Abstract. Streamflow modeling is an enormously challeng- ing problem, due to the complex and nonlinear interactions between climate inputs and landscape characteristics over a wide range of spatial and temporal scales. A basic idea in streamflow studies is to establish connections that generally exist, but attempts to identify such connections are largely dictated by the problem at hand and the system components in place. While numerous approaches have been proposed in the literature, our understanding of these connections re- mains far from adequate. The present study introduces the theory of networks, in particular complexnetworks, to exam- ine the connections in streamflow dynamics, with a partic- ular focus on spatial connections. Monthly streamflow data observed over a period of 52 years from a large network of 639 monitoring stations in the contiguous US are stud- ied. The connections in this streamflow network are exam- ined primarily using the concept of clustering coefficient, which is a measure of local density and quantifies the net- work’s tendency to cluster. The clustering coefficient analy- sis is performed with several different threshold levels, which are based on correlations in streamflow data between the sta- tions. The clustering coefficient values of the 639 stations are used to obtain important information about the connections in the network and their extent, similarity, and differences be- tween stations/regions, and the influence of thresholds. The relationship of the clustering coefficient with the number of links/actual links in the network and the number of neighbors is also addressed. The results clearly indicate the usefulness of the network-based approach for examining connections in streamflow, with important implications for interpolation and extrapolation, classification of catchments, and predictions in ungaged basins.
This paper studied the cluster synchronization of directed complexnetworks with time delays. It is different from undirected networks, the coupling configuration matrix of directed networks cannot be assumed as symmetric or irreducible. In order to achieve cluster synchronization, this paper uses an adaptive controller on each node and an adaptive feedback strategy on the nodes which in-degree is zero. Numerical example is provided to show the effectiveness of main theory. This method is also effective when the number of clusters is unknown. Thus, it can be used in the community recognizing of directed complexnetworks.
A major issue with correlation is that it cannot detect causality, although, of course, it signals comovement. In the literature on complexnetworks, some attempts have been made to correct this. Various measures have been proposed, e.g., imposing thresholds for correlation, using partial or weighted correlation which try to eliminate spurious correlations. However, up to this moment, except for , we are not aware of constructing complexnetworks which really seek causality in a statistical sense.
Humans need to interact socially with others and the environment. These interactions lead to complex systems that elude naïve and casuistic tools for understand these explanations. One way is to search for mechanisms and patterns of behavior in our activities. In this thesis, we focused on players’ interactions in team sports performance and how using complex systems tools, notably complexnetworks theory and tools, can contribute to Performance Analysis. We began by exploring Network Theory, specifically Social Network Analysis (SNA), first applied to Volleyball (experimental study) and then on soccer (2014 World Cup). The achievements with SNA proved limited in relevant scenarios (e.g., dynamics of networks on n-ary interactions) and we moved to other theories and tools from complexnetworks in order to tap into the dynamics on/off networks. In our state-of-the-art and review paper we took an important step to move from SNA to ComplexNetworks Analysis theories and tools, such as Hypernetworks Theory and their structural Multilevel analysis. The method paper explored the Multilevel Hypernetworks Approach to Performance Analysis in soccer matches (English Premier League 2010-11) considering n-ary cooperation and competition interactions between sets of players in different levels of analysis. We presented at an international conference the mathematical formalisms that can express the players’ relationships and the statistical distributions of the occurrence of the sets and their ranks, identifying power law statistical distributions regularities and design (found in some particular exceptions), influenced by coaches’ pre-match arrangement and soccer rules.
Due to the importance of clustering in the complexnetworks, we must explore the process of link clustering in the networks carefully. Since links tend to cluster in the communities, it provides researchers with the possibility to explore any underlying correlations between network community structure and link formation. Cannistraci et al.  and Yan et al.  had noticed the significance of community structure and proposed community- related link prediction approaches respectively. For example, Cannistraci et al. proposed a new paradigm to support link formation called local community paradigm (LCP), which emphasizes the role of the local network community structure in link formation. Their previous works have given us good hints to further study on this problem. In this article, by studying the theory of network partitioning, we present a novel network partitioning algorithm called Fast probability Block Model (FBM) which is based on the greedy strategy. We assess the performance of our algorithm by applying it to the problem of missing link prediction on various real-world networks. Experimental results show that our algorithm improves both prediction accuracy and computational efficiency compared with conventional methods. Meanwhile, by analyzing links having high connection likelihoods in the communities, we find that these links tend to cluster and form cliques. We therefore conclude three principles to demon- strate such mechanism of link formation. The experimental results verify empirically that the mechanism can be well captured by our approach.
measures of proximity among units different from their distances [13–15]. However, to justify the usage of any of these alternative parameters is difficult. A few years ago, we devised a valid strategy to solve the ties in proximity problem . The first step consists in generating a large number of alternative, mathematically equivalent partitions of the network using the distances among the units (primary distances, according to our nomenclature) and conventional (e. g. average linkage) hierarchical clustering. The results are then averaged to obtain a weighted distance measure for each pair of units (secondary distance). This distance corresponds to the fraction of alternative partitions in which two units are assigned to different clusters. Finally, a dendrogram is obtained from the matrix of secondary distances. This strategy, which we called iterative cluster analysis, has already empirically demon- strated its usefulness. High-quality dendrograms have been obtained from complexnetworks derived from different types of biological data [16–18]. However, performing iterative hierarchi- cal clustering has been so far hampered by the intrinsic slowness of obtaining a representative set of partitions. For example, our original program, UVCluster , runs in O(n 3 ) time, n being the number of nodes. For this reason, the largest analysis published so far corresponds to a network with just 632 units .
Liu and others investigated a novel adaptive feedback control method to simultaneously iden- tify the unknown or uncertain time delay complexnetworks structure or system parameters . Chen and others described how a network can practically be identified by an adaptive- feedback control algorithm . They found that the linear independence condition of the coupling terms proposed in this brief is necessary and sufficient for network identification, and synchronization is a property of a dynamical network that makes identification of the topology of the network impossible. Che and others studied two kinds of synchronization based topol- ogy identification of uncertain complexnetworks with time delay [31, 32]. They used stable lag synchronization and stable anticipatory synchronization between drive and response system to identify the unknown complexnetworks with time delay, respectively. In their studies, the adaptive control technique was used to make the network achieve synchronization. They con- sidered an unknown complex network as a drive system. In order to identify the topology and system parameters, they designed a response network with an adaptive controller. Based on Lyapunov theory, the unknown topology and the uncertain system parameters can be identi- fied when the lag/anticipatory synchronization is achieved.
Various types of complexnetworks such as World Wide Web, citations and maps can be mapped on to Bose gas. In this paper it is shown that the neural network involved in consciousness can be mapped on to a Bose gas and may undergo Bose-Einstein condensation. The neurons are taken as nodes and open gap junctions as links in this study. The neuron with maximum open gap junctions has been taken as node with maximum fitness.
Parrondo’s games were first constructed using a simple tossing scenario, which demonstrates the following paradoxical situation: in sequences of games, a winning expectation may be obtained by playing the games in a random order, although each game (game A or game B) in the sequence may result in losing when played individually. The available Parrondo’s games based on the spatial niche (the neighboring environment) are applied in the regular networks. The neighbors of each node are the same in the regular graphs, whereas they are different in the complexnetworks. Here, Parrondo’s model based on complexnetworks is proposed, and a structure of game B applied in arbitrary topologies is constructed. The results confirm that Parrondo’s paradox occurs. Moreover, the size of the region of the parameter space that elicits Parrondo’s paradox depends on the heterogeneity of the degree distributions of the networks. The higher heterogeneity yields a larger region of the parameter space where the strong paradox occurs. In addition, we use scale-free networks to show that the network size has no significant influence on the region of the parameter space where the strong or weak Parrondo’s paradox occurs. The region of the parameter space where the strong Parrondo’s paradox occurs reduces slightly when the average degree of the network increases.
In recent years, graph theory has been widely employed to probe several language proper- ties. More specifically, the so-called word adjacency model has been proven useful for tack- ling several practical problems, especially those relying on textual stylistic analysis. The most common approach to treat texts as networks has simply considered either large pieces of texts or entire books. This approach has certainly worked well—many informative discoveries have been made this way—but it raises an uncomfortable question: could there be important topological patterns in small pieces of texts? To address this problem, the to- pological properties of subtexts sampled from entire books was probed. Statistical analyses performed on a dataset comprising 50 novels revealed that most of the traditional topologi- cal measurements are stable for short subtexts. When the performance of the authorship recognition task was analyzed, it was found that a proper sampling yields a discriminability similar to the one found with full texts. Surprisingly, the support vector machine classification based on the characterization of short texts outperformed the one performed with entire books. These findings suggest that a local topological analysis of large documents might improve its global characterization. Most importantly, it was verified, as a proof of principle, that short texts can be analyzed with the methods and concepts of complexnetworks. As a consequence, the techniques described here can be extended in a straightforward fashion to analyze texts as time-varying complexnetworks.
More recently, growing attention has also been fo- cused on the investigation of dynamic unfolding in systems underlain by specific types of networks, an example being how neuronal activity depends on specific types of connec- tivity between neurons (Costa and Sporns, 2005). Ulti- mately, efforts will converge on the consideration of the interplay between such dynamics and the dynamics of the evolution of the networks. One of the reasons for the im- pressive advance and popularization of complexnetworks research in the brief period since the application of this methodology to science and technology consists of their in- trinsic potential to represent virtually any system composed of discrete elements. Fortunately, most natural and biologi- cal systems are indeed discrete in nature and can be repre- sented as networks. For instance, in a protein-protein interaction network, each protein is represented as a node, or vertex, while the possible interactions between proteins are expressed as links, or edges, between respective nodes. Similarly, metabolic pathways can be represented as net- works formed by metabolites, reactions and enzymes con- nected by two types of relationship, mass flow and catalytic regulation (Jeong et al., 2000), while transcriptional regula- tions can be naturally represented by complexnetworks
This complex network strategy is important in WSNs be- cause a high cluster coefficient avoids the data delivery delay and unnecessary energy consumption by concentrating the data sensing in a given hub node. Interferences and link layer pro- cessing are avoided when different communication frequencies are used between hubs. The low average shortest path length avoids, mainly, the data delivery delay but on the other hand more local energy is consumed. This discussion shows the truthfulness of the Main hypothesis presented in Section II. Finally, features of more specific complexnetworks, e.g. small world or scale free networks, can be easily incorporated in the design of WSNs.
In this work, we have given an overview of the rationale, design and implementation of the program NEXCADE that can assist in analysis of perturbations and assessment of the consequences of perturbations on complex systems defined by networks that can be expressed as interconnected matrices of interactions. It enables users to assess the outcome of seemingly minor events such as a random gene mutation or metabolic fluctuation, which once set in motion, may become explosive and in extreme cases, lead to irreversible collapse through a cascade of detrimental affects. Although such analyses are now being used routinely in diverse areas of scientific research, a large number of potential users are unable to use these methods for analysis of their own datasets for lack of mathematical and/or computational skills. NEXCADE bridges this gap in a simple user friendly way. To demonstrate its generality and use in a variety of different scenarios, we have applied NEXCADE to several reported social, ecological and biochemical networks, providing a glimpse of the applications that NEXCADE can be used for. We anticipate that it can have wide- ranging benefits to the scientific community and would facilitate risk assessment and threat based management studies in complex network analysis.
Despite the above-mentioned developments and applications, studies on graph the- ory, including random graph theory, had some major deficiencies. First, the studies largely focused on networks that are regular, simple, small, and static. As a result, they are generally unsuitable for examining real networks, as such networks are of- ten highly irregular, complex, large, and dynamically evolving in time. Second, even
and group behavior , efficacy of viral product recommenda- tion , global properties of email messages [34,19], blog posts  as well as the identification of influential blogs [10,18]. Many of these studies did not clearly mention the basic structure of their networks but rather had to be understood from the flow of Figure 2. Erdo˝s–Re´nyi (Random) Network. Figure 2 represents the Erdo¨s–Rnyi network formed with the p = 0.1. The network consists of a source, target and intermediate laid randomly in the network. Figure 2a represents the degree centrality of the individual nodes according to the size and color variation. Nodes (blue) have the highest degree centrality and thus have the largest size in the network where as the nodes (red) have the smallest value of degree centrality in the network. Figure 2b represents betweenness centrality of the nodes in the network. Nodes (blue) have the highest betweenness centrality and have the largest size in the network as the betweenness value decreases so the size and also the color changes ultimately to red. Figure 2c and figure 2d represents closeness centrality and eccentricity centrality of nodes of this network. Both of the centralities are analyzed on this network, the highest value nodes are represented as the largest nodes in blue color. To see the central node in the network or to observe which node is most eccentric in the network, reciprocal of these values is taken. Here, smaller the size of a node is more central and eccentric in the network. Figure 2e represents the eigenvector centrality of the nodes in the network. The highest value nodes are represented in blue color where as nodes with lowest values are represented in red color. (a) Degree Centrality. (b) Betweenness Centrality. (c) Closeness Centrality. (d) Eccentricity Centrality. (e) Eigenvector Centrality.
Networks represent a central model for the description of complex phenomena and they have been studied independently in many different fields such as mathematics, neuroscience, biology, epidemiology, sociology, social-psychology and economy. Recent research trends suggest the emergence of the new science of networks as a field by itself, pioneered by the work of Barabasi  and Watts . Typical examples of complexnetworks in nature and society include metabolic networks, the immune system, the brain, human social networks, communication and transport networks, the Internet and the World Wide Web (WWW). The basic unit of the system is reduced to simple nodes (or vertices) connected by edges (or links) depicting their pairwise relationships. The complexity of real networks is given by non-trivial topological features such as skewed degree distribution, high clustering coefficient and hierarchical structure. Furthermore, local interac- tions between simple components bring forth a complex global behavior in a non-trivial manner . The most studied features of real-world complexnetworks include degree distribution, average distance between vertices, network transitivity and community structure [1,4-7]. The focus of the current study is the community structure problem in dynamic complexnetworks.
Abstract: The complexnetworks have been gaining increasing research attention because of their potential applications in many real-world systems from a variety of fields such as biology, social systems, linguistic networks, and technological systems. In this paper, the problem of stochastic synchronization analysis is investigated for a new array of coupled discrete time stochastic complexnetworks with randomly occurred nonlinearities (RONs) and time delays. The discrete-time complexnetworks under consideration are subject to: 1) stochastic nonlinearities that occur according to the Bernoulli distributed white noise sequences; 2) stochastic disturbances that enter the coupling term, the delayed coupling term as well as the overall network; and 3) time delays that include both the discrete and distributed ones. Note that the newly introduced RONs and the multiple stochastic disturbances can better reflect the dynamical behaviors of coupled complexnetworks whose information transmission process is affected by a noisy environment. By constructing a novel Lyapunov-like matrix functional, the idea of delay fractioning is applied to deal with the addressed synchronization analysis problem. By employing a combination of the linear matrix inequality (LMI) techniques, the free-weighting matrix method and stochastic analysis theories, several delay-dependent sufficient conditions are obtained which ensure the asymptotic synchronization in the mean square sense for the discrete-time stochastic complexnetworks with time delays. The criteria derived are characterized in terms of LMIs whose solution can be solved by utilizing the standard numerical software. While these solvers are significantly faster than classical convex optimization algorithms, it should be kept in mind that the complexity of LMI computations remains higher than that of solving, say, a Riccati equation. For instance, problems with a thousand design variables typically take over an hour on today’s workstations. However, this thesis proposes LMI optimization technique to solve this problem. The advantage of the proposed approach is that resulting stability criterion can be used efficiently via existing numerical convex optimization algorithms such as the interior-point algorithms for solving LMIs
To analyze and quantify the connectivity of the emergent networks we used graph theory methods  and small-world networks analyses . These were implemented using functions from the ComplexNetworks Analysis Package . In the current study we calculated for each network the node degree distribution, betweenesss centrality, shortest path and clustering coefficient for the quantification of the small-world properties. These measures were calculated for each node, and averaged over all nodes to estimate the global characteristics of each network. The degree of a node is the number of connections that link it to the rest of the network, and the degree distribution refers to the degrees of all nodes in the network. Whereas random networks have a symmetrically-centered Gaussian degree distribution, complexnetworks generally have non-Gaussian degree distributions, often with a long tail towards high degrees. In scale-free networks the degree distribution follows a power law . The clustering coefficient is calculated as the number of connections that exist between the nearest neighbors of a node as a proportion of the maximum number of possible connections. High clustering is taken to reflect high local efficiency of information transfer and robustness . The shortest path measure is the minimum number of edges that must be traversed to go from one node to another. Short mean path-lengths reflect high global efficiency of parallel information transfer . Betweeness is a measure of centrality indicating the relative importance of a vertex within the network. The betweeness of vertex v is the proportion of shortest paths between every two vertices in the network that pass through v (see  for a review).