Functional Brain Networks: beyond the small-world paradigm

This contribution reviews the current state of art comprising the application of Complex Networks Theory to the analysis of functional brain networks. We briefly overview the main advances in this field during the last decade and we explain how graph analysis has increased our knowledge about how the brain behaves when performing a specific task or how it fails when a certain pathology arises. We also show the limitations of this kind of analysis, which have been a source of confusion and misunderstanding when interpreting the results obtained. Finally, we discuss about a possible direction to follow in the next years.

The Influence of Communication Range on Connectivity for Resilient Wireless Sensor Networks Using a Probabilistic Approach.

Wireless sensor networks (WSNs) consist of thousands of nodes that need to communicate with each other. However, it is possible that some nodes are isolated from other nodes due to limited communication range. This paper focuses on the influence of communication range on the probability that all nodes are connected under two conditions, respectively: (1) all nodes have the same communication range, and (2) communication range of each node is a random variable. In the former case, this work proves that, for 0menor queepsmenor quee^(-1) , if the probability of the network being connected is 0.36eps , by means of increasing communication range by constant C(eps) , the probability of network being connected is at least 1-eps. Explicit function C(eps) is given. It turns out that, once the network is connected, it also makes the WSNs resilient against nodes failure. In the latter case, this paper proposes that the network connection probability is modeled as Cox process. The change of network connection probability with respect to distribution parameters and resilience performance is presented. Finally, a method to decide the distribution parameters of node communication range in order to satisfy a given network connection probability is developed.

Collective Organization of Complex Networks: Emergent Scales, Vulnerability, and Targeting of Desired Dynamics

Cuando una colectividad de sistemas dinámicos acoplados mediante una estructura irregular de interacciones evoluciona, se observan dinámicas de gran complejidad y fenómenos emergentes imposibles de predecir a partir de las propiedades de los sistemas individuales. El objetivo principal de esta tesis es precisamente avanzar en nuestra comprensión de la relación existente entre la topología de interacciones y las dinámicas colectivas que una red compleja es capaz de mantener. Siendo este un tema amplio que se puede abordar desde distintos puntos de vista, en esta tesis se han estudiado tres problemas importantes dentro del mismo que están relacionados entre sí. Por un lado, en numerosos sistemas naturales y artificiales que se pueden describir mediante una red compleja la topología no es estática, sino que depende de la dinámica que se desarrolla en la red: un ejemplo son las redes de neuronas del cerebro. En estas redes adaptativas la propia topología emerge como consecuencia de una autoorganización del sistema. Para conocer mejor cómo pueden emerger espontáneamente las propiedades comúnmente observadas en redes reales, hemos estudiado el comportamiento de sistemas que evolucionan según reglas adaptativas locales con base empírica. Nuestros resultados numéricos y analíticos muestran que la autoorganización del sistema da lugar a dos de las propiedades más universales de las redes complejas: a escala mesoscópica, la aparición de una estructura de comunidades, y, a escala macroscópica, la existencia de una ley de potencias en la distribución de las interacciones en la red. El hecho de que estas propiedades aparecen en dos modelos con leyes de evolución cuantitativamente distintas que siguen unos mismos principios adaptativos sugiere que estamos ante un fenómeno que puede ser muy general, y estar en el origen de estas propiedades en sistemas reales. En segundo lugar, proponemos una medida que permite clasificar los elementos de una red compleja en función de su relevancia para el mantenimiento de dinámicas colectivas. En concreto, estudiamos la vulnerabilidad de los distintos elementos de una red frente a perturbaciones o grandes fluctuaciones, entendida como una medida del impacto que estos acontecimientos externos tienen en la interrupción de una dinámica colectiva. Los resultados que se obtienen indican que la vulnerabilidad dinámica es sobre todo dependiente de propiedades locales, por tanto nuestras conclusiones abarcan diferentes topologías, y muestran la existencia de una dependencia no trivial entre la vulnerabilidad y la conectividad de los elementos de una red. Finalmente, proponemos una estrategia de imposición de una dinámica objetivo genérica en una red dada e investigamos su validez en redes con diversas topologías que mantienen regímenes dinámicos turbulentos. Se obtiene como resultado que las redes heterogéneas (y la amplia mayora de las redes reales estudiadas lo son) son las más adecuadas para nuestra estrategia de targeting de dinámicas deseadas, siendo la estrategia muy efectiva incluso en caso de disponer de un conocimiento muy imperfecto de la topología de la red. Aparte de la relevancia teórica para la comprensión de fenómenos colectivos en sistemas complejos, los métodos y resultados propuestos podrán dar lugar a aplicaciones en sistemas experimentales y tecnológicos, como por ejemplo los sistemas neuronales in vitro, el sistema nervioso central (en el estudio de actividades síncronas de carácter patológico), las redes eléctricas o los sistemas de comunicaciones. ABSTRACT The time evolution of an ensemble of dynamical systems coupled through an irregular interaction scheme gives rise to dynamics of great of complexity and emergent phenomena that cannot be predicted from the properties of the individual systems. The main objective of this thesis is precisely to increase our understanding of the interplay between the interaction topology and the collective dynamics that a complex network can support. This is a very broad subject, so in this thesis we will limit ourselves to the study of three relevant problems that have strong connections among them. First, it is a well-known fact that in many natural and manmade systems that can be represented as complex networks the topology is not static; rather, it depends on the dynamics taking place on the network (as it happens, for instance, in the neuronal networks in the brain). In these adaptive networks the topology itself emerges from the self-organization in the system. To better understand how the properties that are commonly observed in real networks spontaneously emerge, we have studied the behavior of systems that evolve according to local adaptive rules that are empirically motivated. Our numerical and analytical results show that self-organization brings about two of the most universally found properties in complex networks: at the mesoscopic scale, the appearance of a community structure, and, at the macroscopic scale, the existence of a power law in the weight distribution of the network interactions. The fact that these properties show up in two models with quantitatively different mechanisms that follow the same general adaptive principles suggests that our results may be generalized to other systems as well, and they may be behind the origin of these properties in some real systems. We also propose a new measure that provides a ranking of the elements in a network in terms of their relevance for the maintenance of collective dynamics. Specifically, we study the vulnerability of the elements under perturbations or large fluctuations, interpreted as a measure of the impact these external events have on the disruption of collective motion. Our results suggest that the dynamic vulnerability measure depends largely on local properties (our conclusions thus being valid for different topologies) and they show a non-trivial dependence of the vulnerability on the connectivity of the network elements. Finally, we propose a strategy for the imposition of generic goal dynamics on a given network, and we explore its performance in networks with different topologies that support turbulent dynamical regimes. It turns out that heterogeneous networks (and most real networks that have been studied belong in this category) are the most suitable for our strategy for the targeting of desired dynamics, the strategy being very effective even when the knowledge on the network topology is far from accurate. Aside from their theoretical relevance for the understanding of collective phenomena in complex systems, the methods and results here discussed might lead to applications in experimental and technological systems, such as in vitro neuronal systems, the central nervous system (where pathological synchronous activity sometimes occurs), communication systems or power grids.

Localization and Tracking in Energy-Constrained Networks with Dynamic Connectivity

Systems used for target localization, such as goods, individuals, or animals, commonly rely on operational means to meet the final application demands. However, what would happen if some means were powered up randomly by harvesting systems? And what if those devices not randomly powered had their duty cycles restricted? Under what conditions would such an operation be tolerable in localization services? What if the references provided by nodes in a tracking problem were distorted? Moreover, there is an underlying topic common to the previous questions regarding the transfer of conceptual models to reality in field tests: what challenges are faced upon deploying a localization network that integrates energy harvesting modules? The application scenario of the system studied is a traditional herding environment of semi domesticated reindeer (Rangifer tarandus tarandus) in northern Scandinavia. In these conditions, information on approximate locations of reindeer is as important as environmental preservation. Herders also need cost-effective devices capable of operating unattended in, sometimes, extreme weather conditions. The analyses developed are worthy not only for the specific application environment presented, but also because they may serve as an approach to performance of navigation systems in absence of reasonably accurate references like the ones of the Global Positioning System (GPS). A number of energy-harvesting solutions, like thermal and radio-frequency harvesting, do not commonly provide power beyond one milliwatt. When they do, battery buffers may be needed (as it happens with solar energy) which may raise costs and make systems more dependent on environmental temperatures. In general, given our problem, a harvesting system is needed that be capable of providing energy bursts of, at least, some milliwatts. Many works on localization problems assume that devices have certain capabilities to determine unknown locations based on range-based techniques or fingerprinting which cannot be assumed in the approach considered herein. The system presented is akin to range-free techniques, but goes to the extent of considering very low node densities: most range-free techniques are, therefore, not applicable. Animal localization, in particular, uses to be supported by accurate devices such as GPS collars which deplete batteries in, maximum, a few days. Such short-life solutions are not particularly desirable in the framework considered. In tracking, the challenge may times addressed aims at attaining high precision levels from complex reliable hardware and thorough processing techniques. One of the challenges in this Thesis is the use of equipment with just part of its facilities in permanent operation, which may yield high input noise levels in the form of distorted reference points. The solution presented integrates a kinetic harvesting module in some nodes which are expected to be a majority in the network. These modules are capable of providing power bursts of some milliwatts which suffice to meet node energy demands. The usage of harvesting modules in the aforementioned conditions makes the system less dependent on environmental temperatures as no batteries are used in nodes with harvesters--it may be also an advantage in economic terms. There is a second kind of nodes. They are battery powered (without kinetic energy harvesters), and are, therefore, dependent on temperature and battery replacements. In addition, their operation is constrained by duty cycles in order to extend node lifetime and, consequently, their autonomy. There is, in turn, a third type of nodes (hotspots) which can be static or mobile. They are also battery-powered, and are used to retrieve information from the network so that it is presented to users. The system operational chain starts at the kinetic-powered nodes broadcasting their own identifier. If an identifier is received at a battery-powered node, the latter stores it for its records. Later, as the recording node meets a hotspot, its full record of detections is transferred to the hotspot. Every detection registry comprises, at least, a node identifier and the position read from its GPS module by the battery-operated node previously to detection. The characteristics of the system presented make the aforementioned operation own certain particularities which are also studied. First, identifier transmissions are random as they depend on movements at kinetic modules--reindeer movements in our application. Not every movement suffices since it must overcome a certain energy threshold. Second, identifier transmissions may not be heard unless there is a battery-powered node in the surroundings. Third, battery-powered nodes do not poll continuously their GPS module, hence localization errors rise even more. Let's recall at this point that such behavior is tight to the aforementioned power saving policies to extend node lifetime. Last, some time is elapsed between the instant an identifier random transmission is detected and the moment the user is aware of such a detection: it takes some time to find a hotspot. Tracking is posed as a problem of a single kinetically-powered target and a population of battery-operated nodes with higher densities than before in localization. Since the latter provide their approximate positions as reference locations, the study is again focused on assessing the impact of such distorted references on performance. Unlike in localization, distance-estimation capabilities based on signal parameters are assumed in this problem. Three variants of the Kalman filter family are applied in this context: the regular Kalman filter, the alpha-beta filter, and the unscented Kalman filter. The study enclosed hereafter comprises both field tests and simulations. Field tests were used mainly to assess the challenges related to power supply and operation in extreme conditions as well as to model nodes and some aspects of their operation in the application scenario. These models are the basics of the simulations developed later. The overall system performance is analyzed according to three metrics: number of detections per kinetic node, accuracy, and latency. The links between these metrics and the operational conditions are also discussed and characterized statistically. Subsequently, such statistical characterization is used to forecast performance figures given specific operational parameters. In tracking, also studied via simulations, nonlinear relationships are found between accuracy and duty cycles and cluster sizes of battery-operated nodes. The solution presented may be more complex in terms of network structure than existing solutions based on GPS collars. However, its main gain lies on taking advantage of users' error tolerance to reduce costs and become more environmentally friendly by diminishing the potential amount of batteries that can be lost. Whether it is applicable or not depends ultimately on the conditions and requirements imposed by users' needs and operational environments, which is, as it has been explained, one of the topics of this Thesis.

Anatomo-functional organization in brain networks

There are several studies focused on comparing rsFC networks with their structural substrate \cite{hagmann2008, honey2010}. However an accurate description of how anatomo-­functional connections are organized, both at physical and topological levels, is still to be defined. Here we present an approach to quantify the anatomo-functional organization and discuss its consistency.

Continuous and lurching traveling pulses in neuronal networks with delay and spatially decaying connectivity

Propagation of discharges in cortical and thalamic systems, which is used as a probe for examining network circuitry, is studied by constructing a one-dimensional model of integrate-and-fire neurons that are coupled by excitatory synapses with delay. Each neuron fires only one spike. The velocity and stability of propagating continuous pulses are calculated analytically. Above a certain critical value of the constant delay, these pulses lose stability. Instead, lurching pulses propagate with discontinuous and periodic spatio-temporal characteristics. The parameter regime for which lurching occurs is strongly affected by the footprint (connectivity) shape; bistability may occur with a square footprint shape but not with an exponential footprint shape. For strong synaptic coupling, the velocity of both continuous and lurching pulses increases logarithmically with the synaptic coupling strength gsyn for an exponential footprint shape, and it is bounded for a step footprint shape. We conclude that the differences in velocity and shape between the front of thalamic spindle waves in vitro and cortical paroxysmal discharges stem from their different effective delay; in thalamic networks, large effective delay between inhibitory neurons arises from their effective interaction via the excitatory cells which display postinhibitory rebound.

Dynamic causal modeling of load-dependent modulation of effective connectivity within the verbal working memory network

Neuroimaging studies have consistently shown that working memory (WM) tasks engage a distributed neural network that primarily includes the dorsolateral prefrontal cortex, the parietal cortex, and the anterior cingulate cortex. The current challenge is to provide a mechanistic account of the changes observed in regional activity. To achieve this, we characterized neuroplastic responses in effective connectivity between these regions at increasing WM loads using dynamic causal modeling of functional magnetic resonance imaging data obtained from healthy individuals during a verbal n-back task. Our data demonstrate that increasing memory load was associated with (a) right-hemisphere dominance, (b) increasing forward (i.e., posterior to anterior) effective connectivity within the WM network, and (c) reduction in individual variability in WM network architecture resulting in the right-hemisphere forward model reaching an exceedance probability of 99% in the most demanding condition. Our results provide direct empirical support that task difficulty, in our case WM load, is a significant moderator of short-term plasticity, complementing existing theories of task-related reduction in variability in neural networks. Hum Brain Mapp, 2013. © 2013 Wiley Periodicals, Inc.

Hypnotic induction is followed by state-like changes in the organization of EEG functional connectivity in the theta and beta frequency bands in high-hypnotically susceptible individuals

Altered state theories of hypnosis posit that a qualitatively distinct state of mental processing, which emerges in those with high hypnotic susceptibility following a hypnotic induction, enables the generation of anomalous experiences in response to specific hypnotic suggestions. If so then such a state should be observable as a discrete pattern of changes to functional connectivity (shared information) between brain regions following a hypnotic induction in high but not low hypnotically susceptible participants. Twenty-eight channel EEG was recorded from 12 high susceptible (highs) and 11 low susceptible (lows) participants with their eyes closed prior to and following a standard hypnotic induction. The EEG was used to provide a measure of functional connectivity using both coherence (COH) and the imaginary component of coherence (iCOH), which is insensitive to the effects of volume conduction. COH and iCOH were calculated between all electrode pairs for the frequency bands: delta (0.1-3.9 Hz), theta (4-7.9 Hz) alpha (8-12.9 Hz), beta1 (13-19.9 Hz), beta2 (20-29.9 Hz) and gamma (30-45 Hz). The results showed that there was an increase in theta iCOH from the pre-hypnosis to hypnosis condition in highs but not lows with a large proportion of significant links being focused on a central-parietal hub. There was also a decrease in beta1 iCOH from the pre-hypnosis to hypnosis condition with a focus on a fronto-central and an occipital hub that was greater in high compared to low susceptibles. There were no significant differences for COH or for spectral band amplitude in any frequency band. The results are interpreted as indicating that the hypnotic induction elicited a qualitative change in the organization of specific control systems within the brain for high as compared to low susceptible participants. This change in the functional organization of neural networks is a plausible indicator of the much theorized "hypnotic-state". © 2014 Jamieson and Burgess.

Brain oscillations and connectivity in autism spectrum disorders (ASD):new approaches to methodology, measurement and modelling

Although atypical social behaviour remains a key characterisation of ASD, the presence ofsensory and perceptual abnormalities has been given a more central role in recentclassification changes. An understanding of the origins of such aberrations could thus prove afruitful focus for ASD research. Early neurocognitive models of ASD suggested that thestudy of high frequency activity in the brain as a measure of cortical connectivity mightprovide the key to understanding the neural correlates of sensory and perceptual deviations inASD. As our review shows, the findings from subsequent research have been inconsistent,with a lack of agreement about the nature of any high frequency disturbances in ASD brains.Based on the application of new techniques using more sophisticated measures of brainsynchronisation, direction of information flow, and invoking the coupling between high andlow frequency bands, we propose a framework which could reconcile apparently conflictingfindings in this area and would be consistent both with emerging neurocognitive models ofautism and with the heterogeneity of the condition.

Integrating functional connectivity and climate change in the design of protected area networks

Le rapide déclin actuel de la biodiversité est inquiétant et les activités humaines en sont la cause directe. De nombreuses aires protégées ont été mises en place pour contrer cette perte de biodiversité. Afin de maximiser leur efficacité, l’amélioration de la connectivité fonctionnelle entre elles est requise. Les changements climatiques perturbent actuellement les conditions environnementales de façon globale. C’est une menace pour la biodiversité qui n’a pas souvent été intégrée lors de la mise en place des aires protégées, jusqu’à récemment. Le mouvement des espèces, et donc la connectivité fonctionnelle du paysage, est impacté par les changements climatiques et des études ont montré qu’améliorer la connectivité fonctionnelle entre les aires protégées aiderait les espèces à faire face aux impacts des changements climatiques. Ma thèse présente une méthode pour concevoir des réseaux d’aires protégées tout en tenant compte des changements climatiques et de la connectivité fonctionnelle. Mon aire d’étude est la région de la Gaspésie au Québec (Canada). La population en voie de disparition de caribou de la Gaspésie-Atlantique (Rangifer tarandus caribou) a été utilisée comme espèce focale pour définir la connectivité fonctionnelle. Cette petite population subit un déclin continu dû à la prédation et la modification de son habitat, et les changements climatiques pourraient devenir une menace supplémentaire. J’ai d’abord construit un modèle individu-centré spatialement explicite pour expliquer et simuler le mouvement du caribou. J’ai utilisé les données VHF éparses de la population de caribou et une stratégie de modélisation patron-orienté pour paramétrer et sélectionner la meilleure hypothèse de mouvement. Mon meilleur modèle a reproduit la plupart des patrons de mouvement définis avec les données observées. Ce modèle fournit une meilleure compréhension des moteurs du mouvement du caribou de la Gaspésie-Atlantique, ainsi qu’une estimation spatiale de son utilisation du paysage dans la région. J’ai conclu que les données éparses étaient suffisantes pour ajuster un modèle individu-centré lorsqu’utilisé avec une modélisation patron-orienté. Ensuite, j’ai estimé l’impact des changements climatiques et de différentes actions de conservation sur le potentiel de mouvement du caribou. J’ai utilisé le modèle individu-centré pour simuler le mouvement du caribou dans des paysages hypothétiques représentant différents scénarios de changements climatiques et d’actions de conservation. Les actions de conservation représentaient la mise en place de nouvelles aires protégées en Gaspésie, comme définies par le scénario proposé par le gouvernement du Québec, ainsi que la restauration de routes secondaires à l’intérieur des aires protégées. Les impacts des changements climatiques sur la végétation, comme définis dans mes scénarios, ont réduit le potentiel de mouvement du caribou. La restauration des routes était capable d’atténuer ces effets négatifs, contrairement à la mise en place des nouvelles aires protégées. Enfin, j’ai présenté une méthode pour concevoir des réseaux d’aires protégées efficaces et j’ai proposé des nouvelles aires protégées à mettre en place en Gaspésie afin de protéger la biodiversité sur le long terme. J’ai créé de nombreux scénarios de réseaux d’aires protégées en étendant le réseau actuel pour protéger 12% du territoire. J’ai calculé la représentativité écologique et deux mesures de connectivité fonctionnelle sur le long terme pour chaque réseau. Les mesures de connectivité fonctionnelle représentaient l’accès général aux aires protégées pour le caribou de la Gaspésie-Atlantique ainsi que son potentiel de mouvement à l’intérieur. J’ai utilisé les estimations de potentiel de mouvement pour la période de temps actuelle ainsi que pour le futur sous différents scénarios de changements climatiques pour représenter la connectivité fonctionnelle sur le long terme. Le réseau d’aires protégées que j’ai proposé était le scénario qui maximisait le compromis entre les trois caractéristiques de réseau calculées. Dans cette thèse, j’ai expliqué et prédit le mouvement du caribou de la Gaspésie-Atlantique sous différentes conditions environnementales, notamment des paysages impactés par les changements climatiques. Ces résultats m’ont aidée à définir un réseau d’aires protégées à mettre en place en Gaspésie pour protéger le caribou au cours du temps. Je crois que cette thèse apporte de nouvelles connaissances sur le comportement de mouvement du caribou de la Gaspésie-Atlantique, ainsi que sur les actions de conservation qui peuvent être prises en Gaspésie afin d’améliorer la protection du caribou et de celle d’autres espèces. Je crois que la méthode présentée peut être applicable à d’autres écosystèmes aux caractéristiques et besoins similaires.

Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models

The presence of gap junction coupling among neurons of the central nervous systems has been appreciated for some time now. In recent years there has been an upsurge of interest from the mathematical community in understanding the contribution of these direct electrical connections between cells to large-scale brain rhythms. Here we analyze a class of exactly soluble single neuron models, capable of producing realistic action potential shapes, that can be used as the basis for understanding dynamics at the network level. This work focuses on planar piece-wise linear models that can mimic the firing response of several different cell types. Under constant current injection the periodic response and phase response curve (PRC) is calculated in closed form. A simple formula for the stability of a periodic orbit is found using Floquet theory. From the calculated PRC and the periodic orbit a phase interaction function is constructed that allows the investigation of phase-locked network states using the theory of weakly coupled oscillators. For large networks with global gap junction connectivity we develop a theory of strong coupling instabilities of the homogeneous, synchronous and splay state. For a piece-wise linear caricature of the Morris-Lecar model, with oscillations arising from a homoclinic bifurcation, we show that large amplitude oscillations in the mean membrane potential are organized around such unstable orbits.

Human brain regions required for the dividing and switching of attention between two features of a single object

This combined PET and ERP study was designed to identify the brain regions activated in switching and divided attention between different features of a single object using matched sensory stimuli and motor response. The ERP data have previously been reported in this journal [64]. We now present the corresponding PET data. We identified partially overlapping neural networks with paradigms requiring the switching or dividing of attention between the elements of complex visual stimuli. Regions of activation were found in the prefrontal and temporal cortices and cerebellum. Each task resulted in different prefrontal cortical regions of activation lending support to the functional subspecialisation of the prefrontal and temporal cortices being based on the cognitive operations required rather than the stimuli themselves.

Deployment and connectivity repair of a sensor net with a flying robot

We consider multi-robot systems that include sensor nodes and aerial or ground robots networked together. Such networks are suitable for tasks such as large-scale environmental monitoring or for command and control in emergency situations. We present a sensor network deployment method using autonomous aerial vehicles and describe in detail the algorithms used for deployment and for measuring network connectivity and provide experimental data collected from field trials. A particular focus is on determining gaps in connectivity of the deployed network and generating a plan for repair, to complete the connectivity. This project is the result of a collaboration between three robotics labs (CSIRO, USC, and Dartmouth). © Springer-Verlag Berlin/Heidelberg 2006.

Using complex network metrics to predict the persistence of metapopulations with asymmetric connectivity patterns

Almost all metapopulation modelling assumes that connectivity between patches is only a function of distance, and is therefore symmetric. However, connectivity will not depend only on the distance between the patches, as some paths are easy to traverse, while others are difficult. When colonising organisms interact with the heterogeneous landscape between patches, connectivity patterns will invariably be asymmetric. There have been few attempts to theoretically assess the effects of asymmetric connectivity patterns on the dynamics of metapopulations. In this paper, we use the framework of complex networks to investigate whether metapopulation dynamics can be determined by directly analysing the asymmetric connectivity patterns that link the patches. Our analyses focus on “patch occupancy” metapopulation models, which only consider whether a patch is occupied or not. We propose three easily calculated network metrics: the “asymmetry” and “average path strength” of the connectivity pattern, and the “centrality” of each patch. Together, these metrics can be used to predict the length of time a metapopulation is expected to persist, and the relative contribution of each patch to a metapopulation’s viability. Our results clearly demonstrate the negative effect that asymmetry has on metapopulation persistence. Complex network analyses represent a useful new tool for understanding the dynamics of species existing in fragmented landscapes, particularly those existing in large metapopulations.