Dynamical modeling and analysis of large cellular regulatory networks.

The dynamical analysis of large biological regulatory networks requires the development of scalable methods for mathematical modeling. Following the approach initially introduced by Thomas, we formalize the interactions between the components of a network in terms of discrete variables, functions, and parameters. Model simulations result in directed graphs, called state transition graphs. We are particularly interested in reachability properties and asymptotic behaviors, which correspond to terminal strongly connected components (or "attractors") in the state transition graph. A well-known problem is the exponential increase of the size of state transition graphs with the number of network components, in particular when using the biologically realistic asynchronous updating assumption. To address this problem, we have developed several complementary methods enabling the analysis of the behavior of large and complex logical models: (i) the definition of transition priority classes to simplify the dynamics; (ii) a model reduction method preserving essential dynamical properties, (iii) a novel algorithm to compact state transition graphs and directly generate compressed representations, emphasizing relevant transient and asymptotic dynamical properties. The power of an approach combining these different methods is demonstrated by applying them to a recent multilevel logical model for the network controlling CD4+ T helper cell response to antigen presentation and to a dozen cytokines. This model accounts for the differentiation of canonical Th1 and Th2 lymphocytes, as well as of inflammatory Th17 and regulatory T cells, along with many hybrid subtypes. All these methods have been implemented into the software GINsim, which enables the definition, the analysis, and the simulation of logical regulatory graphs.

Differentiation of heroin and cocaine using dual-energy CT-an experimental study.

OBJECTIVE: To evaluate if heroin and cocaine can be distinguished using dual-energy CT. MATERIALS AND METHODS: Twenty samples of heroin and cocaine at different concentrations and standardized compression (SC) were scanned in dual-energy mode on a newest generation Dual Energy 64-row MDCT scanner. CT number, spectral graphs, and dual-energy index (DEI) were evaluated. Results were prospectively tested on six original samples from a body packer. Wilcoxon's test was used for statistical evaluation. RESULTS: Values are given as median and range. Under SC, the CT number of cocaine samples (-29.87 Hounsfield unit (HU) [-125.85; 16.16 HU]) was higher than the CT number of heroin samples (-184.37 HU [-199.81; -159.25 HU]; p < 0.01). Slope of spectral curves for cocaine was -2.36 HU/keV [-7.15; -0.67 HU/keV], and for heroin, 1.75 HU/keV [1.28; 2.5 HU/keV] (p < 0.01). DEI was 0.0352 [0.0081; 0.0528] for cocaine and significantly higher than for heroin samples (-0.0127 [-0.0097; -0.0159]; p < 0.001). While CT number was inconclusive, all six original packs were correctly classified after evaluation of the spectral curve and DEI. In contrast to the CT number, slope of the spectral curve and DEI were independent of concentration and compression. CONCLUSION: The slope of the spectral curve and the DEI from dual-energy CT data can be used to distinguish heroin and cocaine in vitro; these results are independent of compression and concentration in the measured range.

Artificial evolution on network structures : how time and space influence dynamics

Abstract The main objective of this work is to show how the choice of the temporal dimension and of the spatial structure of the population influences an artificial evolutionary process. In the field of Artificial Evolution we can observe a common trend in synchronously evolv¬ing panmictic populations, i.e., populations in which any individual can be recombined with any other individual. Already in the '90s, the works of Spiessens and Manderick, Sarma and De Jong, and Gorges-Schleuter have pointed out that, if a population is struc¬tured according to a mono- or bi-dimensional regular lattice, the evolutionary process shows a different dynamic with respect to the panmictic case. In particular, Sarma and De Jong have studied the selection pressure (i.e., the diffusion of a best individual when the only selection operator is active) induced by a regular bi-dimensional structure of the population, proposing a logistic modeling of the selection pressure curves. This model supposes that the diffusion of a best individual in a population follows an exponential law. We show that such a model is inadequate to describe the process, since the growth speed must be quadratic or sub-quadratic in the case of a bi-dimensional regular lattice. New linear and sub-quadratic models are proposed for modeling the selection pressure curves in, respectively, mono- and bi-dimensional regu¬lar structures. These models are extended to describe the process when asynchronous evolutions are employed. Different dynamics of the populations imply different search strategies of the resulting algorithm, when the evolutionary process is used to solve optimisation problems. A benchmark of both discrete and continuous test problems is used to study the search characteristics of the different topologies and updates of the populations. In the last decade, the pioneering studies of Watts and Strogatz have shown that most real networks, both in the biological and sociological worlds as well as in man-made structures, have mathematical properties that set them apart from regular and random structures. In particular, they introduced the concepts of small-world graphs, and they showed that this new family of structures has interesting computing capabilities. Populations structured according to these new topologies are proposed, and their evolutionary dynamics are studied and modeled. We also propose asynchronous evolutions for these structures, and the resulting evolutionary behaviors are investigated. Many man-made networks have grown, and are still growing incrementally, and explanations have been proposed for their actual shape, such as Albert and Barabasi's preferential attachment growth rule. However, many actual networks seem to have undergone some kind of Darwinian variation and selection. Thus, how these networks might have come to be selected is an interesting yet unanswered question. In the last part of this work, we show how a simple evolutionary algorithm can enable the emrgence o these kinds of structures for two prototypical problems of the automata networks world, the majority classification and the synchronisation problems. Synopsis L'objectif principal de ce travail est de montrer l'influence du choix de la dimension temporelle et de la structure spatiale d'une population sur un processus évolutionnaire artificiel. Dans le domaine de l'Evolution Artificielle on peut observer une tendence à évoluer d'une façon synchrone des populations panmictiques, où chaque individu peut être récombiné avec tout autre individu dans la population. Déjà dans les année '90, Spiessens et Manderick, Sarma et De Jong, et Gorges-Schleuter ont observé que, si une population possède une structure régulière mono- ou bi-dimensionnelle, le processus évolutionnaire montre une dynamique différente de celle d'une population panmictique. En particulier, Sarma et De Jong ont étudié la pression de sélection (c-à-d la diffusion d'un individu optimal quand seul l'opérateur de sélection est actif) induite par une structure régulière bi-dimensionnelle de la population, proposant une modélisation logistique des courbes de pression de sélection. Ce modèle suppose que la diffusion d'un individu optimal suit une loi exponentielle. On montre que ce modèle est inadéquat pour décrire ce phénomène, étant donné que la vitesse de croissance doit obéir à une loi quadratique ou sous-quadratique dans le cas d'une structure régulière bi-dimensionnelle. De nouveaux modèles linéaires et sous-quadratique sont proposés pour des structures mono- et bi-dimensionnelles. Ces modèles sont étendus pour décrire des processus évolutionnaires asynchrones. Différentes dynamiques de la population impliquent strategies différentes de recherche de l'algorithme résultant lorsque le processus évolutionnaire est utilisé pour résoudre des problèmes d'optimisation. Un ensemble de problèmes discrets et continus est utilisé pour étudier les charactéristiques de recherche des différentes topologies et mises à jour des populations. Ces dernières années, les études de Watts et Strogatz ont montré que beaucoup de réseaux, aussi bien dans les mondes biologiques et sociologiques que dans les structures produites par l'homme, ont des propriétés mathématiques qui les séparent à la fois des structures régulières et des structures aléatoires. En particulier, ils ont introduit la notion de graphe sm,all-world et ont montré que cette nouvelle famille de structures possède des intéressantes propriétés dynamiques. Des populations ayant ces nouvelles topologies sont proposés, et leurs dynamiques évolutionnaires sont étudiées et modélisées. Pour des populations ayant ces structures, des méthodes d'évolution asynchrone sont proposées, et la dynamique résultante est étudiée. Beaucoup de réseaux produits par l'homme se sont formés d'une façon incrémentale, et des explications pour leur forme actuelle ont été proposées, comme le preferential attachment de Albert et Barabàsi. Toutefois, beaucoup de réseaux existants doivent être le produit d'un processus de variation et sélection darwiniennes. Ainsi, la façon dont ces structures ont pu être sélectionnées est une question intéressante restée sans réponse. Dans la dernière partie de ce travail, on montre comment un simple processus évolutif artificiel permet à ce type de topologies d'émerger dans le cas de deux problèmes prototypiques des réseaux d'automates, les tâches de densité et de synchronisation.

The evolution of helping and harming on graphs: the return of the inclusive fitness effect.

Evolutionary graph theory has been proposed as providing new fundamental rules for the evolution of co-operation and altruism. But how do these results relate to those of inclusive fitness theory? Here, we carry out a retrospective analysis of the models for the evolution of helping on graphs of Ohtsuki et al. [Nature (2006) 441, 502] and Ohtsuki & Nowak [Proc. R. Soc. Lond. Ser. B Biol. Sci (2006) 273, 2249]. We show that it is possible to translate evolutionary graph theory models into classical kin selection models without disturbing at all the mathematics describing the net effect of selection on helping. Model analysis further demonstrates that costly helping evolves on graphs through limited dispersal and overlapping generations. These two factors are well known to promote relatedness between interacting individuals in spatially structured populations. By allowing more than one individual to live at each node of the graph and by allowing interactions to vary with the distance between nodes, our inclusive fitness model allows us to consider a wider range of biological scenarios leading to the evolution of both helping and harming behaviours on graphs.

Comparing Multilevel Clustering Methods on Weighted Graphs : The Case of Worldwide Air Passenger Traffic 2000-2004

Specific properties emerge from the structure of large networks, such as that of worldwide air traffic, including a highly hierarchical node structure and multi-level small world sub-groups that strongly influence future dynamics. We have developed clustering methods to understand the form of these structures, to identify structural properties, and to evaluate the effects of these properties. Graph clustering methods are often constructed from different components: a metric, a clustering index, and a modularity measure to assess the quality of a clustering method. To understand the impact of each of these components on the clustering method, we explore and compare different combinations. These different combinations are used to compare multilevel clustering methods to delineate the effects of geographical distance, hubs, network densities, and bridges on worldwide air passenger traffic. The ultimate goal of this methodological research is to demonstrate evidence of combined effects in the development of an air traffic network. In fact, the network can be divided into different levels of âeurooecohesionâeuro, which can be qualified and measured by comparative studies (Newman, 2002; Guimera et al., 2005; Sales-Pardo et al., 2007).

Error and attack tolerance of layered complex networks.

Many complex systems may be described by not one but a number of complex networks mapped on each other in a multi-layer structure. Because of the interactions and dependencies between these layers, the state of a single layer does not necessarily reflect well the state of the entire system. In this paper we study the robustness of five examples of two-layer complex systems: three real-life data sets in the fields of communication (the Internet), transportation (the European railway system), and biology (the human brain), and two models based on random graphs. In order to cover the whole range of features specific to these systems, we focus on two extreme policies of system's response to failures, no rerouting and full rerouting. Our main finding is that multi-layer systems are much more vulnerable to errors and intentional attacks than they appear from a single layer perspective.

Evolutionary games on complex networks : the emergence and maintenance of cooperation

Abstract The object of game theory lies in the analysis of situations where different social actors have conflicting requirements and where their individual decisions will all influence the global outcome. In this framework, several games have been invented to capture the essence of various dilemmas encountered in many common important socio-economic situations. Even though these games often succeed in helping us understand human or animal behavior in interactive settings, some experiments have shown that people tend to cooperate with each other in situations for which classical game theory strongly recommends them to do the exact opposite. Several mechanisms have been invoked to try to explain the emergence of this unexpected cooperative attitude. Among them, repeated interaction, reputation, and belonging to a recognizable group have often been mentioned. However, the work of Nowak and May (1992) showed that the simple fact of arranging the players according to a spatial structure and only allowing them to interact with their immediate neighbors is sufficient to sustain a certain amount of cooperation even when the game is played anonymously and without repetition. Nowak and May's study and much of the following work was based on regular structures such as two-dimensional grids. Axelrod et al. (2002) showed that by randomizing the choice of neighbors, i.e. by actually giving up a strictly local geographical structure, cooperation can still emerge, provided that the interaction patterns remain stable in time. This is a first step towards a social network structure. However, following pioneering work by sociologists in the sixties such as that of Milgram (1967), in the last few years it has become apparent that many social and biological interaction networks, and even some technological networks, have particular, and partly unexpected, properties that set them apart from regular or random graphs. Among other things, they usually display broad degree distributions, and show small-world topological structure. Roughly speaking, a small-world graph is a network where any individual is relatively close, in terms of social ties, to any other individual, a property also found in random graphs but not in regular lattices. However, in contrast with random graphs, small-world networks also have a certain amount of local structure, as measured, for instance, by a quantity called the clustering coefficient. In the same vein, many real conflicting situations in economy and sociology are not well described neither by a fixed geographical position of the individuals in a regular lattice, nor by a random graph. Furthermore, it is a known fact that network structure can highly influence dynamical phenomena such as the way diseases spread across a population and ideas or information get transmitted. Therefore, in the last decade, research attention has naturally shifted from random and regular graphs towards better models of social interaction structures. The primary goal of this work is to discover whether or not the underlying graph structure of real social networks could give explanations as to why one finds higher levels of cooperation in populations of human beings or animals than what is prescribed by classical game theory. To meet this objective, I start by thoroughly studying a real scientific coauthorship network and showing how it differs from biological or technological networks using divers statistical measurements. Furthermore, I extract and describe its community structure taking into account the intensity of a collaboration. Finally, I investigate the temporal evolution of the network, from its inception to its state at the time of the study in 2006, suggesting also an effective view of it as opposed to a historical one. Thereafter, I combine evolutionary game theory with several network models along with the studied coauthorship network in order to highlight which specific network properties foster cooperation and shed some light on the various mechanisms responsible for the maintenance of this same cooperation. I point out the fact that, to resist defection, cooperators take advantage, whenever possible, of the degree-heterogeneity of social networks and their underlying community structure. Finally, I show that cooperation level and stability depend not only on the game played, but also on the evolutionary dynamic rules used and the individual payoff calculations. Synopsis Le but de la théorie des jeux réside dans l'analyse de situations dans lesquelles différents acteurs sociaux, avec des objectifs souvent conflictuels, doivent individuellement prendre des décisions qui influenceront toutes le résultat global. Dans ce cadre, plusieurs jeux ont été inventés afin de saisir l'essence de divers dilemmes rencontrés dans d'importantes situations socio-économiques. Bien que ces jeux nous permettent souvent de comprendre le comportement d'êtres humains ou d'animaux en interactions, des expériences ont montré que les individus ont parfois tendance à coopérer dans des situations pour lesquelles la théorie classique des jeux prescrit de faire le contraire. Plusieurs mécanismes ont été invoqués pour tenter d'expliquer l'émergence de ce comportement coopératif inattendu. Parmi ceux-ci, la répétition des interactions, la réputation ou encore l'appartenance à des groupes reconnaissables ont souvent été mentionnés. Toutefois, les travaux de Nowak et May (1992) ont montré que le simple fait de disposer les joueurs selon une structure spatiale en leur permettant d'interagir uniquement avec leurs voisins directs est suffisant pour maintenir un certain niveau de coopération même si le jeu est joué de manière anonyme et sans répétitions. L'étude de Nowak et May, ainsi qu'un nombre substantiel de travaux qui ont suivi, étaient basés sur des structures régulières telles que des grilles à deux dimensions. Axelrod et al. (2002) ont montré qu'en randomisant le choix des voisins, i.e. en abandonnant une localisation géographique stricte, la coopération peut malgré tout émerger, pour autant que les schémas d'interactions restent stables au cours du temps. Ceci est un premier pas en direction d'une structure de réseau social. Toutefois, suite aux travaux précurseurs de sociologues des années soixante, tels que ceux de Milgram (1967), il est devenu clair ces dernières années qu'une grande partie des réseaux d'interactions sociaux et biologiques, et même quelques réseaux technologiques, possèdent des propriétés particulières, et partiellement inattendues, qui les distinguent de graphes réguliers ou aléatoires. Entre autres, ils affichent en général une distribution du degré relativement large ainsi qu'une structure de "petit-monde". Grossièrement parlant, un graphe "petit-monde" est un réseau où tout individu se trouve relativement près de tout autre individu en termes de distance sociale, une propriété également présente dans les graphes aléatoires mais absente des grilles régulières. Par contre, les réseaux "petit-monde" ont, contrairement aux graphes aléatoires, une certaine structure de localité, mesurée par exemple par une quantité appelée le "coefficient de clustering". Dans le même esprit, plusieurs situations réelles de conflit en économie et sociologie ne sont pas bien décrites ni par des positions géographiquement fixes des individus en grilles régulières, ni par des graphes aléatoires. De plus, il est bien connu que la structure même d'un réseau peut passablement influencer des phénomènes dynamiques tels que la manière qu'a une maladie de se répandre à travers une population, ou encore la façon dont des idées ou une information s'y propagent. Ainsi, durant cette dernière décennie, l'attention de la recherche s'est tout naturellement déplacée des graphes aléatoires et réguliers vers de meilleurs modèles de structure d'interactions sociales. L'objectif principal de ce travail est de découvrir si la structure sous-jacente de graphe de vrais réseaux sociaux peut fournir des explications quant aux raisons pour lesquelles on trouve, chez certains groupes d'êtres humains ou d'animaux, des niveaux de coopération supérieurs à ce qui est prescrit par la théorie classique des jeux. Dans l'optique d'atteindre ce but, je commence par étudier un véritable réseau de collaborations scientifiques et, en utilisant diverses mesures statistiques, je mets en évidence la manière dont il diffère de réseaux biologiques ou technologiques. De plus, j'extrais et je décris sa structure de communautés en tenant compte de l'intensité d'une collaboration. Finalement, j'examine l'évolution temporelle du réseau depuis son origine jusqu'à son état en 2006, date à laquelle l'étude a été effectuée, en suggérant également une vue effective du réseau par opposition à une vue historique. Par la suite, je combine la théorie évolutionnaire des jeux avec des réseaux comprenant plusieurs modèles et le réseau de collaboration susmentionné, afin de déterminer les propriétés structurelles utiles à la promotion de la coopération et les mécanismes responsables du maintien de celle-ci. Je mets en évidence le fait que, pour ne pas succomber à la défection, les coopérateurs exploitent dans la mesure du possible l'hétérogénéité des réseaux sociaux en termes de degré ainsi que la structure de communautés sous-jacente de ces mêmes réseaux. Finalement, je montre que le niveau de coopération et sa stabilité dépendent non seulement du jeu joué, mais aussi des règles de la dynamique évolutionnaire utilisées et du calcul du bénéfice d'un individu.

Relative and absolute reliability of the clinical version of the Narrow Path Walking Test (NPWT) under single and dual task conditions.

Decline in gait stability has been associated with increased fall risk in older adults. Reliable and clinically feasible methods of gait instability assessment are needed. This study evaluated the relative and absolute reliability and concurrent validity of the testing procedure of the clinical version of the Narrow Path Walking Test (NPWT) under single task (ST) and dual task (DT) conditions. Thirty independent community-dwelling older adults (65-87 years) were tested twice. Participants were instructed to walk within the 6-m narrow path without stepping out. Trial time, number of steps, trial velocity, number of step errors, and number of cognitive task errors were determined. Intraclass correlation coefficients (ICCs) were calculated as indices of agreement, and a graphic approach called "mountain plot" was applied to help interpret the direction and magnitude of disagreements between testing procedures. Smallest detectable change and smallest real difference (SRD) were computed to determine clinically relevant improvement at group and individual levels, respectively. Concurrent validity was assessed using Performance Oriented Mobility Assessment Tool (POMA) and the Short Physical Performance Battery (SPPB). Test-retest agreement (ICC1,2) varied from 0.77 to 0.92 in ST and from 0.78 to 0.92 in DT conditions, with no apparent systematic differences between testing procedures demonstrated by the mountain plot graphs. Smallest detectable change and smallest real change were small for motor task performance and larger for cognitive errors. Significant correlations were observed for trial velocity and trial time with POMA and SPPB. The present results indicate that the NPWT testing procedure is highly reliable and reproducible.

[(18)F]FDG-PET Standard Uptake Value as a Metabolic Predictor of Bone Marrow Response to Radiation: Impact on Acute and Late Hematological Toxicity in Cervical Cancer Patients Treated With Chemoradiation Therapy.

PURPOSE: To quantify the relationship between bone marrow (BM) response to radiation and radiation dose by using (18)F-labeled fluorodeoxyglucose positron emission tomography [(18)F]FDG-PET standard uptake values (SUV) and to correlate these findings with hematological toxicity (HT) in cervical cancer (CC) patients treated with chemoradiation therapy (CRT). METHODS AND MATERIALS: Seventeen women with a diagnosis of CC were treated with standard doses of CRT. All patients underwent pre- and post-therapy [(18)F]FDG-PET/computed tomography (CT). Hemograms were obtained before and during treatment and 3 months after treatment and at last follow-up. Pelvic bone was autosegmented as total bone marrow (BMTOT). Active bone marrow (BMACT) was contoured based on SUV greater than the mean SUV of BMTOT. The volumes (V) of each region receiving 10, 20, 30, and 40 Gy (V10, V20, V30, and V40, respectively) were calculated. Metabolic volume histograms and voxel SUV map response graphs were created. Relative changes in SUV before and after therapy were calculated by separating SUV voxels into radiation therapy dose ranges of 5 Gy. The relationships among SUV decrease, radiation dose, and HT were investigated using multiple regression models. RESULTS: Mean relative pre-post-therapy SUV reductions in BMTOT and BMACT were 27% and 38%, respectively. BMACT volume was significantly reduced after treatment (from 651.5 to 231.6 cm(3), respectively; P<.0001). BMACT V30 was significantly correlated with a reduction in BMACT SUV (R(2), 0.14; P<.001). The reduction in BMACT SUV significantly correlated with reduction in white blood cells (WBCs) at 3 months post-treatment (R(2), 0.27; P=.04) and at last follow-up (R(2), 0.25; P=.04). Different dosimetric parameters of BMTOT and BMACT correlated with long-term hematological outcome. CONCLUSIONS: The volumes of BMTOT and BMACT that are exposed to even relatively low doses of radiation are associated with a decrease in WBC counts following CRT. The loss in proliferative BM SUV uptake translates into low WBC nadirs after treatment. These results suggest the potential of intensity modulated radiation therapy to spare BMTOT to reduce long-term hematological toxicity.

Temporal distribution and magnitude of the vulnerability period around stroke depend on stroke subtype.

Background: We aimed to analyze the rate and time distribution of pre- and post-morbid cerebrovascular events in a single ischemic stroke population, and whether these depend on the etiology of the index stroke. Methods: In 2,203 consecutive patients admitted to a single stroke center registry (ASTRAL), the ischemic stroke that led to admission was considered the index event. Frequency distribution and cumulative relative distribution graphs of the most recent and first recurrent event (ischemic stroke, transient ischemic attack, intracranial or subarachnoid hemorrhage) were drawn in weekly and daily intervals for all strokes and for all stroke types. Results: The frequency of events at identical time points before and after the index stroke was mostly reduced in the first week after (vs. before) stroke (1.0 vs. 4.2%, p < 0.001) and the first month (2.7 vs. 7.4%, p < 0.001), and then ebbed over the first year (8.4 vs. 13.1%, p < 0.001). On daily basis, the peak frequency was noticed at day -1 (1.6%) with a reduction to 0.7% on the index day and 0.17% 24 h after. The event rate in patients with atherosclerotic stroke was particularly high around the index event, but 1-year cumulative recurrence rate was similar in all stroke types. Conclusions: We confirm a short window of increased vulnerability in ischemic stroke and show a 4-, 3- and 2-fold reduction in post-stroke events at 1 week, 1 month and 1 year, respectively, compared to identical pre-stroke periods. This break in the 'stroke wave' is particularly striking after atherosclerotic and lacunar strokes.

Extreme prematurity and intra uterine growth restriction effects in brain network topology at school age

Higher risk for long-term behavioral and emotional sequelae, with attentional problems (with or without hyperactivity) is now becoming one of the hallmarks of extreme premature (EP) birth and birth after pregancy conditions leading to poor intra uterine growth restriction (IUGR) [1,2]. However, little is know so far about the neurostructural basis of these complexe brain functional abnormalities that seem to have their origins in early critical periods of brain development. The development of cortical axonal pathways happens in a series of sequential events. The preterm phase (24-36 post conecptional weeks PCW) is known for being crucial for growth of the thalamocortical fiber bundles as well as for the development of long projectional, commisural and projectional fibers [3]. Is it logical to expect, thus, that being exposed to altered intrauterine environment (altered nutrition) or to extrauterine environment earlier that expected, lead to alterations in the structural organization and, consequently, alter the underlying white matter (WM) structure. Understanding rate and variability of normal brain development, and detect differences from typical development may offer insight into the neurodevelopmental anomalies that can be imaged at later stages. Due to its unique ability to non-invasively visualize and quantify in vivo white matter tracts in the brain, in this study we used diffusion MRI (dMRI) tractography to derive brain graphs [4,5,6]. This relatively simple way of modeling the brain enable us to use graph theory to study topological properties of brain graphs in order to study the effects of EP and IUGR on childrens brain connectivity at age 6 years old.