This repo is where I am going to be pushing some general implementations of models from the advanced algorithms two module to. Also decided I am going to be slapping some general notes in this README file.

• An implementation for the generation of the ER Random, complex network model.
• Some notes about structural properties of:
  • ER-Random Networks.
• Hopefully I won't forget to add more here when I actually commit other code.. (I am only human!)

• Average Path Length (size)
• Global Clustering Coefficient (density)
• Degree Distribution (connectivity)
• Topological Characteristics:
  • Centrality (Degree, Closeness, Betweenness)
  • Mixing Patterns
• Classification & properties of different network models:
  • ER-Random
  • Small World:
    • Watts-Strogatz
    • Newman-Watts
  • Scale Free:
    • Barabási and Albert (BA) Model
    • BA Variant: Rewiring
    • BA Variant: Competition

Check out the ER-Random folder in the Repo.

• AVG Node Degree: < k > = 2M / N = p(N - 1) ~= pN.
• AVG Path Length: L ~ ln(N) / ln(< k >).
  • L increases logarithmically with the size of the network (Small World Property)
  • ln(N) grows much slower than N (see plot How ln(N) varies as N increases.png in repo) -> a very large sized usually has a relatively small L. Additional reasoning: The numerator for L is growing very slowly and the denominator for L is decreasing, therefore L is increasing as N is increasing. Therefore as more nodes are added to the network, the average path length is decreasing.
• Clustering Co-Efficient: C = p ~= / N << 1.
• Node Degree Distribution follows a Poisson distribution. (Need to think about this one a little more..)
• ER Random graphs are not small world networks, even though they exhibit a small L. This is because they generally do not have a large clustering co-efficient, it is fairly small and C ~ p, which makes sense if you take a look at the figureOne in the repo.
• ER Random graphs have a small clustering coefficient (this is a distinguishing characteristic).

Its also important to note that these types of graphs have an emergent feature, in this case, you generally see the appearance of cycles or a giant component at some threshold value of p.

Check out the Small-World-Network folder in the Repo.

Small world networks are more like real world networks, two distinguishing characteristics of small world networks are as follows:

• Short AVG path length.
• High clustering coefficient.

And there exist two primary models that we need to know about (for the exam anyway):

• Watts-Strogatz Model
• Newman-Watts variant of the Watts-Strogatz Model.

1. Initialise a regular nearest-neighbour coupled network of N nodes arranged in a ring, in which each node i is symmetrically connected to its nearest neighbours i ± 1, i ± 2, ..., i ± K/2 with K an even integer (being the number of adjacent nodes to i). The number of links is NK/2.
2. Randomisation: Randomly rewire each link of the network with probability p: a. For every node i (i = 1,...,N), each link connected to a clockwise (or counter-clockwise) neighbour of i is rewired to a randomly chosen node (from anywhere in the network) with a probability of p. b. The link will be preserved with a probability 1 - p.

Implementation!

• My Personal Website
• My Blog

• Eva Navarro Lopez (2015), Advanced Algorithms 2 Lecture Slides, University of Manchester.
• Alain Barrat, Marc Barthelemy, Alessando Vespignani (2008), Dynamical Processes on Complex Networks, Cambridge University Press.