Communities. Questions 1.What is a community (intuitively)? Examples and fundamental hypothesis 2.What do we really mean by communities? Basic definitions.

Communities

Questions

1. What is a community (intuitively)? Examples and fundamental hypothesis

2. What do we really mean by communities? Basic definitions

3. Graph partitioning and its computational complexity4. Hierarchical clustering: Ravasz algorithm and its

computational complexity5. Hierarchical clustering: Girvan-Newman algorithm

and its complexity6. Hierarchy in real networks7. Modularity

Introduction

Section 1

Section 1 Introduction: Belgium


Same area as Massachusetts (~12,000 sq miles)Same population as Ohio (~11.5 millions )


V.D. Blondel et al, J. Stat. Mech. P10008 (2008).

A.-L. Barabási, Network Science: Communities.

Examples of communities

Section 2

Section 2 Zachary’s Karate Club

W.W. Zachary, J. Anthropol. Res. 33:452-473 (1977).


Section 2 Zachary’s Karate Club

Citation history of the Zachary’s Karate club paper

W.W. Zachary, J. Anthropol. Res. 33:452-473 (1977).


Section 2 Zachary Karate Club Club

The first scientist at any conference on networks who uses Zachary's karate club as an example is inducted into the Zachary Karate Club Club, and awarded a prize.

Chris Moore (9 May 2013).Mason Porter (NetSci, June 2013).Yong-Year Ahn (Oxford University, July 2013)Marián Boguñá (ECCS, September 2013).Mark Newman (Netsci, June 2014)

http://networkkarate.tumblr.com/)

http://networkkarate.tumblr.com/

Section 2 Auxiliary information

Karate Club: Breakup of the club

Belgian Phone Data:Language spoken

Section 2 Biological Modules

E. Ravasz et al., Science 297 (2002).


Basics of communities

Section 3

Section 2 Communities


We focus on the mesoscopic scale of the network

Microscopic Mesoscopic Macroscopic

Section 2 Fundamental Hypothesis


H1: A network’s community structure is uniquely encoded in its wiring diagram

Section 3 Basics of Communities

H2: Connectedness Hypothesis

A community corresponds to a connected subgraph.

H3: Density Hypothesis

Communities correspond to locally dense neighborhoods of a network.



H2: Connectedness Hypothesis

A community corresponds to a connected subgraph.

H3: Density Hypothesis

Communities correspond to locally dense neighborhoods of a network.



Cliques as communities

A clique is a complete subgraph of k-nodes

R.D. Luce & A.D. Perry, Psychometrika 14 (1949)



• Triangles are frequent; larger cliques are rare.

• Communities do not necessarily correspond to complete subgraphs, as many of their nodes do not link directly to each other.

• Finding the cliques of a network is computationally rather demanding, being a so-called NP-complete problem.

Cliques as communities


Consider a connected subgraph C of Nc nodes

Internal degree, kiint : set of links of node i that connects

to other nodes of the same community C.

External degree kiext: the set of links of node i that

connects to the rest of the network.

If kiext=0: all neighbors of i belong to C, and C is a good

community for i.

If kiint=0, all neighbors of i belong to other communities,

then i should be assigned to a different community.

Strong and weak communities



Strong community: Each node of C has more links within the community than with the rest of the graph.

Weak community: The total internal degree of C exceeds its total external degree,

Clique Strong WeakA.-L. Barabási, Network Science: Communities.

Section 3 Number of Partitions

How many ways can we partition a network into 2 communities?

Divide a network into two equal non-overlapping subgraphs, such that the number of links between the nodes in the two groups is minimized.

Two subgroups of size n1 and n2. Total number of combinations:

N=10 256 partitions (1 ms)

N=100 1026 partitions (1021 years)

Graph bisection


Section 3 Graph Partitions (history)

2.5 billion transistors

partition the full wiring diagram of an integrated circuit into smaller subgraphs, so that they minimize the number of connections between them.

Graph Partitioning

Section 3 Graph Partitions (history)

Kerninghan-Lin Algorithm for graph bisection

• Partition a network into two groups of predefined size. This partition is called cut.

• Inspect each a pair of nodes, one from each group. Identify the pair that results in the largest reduction of the cut size (links between the two groups) if we swap them

• Swap them. • If no pair deduces the cut size, we swap the

pair that increases the cut size the least. • The process is repeated until each node is

moved once.

Section 3 Number of communities

Community detection

The number and size of the communities are unknown at the beginning.

PartitionDivision of a network into groups of nodes, so that each node belongs to one group.

Bell Number: number of possible partitions of N nodes


Hierarchical Clustering

Section 4

Section 4 Hierarchical Clustering

Agglomerative algorithms merge nodes and communities with high similarity.

Divisive algorithms split communities by removing links that connect nodes with low similarity.

1. Build a similarity matrix for the network

2. Similarity matrix: how similar two nodes are to each other we need to determine from the adjacency matrix

3. Hierarchical clustering iteratively identifies groups of nodes with high similarity, following one of two distinct strategies:

Hierarchical tree or dendrogram: visualize the history of the merging or splitting process the algorithm follows. Horizontal cuts of this tree offer various community partitions.

4.

Section 4 Agglomerative Algorithms

Step 1: Define the Similarity Matrix (Ravasz algorithm)

• High for node pairs that likely belong to the same community, low for those that likely belong to different communities.

• Nodes that connect directly to each other and/or share multiple neighbors are more likely to belong to the same dense local neighborhood, hence their similarity should be large.

Topological overlap matrix:

JN(i,j): number of common neighbors of node i and j; (+1) if there is a direct link between i and j;



Agglomerative algorithms merge nodes and communities with high similarity.




Step 2: Decide Group Similarity

• Groups are merged based on their mutual similarity through single, complete or average cluster linkage


Step 3: Apply Hierarchical Clustering

• Assign each node to a community of its own and evaluate the similarity for all node pairs. The initial similarities between these “communities” are simply the node similarities.

• Find the community pair with the highest similarity and merge them to form a single community.

• Calculate the similarity between the new community and all other communities.

• Repeat from Step 2 until all nodes are merged into a single community.

Step 4: Build Dendrogram

• Describes the precise order in which the nodes are assigned to communities.




Computational complexity:

• Step 1 (calculation similarity matrix): • Step 2-3 (group similarity): • Step 4 (dendrogram):



Section 4 Divisive Algorithms

Step 1: Define a Centrality Measure (Girvan-Newman algorithm)

• Link betweenness is the number of shortest paths between all node pairs that run along a link.

• Random-walk betweenness. A pair of nodes m and n are chosen at random. A walker starts at m, following each adjacent link with equal probability until it reaches n. Random walk betweenness xij is the probability that the link i→j was crossed by the walker after averaging over all possible choices for the starting nodes m and n

Divisive algorithms split communities by removing links that connect nodes with low similarity.

M. Girvan & M.E.J. Newman, PNAS 99 (2002).





Step 2: Hierarchical Clustering

a) Compute of the centrality of each link.

b) Remove the link with the largest centrality; in case of a tie, choose one randomly.

c) Recalculate the centrality of each link for the altered network.

d) Repeat until all links are removed (yields a dendrogram).




Step 2: Hierarchical Clustering

a) Compute of the centrality of each link.

b) Remove the link with the largest centrality; in case of a tie, choose one randomly.

c) Recalculate the centrality of each link for the altered network.

d) Repeat until all links are removed (yields a dendrogram).

Section 4 Divisive Algorithm




• Step 1a (calculation betweenness centrality):

• Step 1b (Recalculation of betweenness centrality for all links):

for sparse networks

Section 4 Hierarchy in networks


(1) Scale-free property

The obtained network is scale-free, its degree distribution following a power-law with

E. Ravasz & A.-L. Barabási, PRE 67 (2003).



(1) Scale-free property

The obtained network is scale-free, its degree distribution following a power-law with




(2) Clustering coefficient scaling with k

Small k nodes:*high clustering coefficient; *their neighbors tend to link to each other in highly interlinked, compact communities.

High k nodes (hubs):*small clustering coefficient; *connect independent communities.




(3) Clustering coefficient independent of N




(3) Clustering coefficient independent of N



2. Scaling clustering coefficient (DGM)

1. Scale-free 3. Clustering coefficient independent of N

x





Section 4 Hierarchy in real networks

POWER GRID INTERNET

Section 4 Ambiguity in Hierarchical clustering


Where to “cut”?

Phylogenetic dendrograms

In bioinformatrics, clusters and dendrograms have been studied for a long time.

For example, the sequences of the same protein or gene in different species areselected, and compared with each other.


A similarity matrix is constructed between these sequences, by looking at how many aminoacids/nucleotides stay in place


A similarity matrix is constructed between these sequences, by looking at how many aminoacids/nucleotides stay in place



Modularity

Section 4

Section 4 Modularity

MEJ Newman, PNAS 103 (2006).


H4: Random Hypothesis

Randomly wired networks are not expected to have a community structure.




Imagine a partition in nc communities

Modularity







Modularity

Original data







Modularity

Original data Expected connections, a model







Modularity


Relative to a specific partition







Modularity


Relative to a specific partition

Modularity is a measure associated to a partition

Random network




Another way of writing M



where LC is the number of links within C. In a similar fashion, the second term becomes

We can rewrite the first term as

Finally we get:




H5: Maximal Modularity Hypothesis

The partition with the maximum modularity M for a given network offers the optimal community structure




H5: Maximal Modularity Hypothesis

The partition with the maximum modularity M for a given network offers the optimal community structure

Find

Goal

that maximizes M


• Optimal partition, that maximizes the modularity.

• Sub-optimal but positive modularity.

• Negative Modularity: If we assign each node to a different community.

• Zero modularity: Assigning all nodes to the same community, independent of the network structure.

• Modularity is size dependent

Which partition ?


Section 4 Modularity based community identification

A greedy algorithm, which iteratively joins nodes if the move increases the new partition’s modularity.

Step 1. Assign each node to a community of its own. Hence we start with N communities.

Step 2. Inspect each pair of communities connected by at least one link and compute the modularity variation obtained if we merge these two communities.

Step 3. Identify the community pairs for which ΔM is the largest and merge them. Note that modularity of a particular partition is always calculated from the full topology of the network.

Step 4. Repeat step 2 until all nodes are merged into a single community.

Step 5. Record for each step and select the partition for which the modularity is maximal.

MEJ Newman, PRE 69 (2004).



Which partition ?


Modularity can be used to compare different partitions provided by other algorithms, like hierarchical clustering

It can be used to design new algorithms, aiming at maximizing M

Section 4 Modularity for the Girvan-Newman

Which partition ?






• Step 1-2 (calculation of ΔM for L links ): • Step 3 (matrix update): • Step 4 (N-1 community merges):

for sparse networks





• Step 1-2 (calculation of ΔM for L links ): • Step 3 (matrix update): • Step 4 (N-1 community merges):

for sparse networks

Section 4 Limits of Modularity


kA and kB total degree in A and B

A B

Resolution limit




If and

A B

Resolution limit




If and

A B

We merge A and B to maximize modularity.

Resolution limit




If and

Assuming

A B


Resolution limit




If and

Assuming

Modularity has a resolution limit, as it cannot detect communities smaller than this size.

A B


Resolution limit



One maximum?


Null models

Expected connections, a model

can take into account weights

can take into account directions

can take into account attributes or space

S. Fortunato, Phys. Rep. 486 (2010)

S. Fortunato, Phys. Rep. 486 (2010)

P. Expert el al., PNAS 108 (2011)

Section 5 Online Resources (Modularity)

Gephi

NetworkX

R assigns self-loops to nodes to increase or decrease the aversion of nodes to form communities

Finds the partition that maximizes modularity (considers weights and direction)

Calculates the modularity of the partition you provide

Section 4 Online Resources (1)

The greedy algorithm is neither particularly fast nor particularly successful at maximizing M.

Scalability: Due to the sparsity of the adjacency matrix, the update of the matrix involves a large number of useless operations. The use of data structures for sparse matrices can decrease the complexity of the computational algorithm to , which allows us to analyze is of networks up to nodes. See"Fast Modularity" Community Structure Inference Algorithm http://cs.unm.edu/~aaron/research/fastmodularity.htm for the code.

A fast greedy algorithm was proposed by Blondel and collaborators, that can process networks with millions of nodes. For the description of the algorithm seeLouvain method: Finding communities in large networkshttps://sites.google.com/site/findcommunities/ for the code.

http://cs.unm.edu/~aaron/research/fastmodularity.htm

http://cs.unm.edu/~aaron/research/fastmodularity.htm

https://sites.google.com/site/findcommunities/



Communities. Questions 1.What is a community (intuitively)? Examples and fundamental hypothesis 2.What do we really mean by communities? Basic definitions.

Documents

belgium belgium

millions belgium

parts of belgium

ethnic groups

small communities

groups of individuals

huge communities

smaller group of communities