02-CN&S2025-ComplexNetworkMetricsRedesComplejas.pdf

Cinvestav
Guadalajara
Lecture 02: Complex Network Metrics
Arturo Díaz Pérez
Centro de Investigación y de Estudios Avanzados del IPN
Unidad Guadaljara
adiaz@cinvestav.mx

1. Complex Networks
2. Metrics
3. Degree and average degree
4. Small World
5. Hierarchy Metrics
6. Centrality Metrics
7. Metrics Correlation
Contenido
02-ComplexNetworksMetrics
Análisis de Datos, Redes Complejas y Seguridad 2025 2

Are Complex Networks Random?
4
Meme diffusion related
to the 2011 Arab
Spring from the #Egypt
hashtag.
Credit: Indiana
University
Análisis de Datos, Redes Complejas y Seguridad 2025

Are Complex Networks Random?
Communities
Small-world Scale-free

Redes Complejas
Definición Formal de Redes Complejas
Una red compleja 𝐺 = (𝑉, 𝐸) es un grafo con un
conjunto de vértices 𝑉 y un conjunto de aristas 𝐸
con:
• 𝑛 = |𝑉| y 𝑚 = 𝐸 en el orden de miles y millones.
• Bajo grado promedio: 𝑘 ≪ 𝑛.
• Baja densidad: 𝑑 ≪ 1.
• Longitud de caminos promedio baja (small-world):
𝐿 ≪ 𝑛
• Distribución libre de escala: 𝑃(𝑘)~𝑘−𝛼
.
• Promedio alto de coeficiente de clustering:
1
n
≪
𝐶𝐶 < 1.
Communities
Small-world Scale-free (Power Law)
Degree Distribution

Classification of Complex Network Metrics

• The representation of the essential properties of complex networks in a compact fashion. In
this way, it is possible to focus on a set of statistics of interest instead of trying to decipher the
structure of the whole graph.
• The differentiation among distinct classes of complex networks by measuring the set of
invariant statistical properties that characterize the members of a particular graph class.
• The design of structure-aware algorithms capable of determining the elements of the graph
that show a property of interest for a particular application domain.
Complex Network Metrics Applications

Local, Global, and Scaling Complex Network Metrics
• Much of the work in Network Science has been dedicated to the characterization of the
structure of graphs in terms of complex network metrics
Scaling of metrics
Avg.
Betweenness
Cent. with k
Cum. Degree
Dist. with k
Avg.
Neighbor
Conn. with k
Global metrics
Edges Assortativity
Coef.
Nodes
Max.
Degree
Diameter
Hierarchical
Degree
Local metrics
Local
Clustering
Betweenness
Cent.
Node
Degree

Clustering
Betweenness
Degree Max. degree Shortest path
Assortativity Straightness
Deg. Scaling
Local metric:
average over all nodes
Global metric
Common Complex Network Metrics

Degree and Average Degree

Density of a Network

Average Path Length of Networks

Local Clustering Coefficient of a Vertex

• A small-world network is defined to be a network where the typical distance L between two
randomly chosen nodes (the number of steps required) grows proportionally to the logarithm
of the number of nodes in the network
Small-world Networks

Small-world network example
Hubs are bigger than other nodes Average
degree= 3.833
Average shortest path length = 1.803
Clustering coefficient = 0.522
Random graph example
Average degree = 2.833
Average shortest path length = 2.109
Clustering coefficient = 0.167
Small-world vs. Random Networks

Mundo Pequeño
Small-World Property
Las redes tiene pocos caminos-cortos de
“longitud larga” y muchos caminos cortos
entre la mayoría de los pares de nodos,
usualmente creados por ”hubs”
High L, High C Low L, Low C
Low L, High C

Complex Network Measurements

Virtual Hierarchy in Complex Networks

Virtual Hierarchy of the Oregon AS Network

Hierarchical Coefficient at Distance l

Betweenness Centrality

Central Point Dominance

Closeness Centrality

Centrality Metrics
A) Betweenness centrality
B) Closeness centrality
C) Eigenvector centrality
D) Degree centrality
F) Harmonic centrality
E) Katz centrality

Red Eléctrica
Red eléctrica de 4602 nodos
Comunidades
detectadas
50

Centralidad de Grados y de Intermediación

Centralidad de Cercanía y de Excentricidad

Centralidad de Intermediación de Aristas

Correlation Patterns of CN Metrics
Research Question
How to methodologically obtain a set of non-
redundant complex network metrics on a
representative ensemble of complex
networks?

Análisis de Datos, Redes Complejas y Seguridad 2025
Autonomous System Networks
• An Autonomous System (AS) is a group of
hundreds or millions of host IP addresses that share
common routing policies.
• AS interact with each other through a massive
network of thousand of links to form an AS
Network (ASN).
• Internet measurement may be considered a big
data problem that involves topology
measurements and links discovery [Cho 2012].
• The topology of ASNs is far from being random and
presents properties that encode details about its
functional behavior.
Scale-free
Rich-club
Hierarchical
Disassortative
AS
AS
AS AS
AS
AS AS path
37
02-ComplexNetworksMetrics 37

ASN Datasets
⧫ RV/RIPE Dataset: It includes Border Gateway Protocol (BGP) AS-paths obtained from raw BGP table dumps from
two major publicly available collectors: Route Views (RV) and RIPE. RV/ RIPE ASN’s include large/small transit
providers, content/access/hosting providers and Enterprise networks AS’s; and focuses on the modeling of
customer-provider links. The dataset includes 51 ASN’s corresponding to the evolution of the Internet from
January 1998 to January 2010. The smallest ASN contains 3,247 nodes and 5,646 edges, while the largest one has
33,796 nodes and 94,394 edges.
⧫ CAIDA Dataset: It includes ASN’s derived from RV BGP table snapshots. The CAIDA dataset models customer-
provider, peer-to-peer, and sibling-to-sibling AS relationships. It is composed of 61 networks that include ASN’s
from January 2004 to November 2007. The smallest ASN has 8,020 nodes and 18,203 edges, while the largest
one has 26,389 nodes and 52,861 edges.
⧫ DIMES Dataset: Mid-level modeling of the Internet where each node represents a small AS or a Point of
Presence (PoP) of a large/medium size AS. The dataset was built by exploiting a distributed approach where a
large community of host nodes run lightweight measurement agents in background. The DIMES dataset is
composed of 60 networks that include ASN’s from January 2007 to April 2012. The smallest giant component has
16,029 nodes and 27,620 edges, while the largest one has 28,035 nodes and 108,373 edges.
⧫ INET3 Dataset: INET3 is an Internet topology generator that produces random networks that resemble the
topology of the Internet from November 1997 to Feb 2002, and beyond, according to raw BGP tables from The
National Laboratory for Applied Network Research (NLANR) and the RV project (University of Michigan, 2002).
For the INET3 dataset 51 ASN’s were generated with approximately the same number of vertices than the ASN’s
in the RV/RIPE dataset, and the default values were used for the model parameters. The number of edges is
decided by the generator. 02-ComplexNetworksMetrics

Selection and normalization of ASN metrics
• Procedure in [Bonouva and de Weck 2012] to make measurements independent
from network sizes.

Pair-wise metric correlations

Correlations on the RV-RIPE ASNs
• The three-vertex cluster represents metrics that express properties of the most central
vertex.
• Most of the metrics in the six-vertex cluster describe either density or shortest-path
properties of ASNs.
Correlation heat map (0 ≤ C ≤ 1) Correlation graph (0.9 ≤ C ≤ 1)

1 Correlation graphs
Correlation Graphs and Metrics Selection
Garcia-Robledo A., Diaz-Perez A., Morales-Luna A., Correlation Analysis of Complex Network Metrics on the Topology of the Internet, CEWIT'13, Melville NY, 2013

Metrics Selection Frequency
2 Metrics selection frequency

1
2
Validation of Non-redundant Metrics
Unsupervised learning
Supervised learning

PCA visualization of ASN datasets with non-redundant metrics
Garcia-Robledo A., Diaz-Perez A., Morales-Luna A., Characterization and Coarsening of Autonomous System Networks: Measuring and
Simplifying the Internet, Book chapter in Advanced Methods for Complex Network Analysis, IGI Global, 2016

PCA visualization of the RV/RIPE dataset

PCA visualization of the DIMES Dataset

PCA visualization of the CAIDA Dataset

Further Reading
• Garcia-Robledo A., Diaz-Perez A., Morales-Luna A., Correlation Analysis of Complex Network
Metrics on the Topology of the Internet, CEWIT'13, Melville NY, 2013
• Garcia-Robledo A., Diaz-Perez A., Morales-Luna A., Characterization and Coarsening of
Autonomous System Networks: Measuring and Simplifying the Internet, Book chapter in
Advanced Methods for Complex Network Analysis, IGI Global, 2016
• Costa, L. D. F., Rodrigues, F. A., Travieso, G., & Villas Boas, P. R. (2007). Characterization of
complex networks: A survey of measurements. Advances in physics, 56(1), 167-242.
• Visit https://www.opte.org/the-internet

Fin de la Sesión 02

02-CN&S2025-ComplexNetworkMetricsRedesComplejas.pdf

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