Groups of verticesand

Core-periphery structure

By: Ralucca Gera, NPS

Why?

• Mostly observed real networks have:– Heavy tail (powerlaw most probably, exponential)– High clustering (high number of triangles

especially in social networks, lower count otherwise)

– Small average path (usually small diameter)– Communities/periphery/hierarchy– Homophily and assortative mixing (similar nodes

tend to be adjacent)• Where does the structure come from? How do

we model it? 2

Macro and Meso Scale properties

• Macro Scale properties (using all the interactions):– Small world (small average path, high clustering)– Powerlaw degree distr. (generally pref. attachment)

• Meso Scale properties applying to groups (using k-clique, k-core, k-plex):– Community structure– Core-periphery structure

• Micro Scale properties applying to small units:– Edge properties (such as who it connects, being a

bridge)– Node properties (such as degree, cut-vertex) 3

Some local and global metrics pertaining to structure of networks

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Structure they capture Local Metrics Global metrics

Direct influenceGeneral feel for the distribution of the edges

Vertex degree, in and out degree

Degree distribution

Closeness, distance between nodes Geodesic (path)Distance (numerical value)

Diameter, radius, average path length

Connectedness of the networkHow critical are vertices to the connectedness of the graph?How much damage can a network take before disconnecting?

Existence of a bridgeExistence of a cut vertex

Cut setsDegree distribution

Tight node/edge neighborhoods Clique, plex, core,community,k-dense (for edges)

Community detection

Centrality and influence Degree centrality Betweenness, eigenvector, PageRank, hub and authorities

Source: Guido Caldarelli, Communities and Clustering in Some social NetworksNetSci 2007 New York, May 20th 2007

0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 0

0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0

0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1

0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1

0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0

In a very clustered graph,the adjacency matrix can beput in a block form.

Clusters and matrices

Definitions

• Newman’s book uses k-component, k-cliques, k-plexes and k-cores to refer to a set of vertices with some properties.

• In graph theory (and research papers) we use a clique to be the set of vertices and edges, so a clique is actually a graph (or subgraph)

• Either way it works, the graph captures more information.

Groups and subgroups identifications

Some common approaches to subgroup identification and analysis:• Cliques• Cores• Plexes• Denseness• Components (and k-components)• Community detection

COMPONENTS AND K-COMPONENTS

Section 7.8.2

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Components

• Recall that a graph is k-connected/k-component if it can be disconnected by removal of k vertices, and no k-1 vertices can disconnect it.

• Component is a maximal size connected subgraph• A k-component (k-connected component) is a

connected maximal subgraph that can be disconnected (or we’re left with a ) by removal of k vertices, and no k-1 vertices can disconnect it.

• Alternatively: A k-component is a connected maximal subgraph such that there are k-vertex-independent paths between any two vertices

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In class exercise

• The k-component tells how robust a graph or subgraph is.

• Identify a subgraphthat is either a:– 1-connected– 2-connected– 3-connected– 4-connected

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K-CLIQUE, K-PLEX AND K-CORE

Section 7.8.1

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k-cliques

• A clique of size : a subset of nodes, with every node adjacent to every other member of the subset (all one of them)

• We usually search for the maximum clique• Hard to find (decision problem for the clique number is

NP-Complete)• Why is it hard to use this concept on real networks?

– Because one might not infer/know all the edges of the true network, so clique may exist but it may not be captured in the data to be analyzed

– A relaxed version of a clique might be just as useful in large networks.

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In class exercise

• A clique of size : a subset of nodes, with every node connected to every other member of the subset.

• Identify a:– 1-clique– 2-clique– 3-clique– 4-clique

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k-plex

• A -plex : maximal subset of nodes, with , where is the subgraph

induced by S. • What is a 1-plex?

: clique: “approximate clique”

• Idea: missing a few edges to be a cliqueor fewer edges per vertex are allowed to be

missing– Useful in identifying subgroups with small diameter,

(possible cliques in the ground truth network). 14

k-plex

• A -plex : maximal subset of nodes, with , where is the

subgraph induced by S, and .

• For what values of is the subgraph

a -plex?

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In class exercise

• A -plex : maximal subset of nodes, with , where is the

subgraph induced by S, and . • Identify a:

– 1-plex– 2-plex– 3-plex– 4-plex

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k-core

-core: maximal subset of nodes, , with , where is the subgraph

induced by S.• Idea: enough edges are present to make a

group strong, not worrying about diameter of .

• What is the relation between and if S is a -core and -plex?

– If S is a -core, then S is a )-plex

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k-core

• A -core of size n: maximal subset of nodes with , where is the subgraph induced by .

• Approach: eliminate lower-order cores until relatively dense subgroups are identified. 18

In class exercise

• A -core of size n: maximal subset of nodes with , where is the subgraph induced by .

• Identify a:– 1-core– 2-core– 3-core– 4-core

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k-dense

• A k- dense sub-graph is a group of vertices, in which each pair of vertices {i, j} has at least

-2 common neighbors.

Idea: pairwise friends ( –dense looks at edges rather than vertices in making them part of the group)

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k-dense

• A k- dense sub-graph is a group of vertices, in which each pair of vertices {i, j} has at least

-2 common neighbors.

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In class exercise

• A k- dense sub-graph is a group of vertices, in which each pair of vertices {i, j} has at least k-2 common neighbors.

• Identify a:– 1-dense– 2-dense– 3-dense– 4-dense

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Connections between them

• k - clique k - dense k – core

(Source: Saito et. Al -2008)

Most networks have only one core.23

Other extensions• https://academic.oup.com/comnet/article/doi/10.1093/comnet/cnt016/2392115/Structure-and-dynamics-of-core-periphery-networks

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Using them globally

Cliques, plexes and cores

• clique of size : maximal subset of nodes, with every node adjacent to every other member of the subset

-plex of size : maximal subset of nodes, with every node adjacent to at least other members of the subset

: clique: “approximate clique”

-core: maximal subset of nodes, with every node adjacent to at least others in the subset

• A - dense sub-graph is a group of vertices, in which each pair of vertices {i, j} has at least common neighbors.

Communities vs. core/dense/clique

• K-core/plex/dense/clique: look inside the group of nodes

• Communities look both at internal and external ties(high internal and low external ties)

• Core-peripherydecomposition also looking atinternal and ext.to the core (doesn’thave to be a clique)

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K-core (k-shell) decomposition

http://3.bp.blogspot.com/-TIjz3nstWD0/ToGwUGivEjI/AAAAAAAAsWw/etkwklnPNw4/s1600/k-cores.png

Generally, but not well defined: the core of the network (the -core for the largest ) and the periphery (everything else).

There are modifications where several top values of make the core.

Core-periphery adjacency matrix

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dark blue = 1 (adjacent)white = 0 (nonadjacent)

Deciding on core-periphery

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How to decide if a network has core-periphery structure? • Not well defined either, but generally the

density of the -core must be high:• Checked by the high correlation,

,where is the (i,j) adjacency matrix entry, and

http://www.sciencedirect.com/science/article/pii/S0378873399000192

Extensions of core-periphery?!

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Limitation: • There are just two classes of nodes: core

and periphery. • Is a three-class partition consisting of

core, semiperiphery, and periphery more realistic?

• Or even partitioning with more classes?• The problem becomes more difficult as

the number of classes is increased, and good justification is needed.

http://www.sciencedirect.com/science/article/pii/S0378873399000192

Possible structures

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From Aaron Clauset and Mason Porter

dark shade = 0 (nonadjacent)light shade = 1 (adjacent)

References

• M. E. Newman, Analysis of weighted networks Physical Review E, vol. 70, no. 5, 2004.

• Borgatti, Stephen P., and Martin G. Everett. "Models of core/periphery structures“ Social networks 21.4 (2000): 375-395.

• Csermely, Peter, et al. "Structure and dynamics of core/periphery networks.“ Journal of Complex Networks 1.2 (2013): 93-123.

• Kitsak, Maksim, et al. "Identification of influential spreaders in complex networks." Nature Physics 6.11 (2010): 888-893

• S. B. Seidman, Network structure and minimum degree, Social networks, vol. 5, no. 3, pp. 269287, 1983

• Borgatti, Stephen P., and Martin G. Everett. "Models of core/periphery structures." Social networks 21.4 (2000): 375-395.

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