Aspirational Block Program Block Syaldey District - Almora
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;
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.
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
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.