Graphs can be classified into:
Undirected: edges do not have a direction.
Directed: edges have a direction.
Cyclic: contain cycles.
Acyclic: do not contain cycles.
Digraph: directed graph.
Weighted: edges have weights.
Unweighted: edges do not have weights.
Sparse: few edges compared to vertices.
Dense: many edges compared to vertices.
Definition: Large graphs contain thousands of vertices and tens of thousands of edges, making visualization and analysis challenging.
Applications:
Represent biological, technical, and social networks.
Analysis provides insights into structure and behavior.
Example: Protein interaction networks offer insights into health and disease states of organisms.
Not randomly connected; exhibit specific patterns.
Scale-Free Networks: Few vertices have many connections; most have few connections.
Small-World Networks: Small average distances between vertices despite a large number of nodes.
Basic models for analyzing large networks:
Random Networks
Small-World Networks
Scale-Free Networks
Introduced by Erdős and Rényi, these graphs are formed by adding edges randomly based on a probability.
Complete graphs emerge when p = 1. The Python library NetworkX includes functions for creating these graphs.
Characterized by short distances between nodes and high clustering coefficients.
The property holds for networks like protein interactions and the internet.
Mean distance reportedly approximates O(log(n)) as nodes increase.
Graphs exhibit a balance between randomness and order. High clustering allows many connections with few hops.
Examples in various domains: Social networks, road maps, and networks in biology.
Models analyzed by Watts and Strogatz highlight parameters conducive to small-world formation with varying values for p.
Defined by a degree distribution following a power law, assisting in identifying nodes of greater connectivity.
Proposed by Barabási and Albert focusing on growth and preferential attachment:
Growth: Network nodes increase over time.
Preferential attachment: Newer nodes preferentially link to high-degree nodes, underlying the principle that “the rich get richer.”
Utilized to create synthetic complex networks replicating real-world characteristics like scale-free and small-world properties through the Kronecker product.
Python's NumPy library aids in constructing Kronecker graphs, which mimic various complex networks.
Defined by internal and external degrees, emphasizing the cohesion of nodes within communities.
Internal degree: Connections within the community.
Community degree: Sum of node degrees in a community.
Density reflects potential links to gauge community strength.
Strong communities: Nodes have more connections internally than externally.
Weak communities: Total internal node degrees exceed those to the rest of the network.
A partition uniquely groups nodes into a community, with a Bell number indicating possible configurations.
Overlapping communities: Reflect social networks where nodes belong to multiple groups simultaneously.
Defined by identifying a cut separating subnetworks, fundamental in community detection frameworks.
Kernighan-Lin Algorithm: An iterative method that seeks to minimize cut size while maintaining cluster sizes.
Bridge-Removal: Identifies links whose removal disconnects communities, simplifying community detection through analyzes of connected components.
It assesses the quality of a community partition relative to a randomly constructed baseline, highlighting links significant to clusters versus expected random outcomes.
The algorithm can efficiently compute partitions maximizing modularity using the NetworkX library.
KG_Lecture_IV_2024-26
Graphs can be classified into:
Undirected: edges do not have a direction.
Directed: edges have a direction.
Cyclic: contain cycles.
Acyclic: do not contain cycles.
Digraph: directed graph.
Weighted: edges have weights.
Unweighted: edges do not have weights.
Sparse: few edges compared to vertices.
Dense: many edges compared to vertices.
Definition: Large graphs contain thousands of vertices and tens of thousands of edges, making visualization and analysis challenging.
Applications:
Represent biological, technical, and social networks.
Analysis provides insights into structure and behavior.
Example: Protein interaction networks offer insights into health and disease states of organisms.
Not randomly connected; exhibit specific patterns.
Scale-Free Networks: Few vertices have many connections; most have few connections.
Small-World Networks: Small average distances between vertices despite a large number of nodes.
Basic models for analyzing large networks:
Random Networks
Small-World Networks
Scale-Free Networks
Introduced by Erdős and Rényi, these graphs are formed by adding edges randomly based on a probability.
Complete graphs emerge when p = 1. The Python library NetworkX includes functions for creating these graphs.
Characterized by short distances between nodes and high clustering coefficients.
The property holds for networks like protein interactions and the internet.
Mean distance reportedly approximates O(log(n)) as nodes increase.
Graphs exhibit a balance between randomness and order. High clustering allows many connections with few hops.
Examples in various domains: Social networks, road maps, and networks in biology.
Models analyzed by Watts and Strogatz highlight parameters conducive to small-world formation with varying values for p.
Defined by a degree distribution following a power law, assisting in identifying nodes of greater connectivity.
Proposed by Barabási and Albert focusing on growth and preferential attachment:
Growth: Network nodes increase over time.
Preferential attachment: Newer nodes preferentially link to high-degree nodes, underlying the principle that “the rich get richer.”
Utilized to create synthetic complex networks replicating real-world characteristics like scale-free and small-world properties through the Kronecker product.
Python's NumPy library aids in constructing Kronecker graphs, which mimic various complex networks.
Defined by internal and external degrees, emphasizing the cohesion of nodes within communities.
Internal degree: Connections within the community.
Community degree: Sum of node degrees in a community.
Density reflects potential links to gauge community strength.
Strong communities: Nodes have more connections internally than externally.
Weak communities: Total internal node degrees exceed those to the rest of the network.
A partition uniquely groups nodes into a community, with a Bell number indicating possible configurations.
Overlapping communities: Reflect social networks where nodes belong to multiple groups simultaneously.
Defined by identifying a cut separating subnetworks, fundamental in community detection frameworks.
Kernighan-Lin Algorithm: An iterative method that seeks to minimize cut size while maintaining cluster sizes.
Bridge-Removal: Identifies links whose removal disconnects communities, simplifying community detection through analyzes of connected components.
It assesses the quality of a community partition relative to a randomly constructed baseline, highlighting links significant to clusters versus expected random outcomes.
The algorithm can efficiently compute partitions maximizing modularity using the NetworkX library.