Subgraph centrality in complex networks

Using network centrality measures to manage landscape connectivity

We use a graph-theoretical landscape modeling approach to investigate how to identify central patches in the landscape as well as how these central patches influence (1) organism movement within the local neighborhood and (2) the dispersal of organisms beyond the local neighborhood. Organism movements were theoretically estimated based on the spatial configuration of the habitat patches in the studied landscape. We find that centrality depends on the way the graph-theoretical model of habitat patches is constructed, although even the simplest network representation, not taking strength and directionality of potential organisms flows into account, still provides a coarse-grained assessment of the most important patches according to their contribution to landscape connectivity. Moreover, we identify (at least) two general classes of centrality. One accounts for the local flow of organisms in the neighborhood of a patch, and the other accounts for the ability to maintain connectivity beyond the scale of the local neighborhood. Finally, we study how habitat patches with high scores on different network centrality measures are distributed in a fragmented agricultural landscape in Madagascar. Results show that patches with high degree and betweenness centrality are widely spread, while patches with high subgraph and closeness centrality are clumped together in dense clusters. This finding may enable multispecies analyses of single-species network models.

Spectral centrality measures in complex networks

Complex networks are characterized by heterogeneous distributions of the degree of nodes, which produce a large diversification of the roles of the nodes within the network. Several centrality measures have been introduced to rank nodes based on their topological importance within a graph. Here we review and compare centrality measures based on spectral properties of graph matrices. We shall focus on PageRank (PR), eigenvector centrality (EV), and the hub and authority scores of the HITS algorithm. We derive simple relations between the measures and the (in)degree of the nodes, in some limits. We also compare the rankings obtained with different centrality measures.

Ranking vertices or edges of a network by loops: a new approach

We introduce loop ranking, a new ranking measure based on the detection of closed paths, which can be computed in an efficient way. We analyze it with respect to several ranking measures which have been proposed in the past, and are widely used to capture the relative importance of the vertices in complex networks. We argue that loop ranking is a very appropriate measure to quantify the role of both vertices and edges in the network traffic.

Why do hubs in the yeast protein interaction network tend to be essential: reexamining the connection between the network topology and essentiality

The centrality-lethality rule, which notes that high-degree nodes in a protein interaction network tend to correspond to proteins that are essential, suggests that the topological prominence of a protein in a protein interaction network may be a good predictor of its biological importance. Even though the correlation between degree and essentiality was confirmed by many independent studies, the reason for this correlation remains illusive. Several hypotheses about putative connections between essentiality of hubs and the topology of protein-protein interaction networks have been proposed, but as we demonstrate, these explanations are not supported by the properties of protein interaction networks. To identify the main topological determinant of essentiality and to provide a biological explanation for the connection between the network topology and essentiality, we performed a rigorous analysis of six variants of the genomewide protein interaction network for Saccharomyces cerevisiae obtained using different techniques. We demonstrated that the majority of hubs are essential due to their involvement in Essential Complex Biological Modules, a group of densely connected proteins with shared biological function that are enriched in essential proteins. Moreover, we rejected two previously proposed explanations for the centrality-lethality rule, one relating the essentiality of hubs to their role in the overall network connectivity and another relying on the recently published essential protein interactions model.

Hubba: hub objects analyzer--a framework of interactome hubs identification for network biology

One major task in the post-genome era is to reconstruct proteomic and genomic interacting networks using high-throughput experiment data. To identify essential nodes/hubs in these interactomes is a way to decipher the critical keys inside biochemical pathways or complex networks. These essential nodes/hubs may serve as potential drug-targets for developing novel therapy of human diseases, such as cancer or infectious disease caused by emerging pathogens. Hub Objects Analyzer (Hubba) is a web-based service for exploring important nodes in an interactome network generated from specific small- or large-scale experimental methods based on graph theory. Two characteristic analysis algorithms, Maximum Neighborhood Component (MNC) and Density of Maximum Neighborhood Component (DMNC) are developed for exploring and identifying hubs/essential nodes from interactome networks. Users can submit their own interaction data in PSI format (Proteomics Standards Initiative, version 2.5 and 1.0), tab format and tab with weight values. User will get an email notification of the calculation complete in minutes or hours, depending on the size of submitted dataset. Hubba result includes a rank given by a composite index, a manifest graph of network to show the relationship amid these hubs, and links for retrieving output files. This proposed method (DMNC || MNC) can be applied to discover some unrecognized hubs from previous dataset. For example, most of the Hubba high-ranked hubs (80% in top 10 hub list, and >70% in top 40 hub list) from the yeast protein interactome data (Y2H experiment) are reported as essential proteins. Since the analysis methods of Hubba are based on topology, it can also be used on other kinds of networks to explore the essential nodes, like networks in yeast, rat, mouse and human. The website of Hubba is freely available at http://hub.iis.sinica.edu.tw/Hubba.

Communicability in complex networks

We propose a new measure of the communicability of a complex network, which is a broad generalization of the concept of the shortest path. According to the new measure, most of the real-world networks display the largest communicability between the most connected (popular) nodes of the network (assortative communicability). There are also several networks with the disassortative communicability, where the most "popular" nodes communicate very poorly to each other. Using this information we classify a diverse set of real-world complex systems into a small number of universality classes based on their structure-dynamic correlation. In addition, the new communicability measure is able to distinguish finer structures of networks, such as communities into which a network is divided. A community is unambiguously defined here as a set of nodes displaying larger communicability among them than to the rest of the nodes in the network.

Graph theory and networks in Biology

A survey of the use of graph theoretical techniques in Biology is presented. In particular, recent work on identifying and modelling the structure of bio-molecular networks is discussed, as well as the application of centrality measures to interaction networks and research on the hierarchical structure of such networks and network motifs. Work on the link between structural network properties and dynamics is also described, with emphasis on synchronisation and disease propagation.

Random walks along the streets and canals in compact cities: spectral analysis, dynamical modularity, information, and statistical mechanics

Different models of random walks on the dual graphs of compact urban structures are considered. Analysis of access times between streets helps to detect the city modularity. The statistical mechanics approach to the ensembles of lazy random walkers is developed. The complexity of city modularity can be measured by an information-like parameter which plays the role of an individual fingerprint of Genius loci. Global structural properties of a city can be characterized by the thermodynamic parameters calculated in the random walk problem.

Topological structural classes of complex networks

We use theoretical principles to study how complex networks are topologically organized at large scale. Using spectral graph theory we predict the existence of four different topological structural classes of networks. These classes correspond, respectively, to highly homogenous networks lacking structural bottlenecks, networks organized into highly interconnected modules with low inter-community connectivity, networks with a highly connected central core surrounded by a sparser periphery, and networks displaying a combination of highly connected groups (quasicliques) and groups of nodes partitioned into disjoint subsets (quasibipartites). Here we show by means of the spectral scaling method that these classes really exist in real-world ecological, biological, informational, technological, and social networks. We show that neither of three network growth mechanisms--random with uniform distribution, preferential attachment, and random with the same degree sequence as real network--is able to reproduce the four structural classes of complex networks. These models reproduce two of the network classes as a function of the average degree but completely fail in reproducing the other two classes of networks.

Relating Estrada index with spectral radius

The Estrada index EE is a recently proposed molecular structure-descriptor, used in the modeling of certain features of the 3D structure of organic molecules, in particular of the degree of folding of proteins and other long-chain biopolymers. The Estrada index is computed from the spectrum of the molecular graph. Therefore, finding its relation with the spectral radius r (= the greatest graph eigenvalue) is of interest, especially because the structure-dependency of r is relatively well understood. In this work, the basic characteristics of the relation between EE and r, which turned out to be much more complicated than initially anticipated, was determined.

Protein bipartivity and essentiality in the yeast protein-protein interaction network

Protein-protein interaction networks (PINs) are structured by means of a few highly connected proteins linked to a large number of less-connected ones. Essential proteins have been found to be more abundant among these highly connected proteins. Here we demonstrate that the likelihood that removal of a protein in a PIN will prove lethal to yeast correlates with the lack of bipartivity of the protein. A protein is bipartite if it can be partitioned in such a way that there are two groups of proteins with intergroup, but not intragroup, interactions. The abundance of essential proteins found among the least bipartite proteins clearly exceeds that found among the most connected ones. For instance, among the top 50 proteins ranked by their lack of bipartivity 62% are essential proteins. However, this percentage is only 38% for proteins ranked according to their number of interactions. Protein bipartivity also surpasses another 5 measures of protein centrality in yeast PIN in identifying essential proteins and doubles the number of essential proteins selected at random. We propose a possible mechanism for the evolution of essential proteins in yeast PIN based on the duplication-divergence scheme. We conclude that a replica protein evolving from a nonbipartite target will also be nonbipartite with high probability. Consequently, these new replicas evolving from nonbipartite (essential) targets will with high probability be essential.

Congestion-gradient driven transport on complex networks

We present a study of transport on complex networks with routing based on local information. Particles hop from one node of the network to another according to a set of routing rules with different degrees of congestion awareness, ranging from random diffusion to rigid congestion-gradient driven flow. Each node can be either source or destination for particles and all nodes have the same routing capacity, which are features of ad hoc wireless networks. It is shown that the transport capacity increases when a small amount of congestion awareness is present in the routing rules, and that it then decreases as the routing rules become too rigid when the flow becomes strictly congestion-gradient driven. Therefore, an optimum value of the congestion awareness exists in the routing rules. It is also shown that, in the limit of a large number of nodes, networks using routing based on local information jam at any nonzero load. Finally, we study the correlation between congestion at node level and a betweenness centrality measure.

Food webs robustness to biodiversity loss: the roles of connectance, expansibility and degree distribution

We analyse the robustness of food webs against species loss by considering the influence of several structural factors of the networks, such as connectance, degree distribution and expansibility. The last concept refers to the absence of structural bottlenecks in the food web, whose removal separate the network into large isolate clusters. In theory networks with identical connectance can display different expansibility characteristics. Using the spectral scaling method we studied 17 food networks and classified them as good expansion (GE) and not-GE networks. The combination of GE properties and degree distribution of species permitted the classification of food webs into six different classes. These classes characterize the differences in robustness of food webs to species loss. While the webs having uniform degree distributions and displaying GE properties are the most robust to species loss, the presence of bottlenecks and skewed distribution of the number of links per species make food webs very vulnerable to primary removal of species.

Folding degrees of azurins and pseudoazurins

A quantitative measure of the degree of folding of azurins and pseudoazurins has been made. We have found that the reduction potential of azurins and pseudoazurins is a function of the contribution to the degree of folding of His117, a key amino acid in electron transfer which is directly bonded to copper in these proteins. The folding degree of His117 explains 95% of the variance in the experimental values of the reduction potential of azurins and pseudoazurins. The change in the folding degree of this amino acid influences several geometric parameters of the main backbones of these proteins. Among them, the angle formed between N(His117)...Cu...S(Cys112), which plays an important role in electron transport, but not the N(His117)...Cu distance, shows some non-linear correlation with the reduction potential of azurins and pseudoazurins. However, it is only able to explain less than 75% in the variance of the reduction potential of these proteins instead of the 95% explained by the folding degree of His117.

Spectral measures of bipartivity in complex networks

We introduce a quantitative measure of network bipartivity as a proportion of even to total number of closed walks in the network. Spectral graph theory is used to quantify how close to bipartite a network is and the extent to which individual nodes and edges contribute to the global network bipartivity. It is shown that the bipartivity characterizes the network structure and can be related to the efficiency of semantic or communication networks, trophic interactions in food webs, construction principles in metabolic networks, or communities in social networks.