Modularity of regular and treelike graphs

Clustering algorithms for large networks typically use modularity values to test which partitions of the vertex set better represent structure in the data. The modularity of a graph is the maximum modularity of a partition. We consider the modularity of two kinds of graphs. For $r$-regular graphs with a given number of vertices, we investigate the minimum possible modularity, the typical modularity, and the maximum possible modularity. In particular, we see that for random cubic graphs the modularity is usually in the interval $(0.666, 0.804)$, and for random $r$-regular graphs with large $r$ it usually is of order $1/\sqrt{r}$. These results help to establish baselines for statistical tests on regular graphs. The modularity of cycles and low degree trees is known to be close to 1: we extend these results to `treelike' graphs, where the product of treewidth and maximum degree is much less than the number of edges. This yields for example the (deterministic) lower bound $0.666$ mentioned above on the modularity of random cubic graphs.Comment: 25 page

20 years of network community detection

A fundamental technical challenge in the analysis of network data is the automated discovery of communities - groups of nodes that are strongly connected or that share similar features or roles. In this commentary we review progress in the field over the last 20 years.Comment: 6 pages, 1 figure. Published in Nature Physic

Structural and dynamical interdependencies in complex networks at meso- and macroscale: nestedness, modularity, and in-block nestedness

Many real systems like the brain are considered to be complex, i.e. they are made of several interacting components and display a collective behaviour that cannot be inferred from how the individual parts behave. They are usually described as networks, with the components represented as nodes and the interactions between them as links. Research into networks mainly focuses on exploring how a network's dynamic behaviour is constrained by the nature and topology of the interactions between its elements. Analyses of this sort are performed on three scales: the microscale, based on single nodes; the macroscale, which explores the whole network; and the mesoscale, which studies groups of nodes. Nonetheless, most studies so far have focused on only one scale, despite increasing evidence suggesting that networks exhibit structure on several scales. In our thesis, we apply structural analysis to a variety of synthetic and empirical networks on multiple scales. We focus on the examination of nested, modular, and in-block nested patterns, and the effects that they impose on each other. Finally, we introduce a theoretical model to help us to better understand some of the mechanisms that enable such patterns to emerge.Molts sistemes, com el cervell o internet, són considerats complexos: sistemes formats per una gran quantitat d'elements que interactuen entre si, que exhibeixen un comportament col·lectiu que no es pot inferir des de les propietats dels seus elements aïllats. Aquests sistemes s'estudien mitjançant xarxes, en les quals els elements constituents són els nodes, i les interaccions entre ells, els enllaços. La recerca en xarxes s'enfoca principalment a explorar com el comportament dinàmic d'una xarxa està definit per la naturalesa i la topologia de les interaccions entre els seus elements. Aquesta anàlisi sovint es fa en tres escales: la microescala, que estudia les propietats dels nodes individuals; la macroescala, que explora les propietats de tota la xarxa, i la mesoescala, basada en les propietats de grups de nodes. No obstant, la majoria dels estudis se centren només en una escala, tot i la creixent evidència que suggereix que les xarxes sovint exhibeixen estructura a múltiples escales. En aquesta tesi estudiarem les propietats estructurals de les xarxes a escala múltiple. Analitzarem les propietats estructurals dels patrons in-block nested i la seva relació amb els patrons niats i modulars. Finalment, introduirem un model teòric per explorar alguns dels mecanismes que permeten l'emergència d'aquests patrons.Muchos sistemas, como el cerebro o internet, son considerados complejos: sistemas formados por una gran cantidad de elementos que interactúan entre sí, que exhiben un comportamiento colectivo que no puede inferirse desde las propiedades de sus elementos aislados. Estos sistemas se estudian mediante redes, en las que los elementos constituyentes son los nodos, y las interacciones entre ellos, los enlaces. La investigación en redes se enfoca principalmente a explorar cómo el comportamiento dinámico de una red está definido por la naturaleza y la topología de las interacciones entre sus elementos. Este análisis a menudo se hace en tres escalas: la microescala, que estudia las propiedades de los nodos individuales; la macroescala, que explora las propiedades de toda la red, y la mesoescala, basada en las propiedades de grupos de nodos. No obstante, la mayoría de los estudios se centran solo en una escala, a pesar de la creciente evidencia que sugiere que las redes a menudo exhiben estructura a múltiples escalas. En esta tesis estudiaremos las propiedades estructurales de las redes a escala múltiple. Analizaremos las propiedades estructurales de los patrones in-block nested y su relación con los patrones anidados y modulares. Finalmente, introduciremos un modelo teórico para explorar algunos de los mecanismos que permiten la emergencia de estos patrones.Tecnologías de la información y de rede

On the topology Of network fine structures

Multi-relational dynamics are ubiquitous in many complex systems like transportations, social and biological. This thesis studies the two mathematical objects that encapsulate these relationships --- multiplexes and interval graphs. The former is the modern outlook in Network Science to generalize the edges in graphs while the latter was popularized during the 1960s in Graph Theory. Although multiplexes and interval graphs are nearly 50 years apart, their motivations are similar and it is worthwhile to investigate their structural connections and properties. This thesis look into these mathematical objects and presents their connections. For example we will look at the community structures in multiplexes and learn how unstable the detection algorithms are. This can lead researchers to the wrong conclusions. Thus it is important to get formalism precise and this thesis shows that the complexity of interval graphs is an indicator to the precision. However this measure of complexity is a computational hard problem in Graph Theory and in turn we use a heuristic strategy from Network Science to tackle the problem. One of the main contributions of this thesis is the compilation of the disparate literature on these mathematical objects. The novelty of this contribution is in using the statistical tools from population biology to deduce the completeness of this thesis's bibliography. It can also be used as a framework for researchers to quantify the comprehensiveness of their preliminary investigations. From the large body of multiplex research, the thesis focuses on the statistical properties of the projection of multiplexes (the reduction of multi-relational system to a single relationship network). It is important as projection is always used as the baseline for many relevant algorithms and its topology is insightful to understand the dynamics of the system.Open Acces