Graphs and networks theory

This chapter discusses graphs and networks theory

Stochastic pair approximation treatment of the noisy voter model

We present a full stochastic description of the pair approximation scheme to study binary-state dynamics on heterogeneous networks. Within this general approach, we obtain a set of equations for the dynamical correlations, fluctuations and finite-size effects, as well as for the temporal evolution of all relevant variables. We test this scheme for a prototypical model of opinion dynamics known as the noisy voter model that has a finite-size critical point. Using a closure approach based on a system size expansion around a stochastic dynamical attractor we obtain very accurate results, as compared with numerical simulations, for stationary and time dependent quantities whether below, within or above the critical region. We also show that finite-size effects in complex networks cannot be captured, as often suggested, by merely replacing the actual system size $N$ by an effective network dependent size $N_{{\rm eff}}

Networks in cognitive science

Networks of interconnected nodes have long played a key role in Cognitive Science, from artificial neural networks to spreading activation models of semantic memory. Recently, however, a new Network Science has been developed, providing insights into the emergence of global, system-scale properties in contexts as diverse as the Internet, metabolic reactions, and collaborations among scientists. Today, the inclusion of network theory into Cognitive Sciences, and the expansion of complex-systems science, promises to significantly change the way in which the organization and dynamics of cognitive and behavioral processes are understood. In this paper, we review recent contributions of network theory at different levels and domains within the Cognitive Sciences.Postprint (author's final draft

Dynamics and collective phenomena of social systems

This thesis focuses on the study of social systems through methods of theoretical physics, in particular proceedings of statistical physics and complex systems, as well as mathematical tools like game theory and complex networks. There already ex- ists predictive and analysis methods to address these problems in sociology, but the contribution of physics provides new perspectives and complementary and powerful tools. This approach is particularly useful in problems involving stochastic aspects and nonlinear dynamics. The contribution of physics to social systems provides not only prediction procedures, but new insights, especially in the study of emergent properties that arise from holistic approaches. We study social systems by introducing different agent-based models (ABM). When possible, the models are analyzed using mathematical methods of physics, in order to achieve analytical solutions. In addition to a theoretical approach, experi- mental treatment is performed via computer simulations both through Monte Carlo methods and deterministic or mixed procedures. This working method has proved very fruitful for the study of several open problems. The book is structured as follows. This introduction presents the mathematical formalisms used in the investigations, which are structured in two parts: in part I we deal with the emergence of cooperation, while in part II we analyze cultural dynamics under the perspective of tolerance

Beyond hairballs: depicting complexity of a kinase-phosphatase network in the budding yeast

Les kinases et les phosphatases (KP) représentent la plus grande famille des enzymes dans la cellule. Elles régulent les unes les autres ainsi que 60 % du protéome, formant des réseaux complexes kinase-phosphatase (KP-Net) jouant un rôle essentiel dans la signalisation cellulaire. Ces réseaux caractérisés d’une organisation de type commandes-exécutions possèdent généralement une structure hiérarchique. Malgré les nombreuse études effectuées sur le réseau KP-Net chez la levure, la structure hiérarchique ainsi que les principes fonctionnels sont toujours peux connu pour ce réseau. Dans ce contexte, le but de cette thèse consistait à effectuer une analyse d’intégration des données provenant de différentes sources avec la structure hiérarchique d’un réseau KP-Net de haute qualité chez la levure, S. cerevisiae, afin de générer des hypothèses concernant les principes fonctionnels de chaque couche de la hiérarchie du réseau KP-Net. En se basant sur une curation de données d’interactions effectuée dans la présente et dans d’autres études, le plus grand et authentique réseau KP-Net reconnu jusqu’à ce jour chez la levure a été assemblé dans cette étude. En évaluant le niveau hiérarchique du KP-Net en utilisant la métrique de la centralisation globale et en élucidant sa structure hiérarchique en utilisant l'algorithme vertex-sort (VS), nous avons trouvé que le réseau KP-Net possède une structure hiérarchique ayant la forme d’un sablier, formée de trois niveaux disjoints (supérieur, central et inférieur). En effet, le niveau supérieur du réseau, contenant un nombre élevé de KPs, était enrichi par des KPs associées à la régulation des signaux cellulaire; le niveau central, formé d’un nombre limité de KPs fortement connectées les unes aux autres, était enrichi en KPs impliquées dans la régulation du cycle cellulaire; et le niveau inférieur, composé d’un nombre important de KPs, était enrichi en KPs impliquées dans des processus cellulaires diversifiés. En superposant une grande multitude de propriétés biologiques des KPs sur le réseau KP-Net, le niveau supérieur était enrichi en phosphatases alors que le niveau inférieur en était appauvri, suggérant que les phosphatases seraient moins régulées par phosphorylation et déphosphorylation que les kinases. De plus, le niveau central était enrichi en KPs représentant des « bottlenecks », participant à plus d’une voie de signalisation, codées par des gènes essentiels et en KPs qui étaient les plus strictement régulées dans l’espace et dans le temps. Ceci implique que les KPs qui jouent un rôle essentiel dans le réseau KP-Net devraient être étroitement contrôlées. En outre, cette étude a montré que les protéines des KPs classées au niveau supérieur du réseau sont exprimées à des niveaux d’abondance plus élevés et à un niveau de bruit moins élevé que celles classées au niveau inférieur du réseau, suggérant que l’expression des enzymes à des abondances élevées invariables au niveau supérieur du réseau KP-Net pourrait être importante pour assurer un système robuste de signalisation. L’étude de l’algorithme VS a montré que le degré des nœuds affecte leur classement dans les différents niveaux d’un réseau hiérarchique sans biaiser les résultats biologiques du réseau étudié. En outre, une analyse de robustesse du réseau KP-Net a montré que les niveaus du réseau KP-Net sont modérément stable dans des réseaux bruités générés par ajout d’arrêtes au réseau KP-Net. Cependant, les niveaux de ces réseaux bruités et de ceux du réseau KP-Net se superposent significativement. De plus, les propriétés topologiques et biologiques du réseau KP-Net étaient retenues dans les réseaux bruités à différents niveaux. Ces résultats indiquant que bien qu’une robustesse partielle de nos résultats ait été observée, ces derniers représentent l’état actuel de nos connaissances des réseaux KP-Nets. Finalement, l’amélioration des techniques dédiées à l’identification des substrats des KPs aideront davantage à comprendre comment les réseaux KP-Nets fonctionnent. À titre d’exemple, je décris, dans cette thèse, une stratégie que nous avons conçu et qui permet à déterminer les interactions KP-substrats et les sous-unités régulatrices sur lesquelles ces interactions dépendent. Cette stratégie est basée sur la complémentation des fragments de protéines basée sur la cytosine désaminase chez la levure (OyCD PCA). L’OyCD PCA représente un essai in vivo à haut débit qui promet une description plus précise des réseaux KP-Nets complexes. En l’appliquant pour déterminer les substrats de la kinase cycline-dépendante de type 1 (Cdk1, appelée aussi Cdc28) chez la levure et l’implication des cyclines dans la phosphorylation de ces substrats par Cdk1, l’essai OyCD PCA a montré un comportement compensatoire collectif des cyclines pour la majorité des substrats. De plus, cet essai a montré que la tubuline- γ est phosphorylée spécifiquement par Clb3-Cdk1, établissant ainsi le moment pendant lequel cet événement contrôle l'assemblage du fuseau mitotique.Kinases and phosphatases (KP) form the largest family of enzymes in living cells. They regulate each other and 60 % of the proteome forming complex kinase-phosphatase networks (KP-Net) essential for cell signaling. Such networks having the command-execution aspect tend to have a hierarchical structure. Despite the extensive study of the KP-Net in the budding yeast, the hierarchical structure as well as the functional principles of this network are still not known. In this context, this thesis aims to perform an integrative analysis of multi-omics data with the hierarchical structure of a bona fide KP-Net in the budding yeast Saccharomyces cerevisiae, in order to generate hypotheses about the functional principles of each layer in the KP-Net hierarchy. Based on a literature curation effort accomplished in this and in other studies, the largest bona fide KP-Net of the S. cerevisiae known to date was assembled in this thesis. By assessing the hierarchical level of the KP-Net using the global reaching centrality and by elucidating the its hierarchical structure using the vertex-sort (VS) algorithm, we found that the KP-Net has a moderate hierarchical structure made of three disjoint layers (top, core and bottom) resembling a bow tie shape. The top layer having a large size was found enriched for signaling regulation; the core layer made of few strongly connected KPs was found enriched mostly for cell cycle regulation; and the bottom layer having a large size was found enriched for diverse biological processes. On overlaying a wide range of KP biological properties on top of the KP-Net hierarchical structure, the top layer was found enriched for and the bottom layer was found depleted for phosphatases, suggesting that phosphatases are less regulated by phosphorylation and dephosphoryation interactions (PDI) than kinases. Moreover, the core layer was found enriched for KPs representing bottlenecks, pathway-shared components, essential genes and for the most tightly regulated KPs in time and space, implying that KPs playing an essential role in the KP-Net should be firmly controlled. Interestingly, KP proteins in the top layer were found more abundant and less noisy than those of the bottom layer, suggesting that availability of enzymes at invariable protein expression level at the top of the network might be important to ensure a robust signaling. Analysis of the VS algorithm showed that node degrees affect their classification in the different layers of a network hierarchical structure without biasing biological results of the sorted network. Robustness analysis of the KP-Net showed that KP-Net layers are moderately stable in noisy networks generated by adding edges to the KP-Net. However, layers of these noisy overlap significantly with those of the KP-Net. Moreover, topological and biological properties of the KP-Net were retained in the noisy networks to different levels. These findings indicate that despite the observed partial robustness of our results, they mostly represent our current knowledge about KP-Nets. Finally, enhancement of techniques dedicated to identify KPs substrates will enhance our understanding about how KP-Nets function. As an example, I describe here a strategy that we devised to help in determining KP-substrate interactions and the regulatory subunits on which these interactions depend. The strategy is based on a protein-fragment complementation assay based on the optimized yeast cytosine deaminase (OyCD PCA). The OyCD PCA represents a large scale in vivo screen that promises a substantial improvement in delineating the complex KP-Nets. We applied the strategy to determine substrates of the cyclin-dependent kinase 1 (Cdk1; also called Cdc28) and cyclins implicated in phosphorylation of these substrates by Cdk1 in S. cerevisiae. The OyCD PCA showed a wide compensatory behavior of cyclins for most of the substrates and the phosphorylation of γ-tubulin specifically by Clb3-Cdk1, thus establishing the timing of the latter event in controlling assembly of the mitotic spindle

Convergence to consensus in heterogeneous groups and the emergence of informal leadership

When group cohesion is essential, groups must have efficient strategies in place for consensus decisionmaking. Recent theoretical work suggests that shared decision-making is often the most efficient way for dealing with both information uncertainty and individual variation in preferences. However, some animal and most human groups make collective decisions through particular individuals, leaders, that have a disproportionate influence on group decision-making. To address this discrepancy between theory and data, we study a simple, but general, model that explicitly focuses on the dynamics of consensus building in groups composed by individuals who are heterogeneous in preferences, certain personality traits (agreeability and persuasiveness), reputation, and social networks. We show that within-group heterogeneity can significantly delay democratic consensus building as well as give rise to the emergence of informal leaders, i.e. individuals with a disproportionately large impact on group decisions. Our results thus imply strong benefits of leadership particularly when groups experience time pressure and significant conflict of interest between members (due to various between-individual differences). Overall, our models shed light on why leadership and decision-making hierarchies are widespread, especially in human groups

Stabilization Bounds for Influence Propagation from a Random Initial State

We study the stabilization time of two common types of influence propagation. In majority processes, nodes in a graph want to switch to the most frequent state in their neighborhood, while in minority processes, nodes want to switch to the least frequent state in their neighborhood. We consider the sequential model of these processes, and assume that every node starts out from a uniform random state. We first show that if nodes change their state for any small improvement in the process, then stabilization can last for up to $\Theta(n^2)$ steps in both cases. Furthermore, we also study the proportional switching case, when nodes only decide to change their state if they are in conflict with a $\frac{1+\lambda}{2}$ fraction of their neighbors, for some parameter $\lambda \in (0,1)$. In this case, we show that if $\lambda < \frac{1}{3}$, then there is a construction where stabilization can indeed last for $\Omega(n^{1+c})$ steps for some constant $c>0$. On the other hand, if $\lambda > \frac{1}{2}$, we prove that the stabilization time of the processes is upper-bounded by $O(n \cdot \log{n})$