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Dept. of Mathematics & Statistics | Vermont Complex Systems CenterVermont Advanced Computing Core | University of Vermont
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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Outline
Overview
MethodsHierarchy by aggregationHierarchy by divisionHierarchy by shufflingSpectral methodsHierarchies & Missing LinksOverlapping communitiesLink-based methodsGeneral structure detection
References
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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Structure detection
▲ Zachary’s karate club [18, 12]
The issue:how do weelucidate theinternal structure oflarge networksacross many scales?
Possible substructures:hierarchies, cliques, rings, …
Plus:All combinations of substructures.
Much focus on hierarchies...
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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Physics Reports 486 (2010) 75–174
Contents lists available at ScienceDirect
Physics Reports
journal homepage: www.elsevier.com/locate/physrep
Community detection in graphs
Santo Fortunato ∗
Complex Networks and Systems Lagrange Laboratory, ISI Foundation, Viale S. Severo 65, 10133, Torino, I, Italy
a r t i c l e i n f o
Article history:
Accepted 5 November 2009
Available online 4 December 2009
editor: I. Procaccia
Keywords:
Graphs
Clusters
Statistical physics
a b s t r a c t
The modern science of networks has brought significant advances to our understanding of
complex systems. One of the most relevant features of graphs representing real systems
is community structure, or clustering, i.e. the organization of vertices in clusters, with
many edges joining vertices of the same cluster and comparatively few edges joining
vertices of different clusters. Such clusters, or communities, can be considered as fairly
independent compartments of a graph, playing a similar role like, e.g., the tissues or the
organs in the human body. Detecting communities is of great importance in sociology,
biology and computer science, disciplines where systems are often represented as graphs.
This problem is very hard and not yet satisfactorily solved, despite the huge effort of a
large interdisciplinary community of scientists working on it over the past few years. We
will attempt a thorough exposition of the topic, from the definition of the main elements
of the problem, to the presentation of most methods developed, with a special focus on
techniques designed by statistical physicists, from the discussion of crucial issues like the
significance of clustering and how methods should be tested and compared against each
other, to the description of applications to real networks.
2. Communities in real-world networks ................................................................................................................................................... 78
3. Elements of community detection......................................................................................................................................................... 82
3.2.2. Local definitions......................................................................................................................................................... 84
3.2.3. Global definitions....................................................................................................................................................... 85
3.2.4. Definitions based on vertex similarity ..................................................................................................................... 86
4. Traditional methods................................................................................................................................................................................ 90
5.1. The algorithm of Girvan and Newman .................................................................................................................................... 97
5.2. Other methods........................................................................................................................................................................... 99
Tend to plainly not work on data sets representingnetworks with known modular structures.
Good at finding cores of well-connected (orsimilar) nodes... but fail to cope well withperipheral, in-between nodes.
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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Hierarchy by division.Top down:..
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Idea: Identify global structure first and recursivelyuncover more detailed structure.
Basic objective: find dominant components thathave significantly more links within than without,as compared to randomized version.
We’ll first work through “Finding and evaluatingcommunity structure in networks” by Newmanand Girvan (PRE, 2004). [12]
See also1. “Scientific collaboration networks. II. Shortest
paths, weighted networks, and centrality” byNewman (PRE, 2001). [10, 11]
2. “Community structure in social and biologicalnetworks” by Girvan and Newman (PNAS, 2002). [7]
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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Hierarchy by division
Idea: Edges that connect communities have higherbetweenness than edges within communities.
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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Hierarchy by division
.One class of structure-detection algorithms:..
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1. Compute edge betweenness for whole network.2. Remove edge with highest betweenness.3. Recompute edge betweenness4. Repeat steps 2 and 3 until all edges are removed.
5 Record whencomponents appear asa function of # edgesremoved.
Red line indicates appearanceof four (4) components at acertain level.
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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.Key element for division approach:..
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Recomputing betweenness. Reason: Possible to have a low betweenness in
links that connect large communities if other linkscarry majority of shortest paths.
.When to stop?:..
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How do we know which divisions are meaningful? Modularity measure: difference in fraction of
within component nodes to that expected forrandomized version:= ∑�[ �� − �2� ]where �� is the fraction of (undirected) edgestravelling between identified communities and ,and �� = ∑� �� is the fraction of edges with atleast one end in community .
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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Hierarchy by division
.Test case:..
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Generate random community-based networks. = ��� with four communities of size 32. Add edges randomly within and across
Maximum modularity ≃ 0.5 obtained when fourcommunities are uncovered.
Further ‘discovery’ of internal structure issomewhat meaningless, as any communities ariseaccidentally.
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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Hierarchy by division
Factions in Zachary’s karate club network. [18]
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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Betweenness for electrons: Unit resistors on each
edge. For every pair of nodes� (source) and � (sink),
set up unit currents inat � and out at �.
Measure absolutecurrent along eachedge ℓ, |�ℓ,��|.
Sum |�ℓ,��| over all pairs of nodes to obtainelectronic betweenness for edge ℓ.
(Equivalent to random walk betweenness.) Contributing electronic betweenness for edge
between nodes and :� elec��,�� = ���|��,�� − ��,��|.
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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Electronic betweenness Define some arbitrary voltage reference. Kirchhoff’s laws: current flowing out of node
must balance:�∑�=1 ��� (�� − ��) = ��� − ���. Between connected nodes, �� = � = ��� = �/���. Between unconnected nodes, �� = ∞ = �/���. We can therefore write:�∑�=1 ���(�� − ��) = ��� − ���. Some gentle jiggery-pokery on the left hand side:∑� ���(�� − ��) = �� ∑� ��� − ∑� �����= �� � − ∑� ����� = ∑� [ ������ − �����]= [( − �) � ]�
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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Electronic betweenness Write right hand side as [�ext]�,�� = ��� − ���, where�ext�� holds external source and sink currents. Matrixingly then:( − �) � = �ext�� . = − � is a beast of some utility—known as the
Laplacian. Solve for voltage vector � by � decomposition
(Gaussian elimination). Do not compute an inverse! Note: voltage offset is arbitrary so no unique
solution. Presuming network has one component, null
space of − � is one dimensional. In fact, �( − �) = { �, ∈ } since ( − �) � = 0.
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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Alternate betweenness measures:.Random walk betweenness:..
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Asking too much: Need full knowledge of networkto travel along shortest paths.
One of many alternatives: consider all randomwalks between pairs of nodes and .
Walks starts at node , traverses the networkrandomly, ending as soon as it reaches .
Record the number of times an edge is followedby a walk.
Consider all pairs of nodes. Random walk betweenness of an edge = absolute
difference in probability a random walk travelsone way versus the other along the edge.
Equivalent to electronic betweenness (see alsodiffusion).
Blau & Schwartz [2], Simmel [16], Breiger [3], Watts etal. [17]; see also Google+ Circles.
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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.Dealing with community overlap:..
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Earlier structure detection algorithms,agglomerative or divisive, force communities to bepurely distinct.
Overlap: Acknowledge nodes can belong tomultiple communities.
Palla et al. [13] detect communities as sets ofadjacent -cliques (must share − � nodes).
One of several issues: how to choose ? Four new quantities:
�, number of a communities a node belongs to. �ov�,�, number of nodes shared between two given
communities, � and �. �com� , degree of community �. �com� , community �’s size.
Associated distributions:>(�), >(�ov�,�), >( com� ), and >(�com� ).
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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Uncovering the overlapping community structure ofcomplex networks in nature and societyGergely Palla1,2, Imre Derenyi2, Illes Farkas1 & Tamas Vicsek1,2
Many complex systems in nature and society can be described interms of networks capturing the intricate web of connectionsamong the units they are made of1–4. A key question is how tointerpret the global organization of such networks as the co-existence of their structural subunits (communities) associatedwith more highly interconnected parts. Identifying these a prioriunknown building blocks (such as functionally related proteins5,6,industrial sectors7 and groups of people8,9) is crucial to theunderstanding of the structural and functional properties ofnetworks. The existing deterministic methods used for large net-works find separated communities, whereas most of the actualnetworks are made of highly overlapping cohesive groups ofnodes. Here we introduce an approach to analysing the mainstatistical features of the interwoven sets of overlapping commu-nities that makes a step towards uncovering the modular structureof complex systems. After defining a set of new characteristicquantities for the statistics of communities, we apply an efficienttechnique for exploring overlapping communities on a large scale.We find that overlaps are significant, and the distributions weintroduce reveal universal features of networks. Our studies ofcollaboration, word-association and protein interaction graphsshow that the web of communities has non-trivial correlations andspecific scaling properties.Most real networks typically contain parts in which the nodes
(units) are more highly connected to each other than to the rest ofthe network. The sets of such nodes are usually called clusters,communities, cohesive groups or modules8,10,11–13; they have nowidely accepted, unique definition. In spite of this ambiguity,the presence of communities in networks is a signature of thehierarchical nature of complex systems5,14. The existing methodsfor finding communities in large networks are useful if the commu-nity structure is such that it can be interpreted in terms of separatedsets of communities (see Fig. 1b and refs 10, 15, 16–18). However,most real networks are characterized by well-defined statistics ofoverlapping and nested communities. This can be illustrated by thenumerous communities that each of us belongs to, including thoserelated to our scientific activities or personal life (school, hobby,family) and so on, as shown in Fig. 1a. Furthermore, members of ourcommunities have their own communities, resulting in an extremelycomplicated web of the communities themselves. This has long beenunderstood by sociologists19 but has never been studied system-atically for large networks. Another, biological, example is that alarge fraction of proteins belong to several protein complexessimultaneously20.In general, each node i of a network can be characterized by a
membership number m i, which is the number of communities thatthe node belongs to. In turn, any two communities a and b can sharesova;b nodes, which we define as the overlap size between thesecommunities. Naturally, the communities also constitute a network,
with the overlaps being their links. The number of such links ofcommunity a can be called its community degree, dcoma : Finally, thesize scoma of any community a can most naturally be defined as thenumber of its nodes. To characterize the community structure of alarge network we introduce the distributions of these four basicquantities. In particular we focus on their cumulative distribution
LETTERS
Figure 1 | Illustration of the concept of overlapping communities. a, Theblack dot in the middle represents either of the authors of this paper, withseveral of his communities around. Zooming in on the scientific communitydemonstrates the nested and overlapping structure of the communities, anddepicting the cascades of communities starting from some membersexemplifies the interwoven structure of the network of communities.b, Divisive and agglomerative methods grossly fail to identify thecommunities when overlaps are significant. c, An example of overlappingk-clique communities at k ¼ 4. The yellow community overlaps the blue onein a single node, whereas it shares two nodes and a link with the green one.These overlapping regions are emphasized in red. Notice that any k-clique(complete subgraph of size k) can be reached only from the k-cliques of thesame community through a series of adjacent k-cliques. Two k-cliques areadjacent if they share k 2 1 nodes.
1Biological Physics Research Group of the Hungarian Academy of Sciences, Pazmany P. stny. 1A, H-1117 Budapest, Hungary. 2Department of Biological Physics, Eotvos University,
“Uncovering the overlapping communitystructure of complex networks in natureand society”Palla et al.,Nature, 435, 814–818, 2005. [13]
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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Figure 1 | Illustration of the concept of overlapping communities. a, Theblack dot in the middle represents either of the authors of this paper, withseveral of his communities around. Zooming in on the scientific communitydemonstrates the nested and overlapping structure of the communities, anddepicting the cascades of communities starting from some membersexemplifies the interwoven structure of the network of communities.b, Divisive and agglomerative methods grossly fail to identify thecommunities when overlaps are significant. c, An example of overlappingk-clique communities at k ¼ 4. The yellow community overlaps the blue onein a single node, whereas it shares two nodes and a link with the green one.These overlapping regions are emphasized in red. Notice that any k-clique(complete subgraph of size k) can be reached only from the k-cliques of thesame community through a series of adjacent k-cliques. Two k-cliques areadjacent if they share k 2 1 nodes.
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
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Figure 2 | The community structure around a particular node in three
different networks. The communities are colour coded, the overlappingnodes and links between them are emphasized in red, and the volume of theballs and the width of the links are proportional to the total number ofcommunities they belong to. For each network the value of k has been set to4. a, The communities of G. Parisi in the co-authorship network of theLos Alamos CondensedMatter archive (for threshold weightw* ¼ 0.75) can
be associated with his fields of interest. b, The communities of the word‘bright’ in the South Florida Free Association norms list (for w* ¼ 0.025)represent the different meanings of this word. c, The communities of theprotein Zds1 in the DIP core list of the protein–protein interactions of S.cerevisiae can be associated with either protein complexes or certainfunctions.
Two tunable parameters: �∗, the link weightthreshold, and , the clique size.
Figure 4 | Statistics of the k-clique communities for three large
networks. The networks are the co-authorship network of the Los AlamosCondensed Matter archive (triangles, k ¼ 6, f* ¼ 0.93), the word-association network of the South Florida Free Association norms (squares,k ¼ 4, f* ¼ 0.67), and the protein interaction network of the yeast S.cerevisiae from the DIP database (circles, k ¼ 4). a, The cumulativedistribution function of the community size follows a power law withexponents between 21 (upper line) and 21.6 (lower line). b, Thecumulative distribution of the community degree starts exponentially andthen crosses over to a power law (with the same exponent as for thecommunity size distribution). c, The cumulative distribution of the overlapsize. d, The cumulative distribution of the membership number.
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
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.A link-based approach:..
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What we know now: Many network analyses profitfrom focusing on links.
Idea: form communities of links rather thancommunities of nodes.
Observation: Links typically of one flavor, whilenodes may have many flavors.
Link communities induce overlapping and stillhierarchically structured communities of nodes.
“Link communities reveal multiscalecomplexity in networks”Ahn, Bagrow, and Lehmann,Nature, 466, 761–764, 2010. [1]
Inertia
Law
Newton
Physics
Lab Biology
Scientiic
Chemical
Chemistry
Science
EinsteinTheory
Hypothesis
Theorem
Gravity
Relativity
Biologist
Smart
Scientist
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Bright
Genius
Intelligence
Clever
Intelligent
Gifted
Wise
Inventor Brilliant
Wisdom
Kinetic
Exceptional
Retarded
Invent
Chemist
Wit
Velocity
Intellect
Cunning
Outfox
FlaskBeaker
Test tube
Experiment
Research
Apple
Weight
Experiment, science
Newton, gravity, apple
Smart, intellect, scientists
Clever, wit
Science, scientists
f
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COcoNuTS
Overview
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Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
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University
Joint appointment
Home and work
Family
Buildings in same neighborhood
21
3
5
3–4
2–4
1–4
2–3
1–2
1–3
4–7
5–6
4–6
4–5
7–9
7–8
8–9
4
6
7
8
9
a b
c
d
f
e
Inertia
Law
Newton
Physics
Lab Biology
Scientiic
Chemical
Chemistry
Science
EinsteinTheory
Hypothesis
Theorem
Gravity
Relativity
Biologist
Smart
Scientist
Sly
Bright
Genius
Intelligence
Clever
Intelligent
Gifted
Wise
Inventor Brilliant
Wisdom
Kinetic
Exceptional
Retarded
Invent
Chemist
Wit
Velocity
Intellect
Cunning
Outfox
FlaskBeaker
Test tube
Experiment
Research
Apple
Weight
Experiment, science
Newton, gravity, apple
Smart, intellect, scientists
Clever, wit
Science, scientists
f
Figure 1 | Overlapping communities lead to dense networks and prevent
the discovery of a single node hierarchy. a, Local structure in manynetworks is simple: an individual node sees the communities it belongs to.b, Complex global structure emerges when every node is in the situationdisplayed in a. c, Pervasive overlap hinders the discovery of hierarchicalorganization because nodes cannot occupy multiple leaves of a nodedendrogram, preventing a single tree from encoding the full hierarchy.d, e, An example showing link communities (colours in d), the link similaritymatrix (e; darker entries show more similar pairs of links) and the linkdendrogram (e). f, Link communities from the full word association networkaround the word ‘Newton’. Link colours represent communities and filledregions provide a guide for the eye. Link communities capture conceptsrelated to science and allow substantial overlap. Note that the words wereproduced by experiment participants during free word associations.
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Note: See details of paper on how to choose linkcommunities well based on partition density �.
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
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Measures
Overlap coverage
Overlap quality
Community coverage
Community quality
Methods
LCGI
––––
LinksClique percolationGreedy modularity Infomap
Other networksSocial networksBiological networks
885,989
6.34⟨k⟩
N 1,042
16.81
1,647
3.06
1,004
16.57
1,213
4.21
2,729
8.92
67,411
8.90
390
38.95
1,219
9.80
5,018
22.02
18,142
5.09
0
1
2
3
4
L C G I L C G I L C G I L C G I L C G I L C G I L C G I L C G I L C G I L C G I L C G I
Figure 2 | Assessing the relevance of link communities using real-world
networks. Composite performance (Methods and SupplementaryInformation) is a data-drivenmeasure of the quality (relevance of discoveredmemberships) and coverage (fraction of network classified) of communityand overlap. Tested algorithms are link clustering, introduced here; cliquepercolation9; greedy modularity optimization26; and Infomap21. Test
networks were chosen for their varied sizes and topologies and to representthe different domains where network analysis is used. Shown for each are thenumber of nodes, N, and the average number of neighbours per node, Ækæ.Link clustering finds the most relevant community structure in real-worldnetworks. AP/MS, affinity-purification/mass spectrometry; LC, literaturecurated; PPI, protein–protein interaction; Y2H, yeast two-hybrid.
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Comparison of structure detection algorithmsusing four measures over many networks.
Revealed communities are matched against‘known’ communities recorded in networkmetadata.
Link approach particularly good for dense,overlapful networks.
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COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
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Threshold, t = 0.20
t =
0.24
t = 0.27
t = 0.27
50 km
a
0.4
D
0.6 0.8 1
d Largest communityLargest
subcommunity
Remaining
hierarchy
t
e
b
c
Word associationMetabolicPhone
Largestcommunity
Secondlargest
Third largest
e
Q/Q
max
0 0.2 0.4 0.6 0.8
Word association
0 0.2 0.4 0.6 0.8
Link dendrogram threshold, t
Metabolic
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8
Phone
ActualControl
Figure 4 | Meaningful communities at multiple levels of the link
dendrogram. a–c, The social network of mobile phone users displays co-located, overlapping communities on multiple scales. a, Heat map of themost likely locations of all users in the region, showing several cities.b, Cutting the dendrogram above the optimum threshold yields small, intra-city communities (insets). c, Below the optimum threshold, the largestcommunities become spatially extended but still show correlation. d, Thesocial network within the largest community in c, with its largestsubcommunity highlighted. The highlighted subcommunity is shown alongwith its link dendrogram and partition density,D, as a function of threshold,t. Link colours correspond to dendrogram branches. e, Community quality,Q, as a function of dendrogram level, compared with random control(Methods).
Performance for test networks.True Partition Chain Ring Tree Grid
100*
10000*
10000*
5000*
5000
*
Tree
Ring
Chain
Grid
Partition
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References I
[1] Y.-Y. Ahn, J. P. Bagrow, and S. Lehmann.Link communities reveal multiscale complexity innetworks.Nature, 466(7307):761–764, 2010. pdf
[2] P. M. Blau and J. E. Schwartz.Crosscutting Social Circles.Academic Press, Orlando, FL, 1984.
[3] R. L. Breiger.The duality of persons and groups.Social Forces, 53(2):181–190, 1974. pdf
[4] A. Capocci, V. Servedio, G. Caldarelli, andF. Colaiori.Detecting communities in large networks.Physica A: Statistical Mechanics and itsApplications, 352:669–676, 2005. pdf
References II[5] A. Clauset, C. Moore, and M. E. J. Newman.
Hierarchical structure and the prediction ofmissing links in networks.Nature, 453:98–101, 2008. pdf
[6] S. Fortunato.Community detection in graphs.Physics Reports, 486:75–174, 2010. pdf
[7] M. Girvan and M. E. J. Newman.Community structure in social and biologicalnetworks.Proc. Natl. Acad. Sci., 99:7821–7826, 2002. pdf
[8] C. Kemp and J. B. Tenenbaum.The discovery of structural form.Proc. Natl. Acad. Sci., 105:10687–10692, 2008.pdf
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References III
[9] D. P. Kiley, A. J. Reagan, L. Mitchell, C. M. Danforth,and P. S. Dodds.The game story space of professional sports:Australian Rules Football.Draft version of the present paper using purerandom walk null model. Available online athttp://arxiv.org/abs/1507.03886v1. AccesssedJanuary 17, 2016, 2015. pdf
[10] M. E. J. Newman.Scientific collaboration networks. II. Shortestpaths, weighted networks, and centrality.Phys. Rev. E, 64(1):016132, 2001. pdf
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Hierarchy by shuffling
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References IV
[11] M. E. J. Newman.Erratum: Scientific collaboration networks. II.Shortest paths, weighted networks, and centrality[Phys. Rev. E 64, 016132 (2001)].Phys. Rev. E, 73:039906(E), 2006. pdf
[12] M. E. J. Newman and M. Girvan.Finding and evaluating community structure innetworks.Phys. Rev. E, 69(2):026113, 2004. pdf
[13] G. Palla, I. Derényi, I. Farkas, and T. Vicsek.Uncovering the overlapping community structureof complex networks in nature and society.Nature, 435(7043):814–818, 2005. pdf
.
COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
................75 of 76
References V
[14] M. Sales-Pardo, R. Guimerà, A. A. Moreira, andL. A. N. Amaral.Extracting the hierarchical organization ofcomplex systems.Proc. Natl. Acad. Sci., 104:15224–15229, 2007.pdf
[15] M. Sales-Pardo, R. Guimerà, A. A. Moreira, andL. A. N. Amaral.Extracting the hierarchical organization ofcomplex systems: Correction.Proc. Natl. Acad. Sci., 104:18874, 2007. pdf
[16] G. Simmel.The number of members as determining thesociological form of the group. I.American Journal of Sociology, 8:1–46, 1902.
.
COcoNuTS
Overview
MethodsHierarchy by aggregation
Hierarchy by division
Hierarchy by shuffling
Spectral methods
Hierarchies & MissingLinks
Overlapping communities
Link-based methods
General structuredetection
References
................76 of 76
References VI
[17] D. J. Watts, P. S. Dodds, and M. E. J. Newman.Identity and search in social networks.Science, 296:1302–1305, 2002. pdf
[18] W. W. Zachary.An information flow model for conflict and fissionin small groups.J. Anthropol. Res., 33:452–473, 1977.