Random graph ensembles with many short loops

Abstract

Networks observed in the real world often have many short loops. This violates the tree-like assumption that underpins the majority of random graph models and most of the methods used for their analysis. In this paper we sketch possible research routes to be explored in order to make progress on networks with many short loops, involving old and new random graph models and ideas for novel mathematical methods. We do not present conclusive solutions of problems, but aim to encourage and stimulate new activity and in what we believe to be an important but under-exposed area of research. We discuss in more detail the Strauss model, which can be seen as the 'harmonic oscillator' of 'loopy' random graphs, and a recent exactly solvable immunological model that involves random graphs with extensively many cliques and short loops.Résumé. Les réseaux observés dans la Nature ont souvent des cycles courts. Ceci contredit le postulat de hiérarchie sur lequel se base la majorité des modèles de réseaux aléatoires et la plupart des méthodes utilisées pour leur analyse. Dans cet article, nous esquissons des directions de recherches possibles, afin de progresser sur les réseaux contenant beaucoup de cycles courts, faisant appel à des modèles de réseaux aléatoires éprouvés ou nouveaux, et des idées pour de nouvelles méthodes mathématiques. Nous ne présentons pas de solutions définitives, mais notre but est d'encourager et de stimuler de nouveaux travaux dans ce que nous croyons être une direction de recherche importante, bien que insuffisamment explorée. Nous discutons en détail le modèle de Strauss, qui peut être interprété comme 'l'oscillateur harmonique' des réseaux aléatoires 'Ãȃ boucles', ainsi qu'un modèle immunologique soluble exactement qui implique des réseaux aléatoires avec de nombreux cliques et cycles courts.

Quantifying randomness in real networks

“…The main caveat of our approach is that we have no proof that our dk -random graph generation algorithms for d =2.1 and d =2.5 sample graphs uniformly at random from the ensemble. The random-graph ensembles and edge-rewiring processes employed here are known to suffer from problems such as degeneracy and hysteresis 35 61 62 . Ideally, we would wish to calculate analytically the exact expected value of a given property in an ensemble.…”

Distribution of shortest cycle lengths in random networks

“…In this paper we have studied configuration model networks in which the DSCL is completely determined by the degree distribution P (k). Recently, other network ensembles were introduced, which include many short cycles, where the cycle lengths are controlled by various constraints [58,59]. It would be interesting to generalize the calculation of the DSCL to such networks.…”

Entropy of microcanonical finite-graph ensembles

“…Thus, the total number of vertices N in a graph is always fixed to the specified number. Note also that although the summation (15) is apparently complicated, this operation can be easily coded using the itertools.product function in Python.…”

“…In [14], Peixoto used a combinatorial approach to evaluate the entropy of stochastic block models. See [15][16][17][18][19][20][21] for other recent related works in the physics community.…”