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Protein Network scale free
Jeong et al, Nature (2001)
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Sex-web
Nodes people (Females Males) Links sexual
relationships
4781 Swedes 18-74 59 response rate.
Liljeros et al. Nature 2001
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Social networks
Classes/departments
guppies
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Network communities from biochemical to social
Tamás Vicsek Dept. of Biological Physics, Eötvös
University, Hungary http//angel.elte.hu/vicsek h
ttp//angel.elte.hu/clustering
Collaborators I. Derényi, I. Farkas, G. Palla
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Why networks (topological features of
interactions)? The simplest (still rich)
approach to complex systems consisting of many
interacting and individually relevant units. The
full problem usually cannot be treated because
of the enormous number of units and their
intricate interactions. Think of, e.g.,
collaboration (30,000 authors), web chat groups
(millions of users), or genetic networks (tens of
thousands of genes). Why communities (densely
interconnected parts)? Internal organization of
large networks Complex systems are typically
hierarchical. The units organize (become more
closely connected) into groups which can
themselves be regarded as units on a higher
level. For example Person-gtgroup-gtdepartment-gtdi
vision-gtcompany-gtindustrial sector
Letter-gtword-gtsentence-gtparagraph-gtsection-gtchapte
r-gtbook
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• Questions
• How can we recover the hierarchy of
  groups/modules/communities in the network if only
  a list of links between pairs of units is given?
• What are their main characteristics?
• Outline
• Basic facts and principles
• k-clique percolation
• Community finding
• Results for protein interaction, word association
  and collaboration networks

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Large scale internal organization of complex
networks - degree distribution, i.e.,
P(d)d-? - communities (clusters, modules,
cohesive groups), very active field,
Wasserman, Newman, Barabási, many
others
Random tree
Random blobs
Deterministic,loops
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Basic observations A large complex network is
bounded to be highly structured This structure is
typically hierarchical (i.e., displays some sort
of self-similarity of the structure)
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Role of overlaps
Is this like a tree? (hierarchical methods)
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Observations of large real networks
suggest Large scale
communities Local organization motifs (typical
subgraphs) To find overlapping communities we
consider connected groups
(clusters) of motifs define a
cluster of adjacent complete subgraps (cliques)
is a community
(simple assumption) Two aspects I)
clique percolation II) communities
in large real networks
overlaps and their
statistics
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Finding communities
a 4-clique
Hierarchical methods
k-clique template rolling
Two nodes belong to the same community if they
can be connected through adjacent k-cliques
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Finding communities
a 4-clique
Hierarchical methods
k-clique template rolling
Two nodes belong to the same community if they
can be connected through adjacent k-cliques
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Finding communities
a 4-clique
Hierarchical methods
k-clique template rolling
Two nodes belong to the same community if they
can be connected through adjacent k-cliques
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Hierarchical versus template rolling clustering
In a hierarchical clustering type analysis
someone can belong to a single community at a
time only. For example, I can be located as a
member of the community physicists, but not, at
the same time, be found as a member of my
community family or friends, etc. Before
we proceed to cure this obvious problem, we
discuss some of the interesting statistical
mechanical aspects of the percolation of
k-cliques.
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CLIQUE PERCOLATION Definitions k-clique
complete subgraph of k vertices k-clique
adjacency two k-cliques share a k-1
clique k-clique walk series of steps to
adjacent k-cliques k-clique cluster set of
vertices of all k-clique walks from a given
k-clique
(E-R percolation
is the k2 case)
Details
D.I., P. G. and T.V., Phys. Rev. Lett. 2005
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Order parameter for clique percolation, k4
Percolation threshold at pc(k)
(k-1)N(-1/(k-1))
The scaling of the relative size of the giant
cluster of k3,4 and 5-cliques at pc
For k ? 3, Nk/Nk(pc) N -k/6 For k gt 3
Nk/Nk(pc) N 1-k/2
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UNCOVERING THE OVERLAPPING COMMUNITY STRUCTURE OF
COMPLEX NETWORKS IN NATURE AND SOCIETY
P.
G., D. I., I. F and T.V., Nature
2005 Definitions An order k community is a
k-clique percolation cluster Such
communities/clusters obviously can overlap This
is why a lot of new interesting questions can be
posed New fundamental quantities
(cumulative distributions) defined P(dcom)
community degree distribution P(m)
membership number distribution P(sov)
community overlap distribution P(s)
community size distribution (not new)
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DATA cond-mat (electronic preprints, about
30,000 authors) proten-protein (DIP database,
yeast, 2,600 nodes) word association (sets of
words associated with given
words, questionnaire, 10,600
words) large data sets, with N
10,000 or more efficient algorithm is needed!
Method
RESULTS
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A simple illustration of the extraction of the
communities at k 4 using the clique overlap
matrix. Top left picture shows the graph in which
the different cliques are marked by different
colors. The corresponding clique overlap matrix
is shown in the top right corner. To obtain the
communities at k 4, we delete those
off-diagonal elements, which are smaller than 3
and also those diagonal elements which
are smaller than 4, resulting in the matrix shown
in the bottom left of the gure. The clusters
(communities) corresponding to this matrix are
shown in the bottom right.
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Efficient algorithm, user friendly
software downloadable from http//angel.elte.hu/c
lustering/
CPU time in hours
Number of edges
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Visualization of the communities of a node
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Web of networks Each node is a
community Nodes are weighted for community
size Links are weighted for overlap size DIP
data base of protein interactions (S.
cerevisiase, a yeast) The other networks
we analysed are much larger!!
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Community size distribution Community
degree distribution Combination of exponential
and power law! Emergence of a new feature as
going up to the next level
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.

Community overlap size membership
number
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Dedicated home page (software, papers, data)
http//angel.elte.hu/clust
ering/
Home
Screen shots
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Outlook Networks of networks - hierarchical
aspects - correlations, clustering, etc.,
i.e., everything you can do for vertices -
applications, such as, e.g., protein function
prediction
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music
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The number of vertices in the largest component
As N grows the width of the quickly growing
region decays as 1/N1/2
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Evolution of the social network of scientific
collaborations
A.-L. B., H.J, Z.N., E.R., A. S., T. V. (Physica
A, 2002)
The Erdos graph and the Erdos number (Ei2,W8,BG
4)
1976
L. Lovász
1979
B. Bollobás
Data collaboration graphs in (M) Mathematics and
(NS) Neuroscience
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Collaboration network
due to growth and preferential attachment
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Internal preferential attachment
Collaboration network
Measured data shows
Attachment rate
Due to preferential growth and internal
reorganization a complex network with all sorts
of communities of collaborators are formed (e.g.,
due to specific topics or geographical reasons)
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Deterministic scale-free trees
1 n edges 2 each is replaced by n
edges 3 from every m-th free end n new
edges Here n4, m2 P(k/4)8P(k) P(k)
k-? ? 3/2 1
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Inhomogeneity in the local flux distribution
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Order parameter for clique percolation, k3, k4
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The scaling of the relative size of the giant
cluster of k-cliques at pc
For k ? 3, Nk/Nk(pc) N -k/6 For k gt 3
Nk/Nk(pc) N 1-k/2
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A-L Barabási
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