Statistical significance of rich-club phenomena in complex networks

We propose that the rich-club phenomena in complex networks should be defined in the spirit of bootstrapping, in which a null model is adopted to assess the statistical significance of the rich-club detected. Our method can be served as a definition of rich-club phenomenon and is applied to analyzing three real networks and three model networks. The results improve significantly compared with previously reported results. We report a dilemma with an exceptional example, showing that there does not exist an omnipotent definition for the rich-club phenomenon.Comment: 3 Revtex pages + 5 figure

Coarse-Graining and Self-Dissimilarity of Complex Networks

Can complex engineered and biological networks be coarse-grained into smaller and more understandable versions in which each node represents an entire pattern in the original network? To address this, we define coarse-graining units (CGU) as connectivity patterns which can serve as the nodes of a coarse-grained network, and present algorithms to detect them. We use this approach to systematically reverse-engineer electronic circuits, forming understandable high-level maps from incomprehensible transistor wiring: first, a coarse-grained version in which each node is a gate made of several transistors is established. Then, the coarse-grained network is itself coarse-grained, resulting in a high-level blueprint in which each node is a circuit-module made of multiple gates. We apply our approach also to a mammalian protein-signaling network, to find a simplified coarse-grained network with three main signaling channels that correspond to cross-interacting MAP-kinase cascades. We find that both biological and electronic networks are 'self-dissimilar', with different network motifs found at each level. The present approach can be used to simplify a wide variety of directed and nondirected, natural and designed networks.Comment: 11 pages, 11 figure

Subgraphs and network motifs in geometric networks

Many real-world networks describe systems in which interactions decay with the distance between nodes. Examples include systems constrained in real space such as transportation and communication networks, as well as systems constrained in abstract spaces such as multivariate biological or economic datasets and models of social networks. These networks often display network motifs: subgraphs that recur in the network much more often than in randomized networks. To understand the origin of the network motifs in these networks, it is important to study the subgraphs and network motifs that arise solely from geometric constraints. To address this, we analyze geometric network models, in which nodes are arranged on a lattice and edges are formed with a probability that decays with the distance between nodes. We present analytical solutions for the numbers of all 3 and 4-node subgraphs, in both directed and non-directed geometric networks. We also analyze geometric networks with arbitrary degree sequences, and models with a field that biases for directed edges in one direction. Scaling rules for scaling of subgraph numbers with system size, lattice dimension and interaction range are given. Several invariant measures are found, such as the ratio of feedback and feed-forward loops, which do not depend on system size, dimension or connectivity function. We find that network motifs in many real-world networks, including social networks and neuronal networks, are not captured solely by these geometric models. This is in line with recent evidence that biological network motifs were selected as basic circuit elements with defined information-processing functions.Comment: 9 pages, 6 figure

Patterns of Interactions in Complex Social Networks Based on Coloured Motifs Analysis

Coloured network motifs are small subgraphs that enable to discover and interpret the patterns of interaction within the complex networks. The analysis of three-nodes motifs where the colour of the node reflects its high – white node or low – black node centrality in the social network is presented in the paper. The importance of the vertices is assessed by utilizing two measures: degree prestige and degree centrality. The distribution of motifs in these two cases is compared to mine the interconnection patterns between nodes. The analysis is performed on the social network derived from email communication

Potts Model On Random Trees

We study the Potts model on locally tree-like random graphs of arbitrary degree distribution. Using a population dynamics algorithm we numerically solve the problem exactly. We confirm our results with simulations. Comparisons with a previous approach are made, showing where its assumption of uniform local fields breaks down for networks with nodes of low degree.Comment: 10 pages, 3 figure

The role of mentorship in protege performance

The role of mentorship on protege performance is a matter of importance to academic, business, and governmental organizations. While the benefits of mentorship for proteges, mentors and their organizations are apparent, the extent to which proteges mimic their mentors' career choices and acquire their mentorship skills is unclear. Here, we investigate one aspect of mentor emulation by studying mentorship fecundity---the number of proteges a mentor trains---with data from the Mathematics Genealogy Project, which tracks the mentorship record of thousands of mathematicians over several centuries. We demonstrate that fecundity among academic mathematicians is correlated with other measures of academic success. We also find that the average fecundity of mentors remains stable over 60 years of recorded mentorship. We further uncover three significant correlations in mentorship fecundity. First, mentors with small mentorship fecundity train proteges that go on to have a 37% larger than expected mentorship fecundity. Second, in the first third of their career, mentors with large fecundity train proteges that go on to have a 29% larger than expected fecundity. Finally, in the last third of their career, mentors with large fecundity train proteges that go on to have a 31% smaller than expected fecundity.Comment: 23 pages double-spaced, 4 figure

Evaluating Local Community Methods in Networks

We present a new benchmarking procedure that is unambiguous and specific to local community-finding methods, allowing one to compare the accuracy of various methods. We apply this to new and existing algorithms. A simple class of synthetic benchmark networks is also developed, capable of testing properties specific to these local methods.Comment: 8 pages, 9 figures, code included with sourc

Stops making sense: translational trade-offs and stop codon reassignment

Background Efficient gene expression involves a trade-off between (i) premature termination of protein synthesis; and (ii) readthrough, where the ribosome fails to dissociate at the terminal stop. Sense codons that are similar in sequence to stop codons are more susceptible to nonsense mutation, and are also likely to be more susceptible to transcriptional or translational errors causing premature termination. We therefore expect this trade-off to be influenced by the number of stop codons in the genetic code. Although genetic codes are highly constrained, stop codon number appears to be their most volatile feature. Results In the human genome, codons readily mutable to stops are underrepresented in coding sequences. We construct a simple mathematical model based on the relative likelihoods of premature termination and readthrough. When readthrough occurs, the resultant protein has a tail of amino acid residues incorrectly added to the C-terminus. Our results depend strongly on the number of stop codons in the genetic code. When the code has more stop codons, premature termination is relatively more likely, particularly for longer genes. When the code has fewer stop codons, the length of the tail added by readthrough will, on average, be longer, and thus more deleterious. Comparative analysis of taxa with a range of stop codon numbers suggests that genomes whose code includes more stop codons have shorter coding sequences. Conclusions We suggest that the differing trade-offs presented by alternative genetic codes may result in differences in genome structure. More speculatively, multiple stop codons may mitigate readthrough, counteracting the disadvantage of a higher rate of nonsense mutation. This could help explain the puzzling overrepresentation of stop codons in the canonical genetic code and most variants

Invariant Distribution of Promoter Activities in Escherichia coli

Cells need to allocate their limited resources to express a wide range of genes. To understand how Escherichia coli partitions its transcriptional resources between its different promoters, we employ a robotic assay using a comprehensive reporter strain library for E. coli to measure promoter activity on a genomic scale at high-temporal resolution and accuracy. This allows continuous tracking of promoter activity as cells change their growth rate from exponential to stationary phase in different media. We find a heavy-tailed distribution of promoter activities, with promoter activities spanning several orders of magnitude. While the shape of the distribution is almost completely independent of the growth conditions, the identity of the promoters expressed at different levels does depend on them. Translation machinery genes, however, keep the same relative expression levels in the distribution across conditions, and their fractional promoter activity tracks growth rate tightly. We present a simple optimization model for resource allocation which suggests that the observed invariant distributions might maximize growth rate. These invariant features of the distribution of promoter activities may suggest design constraints that shape the allocation of transcriptional resources