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At least 451 records · Page 25Linked to original sources
PubMed2006

Intrinsic bursters increase the robustness of rhythm generation in an excitatory network.

The pre-Botzinger complex (pBC) is a vital subcircuit of the respiratory central pattern generator. Although the existence of neurons with pacemaker-like bursting properties in this network is not questioned, their role in network rhythmogenesis is unresolved. Modeling is ideally suited to address this debate because of the ease with which biophysical parameters of individual cells and network architecture can be manipulated. We modeled the parameter variability of experimental data from pBC bursting pacemaker and nonpacemaker neurons using a modified version of our previously developed pBC neuron and network models. To investigate the role of pacemakers in networkwide rhythmogenesis, we simulated networks of these neurons and varied the fraction of the population made up of pacemakers. For each number of pacemaker neurons, we varied the amount of tonic drive to the network and measured the frequency of synchronous networkwide bursting produced. Both excitatory networks with all-to-all coupling and sparsely connected networks were explored for several levels of synaptic coupling strength. Networks containing only nonpacemakers were able to produce networkwide bursting, but with a low probability of bursting and low input and output ranges. Our results indicate that inclusion of pacemakers in an excitatory network increases robustness of the network by more than tripling the input and output ranges compared with networks containing no pacemakers. The largest increase in dynamic range occurs when the number of pacemakers in the network is greater than 20% of the population. Experimental tests of our model predictions are proposed.

Afferent Pathways↗
PubMed2004

Discriminative topological features reveal biological network mechanisms.

BACKGROUND: Recent genomic and bioinformatic advances have motivated the development of numerous network models intending to describe graphs of biological, technological, and sociological origin. In most cases the success of a model has been evaluated by how well it reproduces a few key features of the real-world data, such as degree distributions, mean geodesic lengths, and clustering coefficients. Often pairs of models can reproduce these features with indistinguishable fidelity despite being generated by vastly different mechanisms. In such cases, these few target features are insufficient to distinguish which of the different models best describes real world networks of interest; moreover, it is not clear a priori that any of the presently-existing algorithms for network generation offers a predictive description of the networks inspiring them. RESULTS: We present a method to assess systematically which of a set of proposed network generation algorithms gives the most accurate description of a given biological network. To derive discriminative classifiers, we construct a mapping from the set of all graphs to a high-dimensional (in principle infinite-dimensional) "word space". This map defines an input space for classification schemes which allow us to state unambiguously which models are most descriptive of a given network of interest. Our training sets include networks generated from 17 models either drawn from the literature or introduced in this work. We show that different duplication-mutation schemes best describe the E. coli genetic network, the S. cerevisiae protein interaction network, and the C. elegans neuronal network, out of a set of network models including a linear preferential attachment model and a small-world model. CONCLUSIONS: Our method is a first step towards systematizing network models and assessing their predictability, and we anticipate its usefulness for a number of communities.

Animals↗
PubMed2006

ARACNE: an algorithm for the reconstruction of gene regulatory networks in a mammalian cellular context.

BACKGROUND: Elucidating gene regulatory networks is crucial for understanding normal cell physiology and complex pathologic phenotypes. Existing computational methods for the genome-wide "reverse engineering" of such networks have been successful only for lower eukaryotes with simple genomes. Here we present ARACNE, a novel algorithm, using microarray expression profiles, specifically designed to scale up to the complexity of regulatory networks in mammalian cells, yet general enough to address a wider range of network deconvolution problems. This method uses an information theoretic approach to eliminate the majority of indirect interactions inferred by co-expression methods. RESULTS: We prove that ARACNE reconstructs the network exactly (asymptotically) if the effect of loops in the network topology is negligible, and we show that the algorithm works well in practice, even in the presence of numerous loops and complex topologies. We assess ARACNE's ability to reconstruct transcriptional regulatory networks using both a realistic synthetic dataset and a microarray dataset from human B cells. On synthetic datasets ARACNE achieves very low error rates and outperforms established methods, such as Relevance Networks and Bayesian Networks. Application to the deconvolution of genetic networks in human B cells demonstrates ARACNE's ability to infer validated transcriptional targets of the cMYC proto-oncogene. We also study the effects of misestimation of mutual information on network reconstruction, and show that algorithms based on mutual information ranking are more resilient to estimation errors. CONCLUSION: ARACNE shows promise in identifying direct transcriptional interactions in mammalian cellular networks, a problem that has challenged existing reverse engineering algorithms. This approach should enhance our ability to use microarray data to elucidate functional mechanisms that underlie cellular processes and to identify molecular targets of pharmacological compounds in mammalian cellular networks.

Algorithms↗
PubMed2002

[Assessment of hepatitis C networks].

AIMS: To assess the functioning of hepatitis networks. METHODOLOGY: A questionnaire was sent to all reference centers, to ten general hospitals and to twelve private hospitals working in hepatology. The main items included: participation and composition of networks, difficulties and positive or negative experiences, equal access to diagnostic and therapeutic means within the networks. Answers could be accompanied by personal remarks. RESULTS: The response rate was satisfactory: 75%, 70% and 100% respectively for the three groups. More than 80% of the questions were answered. Nearly all of the centers participate in a network. Two thirds of the reference centers have organized a therapeutic committee that includes private units and general hospitals. All groups actively participate in the networks. The positive experiences that were most often described were the success of training sessions especially in relation to equal access, and becoming acquainted with other participants. Most networks did not seem to be interested in screening. For reference centers the number of participating practitioners was too low. The main difficulty was felt to be overcentralization of the public hospital. Most networks are independent and meetings take place in the evening. Most responses were in favor of parity and equal access to diagnosis and therapeutic means within the networks. This only occurred in one third of the networks and if these measures were implemented it would increase network participation and effectiveness. Two thirds of the responses were also favourable to extending therapeutic trials to networks. CONCLUSION: Public and private hepatologists have similar opinions about hepatitis networks. Pooling of activities, responsibilities and means should be implemented and could improve participation in the networks.

Community Networks↗
PubMed2006

Comparison of the European Union Disease Surveillance Networks' websites.

Currently the surveillance of infectious disease in the European Union (EU) is supported by the Basic Surveillance Network (BSN) and other disease specific surveillance networks (DSNs). Each network has its own website. The objective of the current study was to describe the information presented with public access on each website from the perspective of its usefulness for the surveillance of an EU member state. The BSN and the DSNs cited in Decision 2003/542/CE were included. Each website was reviewed and assessed on the inclusion of characteristics from three broad categories: 1) general information, 2) procedures for data collection and 3) data presentation. Ten surveillance network websites were reviewed during the week of 5 December 2005. At least 80% of the 10 networks included a list of participating countries, the contact addresses for the coordinator of the network and the participating country gatekeepers and the network and #39;s objectives. Only one network specified the source and coverage of the data of each country on its website, and seven presented the disease case definition. Raw data were shown on eight websites and only two networks included presentation of elaborated data for the whole of the EU. Four networks included no reports on their websites. The periodicity of presentation for both raw data and elaborated data varied greatly between networks. The publicly available information on the 10 network websites studied was not homogeneous. We recommend that all networks present a basic set of characteristics on their websites, including case definitions, procedures used for data collection and periodic reports covering elaborated data for the entire EU.

Communicable Diseases↗
PubMed1995

Changes in network topology during the replication of kinetoplast DNA.

Kinetoplast DNA of Crithidia fasciculata is a network containing several thousand topologically interlocked DNA minicircles. In the prereplicative Form I network, each of the 5000 minicircles is intact and linked to an average of three neighbors (i.e. the minicircle valence is 3). Replication involves the release of minicircles from the interior of the network, the synthesis of nicked or gapped progeny minicircles and the attachment of the progeny to the network periphery. The ultimate result is a Form II network of 10,000 nicked or gapped minicircles. Our measurements of minicircle valence and density, and the network's surface area, revealed striking changes in network topology during replication. During the S phase, the peripheral newly replicated minicircles have a density twice that of minicircles in Form I networks, which suggests that the valence might be as high as 6. Most of the holes in the central region that occur from the removal of intact minicircles are repaired so that the central density and valence remain the same, as in prereplicative networks. When minicircle replication is complete at the end of the S phase, the isolated network has the surface area of a prereplicative network, despite having twice the number of minicircles. During the G2 phase, the Form II network undergoes a remodeling in which the area doubles and the valence is reduced to 3. Finally, the interruptions in the minicircles are repaired and the double-sized network splits in two.

Animals↗
PubMed2002

A provisional regulatory gene network for specification of endomesoderm in the sea urchin embryo.

We present the current form of a provisional DNA sequence-based regulatory gene network that explains in outline how endomesodermal specification in the sea urchin embryo is controlled. The model of the network is in a continuous process of revision and growth as new genes are added and new experimental results become available; see http://www.its.caltech.edu/~mirsky/endomeso.htm (End-mes Gene Network Update) for the latest version. The network contains over 40 genes at present, many newly uncovered in the course of this work, and most encoding DNA-binding transcriptional regulatory factors. The architecture of the network was approached initially by construction of a logic model that integrated the extensive experimental evidence now available on endomesoderm specification. The internal linkages between genes in the network have been determined functionally, by measurement of the effects of regulatory perturbations on the expression of all relevant genes in the network. Five kinds of perturbation have been applied: (1) use of morpholino antisense oligonucleotides targeted to many of the key regulatory genes in the network; (2) transformation of other regulatory factors into dominant repressors by construction of Engrailed repressor domain fusions; (3) ectopic expression of given regulatory factors, from genetic expression constructs and from injected mRNAs; (4) blockade of the beta-catenin/Tcf pathway by introduction of mRNA encoding the intracellular domain of cadherin; and (5) blockade of the Notch signaling pathway by introduction of mRNA encoding the extracellular domain of the Notch receptor. The network model predicts the cis-regulatory inputs that link each gene into the network. Therefore, its architecture is testable by cis-regulatory analysis. Strongylocentrotus purpuratus and Lytechinus variegatus genomic BAC recombinants that include a large number of the genes in the network have been sequenced and annotated. Tests of the cis-regulatory predictions of the model are greatly facilitated by interspecific computational sequence comparison, which affords a rapid identification of likely cis-regulatory elements in advance of experimental analysis. The network specifies genomically encoded regulatory processes between early cleavage and gastrula stages. These control the specification of the micromere lineage and of the initial veg(2) endomesodermal domain; the blastula-stage separation of the central veg(2) mesodermal domain (i.e., the secondary mesenchyme progenitor field) from the peripheral veg(2) endodermal domain; the stabilization of specification state within these domains; and activation of some downstream differentiation genes. Each of the temporal-spatial phases of specification is represented in a subelement of the network model, that treats regulatory events within the relevant embryonic nuclei at particular stages.

Animals↗
PubMed2003

Stoichiometric design of metabolic networks: multifunctionality, clusters, optimization, weak and strong robustness.

Starting from a limited set of reactions describing changes in the carbon skeleton of biochemical compounds complete sets of metabolic networks are constructed. The networks are characterized by the number and types of participating reactions. Elementary networks are defined by the condition that a specific chemical conversion can be performed by a set of given reactions and that this ability will be lost by elimination of any of these reactions. Groups of networks are identified with respect to their ability to perform a certain number of metabolic conversions in an elementary way which are called the network's functions. The number of the network functions defines the degree of multifunctionality. Transitions between networks and mutations of networks are defined by exchanges of single reactions. Different mutations exist such as gain or loss of function mutations and neutral mutations. Based on these mutations neighbourhood relations between networks are established which are described in a graph theoretical way. Basic properties of these graphs are determined such as diameter, connectedness, distance distribution of pairs of vertices. A concept is developed to quantify the robustness of networks against changes in their stoichiometry where we distinguish between strong and weak robustness. Evolutionary algorithms are applied to study the development of network populations under constant and time dependent environmental conditions. It is shown that the populations evolve toward clusters of networks performing a common function and which are closely neighboured. Under changing environmental conditions multifunctional networks prove to be optimal and will be selected.

Algorithms↗
PubMed1999

Long-term depression of perforant path excitatory postsynaptic potentials following synchronous network bursting in area CA3 of immature hippocampus.

Various forms of synaptic long-term potentiation and depression have been studied in detail in hippocampus and neocortex. Each are produced by specific patterns of synaptic activation and rely on electrical stimulation of afferents for their induction. Few studies have explored the ability of activity produced by cortical networks themselves to generate similar long-term changes in synaptic efficacy. Experiments have shown that periods of synchronized network bursting in both slices and dissociated cultures of hippocampus can lead to persistent network discharges. Conversely, one study has reported that network discharging can disrupt the induction of long-term potentiation in hippocampus. Whether long-term depression can be induced by synchronous network discharging is unknown. In experiments reported here, we examined the long-term effects of synchronized activity within hippocampal CA3 networks on synaptic potentials produced by the converging perforant path. Prior to induction of network bursting, slices were incubated in a perfusate containing picrotoxin and elevated (4 mM) Ca2+ and Mg2+. To induce network discharges, the concentration of both divalent cations were reduced to normal levels (1.5 mM). Following 20 min of network bursting, perforant path synapses, that did not participate in network discharging, underwent a 30% non-decrementing long-term depression. At the same time, synchronized network discharges, that were absent prior to induction, persisted upon return to preinduction conditions. The antagonist of the N-methyl-D-aspartate receptor, D(-)2-amino-5-phosphonovalerate, blocked both long-term depression of perforant path excitatory postsynaptic potentials and persistent network discharging. Results suggest that activity generated by hippocampal networks is able to produce long-term depression of non-coincidentally active synapses.

2-Amino-5-phosphonovalerate↗
PubMed1998

How embedded memory in recurrent neural network architectures helps learning long-term temporal dependencies.

Learning long-term temporal dependencies with recurrent neural networks can be a difficult problem. It has recently been shown that a class of recurrent neural networks called NARX networks perform much better than conventional recurrent neural networks for learning certain simple long-term dependency problems. The intuitive explanation for this behavior is that the output memories of a NARX network can be manifested as jump-ahead connections in the time-unfolded network. These jump-ahead connections can propagate gradient information more efficiently, thus reducing the sensitivity of the network to long-term dependencies. This work gives empirical justification to our hypothesis that similar improvements in learning long-term dependencies can be achieved with other classes of recurrent neural network axchitectures simply by increasing the order of the embedded memory. In particular we explore the impact of learning simple long-term dependency problems on three classes of recurrent neural network architectures: globally recurrent networks, locally recurrent networks, and NARX (output feedback) networks.Comparing the performance of these architectures with different orders of embedded memory on two simple long-term dependencies problems shows that all of these classes of network architectures demonstrate significant improvement on learning long-term dependencies when the orders of embedded memory are increased. These results can be important to a user comfortable with a specific recurrent neural network architecture because simply increasing the embedding memory order of that architecture will make it more robust to the problem of long-term dependency learning.

Journal Article↗
PubMed1990

Interactions of neural networks: models for distraction and concentration.

We present a model of neural group interactions, which are projections from one neural network (network B) of McCulloch-Pitts neurons connected via a Hebbian rule, to another network (network A) of the same structure. We first consider the case in which the projecting network B is in a pattern different from the initial attracting state of network A. A critical projecting strength lambda c is found such that for lambda below this value there exists a noise threshold sigma lambda corresponding to each lambda. For the case where lambda less than lambda c and the noise level sigma less than sigma lambda, there are two possible retrievals, with different probabilities: the initial attracting state of network A and the projecting pattern. If lambda less than lambda c and sigma greater than sigma lambda, stable states of network A disappear. In the case lambda greater than lambda c, network A is pulled out of its initial basin of attraction and into that of the projecting pattern. This analysis provides a model for distraction. Second-order interactions reduce the distraction. When the projecting network B is in the same pattern as the initial attracting state of network A, the projection acts as an external reinforcement, which enables network A to retrieve in highly noisy conditions. Sharp noise thresholds for nonzero retrievals are shown to be eliminated by the projection. Higher-order connectivity improves the retrieval ability of the network. The second case serves as a model of concentration. We discuss the model of distraction and concentration (i) in connection with common experience of expectation of recognition and (ii) in connection with recent T-maze experiments on infant rats; finally, we suggest a refined version of the Bruner-Potter experiment to test our prediction of the disappearance of hysteresis.

Animals↗
PubMed2005

Manifestation of function-follow-form in cultured neuronal networks.

We expose hidden function-follow-form schemata in the recorded activity of cultured neuronal networks by comparing the activity with simulation results of a new modeling approach. Cultured networks grown from an arbitrary mixture of neuron and glia cells in the absence of external stimulations and chemical cues spontaneously form networks of different sizes (from 50 to several millions of neurons) that exhibit non-arbitrary complex spatio-temporal patterns of activity. The latter is marked by formation of a sequence of synchronized bursting events (SBEs)--short time windows (approximately 200 ms) of rapid neuron firing, separated by longer time intervals (seconds) of sporadic neuron firing. The new dynamical synapse and soma (DSS) model, used here, has been successful in generating sequences of SBEs with the same statistical scaling properties (over six time decades) as those of the small networks. Large networks generate statistically distinct sub-groups of SBEs, each with its own characteristic pattern of neuronal firing ('fingerprint'). This special function (activity) motif has been proposed to emanate from a structural (form) motif--self-organization of the large networks into a fabric of overlapping sub-networks of about 1 mm in size. Here we test this function-follow-form idea by investigating the influence of the connectivity architecture of a model network (form) on the structure of its spontaneous activity (function). We show that a repertoire of possible activity states similar to the observed ones can be generated by networks with proper underlying architecture. For example, networks composed of two overlapping sub-networks exhibit distinct types of SBEs, each with its own characteristic pattern of neuron activity that starts at a specific sub-network. We further show that it is possible to regulate the temporal appearance of the different sub-groups of SBEs by an additional non-synaptic current fed into the soma of the modeled neurons. The ability to regulate the relative temporal ordering of different SBEs might endow the networks with higher plasticity and complexity. These findings call for additional mechanisms yet to be discovered. Recent experimental observations indicate that glia cells coupled to neuronal soma might generate such non-synaptic regulating currents.

Action Potentials↗
PubMed2006

Efficient estimation of graphlet frequency distributions in protein-protein interaction networks.

MOTIVATION: Algorithmic and modeling advances in the area of protein-protein interaction (PPI) network analysis could contribute to the understanding of biological processes. Local structure of networks can be measured by the frequency distribution of graphlets, small connected non-isomorphic induced subgraphs. This measure of local structure has been used to show that high-confidence PPI networks have local structure of geometric random graphs. Finding graphlets exhaustively in a large network is computationally intensive. More complete PPI networks, as well as PPI networks of higher organisms, will thus require efficient heuristic approaches. RESULTS: We propose two efficient and scalable heuristics for finding graphlets in high-confidence PPI networks. We show that both PPI and their model geometric random networks, have defined boundaries that are sparser than the 'inner parts' of the networks. In addition, these networks exhibit 'uniformity' of local structure inside the networks. Our first heuristic exploits these two structural properties of PPI and geometric random networks to find good estimates of graphlet frequency distributions in these networks up to 690 times faster than the exhaustive searches. Our second heuristic is a variant of a more standard sampling technique and it produces accurate approximate results up to 377 times faster than the exhaustive searches. We indicate how the combination of these approaches may result in an even better heuristic. AVAILABILITY: Supplementary information is available at http://www.cs.toronto.edu/~natasha/BIOINF-2005-0946/Supplementary.pdf. Software implementing the algorithms is available at http://www.cs.toronto.edu/~natasha/BIOINF-2005-0946/estimate_grap-hlets.html. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Algorithms↗
PubMed2004

Identification of genetic networks.

In this report, we propose the use of structural equations as a tool for identifying and modeling genetic networks and genetic algorithms for searching the most likely genetic networks that best fit the data. After genetic networks are identified, it is fundamental to identify those networks influencing cell phenotypes. To accomplish this task we extend the concept of differential expression of the genes, widely used in gene expression data analysis, to genetic networks. We propose a definition for the differential expression of a genetic network and use the generalized T2 statistic to measure the ability of genetic networks to distinguish different phenotypes. However, describing the differential expression of genetic networks is not enough for understanding biological systems because differences in the expression of genetic networks do not directly reflect regulatory strength between gene activities. Therefore, in this report we also introduce the concept of differentially regulated genetic networks, which has the potential to assess changes of gene regulation in response to perturbation in the environment and may provide new insights into the mechanism of diseases and biological processes. We propose five novel statistics to measure the differences in regulation of genetic networks. To illustrate the concepts and methods for reconstruction of genetic networks and identification of association of genetic networks with function, we applied the proposed models and algorithms to three data sets.

Algorithms↗
PubMed2004

TopNet: a tool for comparing biological sub-networks, correlating protein properties with topological statistics.

Biological networks are a topic of great current interest, particularly with the publication of a number of large genome-wide interaction datasets. They are globally characterized by a variety of graph-theoretic statistics, such as the degree distribution, clustering coefficient, characteristic path length and diameter. Moreover, real protein networks are quite complex and can often be divided into many sub-networks through systematic selection of different nodes and edges. For instance, proteins can be sub-divided by expression level, length, amino-acid composition, solubility, secondary structure and function. A challenging research question is to compare the topologies of sub- networks, looking for global differences associated with different types of proteins. TopNet is an automated web tool designed to address this question, calculating and comparing topological characteristics for different sub-networks derived from any given protein network. It provides reasonable solutions to the calculation of network statistics for sub-networks embedded within a larger network and gives simplified views of a sub-network of interest, allowing one to navigate through it. After constructing TopNet, we applied it to the interaction networks and protein classes currently available for yeast. We were able to find a number of potential biological correlations. In particular, we found that soluble proteins had more interactions than membrane proteins. Moreover, amongst soluble proteins, those that were highly expressed, had many polar amino acids, and had many alpha helices, tended to have the most interaction partners. Interestingly, TopNet also turned up some systematic biases in the current yeast interaction network: on average, proteins with a known functional classification had many more interaction partners than those without. This phenomenon may reflect the incompleteness of the experimentally determined yeast interaction network.

Algorithms↗
PubMed2007

Outpatient mental health utilization among commercially insured individuals: in- and out-of-network care.

OBJECTIVE: This study examined the rates and correlates of out-of-network outpatient mental health specialty care. RESEARCH DESIGN: Using administrative data from a large insurer, we examine the frequency of out-of-network utilization, analyze demographic and clinical characteristics of individuals receiving out-of-network care, and examine the types of service provided out-of-network. RESULTS: Out-of-network outpatient mental health care was received by 15.4% of adults who used outpatient mental health services, with 11.8% of adult outpatient mental health users receiving only out-of-network care and 3.6% receiving both in-network and out-of-network care. Out-of-network users received significantly more outpatient mental health care than individuals receiving only in-network mental health care. Rates of out-of-network psychotherapy services were substantially greater than for other commonly provided mental health services. CONCLUSION: A significant number of patients covered under this insurer received their outpatient mental health care out-of-network. This is most pronounced for individuals receiving psychotherapy. Further information is needed to improve our understanding of who seeks care from out-of-network providers and why as well as the effect of such care on clinical outcomes.

Adolescent↗
PubMed2002

Stability of shortest paths in complex networks with random edge weights.

We study shortest paths and spanning trees of complex networks with random edge weights. Edges which do not belong to the spanning tree are inactive in a transport process within the network. The introduction of quenched disorder modifies the spanning tree such that some edges are activated and the network diameter is increased. With analytic random-walk mappings and numerical analysis, we find that the spanning tree is unstable to the introduction of disorder and displays a phase-transitionlike behavior at zero disorder strength epsilon =0. In the infinite network-size limit (N--> infinity ), we obtain a continuous transition with the density of activated edges Phi growing like Phi approximately epsilon (1) and with the diameter-expansion coefficient Upsilon growing like Upsilon approximately epsilon (2) in the regular network, and first-order transitions with discontinuous jumps in Phi and Upsilon at epsilon=0 for the small-world (SW) network and the Barabási-Albert scale-free (SF) network. The asymptotic scaling behavior sets in when N>>N(c), where the crossover size scales as N(c) approximately epsilon (-2) for the regular network, N(c) approximately exp(alpha epsilon (-2)) for the SW network, and N(c) approximately exp(alpha|ln epsilon | epsilon (-2)) for the SF network. In a transient regime with N<<N(c), there is an infinite-order transition with Phi approximately Upsilon approximately exp[-alpha/(epsilon (2)ln N)] for the SW network and approximately exp[-alpha/(epsilon (2)ln N/ln ln N)] for the SF network. It shows that the transport pattern is practically most stable in the SF network.

Journal Article↗
PubMed2006

Thin Watts-Strogatz networks.

A modified version of the Watts-Strogatz (WS) network model is proposed, in which the number of shortcuts scales with the network size N as Nalpha, with alpha < 1. In these networks, the ratio of the number of shortcuts to the network size approaches zero as N --> infinity, whereas in the original WS model, this ratio is constant. We call such networks "thin Watts-Strogatz networks." We show that even though the fraction of shortcuts becomes vanishingly small for large networks, they still cause a kind of small-world effect, in the sense that the length L of the network increases sublinearly with the size. We develop a mean-field theory for these networks, which predicts that the length scales as N1-alpha ln N for large N. We also study how a search using only local information works in thin WS networks. We find that the search performance is enhanced compared to the regular network, and we predict that the search time tau scales as N1-alpha/2. These theoretical results are tested using numerical simulations. We comment on the possible relevance of thin WS networks for the design of high-performance low-cost communication networks.

Journal Article↗