The Onset of Geometry in Complex Networks - From Stuctural Properties to Dynamical Processes

Abstract

[eng] Over the past few decades, the use of complex networks to describe the properties of systems of many interacting particles has become widespread in many fields of science. Surprisingly, networks from disparate disciplines share a wide range of basic properties, such as small worldness, high levels of clustering and broad degree distributions. One of the most promising frameworks to explain this observation is that of network geometry, where nodes are assumed to live in some underlying metric space that conditions their connectivity. The fact that this approach can reproduce all the basic network properties and symmetries as well as produce strong results in practical tasks such as community detection and link prediction has led many to wonder if there is a way to determine if real networks are indeed geometric in nature. However, these studies do not contemplate the fact that the transition between non-geometric and geometric networks might not be sharp. In this thesis we study the effect the effect of the underlying metric space on the complex network for different geometric coupling strengths. We show that three different regimes can be identified: In the non-geometric regime, where the coupling is extremely weak, results are similar to those of the configuration model, which is explicitly non-geometric. Increasing the coupling slightly leads us to the quasi-geometric regime, where the scaling of the clustering coefficient with the system size is extremely slow, leading to significant levels of this quantity for finite systems. Additionally, we show that, here, geometric information can be extracted from the topology alone through network embedding, and that it is essential for obtaining self-similar network replicas through geometric renormalization. Finally, we study a large number of empirical networks and show that they are best described in the quasi-geometric regime. Increasing the coupling further leads us the geometric region. This is the regime typically studied past works where the effects of the underlying metric space are strong and where clustering remains finite in the thermodynamic limit. Motivated by these results for single-layer graphs we also study geometric multiplexes. We introduce the mutual network, which is made up of all edges that are shared by all layers. This object allows us to obtain rigorous results on edge overlap as well as mutual clustering. We show that the geometric region can be extended in the mutual case when the coupling of the individual layers to their underlying metric space are of similar strengths. Having extensively investigated the structural properties at various coupling strengths, we lastly turn to dynamical processes running of top of the network. Specifically, we show that the underlying metric space reveals periodic Turing patterns, both in the quasi- and strongly geometric regimes as well as in empirical networks. All these results show that the underlying geometry is essential for understanding complex networks, both from a structural as well as dynamical point of view.

[cat] En les darreres dècades, l'ús de xarxes complexes per descriure les propietats de sistemes amb moltes parts que interactuen ha esdevingut habitual en molts camps de la ciència. Sorprenentment, xarxes de disciplines diverses comparteixen una àmplia gamma de propietats bàsiques, com ara la propietat “small world”, alts nivells d’agrupació i distribucions de grau amplis. Un dels marcs més prometedors per explicar aquesta observació és el de la geometria de xarxes, on es presumeix que els nodes es troben en un espai mètric subjacent que condiciona la seva connectivitat. En aquesta tesi estudiem l'efecte de l'espai mètric subjacent en xarxes complexes per diferents intensitats de l'acoblament geomètric. Identifiquem tres règims diferents: En el règim no geomètric, on l'acoblament és extremadament feble, els resultats són similars als de models explícitament no geomètrics. Un augment lleuger de l'acoblament ens porta al règim quasi-geomètric, on l'escalat del coeficient d’agrupació amb la mida del sistema és molt lent, cosa que genera nivells significatius d'aquesta quantitat per a sistemes finits. A més, mostrem que, aquí, es pot extreure informació geomètrica només de la topologia mitjançant encaix de xarxes, i que és essencial per obtenir rèpliques auto-similars de xarxes mitjançant la renormalització geomètrica. Finalment, estudiem un gran nombre de xarxes empíriques i mostrem que es descriuen millor en el règim quasi-geomètric. Augmentar encara més l'acoblament ens condueix al règim geomètric. Aquest és el règim típicament estudiat en treballs anteriors, on l'agrupació es manté finita en el límit termodinàmic. Motivats per aquests resultats per a grafs monocapa, també estudiem multiplexes geomètriques. Introduïm la xarxa mútua, formada per totes les arestes compartides per totes les capes, i obtenim resultats rigorosos sobre la superposició d’arestes i triangles. Després d'investigar les propietats estructurals a diferents intensitats d'acoblament, ens centrem en els processos dinàmics a la xarxa. Mostrem que l'espai mètric subjacent revela patrons periòdics de Turing, tant en règims quasi-geomètrics i fortament geomètrics com en xarxes empíriques. Aquests resultats mostren que la geometria subjacent és essencial per entendre les xarxes complexes, tant des d'un punt de vista estructural com dinàmic.