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This paper studies $n$-person simultaneous-move games with linear best response function, where individuals interact within a given network structure. This class of games have been used to model various settings, such as, public goods, belief formation, peer effects, and oligopoly. The purpose of this paper is to study the effect of the network structure on Nash equilibrium outcomes of this class of games. Bramoull\'{e} et al. derived conditions for uniqueness and stability of a Nash equilibrium in terms of the smallest eigenvalue of the adjacency matrix representing the network of interactions. Motivated by this result, we study how local structural properties of the network of interactions affect this eigenvalue, influencing game equilibria. In particular, we use algebraic graph theory and convex optimization to derive new bounds on the smallest eigenvalue in terms of the distribution of degrees, cycles, and other relevant substructures. We illustrate our results with numerical simulations involving online social networks.

Complex networks can model the structure and dynamics of different types of systems. It has been shown that they are characterized by a set of measures. In this work, we evaluate the variability of complex networks measures face to perturbations and, for this purpose, we impose controlled perturbations and quantify their effect. We analyze theoretical models (random, small-world and scale-free) and real networks (a collaboration network and a metabolic networks) along with the shortest path length, vertex degree, local cluster coefficient and betweenness centrality measures. In such analysis, we propose the use of three stochastic quantifiers: the Kullback-Leibler divergence and the Jensen-Shannon and Hellinger distances. The sensitivity of these measures was analyzed with respect to the following perturbations: edge addition, edge removal, edge rewiring and node removal, all of them applied at different intensities. The results reveal that the evaluated measures are influenced by these perturbations. Additionally, hypotheses tests were performed to verify the behavior of the degree distribution to identify the intensity of the perturbations that leads to break this property.

Many real-world complex networks are best modeled as bipartite (or 2-mode) graphs, where nodes are divided into two sets with links connecting one side to the other. However, there is currently a lack of methods to analyze properly such graphs as most existing measures and methods are suited to classical graphs. A usual but limited approach consists in deriving 1-mode graphs (called projections) from the underlying bipartite structure, though it causes important loss of information and data storage issues. We introduce here internal links and pairs as a new notion useful for such analysis: it gives insights on the information lost by projecting the bipartite graph. We illustrate the relevance of theses concepts on several real-world instances illustrating how it enables to discriminate behaviors among various cases when we compare them to a benchmark of random networks. Then, we show that we can draw benefit from this concept for both modeling complex networks and storing them in a compact format.

Comment on "Approaching human language with complex networks" by Cong and Liu (Physics of Life Reviews, Volume 11, Issue 4, December 2014, Pages 598-618).

We study the spread of discrete-time epidemics over arbitrary networks for well-known propagation models, namely SIS (susceptible-infected-susceptible), SIR (susceptible-infected-recovered), SIRS (susceptible-infected-recovered-susceptible) and SIV (susceptible-infected-vaccinated). Such epidemics are described by $2^n$- or $3^n$-state Markov chains. Ostensibly, because analyzing such Markov chains is too complicated, their $O(n)$-dimensional nonlinear "mean-field" approximation, and its linearization, are often studied instead. We provide a complete global analysis of the epidemic dynamics of the nonlinear mean-field approximation. In particular, we show that depending on the largest eigenvalue of the underlying graph adjacency matrix and the rates of infection, recovery, and vaccination, the global dynamics takes on one of two forms: either the epidemic dies out, or it converges to another unique fixed point (the so-called endemic state where a constant fraction of the nodes remain infected). A similar result has also been shown in the continuous-time case. We tie in these results with the "true" underlying Markov chain model by showing that the linear model is the tightest upper-bound on the true probabilities of infection that involves only marginals, and that, even though the nonlinear model is not an upper-bound on the true probabilities in general, it does provide an upper-bound on the probability of the chain not being absorbed. As a consequence, we also show that when the disease-free fixed point is globally stable for the mean-field model, the Markov chain has an $O(\log n)$ mixing time, which means the epidemic dies out quickly. We compare and summarize the results on different propagation models.

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We analyze large systems of interacting proteins, using techniques from the non-equilibrium statistical mechanics of disordered many-particle systems. Apart from protein production and removal, the most relevant microscopic processes in the proteome are complex formation and dissociation, and the microscopic degrees of freedom are the evolving concentrations of unbound proteins (in multiple post-translational states) and of protein complexes. Here we only include dimer-complexes, for mathematical simplicity, and we draw the network that describes which proteins are reaction partners from an ensemble of random graphs with an arbitrary degree distribution. We show how generating functional analysis methods can be used successfully to derive closed equations for dynamical order parameters, representing an exact macroscopic description of the complex formation and dissociation dynamics in the infinite system limit. We end this paper with a discussion of the possible routes towards solving the nontrivial order parameter equations, either exactly (in specific limits) or approximately.

We analyze large systems of interacting proteins, using techniques from the non-equilibrium statistical mechanics of disordered many-particle systems. Apart from protein production and removal, the most relevant microscopic processes in the proteome are complex formation and dissociation, and the microscopic degrees of freedom are the evolving concentrations of unbound proteins (in multiple post-translational states) and of protein complexes. Here we only include dimer-complexes, for mathematical simplicity, and we draw the network that describes which proteins are reaction partners from an ensemble of random graphs with an arbitrary degree distribution. We show how generating functional analysis methods can be used successfully to derive closed equations for dynamical order parameters, representing an exact macroscopic description of the complex formation and dissociation dynamics in the infinite system limit. We end this paper with a discussion of the possible routes towards solving the nontrivial order parameter equations, either exactly (in specific limits) or approximately.

Modular structure is ubiquitous in real-world complex networks, and its detection is important because it gives insights in the structure-functionality Modular structure is ubiquitous in real-world complex networks, and its detection is important because it gives insights in the structure-functionality relationship. The standard approach is based on the optimization of a quality function, modularity, which is a relative quality measure for a partition of a network into modules. Recently some authors [1,2] have pointed out that the optimization of modularity has a fundamental drawback: the existence of a resolution limit beyond which no modular structure can be detected even though these modules might have own entity. The reason is that several topological descriptions of the network coexist at different scales, which is, in general, a fingerprint of complex systems. Here we propose a method that allows for multiple resolution screening of the modular structure. The method has been validated using synthetic networks, discovering the predefined structures at all scales. Its application to two real social networks allows to find the exact splits reported in the literature, as well as the substructure beyond the actual split.

Today, the human brain can be studied as a whole. Electroencephalography, magnetoencephalography, or functional magnetic resonance imaging techniques provide functional connectivity patterns between different brain areas, and during different pathological and cognitive neuro-dynamical states. In this Tutorial we review novel complex networks approaches to unveil how brain networks can efficiently manage local processing and global integration for the transfer of information, while being at the same time capable of adapting to satisfy changing neural demands.

Network synchronization is an emerging phenomenon in complex networks. The spectrum of Laplacian matrix will be immensely helpful for getting the network dynamics information. Especially, network synchronizability is characterized by the ratio of second smallest eigen value to largest eigen value of the Laplacian matrix. We study the synchronization of complex networks modeled by regular graphs. We obtained the analytical expressions for network synchronizability for r-nearest neighbor cycle and r-nearest neighbor torus. We have also derived the generalized expression for synchronizability for m-dimensional r-nearest neigh- bor torus. The obtained analytical results agree with the simulation results and shown the effect of network dimension, number of nodes and overhead on syn- chronizability and connectivity in complex networks. This work provides the basic analytical tools for managing and controlling the synchronization in the fi- nite sized complex networks and also given the generalized expressions for eigen values of Laplacian matrix for multi dimensional r- nearest neighbor networks.

In epidemic modeling, the term infection strength indicates the ratio of infection rate and cure rate. If the infection strength is higher than a certain threshold -- which we define as the epidemic threshold - then the epidemic spreads through the population and persists in the long run. For a single generic graph representing the contact network of the population under consideration, the epidemic threshold turns out to be equal to the inverse of the spectral radius of the contact graph. However, in a real world scenario it is not possible to isolate a population completely: there is always some interconnection with another network, which partially overlaps with the contact network. Results for epidemic threshold in interconnected networks are limited to homogeneous mixing populations and degree distribution arguments. In this paper, we adopt a spectral approach. We show how the epidemic threshold in a given network changes as a result of being coupled with another network with fixed infection strength. In our model, the contact network and the interconnections are generic. Using bifurcation theory and algebraic graph theory, we rigorously derive the epidemic threshold in interconnected networks. These results have implications for the broad field of epidemic modeling and control. Our analytical results are supported by numerical simulations.

Methods from statistical physics, such as those involving complex networks, have been increasingly used in quantitative analysis of linguistic phenomena. In this paper, we represented pieces of text with different levels of simplification in co-occurrence networks and found that topological regularity correlated negatively with textual complexity. Furthermore, in less complex texts the distance between concepts, represented as nodes, tended to decrease. The complex networks metrics were treated with multivariate pattern recognition techniques, which allowed us to distinguish between original texts and their simplified versions. For each original text, two simplified versions were generated manually with increasing number of simplification operations. As expected, distinction was easier for the strongly simplified versions, where the most relevant metrics were node strength, shortest paths and diversity. Also, the discrimination of complex texts was improved with higher hierarchical network metrics, thus pointing to the usefulness of considering wider contexts around the concepts. Though the accuracy rate in the distinction was not as high as in methods using deep linguistic knowledge, the complex network approach is still useful for a rapid screening of texts whenever assessing complexity is essential to guarantee accessibility to readers with limited reading ability

The first passage time (FPT) distribution for random walk in complex networks is calculated through an asymptotic analysis. For network with size $N$ and short relaxation time $\tau\ll N$, the computed mean first passage time (MFPT), which is inverse of the decay rate of FPT distribution, is inversely proportional to the degree of the destination. These results are verified numerically for the paradigmatic networks with excellent agreement. We show that the range of validity of the analytical results covers networks that have short relaxation time and high mean degree, which turn out to be valid to many real networks.

Multiple mesoscale numerical weather simulations are conducted to evaluate whether Ensemble Sensitivity Analysis (ESA) is a useful tool determining the sensitivity of a severe downslope windstorm (DSWS) in complex terrain to initial conditions and assimilated observations. A 96-member ensemble is implemented with 1.33 km grid spacing. Sensitive regions are found both upstream and downstream, based on a new forecast metric that indicates the potential for turbulence and strong winds reaching the Earth s surface. Approximating the effects of assimilating a perfect observation at these sensitivity locations, then executing non-linear ensemble forecasts, shows the linear approximations in ESA are reasonable. We analyze the roles of upstream wind and stability structures, and leeside conditions, in determining the strength and propagation of winds down the mountain slope and onto the adjacent plains. Results suggest that ESA is a viable method to identify observation locations to improve forecasts of fine-scale, non-linear, high-impact events such as DSWS. Also, 14 severe DSWS identified by the High Wind Alert System located at the USAF Academy, CO are modeled utilizing a deterministic WRF configuration. Analysis shows that non-wave breaking events account for the strongest DSWS and propagate further away from the mountains than breaking events at this location.

Multiple mesoscale numerical weather simulations are conducted to evaluate whether Ensemble Sensitivity Analysis (ESA) is a useful tool determining the sensitivity of a severe downslope windstorm (DSWS) in complex terrain to initial conditions and assimilated observations. A 96-member ensemble is implemented with 1.33 km grid spacing. Sensitive regions are found both upstream and downstream, based on a new forecast metric that indicates the potential for turbulence and strong winds reaching the Earths surface. Approximating the effects of assimilating a perfect observation at these sensitivity locations, then executing non-linear ensemble forecasts, shows the linear approximations in ESA are reasonable. We analyze the roles of upstream wind and stability structures, and leeside conditions, in determining the strength and propagation of winds down the mountain slope and onto the adjacent plains. Results suggest that ESA is a viable method to identify observation locations to improve forecasts of fine-scale, non-linear, high-impact events such as DSWS. Also, 14 severe DSWS identified by the High Wind Alert System located at the USAF Academy, CO are modeled utilizing a deterministic WRF configuration. Analysis shows that non-wave breaking events account for the strongest DSWS and propagate further away from the mountains than breaking events at this location.

It is becoming more and more clear that complex networks present remarkable large fluctuations. These fluctuations may manifest differently according to the given model. In this paper we re-consider hidden variable models which turn out to be more analytically treatable and for which we have recently shown clear evidence of non-self averaging; the density of a motif being subject to possible uncontrollable fluctuations in the infinite size limit. Here we provide full detailed calculations and we show that large fluctuations are only due to the node hidden variables variability while, in ensembles where these are frozen, fluctuations are negligible in the thermodynamic limit, and equal the fluctuations of classical random graphs. A special attention is paid to the choice of the cut-off: we show that in hidden-variable models, only a cut-off growing as $N^\lambda$ with $\lambda\geq 1$ can reproduce the scaling of a power-law degree distribution. In turn, it is this large cut-off that generates non-self-averaging.

This thesis analyzes the feasibility, efficacy and usability of medical operations working in concert with a Fly-Away-Kit (FLAK) and the forming of Hastily Formed Networks (HFNs) in support of Humanitarian Assistance and Disaster Relief (HA/DR) operations. The initial focus of this research is on the requirements, situation, area of operations, and mission differences between nongovernmental organizations and governmental organizations. The thesis researches and discusses the possibilities for implementing medical technology in the field and the conditions and scenarios in HA/DR that may affect its success. This process will also define the requirements for medical operations as well as facilitate a methodology for ensuring those requirements are met. This thesis investigates the suitability of currently available COTS hardware and software components for medical operations. In addition, it includes a comprehensive review of the value of electronic medical records and telemedicine technologies. Virtually all organizations responding to the December 26, 2004 Southeast Asia tsunami did not have the benefit of large scale medical information technology. For example, the ability to ascertain the real extent of injuries due to the tsunami was hampered by the lack of a central database. Initial media reports claimed a death toll of over 300,000 people, when in fact hindsight now provides a more accurate tally of just over 200,000 dead. This disparity resulted from an archaic system of tracking and accounting. Undoubtedly, humanitarian medical organizations will greatly benefit from the implementation of medical information technology capabilities. This thesis lays the groundwork for further research into medical technologies that can be deployed in the field with humanitarian medical teams in the near future.

We analyze the correlation properties of the Erdos-Renyi random graph (RG) and the Barabasi-Albert scale-free network (SF) under the attack and repair strategy with detrended fluctuation analysis (DFA). The maximum degree k_max, representing the local property of the system, shows similar scaling behaviors for random graphs and scale-free networks. The fluctuations are quite random at short time scales but display strong anticorrelation at longer time scales under the same system size N and different repair probability p_re. The average degree , revealing the statistical property of the system, exhibits completely different scaling behaviors for random graphs and scale-free networks. Random graphs display long-range power-law correlations. Scale-free networks are uncorrelated at short time scales; while anticorrelated at longer time scales and the anticorrelation becoming stronger with the increase of p_re.

The problem is addressed of developing a very general mathematical foundation for networks that permits practical application in the monitoring of large networks such as the internet for both known and unknown attacks, intrusions, worms, viruses, and generally for destructive agents and processes. The PI, under the funding of this grant, has discovered a strong connection between the topological specification of a network in the form of a connection matrix and the branches of mathematics known as continuous group theory and Markov processes. Based upon this research he has proposed that entropy metrics, and the associated cluster analysis of the network so measured by these metrics, can be useful indicators of aberrant processes and behavior. Other team members have obtained important connections using higher order Renyi entropy metrics, and complexity theory to both monitor real networks and to study networks by simulation.

Correlations between intrinsic dynamics and local topology have become a new trend in the study of synchronization in complex networks. In this paper, we investigate in this paradigm the influence of topology on dynamics of networks made up of second-order Kuramoto oscillators. In particular, based on mean-field calculations, we provide a detailed investigation of the recently reported phenomena of cluster explosive synchronization (CES)[Phys. Rev. Lett. 110, 218701 (2013)], analysing the model in scale-free and small-world networks as a function of several topological properties. We show that, in contrast to scale-free networks, the transition to the synchronous state in small-world structures tends to be continuous as the probability of rewiring increases. These results complement the previous findings regarding CES and also fundamentally deepen the understanding of the interplay between topology and dynamics under the constraint of correlating natural frequencies and local structure.

In this paper we derive a compact mathematical formulation describing the dynamics of chemical reaction networks that are complex-balanced and are governed by mass action kinetics. The formulation is based on the graph of (substrate and product) complexes and the stoichiometric information of these complexes, and crucially uses a balanced weighted Laplacian matrix. It is shown that this formulation leads to elegant methods for characterizing the space of all equilibria for complex-balanced networks and for deriving stability properties of such networks. We propose a method for model reduction of complex-balanced networks, which is similar to the Kron reduction method for electrical networks and involves the computation of Schur complements of the balanced weighted Laplacian matrix.

This article reviews and presents various solved and open problems in the development, analysis, and control of epidemic models. We are interested in presenting a relatively concise report for new engineers looking to enter the field of spreading processes on complex networks.

We apply complex network analysis to the state spaces of random Boolean networks (RBNs). An RBN contains $N$ Boolean elements each with $K$ inputs. A directed state space network (SSN) is constructed by linking each dynamical state, represented as a node, to its temporal successor. We study the heterogeneity of an SSN at both local and global scales, as well as sample-to-sample fluctuations within an ensemble of SSNs. We use in-degrees of nodes as a local topological measure, and the path diversity [Phys. Rev. Lett. 98, 198701 (2007)] of an SSN as a global topological measure. RBNs with $2 \leq K \leq 5$ exhibit non-trivial fluctuations at both local and global scales, while K=2 exhibits the largest sample-to-sample, possibly non-self-averaging, fluctuations. We interpret the observed ``multi scale'' fluctuations in the SSNs as indicative of the criticality and complexity of K=2 RBNs. ``Garden of Eden'' (GoE) states are nodes on an SSN that have in-degree zero. While in-degrees of non-GoE nodes for $K>1$ SSNs can assume any integer value between 0 and $2^N$, for K=1 all the non-GoE nodes in an SSN have the same in-degree which is always a power of two.

We study the diffusion of influence in random multiplex networks where links can be of $r$ different types, and for a given content (e.g., rumor, product, political view), each link type is associated with a content dependent parameter $c_i$ in $[0,\infty]$ that measures the relative bias type-$i$ links have in spreading this content. In this setting, we propose a linear threshold model of contagion where nodes switch state if their "perceived" proportion of active neighbors exceeds a threshold \tau. Namely, a node connected to $m_i$ active neighbors and $k_i-m_i$ inactive neighbors via type-$i$ links will turn active if $\sum{c_i m_i}/\sum{c_i k_i}$ exceeds its threshold \tau. Under this model, we obtain the condition, probability and expected size of global spreading events. Our results extend the existing work on complex contagions in several directions by i) providing solutions for coupled random networks whose vertices are neither identical nor disjoint, (ii) highlighting the effect of content on the dynamics of complex contagions, and (iii) showing that content-dependent propagation over a multiplex network leads to a subtle relation between the giant vulnerable component of the graph and the global cascade condition that is not seen in the existing models in the literature.

This article is from Journal of Experimental Botany , volume 60 . Abstract The aim of the study was to describe the molecular and biochemical interactions associated with amino acid biosynthesis and storage protein accumulation in the developing grains of field-grown barley. Our strategy was to analyse the transcription of genes associated with the biosynthesis of storage products during the development of field-grown barley grains using a grain-specific microarray assembled in our laboratory. To identify co-regulated genes, a distance matrix was constructed which enabled the identification of three clusters corresponding to early, middle, and late grain development. The gene expression pattern associated with the clusters was investigated using pathway-specific analysis with specific reference to the temporal expression levels of a range of genes involved mainly in the photosynthesis process, amino acid and storage protein metabolism. It is concluded that the grain-specific microarray is a reliable and cost-effective tool for monitoring temporal changes in the transcriptome of the major metabolic pathways in the barley grain. Moreover, it was sensitive enough to monitor differences in the gene expression profiles of different homologues from the storage protein families. The study described here should provide a strong complement to existing knowledge assisting further understanding of grain development and thereby provide a foundation for plant breeding towards storage proteins with improved nutritional quality.

The degree distribution is an important characteristic of complex networks. In many data analysis applications, the networks should be represented as fixed-length feature vectors and therefore the feature extraction from the degree distribution is a necessary step. Moreover, many applications need a similarity function for comparison of complex networks based on their degree distributions. Such a similarity measure has many applications including classification and clustering of network instances, evaluation of network sampling methods, anomaly detection, and study of epidemic dynamics. The existing methods are unable to effectively capture the similarity of degree distributions, particularly when the corresponding networks have different sizes. Based on our observations about the structure of the degree distributions in networks over time, we propose a feature extraction and a similarity function for the degree distributions in complex networks. We propose to calculate the feature values based on the mean and standard deviation of the node degrees in order to decrease the effect of the network size on the extracted features. The proposed method is evaluated using different artificial and real network datasets, and it outperforms the state of the art methods with respect to the accuracy of the distance function and the effectiveness of the extracted features.

The local structure entropy is a new method which is proposed to identify the influential nodes in the complex networks. In this paper a new form of the local structure entropy of the complex networks is proposed based on the Tsallis entropy. The value of the entropic index $q$ will influence the property of the local structure entropy. When the value of $q$ is equal to 0, the nonextensive local structure entropy is degenerated to a new form of the degree centrality. When the value of $q$ is equal to 1, the nonextensive local structure entropy is degenerated to the existing form of the local structure entropy. We also have find a nonextensive threshold value in the nonextensive local structure entropy. When the value of $q$ is bigger than the nonextensive threshold value, change the value of $q$ will has no influence on the property of the local structure entropy, and different complex networks have different nonextensive threshold value. The results in this paper show that the new nonextensive local structure entropy is a generalised of the local structure entropy. It is more reasonable and useful than the existing one.

This thesis analyzes the feasibility, efficacy and usability of medical operations working in concert with a Fly-Away Kit (FLAK) and the forming of Hastily Formed Networks (HFNs) in support of Humanitarian Assistance and Disaster Relief (HA/DR) operations. The initial focus of this research is on the requirements, situation, area of operations, and mission differences between nongovernmental organizations and governmental organizations. The thesis researches and discusses the possibilities for implementing medical technology in the field and the conditions and scenarios in HA/DR that may affect its success. This process will also define the requirements for medical operations as well as facilitate a methodology for ensuring those requirements are met. This thesis investigates the suitability of currently available COTS hardware and software components for medical operations. In addition, it includes a comprehensive review of the value of electronic medical records and telemedicine technologies. Virtually all organizations responding to the December 26, 2004 Southeast Asia tsunami did not have the benefit of large scale medical information technology. For example, the ability to ascertain the real extent of injuries due to the tsunami was hampered by the lack of a central database. Initial media reports claimed a death toll of over 300,000 people, when in fact hindsight now provides a more accurate tally of just over 200,000 dead. This disparity resulted from an archaic system of tracking and accounting. Undoubtedly, humanitarian medical organizations will greatly benefit from the implementation of medical information technology capabilities. This thesis lays the groundwork for further research into medical technologies that can be deployed in the field with humanitarian medical teams in the near future.

We study complex networks under random matrix theory (RMT) framework. Using nearest-neighbor and next-nearest-neighbor spacing distributions we analyze the eigenvalues of adjacency matrix of various model networks, namely, random, scale-free and small-world networks. These distributions follow Gaussian orthogonal ensemble statistic of RMT. To probe long-range correlations in the eigenvalues we study spectral rigidity via $\Delta_3$ statistic of RMT as well. It follows RMT prediction of linear behavior in semi-logarithmic scale with slope being $\sim 1/\pi^2$. Random and scale-free networks follow RMT prediction for very large scale. Small-world network follows it for sufficiently large scale, but much less than the random and scale-free networks.

Background: How to extract useful information from complex biological networks is a major goal in many fields, especially in genomics and proteomics. We have shown in several works that iterative hierarchical clustering, as implemented in the UVCluster program, is a powerful tool to analyze many of those networks. However, the amount of computation time required to perform UVCluster analyses imposed significant limitations to its use. Methodology/Principal Findings: We describe the suite Jerarca, designed to efficiently convert networks of interacting units into dendrograms by means of iterative hierarchical clustering. Jerarca is divided into three main sections. First, weighted distances among units are computed using up to three different approaches: a more efficient version of UVCluster and two new, related algorithms called RCluster and SCluster. Second, Jerarca builds dendrograms based on those distances, using well-known phylogenetic algorithms, such as UPGMA or Neighbor-Joining. Finally, Jerarca provides optimal partitions of the trees using statistical criteria based on the distribution of intra- and intercluster connections. Outputs compatible with the phylogenetic software MEGA and the Cytoscape package are generated, allowing the results to be easily visualized. Conclusions/Significance: The four main advantages of Jerarca in respect to UVCluster are: 1) Improved speed of a novel UVCluster algorithm; 2) Additional, alternative strategies to perform iterative hierarchical clustering; 3) Automatic evaluation of the hierarchical trees to obtain optimal partitions; and, 4) Outputs compatible with popular software such as MEGA and Cytoscape.

The eigenvalues and eigenvectors of the connectivity matrix of complex networks contain information about its topology and its collective behavior. In particular, the spectral density $\rho(\lambda)$ of this matrix reveals important network characteristics: random networks follow Wigner's semicircular law whereas scale-free networks exhibit a triangular distribution. In this paper we show that the spectral density of hierarchical networks follow a very different pattern, which can be used as a fingerprint of modularity. Of particular importance is the value $\rho(0)$, related to the homeostatic response of the network: it is maximum for random and scale free networks but very small for hierarchical modular networks. It is also large for an actual biological protein-protein interaction network, demonstrating that the current leading model for such networks is not adequate.

In many problems arising in social, technological, and other fields, it is often necessary to analyze populations of individuals interconnected by a network. Real-time analysis of network data is important for detecting anomaly, predicting vulnerability, and assessing the potential impact of interventions in various social and information systems. It is not unusual for network data to be large, dynamic, heterogeneous, noisy and incomplete. Each of these characteristics adds a degree of complexity to the interpretation and analysis of networks. Traditional approaches to network analysis tend to make simplistic assumptions, such as assuming that there is only a single node or edge type, or ignoring the role/mind of nodal actors and the dynamics of the networks. We intend to develop new hierarchical and dynamic Bayesian formalisms and novel graph.

The goals of this project are to improve the understanding of multi-scale Features of networks and to use that understanding to developed improved protocols. The theoretical contributions of this work include sparse models of long-range dependency in network, a stochastic approximation theory for long range dependent processes, techniques to learn the characteristics of losses in wireless networks, stochastic gradient algorithms for wireless ad hoc networks, sampling techniques to estimate the features of complex graphs, and improved models of transmission protocols. The practical contributions include media access control algorithms for maximum throughput in ad hoc networks, protocols for higher throughput and lower delays in WiFi, protocols for generating secret keys in wireless networks.

In this paper we extend the concept of Competitivity Graph to compare series of rankings with ties ({\em partial rankings}). We extend the usual method used to compute Kendall's coefficient for two partial rankings to the concept of evolutive Kendall's coefficient for a series of partial rankings. The theoretical framework consists of a four-layer multiplex network. Regarding the treatment of ties, our approach allows to define a tie between two values when they are close {\em enough}, depending on a threshold. We show an application using data from the Spanish Stock Market; we analyse the series of rankings defined by $25$ companies that have contributed to the IBEX-35 return and volatility values over the period 2003 to 2013.