Configuration model
This flexibility allows the modeler to construct networks with arbitrary degree distributions, making it widely used as a reference model for real-life networks, particularly in social, biological, and technological domains.
The concept of "Configuration Model" was first introduced by Béla Bollobás, who laid the foundation for its application in graph theory and network science[2].
The key advantage of the Configuration Model lies in its ability to decouple the degree sequence from specific edge generation processes.
This variation is referred to as micro-canonical because it defines a uniform probability space where all possible graphs consistent with the given degree sequence are equally likely to be sampled.
Canonical Configuration Models relax the exact degree sequence constraint and preserve it only in expectation.
It assumes that edges between nodes are independent from each other, differently from the micro-canonical model.
The Chung-Lu configuration model, provides the benchmark in the calculation of network modularity.
where: This equation compares the observed network to the configuration model, where edges are distributed based solely on the degree sequence.
As for Chung-Lu configuration model, nodes pairs are assumed to be independent from each other.
This model captures resource competition by enforcing that the sum of interactions across all node pairs is fixed.
Forming an edge between two nodes reduces the availability of resources for other connections, reflecting realistic constraints in systems like social or ecological networks.
The uniform distribution of the matching is an essential property in calculating other features of the generated networks.
The network generation process does not exclude the development of a self-loop or a multi-link.
If we designed the process where self-loops and multi-edges are not allowed, the matching of the stubs would not follow a uniform distribution.
The expected total number of multi-links in a configuration model network would be:
For some power-law degree distributions where the second moment diverges, density of multi-links may not vanish or may do so more slowly than
(the average probability that the neighbors of a node are connected) is computed approximately as follows: where
That means that, the critical threshold solely depends on quantities which are uniquely determined by the degree distribution
or less from that starting node, the set will, with probability tending to 1 as n → ∞, take the form of a tree.
[11] The intuition behind this criterion is that if the giant component (GC) exists, then the average degree of a randomly chosen vertex
, the number of nodes of degree 0 and 2 have no contribution in the existence of the giant component.
tends to infinity, the probability to find two giant components is vanishing.
- the probability that a randomly sampled vertex belongs to a connected component of size
Three general properties of complex networks are heterogeneous degree distribution, short average path length and high clustering.
[1][14][15][16] Having the opportunity to define any arbitrary degree sequence, the first condition can be satisfied by design.
All the networks generated by this model are locally tree-like provided the average of the excess degree distribution is either constant or grows slower than the square root of number of links,
In other words, this model prevents forming substructures such as loops in the large size limit.
[10] In the DCM (directed configuration model),[18] each node is given a number of half-edges called tails and heads.
Then tails and heads are matched uniformly at random to form directed edges.
The size of the giant component,[18][19] the typical distance,[20] and the diameter[21] of DCM have been studied mathematically.
