Tuning clustering in random networks with arbitrary degree distributions

We present a generator of random networks where both the degree-dependent clustering coefficient and the degree distribution are tunable. Following the same philosophy as in the configuration model, the degree distribution and the clustering coefficient for each class of nodes of degree k are fixed ad hoc and a priori. The algorithm generates corresponding topologies by applying first a closure of triangles and second the classical closure of remaining free stubs. The procedure unveils an universal relation among clustering and degree-degree correlations for all networks, where the level of assortativity establishes an upper limit to the level of clustering. Maximum assortativity ensures no restriction on the decay of the clustering coefficient whereas disassortativity sets a stronger constraint on its behavior. Correlation measures in real networks are seen to observe this structural bound.

Class of correlated random networks with hidden variables

We study a class of models of correlated random networks in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices. We find analytical expressions for the main topological properties of these models as a function of the distribution of hidden variables and the probability of connecting vertices. The expressions obtained are checked by means of numerical simulations in a particular example. The general model is extended to describe a practical algorithm to generate random networks with an a priori specified correlation structure. We also present an extension of the class, to map nonequilibrium growing networks to networks with hidden variables that represent the time at which each vertex was introduced in the system.

Undersampled Critical Branching Processes on Small-World and Random Networks Fail to Reproduce the Statistics of Spike Avalanches

The power-law size distributions obtained experimentally for neuronal avalanches are an important evidence of criticality in the brain. This evidence is supported by the fact that a critical branching process exhibits the same exponent t~3=2. Models at criticality have been employed to mimic avalanche propagation and explain the statistics observed experimentally. However, a crucial aspect of neuronal recordings has been almost completely neglected in the models: undersampling. While in a typical multielectrode array hundreds of neurons are recorded, in the same area of neuronal tissue tens of thousands of neurons can be found. Here we investigate the consequences of undersampling in models with three different topologies (two-dimensional, small-world and random network) and three different dynamical regimes (subcritical, critical and supercritical). We found that undersampling modifies avalanche size distributions, extinguishing the power laws observed in critical systems. Distributions from subcritical systems are also modified, but the shape of the undersampled distributions is more similar to that of a fully sampled system. Undersampled supercritical systems can recover the general characteristics of the fully sampled version, provided that enough neurons are measured. Undersampling in two-dimensional and small-world networks leads to similar effects, while the random network is insensitive to sampling density due to the lack of a well-defined neighborhood. We conjecture that neuronal avalanches recorded from local field potentials avoid undersampling effects due to the nature of this signal, but the same does not hold for spike avalanches. We conclude that undersampled branching-process-like models in these topologies fail to reproduce the statistics of spike avalanches.

Undersampled Critical Branching Processes on Small-World and Random Networks Fail to Reproduce the Statistics of Spike Avalanches

The power-law size distributions obtained experimentally for neuronal avalanches are an important evidence of criticality in the brain. This evidence is supported by the fact that a critical branching process exhibits the same exponent t~3=2. Models at criticality have been employed to mimic avalanche propagation and explain the statistics observed experimentally. However, a crucial aspect of neuronal recordings has been almost completely neglected in the models: undersampling. While in a typical multielectrode array hundreds of neurons are recorded, in the same area of neuronal tissue tens of thousands of neurons can be found. Here we investigate the consequences of undersampling in models with three different topologies (two-dimensional, small-world and random network) and three different dynamical regimes (subcritical, critical and supercritical). We found that undersampling modifies avalanche size distributions, extinguishing the power laws observed in critical systems. Distributions from subcritical systems are also modified, but the shape of the undersampled distributions is more similar to that of a fully sampled system. Undersampled supercritical systems can recover the general characteristics of the fully sampled version, provided that enough neurons are measured. Undersampling in two-dimensional and small-world networks leads to similar effects, while the random network is insensitive to sampling density due to the lack of a well-defined neighborhood. We conjecture that neuronal avalanches recorded from local field potentials avoid undersampling effects due to the nature of this signal, but the same does not hold for spike avalanches. We conclude that undersampled branching-process-like models in these topologies fail to reproduce the statistics of spike avalanches.

Undersampled Critical Branching Processes on Small-World and Random Networks Fail to Reproduce the Statistics of Spike Avalanches

The power-law size distributions obtained experimentally for neuronal avalanches are an important evidence of criticality in the brain. This evidence is supported by the fact that a critical branching process exhibits the same exponent t~3=2. Models at criticality have been employed to mimic avalanche propagation and explain the statistics observed experimentally. However, a crucial aspect of neuronal recordings has been almost completely neglected in the models: undersampling. While in a typical multielectrode array hundreds of neurons are recorded, in the same area of neuronal tissue tens of thousands of neurons can be found. Here we investigate the consequences of undersampling in models with three different topologies (two-dimensional, small-world and random network) and three different dynamical regimes (subcritical, critical and supercritical). We found that undersampling modifies avalanche size distributions, extinguishing the power laws observed in critical systems. Distributions from subcritical systems are also modified, but the shape of the undersampled distributions is more similar to that of a fully sampled system. Undersampled supercritical systems can recover the general characteristics of the fully sampled version, provided that enough neurons are measured. Undersampling in two-dimensional and small-world networks leads to similar effects, while the random network is insensitive to sampling density due to the lack of a well-defined neighborhood. We conjecture that neuronal avalanches recorded from local field potentials avoid undersampling effects due to the nature of this signal, but the same does not hold for spike avalanches. We conclude that undersampled branching-process-like models in these topologies fail to reproduce the statistics of spike avalanches.

Generalized percolation in random directed networks

We develop a general theory for percolation in directed random networks with arbitrary two-point correlations and bidirectional edgesthat is, edges pointing in both directions simultaneously. These two ingredients alter the previously known scenario and open new views and perspectives on percolation phenomena. Equations for the percolation threshold and the sizes of the giant components are derived in the most general case. We also present simulation results for a particular example of uncorrelated network with bidirectional edges confirming the theoretical predictions.

Continuum formalism for modeling growing networks with deletion of nodes

We present a continuum formalism for modeling growing random networks under addition and deletion of nodes based on a differential mass balance equation. As examples of its applicability, we obtain new results on the degree distribution for growing networks with a uniform attachment and deletion of nodes, and complete some recent results on growing networks with preferential attachment and uniform removal

Synchronization, diversity, and topology of networks of integrate and fire oscillators

We study synchronization dynamics of a population of pulse-coupled oscillators. In particular, we focus our attention on the interplay between topological disorder and synchronization features of networks. First, we analyze synchronization time T in random networks, and find a scaling law which relates T to network connectivity. Then, we compare synchronization time for several other topological configurations, characterized by a different degree of randomness. The analysis shows that regular lattices perform better than a disordered network. This fact can be understood by considering the variability in the number of links between two adjacent neighbors. This phenomenon is equivalent to having a nonrandom topology with a distribution of interactions and it can be removed by an adequate local normalization of the couplings.

Nature of the Epidemic Threshold for the Susceptible-Infected-Susceptible Dynamics in Networks

We develop an analytical approach to the susceptible-infected-susceptible epidemic model that allows us to unravel the true origin of the absence of an epidemic threshold in heterogeneous networks. We find that a delicate balance between the number of high degree nodes in the network and the topological distance between them dictates the existence or absence of such a threshold. In particular, small-world random networks with a degree distribution decaying slower than an exponential have a vanishing epidemic threshold in the thermodynamic limit.

Modelling disease spread and control in networks: implications for plant sciences

Networks are ubiquitous in natural, technological and social systems. They are of increasing relevance for improved understanding and control of infectious diseases of plants, animals and humans, given the interconnectedness of today's world. Recent modelling work on disease development in complex networks shows: the relative rapidity of pathogen spread in scale-free compared with random networks, unless there is high local clustering; the theoretical absence of an epidemic threshold in scale-free networks of infinite size, which implies that diseases with low infection rates can spread in them, but the emergence of a threshold when realistic features are added to networks (e.g. finite size, household structure or deactivation of links); and the influence on epidemic dynamics of asymmetrical interactions. Models suggest that control of pathogens spreading in scale-free networks should focus on highly connected individuals rather than on mass random immunization. A growing number of empirical applications of network theory in human medicine and animal disease ecology confirm the potential of the approach, and suggest that network thinking could also benefit plant epidemiology and forest pathology, particularly in human-modified pathosystems linked by commercial transport of plant and disease propagules. Potential consequences for the study and management of plant and tree diseases are discussed.

Dynamics of brand competition: Effects of unobserved social networks

Brand competition is modelled using an agent based approach in order to examine the long run dynamics of market structure and brand characteristics. A repeated game is designed where myopic firms choose strategies based on beliefs about their rivals and consumers. Consumers are heterogeneous and can observe neighbour behaviour through social networks. Although firms do not observe them, the social networks have a significant impact on the emerging market structure. Presence of networks tends to polarize market share and leads to higher volatility in brands. Yet convergence in brand characteristics usually happens whenever the market reaches a steady state. Scale-free networks accentuate the polarization and volatility more than small world or random networks. Unilateral innovations are less frequent under social networks.

Signal propagation in Bayesian networks and its relationship with intrinsically multivariate predictive variables

A set of predictor variables is said to be intrinsically multivariate predictive (IMP) for a target variable if all properly contained subsets of the predictor set are poor predictors of the. target but the full set predicts the target with great accuracy. In a previous article, the main properties of IMP Boolean variables have been analytically described, including the introduction of the IMP score, a metric based on the coefficient of determination (CoD) as a measure of predictiveness with respect to the target variable. It was shown that the IMP score depends on four main properties: logic of connection, predictive power, covariance between predictors and marginal predictor probabilities (biases). This paper extends that work to a broader context, in an attempt to characterize properties of discrete Bayesian networks that contribute to the presence of variables (network nodes) with high IMP scores. We have found that there is a relationship between the IMP score of a node and its territory size, i.e., its position along a pathway with one source: nodes far from the source display larger IMP scores than those closer to the source, and longer pathways display larger maximum IMP scores. This appears to be a consequence of the fact that nodes with small territory have larger probability of having highly covariate predictors, which leads to smaller IMP scores. In addition, a larger number of XOR and NXOR predictive logic relationships has positive influence over the maximum IMP score found in the pathway. This work presents analytical results based on a simple structure network and an analysis involving random networks constructed by computational simulations. Finally, results from a real Bayesian network application are provided. (C) 2012 Elsevier Inc. All rights reserved.

Network construction and graph theoretical analysis of functional language networks in pediatric epilepsy

This dissertation introduces a new approach for assessing the effects of pediatric epilepsy on the language connectome. Two novel data-driven network construction approaches are presented. These methods rely on connecting different brain regions using either extent or intensity of language related activations as identified by independent component analysis of fMRI data. An auditory description decision task (ADDT) paradigm was used to activate the language network for 29 patients and 30 controls recruited from three major pediatric hospitals. Empirical evaluations illustrated that pediatric epilepsy can cause, or is associated with, a network efficiency reduction. Patients showed a propensity to inefficiently employ the whole brain network to perform the ADDT language task; on the contrary, controls seemed to efficiently use smaller segregated network components to achieve the same task. To explain the causes of the decreased efficiency, graph theoretical analysis was carried out. The analysis revealed no substantial global network feature differences between the patient and control groups. It also showed that for both subject groups the language network exhibited small-world characteristics; however, the patient's extent of activation network showed a tendency towards more random networks. It was also shown that the intensity of activation network displayed ipsilateral hub reorganization on the local level. The left hemispheric hubs displayed greater centrality values for patients, whereas the right hemispheric hubs displayed greater centrality values for controls. This hub hemispheric disparity was not correlated with a right atypical language laterality found in six patients. Finally it was shown that a multi-level unsupervised clustering scheme based on self-organizing maps, a type of artificial neural network, and k-means was able to fairly and blindly separate the subjects into their respective patient or control groups. The clustering was initiated using the local nodal centrality measurements only. Compared to the extent of activation network, the intensity of activation network clustering demonstrated better precision. This outcome supports the assertion that the local centrality differences presented by the intensity of activation network can be associated with focal epilepsy.^

Network Construction and Graph Theoretical Analysis of Functional Language Networks in Pediatric Epilepsy

This dissertation introduces a new approach for assessing the effects of pediatric epilepsy on the language connectome. Two novel data-driven network construction approaches are presented. These methods rely on connecting different brain regions using either extent or intensity of language related activations as identified by independent component analysis of fMRI data. An auditory description decision task (ADDT) paradigm was used to activate the language network for 29 patients and 30 controls recruited from three major pediatric hospitals. Empirical evaluations illustrated that pediatric epilepsy can cause, or is associated with, a network efficiency reduction. Patients showed a propensity to inefficiently employ the whole brain network to perform the ADDT language task; on the contrary, controls seemed to efficiently use smaller segregated network components to achieve the same task. To explain the causes of the decreased efficiency, graph theoretical analysis was carried out. The analysis revealed no substantial global network feature differences between the patient and control groups. It also showed that for both subject groups the language network exhibited small-world characteristics; however, the patient’s extent of activation network showed a tendency towards more random networks. It was also shown that the intensity of activation network displayed ipsilateral hub reorganization on the local level. The left hemispheric hubs displayed greater centrality values for patients, whereas the right hemispheric hubs displayed greater centrality values for controls. This hub hemispheric disparity was not correlated with a right atypical language laterality found in six patients. Finally it was shown that a multi-level unsupervised clustering scheme based on self-organizing maps, a type of artificial neural network, and k-means was able to fairly and blindly separate the subjects into their respective patient or control groups. The clustering was initiated using the local nodal centrality measurements only. Compared to the extent of activation network, the intensity of activation network clustering demonstrated better precision. This outcome supports the assertion that the local centrality differences presented by the intensity of activation network can be associated with focal epilepsy.

Optimized network structure for full-synchronization

A network of Kuramoto oscillators with different natural frequencies is optimized for enhanced synchronizability. All node inputs are normalized by the node connectivity and some important properties of the network Structure are determined in this case: (i) optimized networks present a strong anti-correlation between natural frequencies of adjacent nodes: (ii) this anti-correlation should be as high as possible since the average path length between nodes is maintained as small as in random networks: and (iii) high anti-correlation is obtained without any relation between nodes natural frequencies and the degree of connectivity. We also propose a network construction model with which it is shown that high anti-correlation and small average paths may be achieved by randomly rewiring a fraction of the links of a totally anti-correlated network, and that these networks present optimal synchronization properties. (C) 2008 Elsevier B.V. All rights reserved.