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Given an ensemble of random graphs with a specific degree distribution, we show that the transformation which converts these graphs to their line (edge-dual) graphs produces an ensemble of graphs with nearly the same degree distribution, but with degree correlations and a much higher clustering coefficient. We also study the percolation properties of these new graphs.
The notion of k-clique percolation in random graphs is introduced, where k is the size of the complete subgraphs whose large scale organizations are analytically and numerically investigated. For the Erdos-Rényi graph of N vertices we obtain that the percolation transition of k-cliques takes place when the probability of two vertices being connected by an edge reaches the threshold p(c) (k) = [(k - 1)N](-1/(k - 1)). At the transition point the scaling of the giant component with N is highly nontrivial and depends on k. We discuss why clique percolation is a novel and efficient approach to the identification of overlapping communities in large real networks.
To characterize the dynamical features of seismicity as a complex phenomenon, the seismic data are mapped to a growing random graph, which is a small-world scale-free network. Here, hierarchical and mixing properties of such a network are studied. The clustering coefficient is found to exhibit asymptotic power-law decay with respect to connectivity, showing hierarchical organization. This structure is supported by not only main shocks but also small shocks, and may have its origin in the combined effect of vertex fitness and deactivation by stress release at faults. The nearest-neighbor average connectivity and the Pearson correlation coefficient are also calculated. It is found that the earthquake network has assortative mixing. This is a main difference of the earthquake network from the Internet with disassortative mixing. Physical implications of these results are discussed.
In this article we study a class of randomly grown graphs that includes some preferential attachment and uniform attachment models, as well as some evolving graph models that have been discussed previously in the literature. The degree distribution is assumed to form a Markov chain; this gives a particularly simple form for a stochastic recursion of the degree distribution. We show that for this class of models the empirical degree distribution tends almost surely and in norm to the expected degree distribution as the size of the graph grows to infinity and we provide a simple asymptotic expression for the expected degree distribution. Convergence of the empirical degree distribution has consequences for statistical analysis of network data in that it allows the full data to be summarized by the degree distribution of the nodes without losing the ability to obtain consistent estimates of parameters describing the network.
Stoichiometrically, exact candidate pathways or mechanisms for deriving the rate law of a catalytic or complex reaction can be determined through the synthesis of networks of plausible elementary reactions constituting such pathways. A rigorous algorithmic method is proposed for executing this synthesis, which is exceedingly convoluted due to its combinatorial complexity. Such a method for synthesizing networks of reaction pathways follows the general framework of a highly exacting combinatorial method established by us for process-network synthesis. It is based on the unique graph-representation in terms of P-graphs, a set of axioms, and a group of combinatorial algorithms. In the method, the inclusion or exclusion of a step of each elementary reaction in the mechanism of concern hinges on the general combinatorial properties of feasible reaction networks. The decisions are facilitated by solving linear programming problems comprising a set of mass-balance constraints to determine the existence or absence of any feasible solution. The search is accelerated further by exploiting the inferences of preceding decisions, thereby eliminating redundancy. As a result, all feasible independent reaction networks, i.e. pathways, are generated only once; the pathways violating any first principle of either stoichiometry or thermodynamics are eliminated. The method is also capable of generating those combinations of independent pathways directly, which are not microscopically reversible. The efficiency and efficacy of the method are demonstrated with the identification of the feasible mechanisms of ammonia synthesis involving as many as 14 known elementary reactions.
We analyze a minimal model of a growing network. At each time step, a new vertex is added; then, with probability delta, two vertices are chosen uniformly at random and joined by an undirected edge. This process is repeated for t time steps. In the limit of large t, the resulting graph displays surprisingly rich characteristics. In particular, a giant component emerges in an infinite-order phase transition at delta=1/8. At the transition, the average component size jumps discontinuously but remains finite. In contrast, a static random graph with the same degree distribution exhibits a second-order phase transition at delta=1/4, and the average component size diverges there. These dramatic differences between grown and static random graphs stem from a positive correlation between the degrees of connected vertices in the grown graph-older vertices tend to have higher degree, and to link with other high-degree vertices, merely by virtue of their age. We conclude that grown graphs, however randomly they are constructed, are fundamentally different from their static random graph counterparts.
BACKGROUND: The development of algorithms to infer the structure of gene regulatory networks based on expression data is an important subject in bioinformatics research. Validation of these algorithms requires benchmark data sets for which the underlying network is known. Since experimental data sets of the appropriate size and design are usually not available, there is a clear need to generate well-characterized synthetic data sets that allow thorough testing of learning algorithms in a fast and reproducible manner. RESULTS: In this paper we describe a network generator that creates synthetic transcriptional regulatory networks and produces simulated gene expression data that approximates experimental data. Network topologies are generated by selecting subnetworks from previously described regulatory networks. Interaction kinetics are modeled by equations based on Michaelis-Menten and Hill kinetics. Our results show that the statistical properties of these topologies more closely approximate those of genuine biological networks than do those of different types of random graph models. Several user-definable parameters adjust the complexity of the resulting data set with respect to the structure learning algorithms. CONCLUSION: This network generation technique offers a valid alternative to existing methods. The topological characteristics of the generated networks more closely resemble the characteristics of real transcriptional networks. Simulation of the network scales well to large networks. The generator models different types of biological interactions and produces biologically plausible synthetic gene expression data.
A road network usually has to fulfill two requirements: (i) it should as far as possible provide direct connections between nodes to avoid large detours; and (ii) the costs for road construction and maintenance, which are assumed proportional to the total length of the roads, should be low. The optimal solution is a compromise between these contradictory demands, which in our model can be weighted by a parameter. The road optimization problem belongs to the class of frustrated optimization problems. In this paper, a special class of evolutionary strategies, such as the Boltzmann and Darwin and mixed strategies, are applied to find differently optimized solutions (graphs of varying density) for the road network, depending on the degree of frustration. We show that the optimization process occurs on two different time scales. In the asymptotic limit, a fixed relation between the mean connection distance (detour) and the total length (costs) of the network exists that defines a range of possible compromises. Furthermore, we investigate the density of states, which describes the number of solutions with a certain fitness value in the stationary regime. We find that the network problem belongs to a class of optimization problems in which more effort in optimization certainly yields better solutions. An analytical approximation for the relation between effort and improvement is derived.
In this Brief Report we present a version of a network growth model, generalized in order to describe the behavior of social networks. The case of study considered is the preprint archive at cul.arxiv.org. Each node corresponds to a scientist, and a link is present whenever two authors wrote a paper together. This graph is a nice example of degree-assortative network, that is, to say a network where sites with similar degree are connected to each other. The model presented is one of the few able to reproduce such behavior, giving some insight on the microscopic dynamics at the basis of the graph structure.
Many real networks are equipped with short diameters, high clustering, and power-law degree distributions. With preferential attachment and network growth, the model by Barabási and Albert simultaneously reproduces these properties, and geographical versions of growing networks have also been analyzed. However, nongrowing networks with intrinsic vertex weights often explain these features more plausibly, since not all networks are really growing. We propose a geographical nongrowing network model with vertex weights. Edges are assumed to form when a pair of vertices are spatially close and/or have large summed weights. Our model generalizes a variety of models as well as the original nongeographical counterpart, such as the unit disk graph, the Boolean model, and the gravity model, which appear in the contexts of percolation, wire communication, mechanical and solid physics, sociology, economy, and marketing. In appropriate configurations, our model produces small-world networks with power-law degree distributions. We also discuss the relation between geography, power laws in networks, and power laws in general quantities serving as vertex weights.
A structure for representing problems in decision analysis and in expert systems, which reason under uncertainty, is the influence diagram or causal network. A causal network consists of an underlying joint probability distribution and a directed acyclic graph in which a propositional variable that represents a marginal distribution is stored at each vertex in the graph. This paper is concerned with two of the problems in applications that use causal networks. The first problem is the determination of the conditional probabilities of the values of remaining propositional variables in the network given that certain variables are instantiated for particular values. This is called probability propagation. The second problem is the determination of the most probable, second most probable, third most probable, and so on sets of values of a particular set of variables (called the explanation set) given that certain variables are instantiated for particular values. This problem is called abductive inference. There exists a class of causal networks in which each variable has only two parents, for which the time is required, by any known method, for probability propagation is exponential relative to the number of vertices in the network. The determination of a new method that would be efficient for all causal networks appears unlikely, because probability propagation has been shown to be #P-complete. In many medical applications, networks are often large and not sparsely connected. Therefore a method for the exact determination of probability values appears unlikely for such applications, and the development of approximation methods seems to be the best solution. The current approximation methods obtain interval bounds for the probability values. When such intervals are obtained, it is not possible in general to rank the alternatives. In this paper, a method is developed for obtaining expected values for the point probabilities from interval constraints on the probabilities. The method is based on an application of the principle of indifference to the probability values themselves. The distributions obtained with the principle of indifference are a generalization of the symmetric Dirichlet distribution in which prior ignorance is assumed.
BACKGROUND: Graph theory provides a computational framework for modeling a variety of datasets including those emerging from genomics, proteomics, and chemical genetics. Networks of genes, proteins, small molecules, or other objects of study can be represented as graphs of nodes (vertices) and interactions (edges) that can carry different weights. SpectralNET is a flexible application for analyzing and visualizing these biological and chemical networks. RESULTS: Available both as a standalone .NET executable and as an ASP.NET web application, SpectralNET was designed specifically with the analysis of graph-theoretic metrics in mind, a computational task not easily accessible using currently available applications. Users can choose either to upload a network for analysis using a variety of input formats, or to have SpectralNET generate an idealized random network for comparison to a real-world dataset. Whichever graph-generation method is used, SpectralNET displays detailed information about each connected component of the graph, including graphs of degree distribution, clustering coefficient by degree, and average distance by degree. In addition, extensive information about the selected vertex is shown, including degree, clustering coefficient, various distance metrics, and the corresponding components of the adjacency, Laplacian, and normalized Laplacian eigenvectors. SpectralNET also displays several graph visualizations, including a linear dimensionality reduction for uploaded datasets (Principal Components Analysis) and a non-linear dimensionality reduction that provides an elegant view of global graph structure (Laplacian eigenvectors). CONCLUSION: SpectralNET provides an easily accessible means of analyzing graph-theoretic metrics for data modeling and dimensionality reduction. SpectralNET is publicly available as both a .NET application and an ASP.NET web application from http://chembank.broad.harvard.edu/resources/. Source code is available upon request.
Despite their apparent simplicity, random Boolean networks display a rich variety of dynamical behaviors. Much work has been focused on the properties and abundance of attractors. The topologies of random Boolean networks with one input per node can be seen as graphs of random maps. We introduce an approach to investigating random maps and finding analytical results for attractors in random Boolean networks with the corresponding topology. Approximating some other non-chaotic networks to be of this class, we apply the analytic results to them. For this approximation, we observe a strikingly good agreement on the numbers of attractors of various lengths. We also investigate observables related to the average number of attractors in relation to the typical number of attractors. Here, we find strong differences that highlight the difficulties in making direct comparisons between random Boolean networks and real systems. Furthermore, we demonstrate the power of our approach by deriving some results for random maps. These results include the distribution of the number of components in random maps, along with asymptotic expansions for cumulants up to the fourth order.
MOTIVATION: The sheer volume and variety of genomic content within microbial communities makes metagenomics a field rich in biomedical knowledge. To traverse these complex communities and their vast unknowns, metagenomic studies often depend on distinct reference databases, such as the Genome Taxonomy Database (GTDB), the Kyoto Encyclopedia of Genes and Genomes (KEGG), and the Bacterial and Viral Bioinformatics Resource Center (BV-BRC), for various analytical purposes. These databases are crucial for the genetic and functional annotation of microbial communities. Nevertheless, the inconsistent nomenclature or identifiers of these databases present challenges for effective integration, representation, and utilization. Knowledge graphs (KGs) offer an appropriate solution by organizing biological entities from different databases to standardized identifiers, allowing their interrelations to be captured into a cohesive network regardless of the naming conventions used in each source. The graph structure not only facilitates the unveiling of hidden patterns but also enriches our biological understanding with deeper insights. Despite KGs having shown potential in various biomedical fields, their application in metagenomics remains underexplored. RESULTS: We present MetagenomicKG, a novel knowledge graph specifically tailored for metagenomic analysis. MetagenomicKG integrates taxonomic, functional, and pathogenesis-related information on the human microbiome sourced from various databases, and further connects these with existing biomedical KGs to expand the biological network. Through various case studies involving the human microbiome, we demonstrate its utility in enabling hypothesis generation regarding the relationships between microbes and diseases, generating sample-specific graph embeddings, and providing robust pathogen prediction. CODE AVAILABILITY: The source code and technical details for constructing the MetagenomicKG and reproducing all analyses are available on GitHub at https://github.com/KoslickiLab/MetagenomicKG. The data used in this manuscript, including the pre-built files and use case input data, are archived on Zenodo with DOI: 10.5281/zenodo.17546861.
The protein interaction network presents one perspective for understanding cellular processes. Recent experiments employing high-throughput mass spectrometric characterizations have resulted in large data sets of physiologically relevant multiprotein complexes. We present a unified representation of such data sets based on an underlying bipartite graph model that is an advance over existing models of the network. Our unified representation allows for weighting of connections between proteins shared in more than one complex, as well as addressing the higher level organization that occurs when the network is viewed as consisting of protein complexes that share components. This representation also allows for the application of the rigorous MinMaxCut graph clustering algorithm for the determination of relevant protein modules in the networks. Statistically significant annotations of clusters in the protein-protein and complex-complex networks using terms from the Gene Ontology indicate that this method will be useful for posing hypotheses about uncharacterized components of protein complexes or uncharacterized relationships between protein complexes.
In intraspecific studies, reticulated graphs are valuable tools for visualization, within a single figure, of alternative genealogical pathways among haplotypes. As available software packages implementing the global maximum parsimony (MP) approach only give the possibility to merge resulting topologies into less-resolved consensus trees, MP has often been neglected as an alternative approach to purely algorithmic (i.e., methods defined solely on the basis of an algorithm) "network" construction methods. Here, we propose to search tree space using the MP criterion and present a new algorithm for uniting all equally most parsimonious trees into a single (possibly reticulated) graph. Using simulated sequence data, we compare our method with three purely algorithmic and widely used graph construction approaches (minimum-spanning network, statistical parsimony, and median-joining network). We demonstrate that the combination of MP trees into a single graph provides a good estimate of the true genealogy. Moreover, our analyses indicate that, when internal node haplotypes are not sampled, the median-joining and MP methods provide the best estimate of the true genealogy whereas the minimum-spanning algorithm shows very poor performances.
A stochastic model for the spread of a sexually transmitted disease (STD) is presented. To reflect varying degrees of promiscuity among individuals it is assumed that the infectivity of any infected individual is proportional to the number of previous contacts the individual has had with other infected individuals. In both the simple single-sex model and in the more complex two-sex model, the tree graphs of the infection exhibit scale-free network behaviour (i.e. power-law behaviour in the upper tail of the degree distribution). The distributions of the size of the infection and of the ring number (distance from the original source of the infection) are determined.
BACKGROUND: Large-scale protein interaction maps provide a new, global perspective with which to analyse protein function. PSIMAP, the Protein Structural Interactome Map, is a database of all the structurally observed interactions between superfamilies of protein domains with known three-dimensional structure in the PDB. PSIMAP incorporates both functional and evolutionary information into a single network. RESULTS: We present a global analysis of PSIMAP using several distinct network measures relating to centrality, interactivity, fault-tolerance, and taxonomic diversity. We found the following results: Centrality: we show that the center and barycenter of PSIMAP do not coincide, and that the superfamilies forming the barycenter relate to very general functions, while those constituting the center relate to enzymatic activity. Interactivity: we identify the P-loop and immunoglobulin superfamilies as the most highly interactive. We successfully use connectivity and cluster index, which characterise the connectivity of a superfamily's neighbourhood, to discover superfamilies of complex I and II. This is particularly significant as the structure of complex I is not yet solved. Taxonomic diversity: we found that highly interactive superfamilies are in general taxonomically very diverse and are thus amongst the oldest. Fault-tolerance: we found that the network is very robust as for the majority of superfamilies removal from the network will not break up the network. CONCLUSIONS: Overall, we can single out the P-loop containing nucleotide triphosphate hydrolases superfamily as it is the most highly connected and has the highest taxonomic diversity. In addition, this superfamily has the highest interaction rank, is the barycenter of the network (it has the shortest average path to every other superfamily in the network), and is an articulation vertex, whose removal will disconnect the network. More generally, we conclude that the graph-theoretic and taxonomic analysis of PSIMAP is an important step towards the understanding of protein function and could be an important tool for tracing the evolution of life at the molecular level.