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PubMed2005

A simple physical model for scaling in protein-protein interaction networks.

It has recently been demonstrated that many biological networks exhibit a "scale-free" topology, for which the probability of observing a node with a certain number of edges (k) follows a power law: i.e., p(k) approximately k(-gamma). This observation has been reproduced by evolutionary models. Here we consider the network of protein-protein interactions (PPIs) and demonstrate that two published independent measurements of these interactions produce graphs that are only weakly correlated with one another despite their strikingly similar topology. We then propose a physical model based on the fundamental principle that (de)solvation is a major physical factor in PPIs. This model reproduces not only the scale-free nature of such graphs but also a number of higher-order correlations in these networks. A key support of the model is provided by the discovery of a significant correlation between the number of interactions made by a protein and the fraction of hydrophobic residues on its surface. The model presented in this paper represents a physical model for experimentally determined PPIs that comprehensively reproduces the topological features of interaction networks. These results have profound implications for understanding not only PPIs but also other types of scale-free networks.

Biophysical Phenomena↗
PubMed2002

Correlated random networks.

We develop a statistical theory of networks. A network is a set of vertices and links given by its adjacency matrix c, and the relevant statistical ensembles are defined in terms of a partition function Z= summation operator exp([-betaH(c)]. The simplest cases are uncorrelated random networks such as the well-known Erdös-Rényi graphs. Here we study more general interactions H(c) which lead to correlations, for example, between the connectivities of adjacent vertices. In particular, such correlations occur in optimized networks described by partition functions in the limit beta--> infinity. They are argued to be a crucial signature of evolutionary design in biological networks.

Brain↗
PubMed2005

Spatial-temporal modeling of malware propagation in networks.

Network security is an important task of network management. One threat to network security is malware (malicious software) propagation. One type of malware is called topological scanning that spreads based on topology information. The focus of this work is on modeling the spread of topological malwares, which is important for understanding their potential damages, and for developing countermeasures to protect the network infrastructure. Our model is motivated by probabilistic graphs, which have been widely investigated in machine learning. We first use a graphical representation to abstract the propagation of malwares that employ different scanning methods. We then use a spatial-temporal random process to describe the statistical dependence of malware propagation in arbitrary topologies. As the spatial dependence is particularly difficult to characterize, the problem becomes how to use simple (i.e., biased) models to approximate the spatially dependent process. In particular, we propose the independent model and the Markov model as simple approximations. We conduct both theoretical analysis and extensive simulations on large networks using both real measurements and synthesized topologies to test the performance of the proposed models. Our results show that the independent model can capture temporal dependence and detailed topology information and, thus, outperforms the previous models, whereas the Markov model incorporates a certain spatial dependence and, thus, achieves a greater accuracy in characterizing both transient and equilibrium behaviors of malware propagation.

Algorithms↗
PubMed2004

A graph grammar approach to artificial life.

We present the high-level language of relational growth grammars (RGGs) as a formalism designed for the specification of ALife models. RGGs can be seen as an extension of the well-known parametric Lindenmayer systems and contain rule-based, procedural, and object-oriented features. They are defined as rewriting systems operating on graphs with the edges coming from a set of user-defined relations, whereas the nodes can be associated with objects. We demonstrate their ability to represent genes, regulatory networks of metabolites, and morphologically structured organisms, as well as developmental aspects of these entities, in a common formal framework. Mutation, crossing over, selection, and the dynamics of a network of gene regulation can all be represented with simple graph rewriting rules. This is demonstrated in some detail on the classical example of Dawkins' biomorphs and the ABC model of flower morphogenesis: other applications are briefly sketched. An interactive program was implemented, enabling the execution of the formalism and the visualization of the results.

Algorithms↗
PubMed2004

Reverse engineering of linking preferences from network restructuring.

We provide a method to deduce the preferences governing the restructuring dynamics of a network from the observed rewiring of the edges. Our approach is applicable for systems in which the preferences can be formulated in terms of a single-vertex energy function with f (k) being the contribution of a node of degree k to the total energy, and the dynamics obeys the detailed balance. The method is first tested by Monte Carlo simulations of restructuring graphs with known energies; then it is used to study variations of real network systems ranging from the coauthorship network of scientific publications to the asset graphs of the New York Stock Exchange. The empirical energies obtained from the restructuring can be described by a universal function f (k) approximately -k ln k , which is consistent with and justifies the validity of the preferential attachment rule proposed for growing networks.

Journal Article↗
PubMedDate not supplied

Addition patterns in carbon allotropes: independence numbers and d-codes in the Klein and related graphs.

The problem of predicting stoichiometries and patterns of chemical addition to a carbon framework, subject solely to the restriction that each addend excludes neighboring sites up to some distance d, is equivalent to determination of d-codes of a graph, and for d = 2 to determination of maximum independent sets. Sizes, symmetries, and numbers of d-codes are found for the all-heptagon Klein graph (prototype for "plumber's nightmare" carbon) and for three related graphs. The independence number of the Klein graph is 23, which increases to 24 for a related, but sterically relaxed, all-heptagon network with the same number of vertices and modified adjacencies. Expansion of the Klein graph and its relaxed analogue by insertion of hexagonal faces to form leapfrog graphs also allows all heptagons to achieve their maximum of 3 addends. Consideration of the pi system that is the complement of the addition pattern imposes a closed-shell requirement on the adjacency spectrum, which typically reduces the size of acceptable independent sets. The closed-shell independence numbers of the Klein graph and its relaxed analogue are 18 and 20, respectively.

Journal Article↗
PubMed2003

Complexity management in visualizing protein interaction networks.

MOTIVATION: Protein-protein interaction networks often consist of thousands of nodes or more. This severely limits the utility of many graph drawing tools because they become too slow for an interactive analysis of the networks and because they produce cluttered drawings with many edge crossings. RESULTS: A new layout algorithm with complexity management operations in visualizing a large-scale protein interaction network was developed and implemented in a program called InterViewer3. InterViewer3 simplifies a complex network by collapsing a group of nodes with the same interacting partners into a composite node and by replacing a clique with a star-shaped subgraph. The experimental results demonstrated that InterViewer3 is one order of magnitude faster than the other drawing programs and that its complexity management is successful.

Algorithms↗
PubMed2005

Mathematical approaches to differentiation and gene regulation.

We consider some mathematical issues raised by the modelling of gene networks. The expression of genes is governed by a complex set of regulations, which is often described symbolically by interaction graphs. These are finite oriented graphs where vertices are the genes involved in the biological system of interest and arrows describe their interactions: a positive (resp. negative) arrow from a gene to another represents an activation (resp. inhibition) of the expression of the latter gene by some product of the former. Once such an interaction graph has been established, there remains the difficult task to decide which dynamical properties of the gene network can be inferred from it, in the absence of precise quantitative data about their regulation. There mathematical tools, among others, can be of some help. In this paper we discuss a rule proposed by Thomas according to which the possibility for the network to have several stationary states implies the existence of a positive circuit in the corresponding interaction graph. We prove that, when properly formulated in rigorous terms, this rule becomes a theorem valid for several different types of formal models of gene networks. This result is already known for models of differential [C. Soulé, Graphic requirements for multistationarity, ComPlexUs 1 (2003) 123-133] or Boolean [E. Rémy, P. Ruet, D. Thieffry, Graphic requirements for multistability and attractive cycles in a boolean dynamical framework, 2005, Preprint] type. We show here that a stronger version of it holds in the differential setup when the decay of protein concentrations is taken into account. This allows us to verify also the validity of Thomas' rule in the context of piecewise-linear models. We then discuss open problems.

Cell Differentiation↗
PubMed2006

GINsim: a software suite for the qualitative modelling, simulation and analysis of regulatory networks.

This paper presents GINsim, a Java software suite devoted to the qualitative modelling, analysis and simulation of genetic regulatory networks. Formally, our approach leans on discrete mathematical and graph-theoretical concepts. GINsim encompasses an intuitive graph editor, enabling the definition and the parameterisation of a regulatory graph, as well as a simulation engine to compute the corresponding qualitative dynamical behaviour. Our computational approach is illustrated by a preliminary model analysis of the inter-cellular regulatory network activating Notch at the dorsal-ventral boundary in the wing imaginal disc of Drosophila. We focus on the cross-regulations between five genes (within and between two cells), which implements the dorsal-ventral border in the developing imaginal disc. Our simulations qualitatively reproduce the wild-type developmental pathway, as well as the outcome of various types of experimental perturbations, such as loss-of-function mutations or ectopically induced gene expression.

Animals↗
PubMed2006

Efficient algorithms for detecting signaling pathways in protein interaction networks.

The interpretation of large-scale protein network data depends on our ability to identify significant substructures in the data, a computationally intensive task. Here we adapt and extend efficient techniques for finding paths and trees in graphs to the problem of identifying pathways in protein interaction networks. We present linear-time algorithms for finding paths and trees in networks under several biologically motivated constraints. We apply our methodology to search for protein pathways in the yeast protein-protein interaction network. We demonstrate that our algorithm is capable of reconstructing known signaling pathways and identifying functionally enriched paths and trees in an unsupervised manner. The algorithm is very efficient, computing optimal paths of length 8 within minutes and paths of length 10 in about three hours.

Algorithms↗
PubMed1998

Classification of temporal patterns in dynamic biological networks.

A general method is presented to classify temporal patterns generated by rhythmic biological networks when synaptic connections and cellular properties are known. The method is discrete in nature and relies on algebraic properties of state transitions and graph theory. Elements of the set of rhythms generated by a network are compared using a metric that quantifies the functional differences among them. The rhythms are then classified according to their location in a metric space. Examples are given, and biological implications are discussed.

Models, Neurological↗
PubMed2006

CoryneRegNet: an ontology-based data warehouse of corynebacterial transcription factors and regulatory networks.

BACKGROUND: The application of DNA microarray technology in post-genomic analysis of bacterial genome sequences has allowed the generation of huge amounts of data related to regulatory networks. This data along with literature-derived knowledge on regulation of gene expression has opened the way for genome-wide reconstruction of transcriptional regulatory networks. These large-scale reconstructions can be converted into in silico models of bacterial cells that allow a systematic analysis of network behavior in response to changing environmental conditions. DESCRIPTION: CoryneRegNet was designed to facilitate the genome-wide reconstruction of transcriptional regulatory networks of corynebacteria relevant in biotechnology and human medicine. During the import and integration process of data derived from experimental studies or literature knowledge CoryneRegNet generates links to genome annotations, to identified transcription factors and to the corresponding cis-regulatory elements. CoryneRegNet is based on a multi-layered, hierarchical and modular concept of transcriptional regulation and was implemented by using the relational database management system MySQL and an ontology-based data structure. Reconstructed regulatory networks can be visualized by using the yFiles JAVA graph library. As an application example of CoryneRegNet, we have reconstructed the global transcriptional regulation of a cellular module involved in SOS and stress response of corynebacteria. CONCLUSION: CoryneRegNet is an ontology-based data warehouse that allows a pertinent data management of regulatory interactions along with the genome-scale reconstruction of transcriptional regulatory networks. These models can further be combined with metabolic networks to build integrated models of cellular function including both metabolism and its transcriptional regulation.

Computer Graphics↗
PubMed1988

Automata with hierarchical control and evolutionary learning.

We propose an automata-theoretical framework for structured hierarchical control, in terms of rules and meta-rules, for sequences of moves on a graph. This leads to a notion of a "universal" hierarchically structured automaton mu which can move on a given graph in such a way as to emulate any automaton which moves on that graph in response to inputs. This emulation is achieved via a mapping of the inputs in the given automaton to those of mu, and we think of such a mapping as an encoding of the given automaton. We see in several examples that efficient encodings of graph-search algorithms correspond to their natural hierarchical structure (in terms of rules and meta-rules), and this leads one to a precise notion of the "depth" of an automaton which moves on a given graph. By way of application, we discuss a proposed structure of a series of stochastic neural networks which can learn, by example, to encode a given sequence of moves on a graph, so that the encoding obtained is structurally the "natural" one for the given sequence of moves. Thus, such a learning system would perform both structural pattern recognition (in terms of "patterns" of moves), and encoding based on a desired outcome.

Algorithms↗
PubMed2006

A note on the spread of worms in scale-free networks.

This paper considers the spread of worms in computer networks using insights from epidemiology and percolation theory. We provide three new results. The first result refines previous work showing that epidemics occur in scale-free graphs more easily because of their structure. We argue, using recent results from random graph theory that for scaling factors between 0 and approximately 3.4875, any computer worm infection of a scale-free network will become an epidemic. Our second result uses this insight to provide a mathematical explanation for the empirical results of Chen and Carley, who demonstrate that the Countermeasure Competing strategy can be more effective for immunizing networks to viruses or worms than traditional approaches. Our third result uses random graph theory to contradict the current supposition that, for very large networks, monocultures are necessarily more susceptible than diverse networks to worm infections.

Computer Communication Networks↗
PubMed2005

Network reachability of real-world contact sequences.

We use real-world contact sequences, time-ordered lists of contacts from one person to another, to study how fast information or disease can spread across network of contacts. Specifically we measure the reachability time--the average shortest time for a series of contacts to spread information between a reachable pair of vertices (a pair where a chain of contacts exists leading from one person to the other)--and the reachability ratio--the fraction of reachable vertex pairs. These measures are studied using conditional uniform graph tests. We conclude, among other things, that the network reachability depends much on a core where the path lengths are short and communication frequent, that clustering of the contacts of an edge in time tends to decrease the reachability, and that the order of the contacts really does make sense for dynamical spreading processes.

Journal Article↗
PubMed2003

Graph invariants for periodic systems: towards predicting physical properties from the hydrogen bond topology of ice.

Ice-Ih consists of a disordered hydrogen-bonded network. The degree of disorder in ice-Ih, and possible phase transitions to an ordered phase have been debated in recent years. The dependence of energy, free energy, and other scalar physical properties on H-bond topology is needed to understand these phenomena. Graph invariants provide a means of linking physical properties to the topology of the H-bond network. We have previously shown the effectiveness of graph invariants for finite water clusters [J.-L. Kuo, J. V. Coe, S. J. Singer, Y. B. Band, and L. Ojamäe, J. Chem. Phys., 114, 2527 (2001)]. In this work, we develop a formalism for the graph invariants of periodic systems. We demonstrate that graph invariants for small unit cells are a subset of the graph invariants of larger unit cells, providing a hierarchy of approximations by which detailed calculations for small unit cells, such as periodic ab initio calculations as they become available, can be used to parametrize the energy of the astronomical number of H-bond arrangements present in large unit cells. We also present graph enumeration results for ice-Ih, analyzing conflicting results that have appeared previously in the literature and furnishing information on the statistical properties of the H-bond network of ice-Ih in the large system limit.

Journal Article↗
PubMed2002

Global organization of the Wordnet lexicon.

The lexicon consists of a set of word meanings and their semantic relationships. A systematic representation of the English lexicon based in psycholinguistic considerations has been put together in the database Wordnet in a long-term collaborative effort. We present here a quantitative study of the graph structure of Wordnet to understand the global organization of the lexicon. Semantic links follow power-law, scale-invariant behaviors typical of self-organizing networks. Polysemy (the ambiguity of an individual word) is one of the links in the semantic network, relating the different meanings of a common word. Polysemous links have a profound impact in the organization of the semantic graph, conforming it as a small world network, with clusters of high traffic (hubs) representing abstract concepts such as line, head, or circle. Our results show that: (i) Wordnet has global properties common to many self-organized systems, and (ii) polysemy organizes the semantic graph in a compact and categorical representation, in a way that may explain the ubiquity of polysemy across languages.

Cluster Analysis↗
PubMed2006

H-CORE: enabling genome-scale Bayesian analysis of biological systems without prior knowledge.

The Bayesian network is a popular tool for describing relationships between data entities by representing probabilistic (in)dependencies with a directed acyclic graph (DAG) structure. Relationships have been inferred between biological entities using the Bayesian network model with high-throughput data from biological systems in diverse fields. However, the scalability of those approaches is seriously restricted because of the huge search space for finding an optimal DAG structure in the process of Bayesian network learning. For this reason, most previous approaches limit the number of target entities or use additional knowledge to restrict the search space. In this paper, we use the hierarchical clustering and order restriction (H-CORE) method for the learning of large Bayesian networks by clustering entities and restricting edge directions between those clusters, with the aim of overcoming the scalability problem and thus making it possible to perform genome-scale Bayesian network analysis without additional biological knowledge. We use simulations to show that H-CORE is much faster than the widely used sparse candidate method, whilst being of comparable quality. We have also applied H-CORE to retrieving gene-to-gene relationships in a biological system (The 'Rosetta compendium'). By evaluating learned information through literature mining, we demonstrate that H-CORE enables the genome-scale Bayesian analysis of biological systems without any prior knowledge.

Algorithms↗