triples (Eq 4) C = 1 N N i=1 Ci (3) C = 3 × number of triangles in the graph number of connected triples in the graph (4)
- 9. Community detection and virus propagation E.S. Silva Community Detection Clustering coefficient Detecting communities Virus propagation and immunization Terminology and basic models In social network References Detecting communities Dendograms; Edge-betweenness (centrality); Max-flow min-cut; Graph partitioning; Bipartite cores; Spectral methods; cross-association: minimum description length and compressability; Random walks; Metric embeddings.
- 10. Community detection and virus propagation E.S. Silva Community Detection Clustering coefficient Detecting communities Virus propagation and immunization Terminology and basic models In social network References Virus propagation and immunization Long standing tradition of mathematical models in epidemiology. Traditional models are based on dividing the population in a small set of compartments with few differential equations describing the transition between the compartments – many analytical results derived from this framework. Application in distinct emerging areas such as information spreading, social contagion, marketing and analysis of online social networks depends on expanding this framework to more complex network topologies using more computational intensive tools.
- 11. Community detection and virus propagation E.S. Silva Community Detection Clustering coefficient Detecting communities Virus propagation and immunization Terminology and basic models In social network References Terminology basic terminology S: Susceptible/healthy I: Infected (and infectious) R: Removed/recovered – the node has immunity for life (or is deceased) V: Vigilant – the node can not be infected (but may lose it’s immunity, depending on the VPM) E: Exposed – the node is not infectious, but it is a carrier of the virus, and it will eventually evolve to the “Infected/Infectious” state.
- 12. Community detection and virus propagation E.S. Silva Community Detection Clustering coefficient Detecting communities Virus propagation and immunization Terminology and basic models In social network References Basic models (a) SIR and SEIV (b) SIR,SIS,SIRS and SEIR
- 13. Community detection and virus propagation E.S. Silva Community Detection Clustering coefficient Detecting communities Virus propagation and immunization Terminology and basic models In social network References Main results
- 14. Community detection and virus propagation E.S. Silva Community Detection Clustering coefficient Detecting communities Virus propagation and immunization Terminology and basic models In social network References Main result main result For S ∗ I2V∗ model with arbitrar undirect graph with adjancy matrix A the sufficient condition for stability is: s < 1 where s is s = λ1 × CVPM
- 15. Community detection and virus propagation E.S. Silva Community Detection Clustering coefficient Detecting communities Virus propagation and immunization Terminology and basic models In social network References Modelling approaches Markov theory; Mean Field (individual, density based); non-markovian simulations;
- 16. Community detection and virus propagation E.S. Silva Community Detection Clustering coefficient Detecting communities Virus propagation and immunization Terminology and basic models In social network References Contagion in social networks
- 17. Community detection and virus propagation E.S. Silva Community Detection Clustering coefficient Detecting communities Virus propagation and immunization Terminology and basic models In social network References References I Duncan J Watts and Steven H Strogatz. Collective dynamics of ‘small-world’networks. nature, 393(6684):440–442, 1998.