1. Groups of vertices and Core-periphery structure
By: Ralucca Gera, NPS

2. Why? Mostly observed real networks have:
Heavy tail (powerlaw, 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?

3. Macro and Meso Scale
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)

4. Some common approaches to subgroup identification:
components (and k-components)
k-plex
k-core
k-dense
k-clique
Core-periphery structure
Community structure

5. Communities with Matrices1
In a very clustered graph, the adjacency matrix can be put in a block form by communities.
Source: Guido Caldarelli, Communities and Clustering in Some social Networks
NetSci 2007 New York, May 20th 2007

6. Some local and global metrics pertaining to structure of networks
Structure they capture
Local Metrics
Global metrics
Direct influence
General 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 network
How critical are vertices to the connectedness of the graph?
How much damage can a network take before disconnecting?
Existence of a bridge
Existence of a cut vertex
Cut sets
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

7. Some common approaches to subgroup identification:
components (and k-components)
k-plex
k-core
k-dense
k-clique
Core-periphery structure
Community structure

8. Components
Recall that a graph is 𝑘-connected or 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 𝐾 1 ) 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

9. In class exercise
The 𝑘-component tells how robust a graph or subgraph is.
Identify a subgraph that is either a:
1-connected
2-connected
3-connected
4-connected

10. Some common approaches to subgroup identification:
components (and k-components)
k-plex
k-core
k-dense
k-clique
Core-periphery structure
Community structure

11. 𝑘-plex
A 𝑘-plex : maximal subset of nodes, with deg 𝐺[𝑆] 𝑣 ≥𝑛−𝑘 , where 𝐺[𝑆] is the subgraph induced by S.
What is a 1-plex?
𝑘=1 : clique
𝑘>1 : “approximate clique”
Idea: missing a few edges to be a clique
𝑘−1 or fewer edges per vertex are allowed to be missing
Useful in identifying subgroups with small diameter, (possible cliques in the ground truth network).

12. k-plex
A 𝑘-plex : maximal subset of nodes, with deg 𝐺[𝑆] 𝑣 ≥𝑛−𝑘 , where 𝐺[𝑆] is the subgraph induced by S, and 𝑘=|𝐺[𝑆]|.
For what values of 𝑘 is the subgraph 𝐺[{2, 3, 4, 5, 6}] a 𝑘 -plex?

13. In class exercise
A 𝑘-plex : maximal subset of nodes, with deg 𝐺[𝑆] 𝑣 ≥𝑛−𝑘 , where 𝐺[𝑆] is the subgraph induced by S, and 𝑘=|𝐺[𝑆]|.
Identify a:
1-plex
2-plex
3-plex
4-plex

14. Some common approaches to subgroup identification:
components (and k-components)
k-plex
k-core
k-dense
k-clique
Core-periphery structure
Community structure

15. k-core
𝐴 𝑘-core: maximal subset of nodes, 𝑆, with deg 𝐺[𝑆] 𝑣 ≥𝑘 , where 𝐺[𝑆] is the subgraph induced by S.
Idea: enough edges are present to make a group strong, not worrying about diameter of 𝐺[𝑆].
What is the relation between 𝑘 and 𝑙 if S is a
𝑘-core and 𝑙-plex?
If S is a 𝑘 -core, then S is a (n−𝑘)-plex

16. k-core
A 𝑘-core of size n: maximal subset of nodes 𝑘 with deg 𝐺[𝑆] 𝑣 ≥𝑘 , where 𝐺[𝑆] is the subgraph induced by 𝑆.
Approach: eliminate lower-order cores until relatively dense subgroups are identified.

17. In class exercise
A 𝑘-core of size n: maximal subset of nodes 𝑘 with deg 𝐺[𝑆] 𝑣 ≥𝑘 , where 𝐺[𝑆] is the subgraph induced by 𝑆.
Identify a:
1-core
2-core
3-core
4-core

18. 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

19. Some common approaches to subgroup identification:
components (and k-components)
k-plex
k-core
k-dense
k-clique
Core-periphery structure
Community structure

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

21. k-dense
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.

22. Some common approaches to subgroup identification:
components (and k-components)
k-plex
k-core
k-dense
k-clique
Core-periphery structure
Community structure

23. k-cliques
A clique of size 𝑘: a subset of 𝑘 nodes, with every node adjacent to every other member of the subset (all 𝑘−1 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.

24. In class exercise
A clique of size 𝑘: a subset of 𝑘 nodes, with every node connected to every other member of the subset.
Identify a:
1-clique
2-clique
3-clique
4-clique

25. Cliques, plexes and cores
clique of size 𝑘: maximal subset of nodes, with every node adjacent to every other member of the subset
𝑘-plex of size 𝑛 : maximal subset of nodes, with every node adjacent to at least 𝑛−𝑘 other members of the subset
𝑘=1 : clique
𝑘>1 : “approximate clique”
𝑘-core: maximal subset of nodes, with every node adjacent to at least 𝑘 others in the subset
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.
Not used so much, rather k-core

26. Some common approaches to subgroup identification:
components (and k-components)
k-plex
k-core
k-dense
k-clique
Core-periphery structure
Community structure

27. 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-periphery decomposition also looking at internal and ext. to the core (doesn’t have to be a clique)

28. K-core (k-shell)
Generally, but not well defined: the core of the network (the 𝑘-core for the largest 𝑘) and the periphery (everything else).
There are modifications where several top values of 𝑘 make the core.

29. Core-periphery
dark blue = 1 (adjacent) white = 0 (nonadjacent)

30. Deciding on core-periphery
How to decide if a network has core-periphery structure?
Not well defined either, but generally the density of the 𝑘-core must be high:
Desired: high correlation, 𝜌 , defined as:
𝜌= 𝑖,𝑗 𝑎 𝑖𝑗 𝛿 𝑖𝑗 ,
where 𝑎 𝑖𝑗 is the (i,j) adjacency matrix entry, and 𝛿 𝑖𝑗 = 1, 𝑖𝑓 𝑛𝑜𝑑𝑒 𝑖 𝑜𝑟 𝑗 𝑖𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑐𝑜𝑟𝑒 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

31. Extensions of core-periphery?!
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.

32. Review of structures! dark shade = 0 (nonadjacent)
light shade = 1 (adjacent)
From Aaron Clauset and Mason Porter

33. 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):
Csermely, Peter, et al. "Structure and dynamics of core/periphery networks.“ Journal of Complex Networks 1.2 (2013):
Kitsak, Maksim, et al. "Identification of influential spreaders in complex networks." Nature Physics 6.11 (2010):
S. B. Seidman, Network structure and minimum degree, Social networks, vol. 5, no. 3, pp , 1983
Borgatti, Stephen P., and Martin G. Everett. "Models of core/periphery structures." Social networks 21.4 (2000):