Complex cooperative networks from evolutionary preferential attachment

In spite of its relevance to the origin of complex networks, the interplay between form and function and its role during network formation remains largely unexplored. While recent studies introduce dynamics by considering rewiring processes of a pre-existent network, we study network growth and formation by proposing an evolutionary preferential attachment model, its main feature being that the capacity of a node to attract new links depends on a dynamical variable governed in turn by the node interactions. As a specific example, we focus on the problem of the emergence of cooperation by analyzing the formation of a social network with interactions given by the Prisoner's Dilemma. The resulting networks show many features of real systems, such as scale-free degree distributions, cooperative behavior and hierarchical clustering. Interestingly, results such as the cooperators being located mostly on nodes of intermediate degree are very different from the observations of cooperative behavior on static networks. The evolutionary preferential attachment mechanism points to an evolutionary origin of scale-free networks and may help understand similar feedback problems in the dynamics of complex networks by appropriately choosing the game describing the interaction of nodes.     

Efficient Network Reconstruction from Dynamical Cascades Identifies Small-World Topology of Neuronal Avalanches

Cascading activity is commonly found in complex systems with directed interactions such as metabolic networks, neuronal networks, or disease spreading in social networks. Substantial insight into a system's organization can be obtained by reconstructing the underlying functional network architecture from the observed activity cascades. Here we focus on Bayesian approaches and reduce their computational demands by introducing the Iterative Bayesian (IB) and Posterior Weighted Averaging (PWA) methods. We introduce a special case of PWA, cast in nonparametric form, which we call the normalized count (NC) algorithm. NC efficiently reconstructs random and small-world functional network topologies and architectures from subcritical, critical, and supercritical cascading dynamics and yields significant improvements over commonly used correlation methods. With experimental data, NC identified a functional and structural small-world topology and its corresponding traffic in cortical networks with neuronal avalanche dynamics.     

Network archaeology: uncovering ancient networks from present-day interactions

What proteins interacted in a long-extinct ancestor of yeast? How have different members of a protein complex assembled together over time? Our ability to answer such questions has been limited by the unavailability of ancestral protein-protein interaction (PPI) networks. To overcome this limitation, we propose several novel algorithms to reconstruct the growth history of a present-day network. Our likelihood-based method finds a probable previous state of the graph by applying an assumed growth model backwards in time. This approach retains node identities so that the history of individual nodes can be tracked. Using this methodology, we estimate protein ages in the yeast PPI network that are in good agreement with sequence-based estimates of age and with structural features of protein complexes. Further, by comparing the quality of the inferred histories for several different growth models (duplication-mutation with complementarity, forest fire, and preferential attachment), we provide additional evidence that a duplication-based model captures many features of PPI network growth better than models designed to mimic social network growth. From the reconstructed history, we model the arrival time of extant and ancestral interactions and predict that complexes have significantly re-wired over time and that new edges tend to form within existing complexes. We also hypothesize a distribution of per-protein duplication rates, track the change of the network''s clustering coefficient, and predict paralogous relationships between extant proteins that are likely to be complementary to the relationships inferred using sequence alone. Finally, we infer plausible parameters for the model, thereby predicting the relative probability of various evolutionary events. The success of these algorithms indicates that parts of the history of the yeast PPI are encoded in its present-day form.     

Assembly rules for protein networks derived from phylogenetic-statistical analysis of whole genomes

Background

We report an analysis of a protein network of functionally linked proteins, identified from a phylogenetic statistical analysis of complete eukaryotic genomes. Phylogenetic methods identify pairs of proteins that co-evolve on a phylogenetic tree, and have been shown to have a high probability of correctly identifying known functional links.

Results

The eukaryotic correlated evolution network we derive displays the familiar power law scaling of connectivity. We introduce the use of explicit phylogenetic methods to reconstruct the ancestral presence or absence of proteins at the interior nodes of a phylogeny of eukaryote species. We find that the connectivity distribution of proteins at the point they arise on the tree and join the network follows a power law, as does the connectivity distribution of proteins at the time they are lost from the network. Proteins resident in the network acquire connections over time, but we find no evidence that 'preferential attachment' – the phenomenon of newly acquired connections in the network being more likely to be made to proteins with large numbers of connections – influences the network structure. We derive a 'variable rate of attachment' model in which proteins vary in their propensity to form network interactions independently of how many connections they have or of the total number of connections in the network, and show how this model can produce apparent power-law scaling without preferential attachment.

Conclusion

A few simple rules can explain the topological structure and evolutionary changes to protein-interaction networks: most change is concentrated in satellite proteins of low connectivity and small phenotypic effect, and proteins differ in their propensity to form attachments. Given these rules of assembly, power law scaled networks naturally emerge from simple principles of selection, yielding protein interaction networks that retain a high-degree of robustness on short time scales and evolvability on longer evolutionary time scales.

     

An assessment of preferential attachment as a mechanism for human sexual network formation

Recent research into the properties of human sexual-contact networks has suggested that the degree distribution of the contact graph exhibits power-law scaling. One notable property of this power-law scaling is that the epidemic threshold for the population disappears when the scaling exponent rho is in the range 2 < rho < or = 3. This property is of fundamental significance for the control of sexually transmitted diseases (STDs) such as HIV/AIDS since it implies that an STD can persist regardless of its transmissibility. A stochastic process, known as preferential attachment, that yields one form of power-law scaling has been suggested to underlie the scaling of sexual degree distributions. The limiting distribution of this preferential attachment process is the Yule distribution, which we fit using maximum likelihood to local network data from samples of three populations: (i) the Rakai district, Uganda; (ii) Sweden; and (iii) the USA. For all local networks but one, our interval estimates of the scaling parameters are in the range where epidemic thresholds exist. The estimate of the exponent for male networks in the USA is close to 3, but the preferential attachment model is a very poor fit to these data. We conclude that the epidemic thresholds implied by this model exist in both single-sex and two-sex epidemic model formulations. A strong conclusion that we derive from these results is that public health interventions aimed at reducing the transmissibility of STD pathogens, such as implementing condom use or high-activity anti-retroviral therapy, have the potential to bring a population below the epidemic transition, even in populations exhibiting large degrees of behavioural heterogeneity.     

Selective integration of multiple biological data for supervised network inference

MOTIVATION: Inferring networks of proteins from biological data is a central issue of computational biology. Most network inference methods, including Bayesian networks, take unsupervised approaches in which the network is totally unknown in the beginning, and all the edges have to be predicted. A more realistic supervised framework, proposed recently, assumes that a substantial part of the network is known. We propose a new kernel-based method for supervised graph inference based on multiple types of biological datasets such as gene expression, phylogenetic profiles and amino acid sequences. Notably, our method assigns a weight to each type of dataset and thereby selects informative ones. Data selection is useful for reducing data collection costs. For example, when a similar network inference problem must be solved for other organisms, the dataset excluded by our algorithm need not be collected. RESULTS: First, we formulate supervised network inference as a kernel matrix completion problem, where the inference of edges boils down to estimation of missing entries of a kernel matrix. Then, an expectation-maximization algorithm is proposed to simultaneously infer the missing entries of the kernel matrix and the weights of multiple datasets. By introducing the weights, we can integrate multiple datasets selectively and thereby exclude irrelevant and noisy datasets. Our approach is favorably tested in two biological networks: a metabolic network and a protein interaction network. AVAILABILITY: Software is available on request.     

Temporal Effects in Trend Prediction: Identifying the Most Popular Nodes in the Future

Prediction is an important problem in different science domains. In this paper, we focus on trend prediction in complex networks, i.e. to identify the most popular nodes in the future. Due to the preferential attachment mechanism in real systems, nodes’ recent degree and cumulative degree have been successfully applied to design trend prediction methods. Here we took into account more detailed information about the network evolution and proposed a temporal-based predictor (TBP). The TBP predicts the future trend by the node strength in the weighted network with the link weight equal to its exponential aging. Three data sets with time information are used to test the performance of the new method. We find that TBP have high general accuracy in predicting the future most popular nodes. More importantly, it can identify many potential objects with low popularity in the past but high popularity in the future. The effect of the decay speed in the exponential aging on the results is discussed in detail.     

Predicting co-complexed protein pairs from heterogeneous data

Proteins do not carry out their functions alone. Instead, they often act by participating in macromolecular complexes and play different functional roles depending on the other members of the complex. It is therefore interesting to identify co-complex relationships. Although protein complexes can be identified in a high-throughput manner by experimental technologies such as affinity purification coupled with mass spectrometry (APMS), these large-scale datasets often suffer from high false positive and false negative rates. Here, we present a computational method that predicts co-complexed protein pair (CCPP) relationships using kernel methods from heterogeneous data sources. We show that a diffusion kernel based on random walks on the full network topology yields good performance in predicting CCPPs from protein interaction networks. In the setting of direct ranking, a diffusion kernel performs much better than the mutual clustering coefficient. In the setting of SVM classifiers, a diffusion kernel performs much better than a linear kernel. We also show that combination of complementary information improves the performance of our CCPP recognizer. A summation of three diffusion kernels based on two-hybrid, APMS, and genetic interaction networks and three sequence kernels achieves better performance than the sequence kernels or diffusion kernels alone. Inclusion of additional features achieves a still better ROC(50) of 0.937. Assuming a negative-to-positive ratio of 600ratio1, the final classifier achieves 89.3% coverage at an estimated false discovery rate of 10%. Finally, we applied our prediction method to two recently described APMS datasets. We find that our predicted positives are highly enriched with CCPPs that are identified by both datasets, suggesting that our method successfully identifies true CCPPs. An SVM classifier trained from heterogeneous data sources provides accurate predictions of CCPPs in yeast. This computational method thereby provides an inexpensive method for identifying protein complexes that extends and complements high-throughput experimental data.     

Age-Dependent Evolution of the Yeast Protein Interaction Network Suggests a Limited Role of Gene Duplication and Divergence

Proteins interact in complex protein–protein interaction (PPI) networks whose topological properties—such as scale-free topology, hierarchical modularity, and dissortativity—have suggested models of network evolution. Currently preferred models invoke preferential attachment or gene duplication and divergence to produce networks whose topology matches that observed for real PPIs, thus supporting these as likely models for network evolution. Here, we show that the interaction density and homodimeric frequency are highly protein age–dependent in real PPI networks in a manner which does not agree with these canonical models. In light of these results, we propose an alternative stochastic model, which adds each protein sequentially to a growing network in a manner analogous to protein crystal growth (CG) in solution. The key ideas are (1) interaction probability increases with availability of unoccupied interaction surface, thus following an anti-preferential attachment rule, (2) as a network grows, highly connected sub-networks emerge into protein modules or complexes, and (3) once a new protein is committed to a module, further connections tend to be localized within that module. The CG model produces PPI networks consistent in both topology and age distributions with real PPI networks and is well supported by the spatial arrangement of protein complexes of known 3-D structure, suggesting a plausible physical mechanism for network evolution.     

Identification of core-attachment complexes based on maximal frequent patterns in protein-protein interaction networks

In this paper, we present a method for core-attachment complexes identification based on maximal frequent patterns (CCiMFP) in yeast protein-protein interaction (PPI) networks. First, we detect subgraphs with high degree as candidate protein cores by mining maximal frequent patterns. Then using topological and functional similarities, we combine highly similar protein cores and filter insignificant ones. Finally, the core-attachment complexes are formed by adding attachment proteins to each significant core. We experimentally evaluate the performance of our method CCiMFP on yeast PPI networks. Using gold standard sets of protein complexes, Gene Ontology (GO), and localization annotations, we show that our method gains an improvement over the previous algorithms in terms of precision, recall, and biological significance of the predicted complexes. The colocalization scores of our predicted complex sets are higher than those of two known complex sets. Moreover, our method can detect GO-enriched complexes with disconnected cores compared with other methods based on the subgraph connectivity.     

Emergence of scale-free distribution in protein–protein interaction networks based on random selection of interacting domain pairs

Recent analyses of biological and artificial networks have revealed a common network architecture, called scale-free topology. The origin of the scale-free topology has been explained by using growth and preferential attachment mechanisms. In a cell, proteins are the most important carriers of function, and are composed of domains as elemental units responsible for the physical interaction between protein pairs. Here, we propose a model for protein–protein interaction networks that reveals the emergence of two possible topologies. We show that depending on the number of randomly selected interacting domain pairs, the connectivity distribution follows either a scale-free distribution, even in the absence of the preferential attachment, or a normal distribution. This new approach only requires an evolutionary model of proteins (nodes) but not for the interactions (edges). The edges are added by means of random interaction of domain pairs. As a result, this model offers a new mechanistic explanation for understanding complex networks with a direct biological interpretation because only protein structures and their functions evolved through genetic modifications of amino acid sequences. These findings are supported by numerical simulations as well as experimental data.     

Though with constraints imposed by endosymbiosis,preferential attachment is still a plausible mechanism responsible for the evolution of the chloroplast metabolic network

Chloroplasts evolved as a result of endosymbiosis, during which sophisticated mechanisms evolved to translocate nucleus‐encoded plastid‐targeted enzymes into the chloroplast to form the chloroplast metabolic network. Given the constraints and complexity of endosymbiosis, will preferential attachment still be a plausible mechanism for chloroplast metabolic network evolution? We answer this question by analysing the metabolic network properties of the chloroplast and a cyanobacterium, Synechococcus sp. WH8102 (syw). First, we found that enzymes related to more ancient pathways are more connected, and synthetases have the highest connectivity. Most of the enzymes shared by the two densest cores between the chloroplast and syw are synthetases. Second, the highly conserved functional modules mainly consist of highly connected enzymes. Finally, isozymes and enzymes from endosymbiotic gene transfer (EGT) were distributed mainly in conserved modules and showed higher connectivity than nonisozymes or non‐EGT enzymes. These results suggest that even with severe evolutionary constraints imposed by endosymbiosis, preferential attachment is still a plausible mechanism responsible for the evolution of the chloroplast metabolic network. However, the current analysis may not completely differentiate whether the chloroplast network properties reflect the evolution of the chloroplast network through preferential attachment or has been inherited from its cyanobacterial ancestor. To fully differentiate these two possibilities, further analyses of the metabolic network structure properties of organisms at various intermediate evolutionary stages between cyanobacteria and the chloroplast are needed.     

An exponential core in the heart of the yeast protein interaction network

Protein interactions in the budding yeast have been shown to form a scale-free network, a feature of other organized networks such as bacterial and archaeal metabolism and the World Wide Web. Here, we study the connections established by yeast proteins and discover a preferential attachment between essential proteins. The essential-essential connections are long ranged and form a subnetwork where the giant component includes 97% of these proteins. Unexpectedly, this subnetwork displays an exponential connectivity distribution, in sharp contrast to the scale-free topology of the complete network. Furthermore, the wide phylogenetic extent of these core proteins and interactions provides evidence that they represent the ancestral state of the yeast protein interaction network. Finally, we propose that this core exponential network may represent a generic scaffold around which organism-specific and taxon-specific proteins and interactions coalesce.     

A Model for Improving the Learning Curves of Artificial Neural Networks

In this article, the performance of a hybrid artificial neural network (i.e. scale-free and small-world) was analyzed and its learning curve compared to three other topologies: random, scale-free and small-world, as well as to the chemotaxis neural network of the nematode Caenorhabditis Elegans. One hundred equivalent networks (same number of vertices and average degree) for each topology were generated and each was trained for one thousand epochs. After comparing the mean learning curves of each network topology with the C. elegans neural network, we found that the networks that exhibited preferential attachment exhibited the best learning curves.     

Discriminating Different Classes of Biological Networks by Analyzing the Graphs Spectra Distribution

The brain''s structural and functional systems, protein-protein interaction, and gene networks are examples of biological systems that share some features of complex networks, such as highly connected nodes, modularity, and small-world topology. Recent studies indicate that some pathologies present topological network alterations relative to norms seen in the general population. Therefore, methods to discriminate the processes that generate the different classes of networks (e.g., normal and disease) might be crucial for the diagnosis, prognosis, and treatment of the disease. It is known that several topological properties of a network (graph) can be described by the distribution of the spectrum of its adjacency matrix. Moreover, large networks generated by the same random process have the same spectrum distribution, allowing us to use it as a “fingerprint”. Based on this relationship, we introduce and propose the entropy of a graph spectrum to measure the “uncertainty” of a random graph and the Kullback-Leibler and Jensen-Shannon divergences between graph spectra to compare networks. We also introduce general methods for model selection and network model parameter estimation, as well as a statistical procedure to test the nullity of divergence between two classes of complex networks. Finally, we demonstrate the usefulness of the proposed methods by applying them to (1) protein-protein interaction networks of different species and (2) on networks derived from children diagnosed with Attention Deficit Hyperactivity Disorder (ADHD) and typically developing children. We conclude that scale-free networks best describe all the protein-protein interactions. Also, we show that our proposed measures succeeded in the identification of topological changes in the network while other commonly used measures (number of edges, clustering coefficient, average path length) failed.     

Modelling dendritic ecological networks in space: an integrated network perspective

Dendritic ecological networks (DENs) are a unique form of ecological networks that exhibit a dendritic network topology (e.g. stream and cave networks or plant architecture). DENs have a dual spatial representation; as points within the network and as points in geographical space. Consequently, some analytical methods used to quantify relationships in other types of ecological networks, or in 2‐D space, may be inadequate for studying the influence of structure and connectivity on ecological processes within DENs. We propose a conceptual taxonomy of network analysis methods that account for DEN characteristics to varying degrees and provide a synthesis of the different approaches within the context of stream ecology. Within this context, we summarise the key innovations of a new family of spatial statistical models that describe spatial relationships in DENs. Finally, we discuss how different network analyses may be combined to address more complex and novel research questions. While our main focus is streams, the taxonomy of network analyses is also relevant anywhere spatial patterns in both network and 2‐D space can be used to explore the influence of multi‐scale processes on biota and their habitat (e.g. plant morphology and pest infestation, or preferential migration along stream or road corridors).     

Is Payoff Necessarily Weighted by Probability When Making a Risky Choice? Evidence from Functional Connectivity Analysis

How people make decisions under risk remains an as-yet-unresolved but fundamental question. Mainstream theories about risky decision making assume that the core processes involved in reaching a risky decision include weighting each payoff or reward magnitude by its probability and then summing the outcomes. However, recently developed theories question whether payoffs are necessarily weighted by probability when making a risky choice. Using functional connectivity analysis, we aimed to provide neural evidence to answer whether this key assumption of computing expectations holds when making a risky choice. We contrasted a trade-off instruction choice that required participants to integrate probability and payoff information with a preferential choice that did not. Based on the functional connectivity patterns between regions in which activity was detected during both of the decision-making tasks, we classified the regions into two networks. One network includes primarily the left and right lateral prefrontal cortices and posterior parietal cortices, which were found to be related to probability in previous reports, and the other network is composed of the bilateral basal ganglia, which have been implicated in payoff. We also found that connectivity between the payoff network and some regions in the probability network (including the left lateral prefrontal cortices and bilateral inferior parietal lobes) were stronger during the trade-off instruction choice task than during the preferential choice task. This indicates that the functional integration between the probability and payoff networks during preferential choice was not as strong as the integration during trade-off instruction choice. Our results provide neural evidence that the weighting process uniformly predicted by the mainstream theory is unnecessary during preferential choice. Thus, our functional integration findings can provide a new direction for the investigation of the principles of risky decision making.     

Direct analysis of continuous relaxation spectra

The dynamic response of a complex biological or chemical system to a perturbation must often be described by an integral over an effectively continuous relaxation spectrum. Because of its well known instability to experimental error, the direct estimation of the spectrum is generally considered unfeasible. However, we show that good estimates can be obtained by constraining the spectrum to be the smoothest one that is consistent with the data. Also constraining the spectrum to be non-negative, if there is a priori knowledge of this, can further increase its accuracy. The method is completely automatic in that no initial estimates or assumptions about the functional form of the spectrum are necessary. Therefore models can be tested more rigorously and objectively since the functional form that they predict for the spectrum need not be assumed at the outset of the analysis as with parameter-fitting procedures. The method is illustrated on simulated data on the photodissociation of CO from heme proteins at low temperatures. The nonuniqueness of the solutions is discussed.     

Human matching behavior in social networks: an algorithmic perspective

We argue that algorithmic modeling is a powerful approach to understanding the collective dynamics of human behavior. We consider the task of pairing up individuals connected over a network, according to the following model: each individual is able to propose to match with and accept a proposal from a neighbor in the network; if a matched individual proposes to another neighbor or accepts another proposal, the current match will be broken; individuals can only observe whether their neighbors are currently matched but have no knowledge of the network topology or the status of other individuals; and all individuals have the common goal of maximizing the total number of matches. By examining the experimental data, we identify a behavioral principle called prudence, develop an algorithmic model, analyze its properties mathematically and by simulations, and validate the model with human subject experiments for various network sizes and topologies. Our results include i) a [Formula: see text]-approximate maximum matching is obtained in logarithmic time in the network size for bounded degree networks; ii) for any constant [Formula: see text], a [Formula: see text]-approximate maximum matching is obtained in polynomial time, while obtaining a maximum matching can require an exponential time; and iii) convergence to a maximum matching is slower on preferential attachment networks than on small-world networks. These results allow us to predict that while humans can find a "good quality" matching quickly, they may be unable to find a maximum matching in feasible time. We show that the human subjects largely abide by prudence, and their collective behavior is closely tracked by the above predictions.