The document discusses various centrality measures used in graph theory to identify influential nodes, including degree centrality, closeness centrality, betweenness centrality, eigenvector centrality, and clustering coefficient. Each measure highlights different aspects of node connectivity and importance, applicable in various contexts like infrastructure robustness and disease spread. Real-life examples include applications in social networks and dolphin interaction studies.

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Centrality Measures of a Graph
Alfonso Meléndez

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Objectives
• Centrality measures are critical in a wide-range of applications that
involve concepts/phenomena such as :
• infrastructure vulnerability and robustness (e.g., in transportation
systems, urban systems, the Internet, etc.)
• spreading or flow (of information or diseases, etc.).
• Essentially, it's about identifying the "important" or "influential"
nodes in the system.
Link to Model

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Centrality Measures

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1. Degree Centrality

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Degree Centrality
• Centrality depends on the degree of node (number of neighbors)
• Intuitively:
• Lots of Friends
• Easy to send or receive information

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Example (degree centrality)
Link to Model

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2. Closeness
Centrality

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Closeness Centrality
• Being the Geographic center makes you important
• You are never far from the action
• Uses shortest-path-lengths
• Farness: Mean distance to other nodes
• Closeness: Inverse of Farness
FarNess:
Closeness:
Normalized Measures

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Example 1 Closeness Centrality
Closeness
Normalized
Link to Model

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3. Betweenness
Centrality

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Betweenness Centrality
• Being in a lot of shortest paths is important
• Intuitively:
• Messages within the network pass through you
• You are a “bridge” between clusters of the network
: Shortest paths from s to t
: Shortest paths from s to t
that go through v

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Betweenness centrality of node C?
Link to Model

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Another Example
• Shortest paths are unique
• By Symmetry there are
• only 3 values
• (corresponding to the
• three colors)
Link to Model

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Calculus are long and tedious by hand

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4. Eigenvector
Centrality

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Improving Degree Centrality
•Degree centrality is miopic.
•Ignores the place of the node in the network
•Maybe the node is adjacent to important nodes

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¿Which node has more power?
2,4 or 6?

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Look at the neighbors
(Links give power)
3
2
1
3
2
3
2
3
2
6
7
8
Link to Excel Model
New Power: Sum of power
of my neighbors

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5. PageRank
Centrality
(For Directed Graphs)
Link To Model Page Rank vs Eigenvector if time
Link to Page-Rank Model

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Summary
1. Degree-centrality: Centrality depends on the degree of a node (number
of neighbors)
2. Closeness-Centrality: Centrality depends on the distance of a node to h
the other nodes
3. Betweenness-centrality: Centrality depends on haoe many shortest
paths pass through a node.
4. Clustering: How nodes cluster locally
5. Eigenvector-Centrality: Importance of a node depends on the
importance of neighbour nodes (recursive definition)

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Example 4
centralities
(15 Minutes)
Link to ppt for the
Google Meet Chat

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Another Example of 4 Centralities
Grandjean 2021
Introduction to
social networks

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Beautiful Example of 4 Centralities

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Netlogo Lab Models
Three Views Four Views
Two Views
One View

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Real Life Applications
Diffusion Vulnerability
Link to Model

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6. Clustering
coefficient

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Clustering
• How connected are the neighbors of a node
• (Friends of my friens are my friends)
• How many traingles are formed by myn neighbors
• Clustering coefficient of a node is the proprtiion of :
• number of links between neighbors
• divided by
• number of possible links

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Examples

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Another Example
1
2 0
4
3
6
5
Link to Model

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Additional Example
Link to Model

31. Case Study Florentine Families
(Familias Florentinas)

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Thanks You
for Your Attention!!!

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Excellent example 4 centralities
(Geometric network)

34. Centrality Measures
Closeness
Betweeness
EigenVector
Degree

35. Centrality Measures

36. Centrality Measures

37. Centrality Measures

38. Centrality Measures

39. BackUp

40. Florentine families
Ver Modelo

41. 9 Medidas
Centrales

42. Ejemplo Delfines

43. Aplicación de Medidas de Centralidad
• Red de Delfines (Mark Newman , Davidb
Lusseau)
• 62 Delfines (Doubtful Sound, Nueva
Zelandia)
• 7 años de “interacción social” de los delfines
• Dos delfines tienen un enlace si fueron
observados juntos más frecuentemente que
lo esperado por azar.
• Enlace al artículo de investigación: enlace

44. Red de Delfines
• Ver Modelo
¿Qué delfín tiene
muchas conexiones?
¿Quíen está en el centro
de la red?
¿Quién actúa como
inteemediario? (puente)

45. Conclusión Estudio Delfines
•La estructura de la red permite identificar delfines
populares y delfines cohesivos.
•La desaparición temporal de SN100 fisionó la red en
dos grupos
•La reaparición de SN100 coincidió con una posterior
unión de los dos grupos
•El alto valor de intermediación de SN100 jugó un
papel importante en mantener la comunidad unida.

46. Back-up

47. Temas Adicionales (no ver en clase)

48. ¿Cuál es el nodo más influyente?
• Los enlaces dan poder • Los enlaces entrantes (in-links)dan poder

49. Respuesta
• Los enlaces dan poder • Los enlaces entrantes (in-links)dan poder

50. Respuesta
• Los enlaces dan poder • Los enlaces entrantes (in-links)dan poder

51. Taller GeoGebra PageRank
• Taller
Geogebra n=9

52. Taller 2 PageRank
• Taller
Networkx
Netlogo(100000 iter)

53. Taller 2 PageRank
• Taller

54. Medidas Centrales Adicionales

55. Jackson Question
• Imagine that you are advertising your new product,
"CourseraPopcorn". You realize that various potential areas where
you will be selling your corn are connected to each other in a
network, and that if you advertise it in one area then it might be
communicated to neighboring areas. You only have the chance to
advertise in one area to seed the market and hope that news
spreads from there. You know the communication network among
all the areas. If you were to have to rely on one centrality measure
to pick the best area, and you did this based on what was found
from the data analysis in video 2.5a, which centrality measure
would you use?
• Eigenvector centrality