Geographical threshold graphs with small-world and scale-free properties

A theory of evolutionary dynamics on any complex population structure reveals stem cell niche architecture as a spatial suppressor of selection

How the spatial arrangement of a population shapes its evolutionary dynamics has been of long-standing interest in population genetics. Most previous studies assume a small number of demes or symmetrical structures that, most often, act as well-mixed populations. Other studies use network theory to study more heterogeneous spatial structures, however they usually assume small, regular networks, or strong constraints on the strength of selection considered. Here we build network generation algorithms, conduct evolutionary simulations and derive general analytic approximations for probabilities of fixation in populations with complex spatial structure. We build a unifying evolutionary theory across network families and derive the relevant selective parameter, which is a combination of network statistics, predictive of evolutionary dynamics. We also illustrate how to link this theory with novel datasets of spatial organization and use recent imaging data to build the cellular spatial networks of the stem cell niches of the bone marrow. Across a wide variety of parameters, we find these networks to be strong suppressors of selection, delaying mutation accumulation in this tissue. We also find that decreases in stem cell population size also decrease the suppression strength of the tissue spatial structure.

Extending the Framework for Developing Intelligent Virtual Environments (FIVE) with Artifacts for Modeling Internet of Things Devices and a New Decentralized Federated Learning Based on Consensus for Dynamic Networks

One of the main lines of research in distributed learning in recent years is the one related to Federated Learning (FL). In this work, a decentralized Federated Learning algorithm based on consensus (CoL) is applied to Wireless Ad-hoc Networks (WANETs), where the agents communicate with other agents to share their learning model as they are available to the wireless connection range. When deploying a set of agents, it is essential to study whether all the WANET agents will be reachable before the deployment. The paper proposes to explore it by generating a simulation close to the real world using a framework (FIVE) that allows the easy development and modification of simulations based on Unity and SPADE agents. A fruit orchard with autonomous tractors is presented as a case study. The paper also presents how and why the concept of artifact has been included in the above-mentioned framework as a way to highlight the importance of some devices used in the environment that have to be located in specific places to ensure the full connection of the system. This inclusion is the first step to allow Digital Twins to be modeled with this framework, now allowing a Digital Shadow of those devices.

Epidemic dynamics on metapopulation networks with node2vec mobility

Metapopulation models have been a powerful tool for both theorizing and simulating epidemic dynamics. In a metapopulation model, one considers a network composed of subpopulations and their pairwise connections, and individuals are assumed to migrate from one subpopulation to another obeying a given mobility rule. While how different mobility rules affect epidemic dynamics in metapopulation models has been studied, there have been relatively few efforts on comparison of the effects of simple (i.e., unbiased) random walks and more complex mobility rules. Here we study a susceptible-infectious-susceptible (SIS) dynamics in a metapopulation model in which individuals obey a parametric second-order random-walk mobility rule called the node2vec. We map the second-order mobility rule of the node2vec to a first-order random walk in a network whose each node is a directed edge connecting a pair of subpopulations and then derive the epidemic threshold. For various networks, we find that the epidemic threshold is large (therefore, epidemic spreading tends to be suppressed) when the individuals infrequently backtrack or infrequently visit the common neighbors of the currently visited and the last visited subpopulations than when the individuals obey the simple random walk. The amount of change in the epidemic threshold induced by the node2vec mobility is in general not as large as, but is sometimes comparable with, the one induced by the change in the diffusion rate for individuals.

Spatial strength centrality and the effect of spatial embeddings on network architecture

For many networks, it is useful to think of their nodes as being embedded in a latent space, and such embeddings can affect the probabilities for nodes to be adjacent to each other. In this paper, we extend existing models of synthetic networks to spatial network models by first embedding nodes in Euclidean space and then modifying the models so that progressively longer edges occur with progressively smaller probabilities. We start by extending a geographical fitness model by employing Gaussian-distributed fitnesses, and we then develop spatial versions of preferential attachment and configuration models. We define a notion of "spatial strength centrality" to help characterize how strongly a spatial embedding affects network structure, and we examine spatial strength centrality on a variety of real and synthetic networks.

Generalised thresholding of hidden variable network models with scale-free property

The hidden variable formalism (based on the assumption of some intrinsic node parameters) turned out to be a remarkably efficient and powerful approach in describing and analyzing the topology of complex networks. Owing to one of its most advantageous property - namely proven to be able to reproduce a wide range of different degree distribution forms - it has become a standard tool for generating networks having the scale-free property. One of the most intensively studied version of this model is based on a thresholding mechanism of the exponentially distributed hidden variables associated to the nodes (intrinsic vertex weights), which give rise to the emergence of a scale-free network where the degree distribution p(k) ~ k^−γ is decaying with an exponent of γ = 2. Here we propose a generalization and modification of this model by extending the set of connection probabilities and hidden variable distributions that lead to the aforementioned degree distribution, and analyze the conditions leading to the above behavior analytically. In addition, we propose a relaxation of the hard threshold in the connection probabilities, which opens up the possibility for obtaining sparse scale free networks with arbitrary scaling exponent.

The Statistical Analysis of Top Hubs in Growing Geographical Networks with Optimal Policy

Many practical networks, such as city networks, road networks and neural networks, usually grow up on basis of topological structures and geographical measures. Big hubs, importance of which have been well known in complex networks, still play crucial roles in growing networks with geographical measures. Therefore, it is very necessary to investigate the underlying mechanisms of statistical features of different top hubs in such networks. Here, we propose a growing network model based on optimal policy in geographical ground. Through the statistics of a great number of geographical networks, we find that the degree and position distributions of top four hubs are diverse between them and closely interrelated with each other, and further gain the relationships between the upper limits of top hubs and the size of networks. Then, the underlying mechanisms are explored. Meanwhile, we are diligent to obtain the corresponding relationships of different spatial distribution areas for different top hubs, and compute their abnormal average degrees at different spatial positions, which show significant differences and imply the advantage of spatial positions and intense competition between top hubs. We hope our results could offer useful inspirations for related practical network studies.

Local Attachment Explains Small World-like Properties of Fibroblastic Reticular Cell Networks in Lymph Nodes

Fibroblastic reticular cells (FRCs) form a cellular network that serves as the structural backbone of lymph nodes and facilitates lymphocyte migration. In mice, this FRC network has been found to have small-world properties. Using a model based on geographical preferential attachment, we simulated the formation of a variety of cellular networks and show that similar small-world properties robustly emerge under such natural conditions. By estimating the parameters of this model, we generated FRC network representations with realistic topological properties. We found that the topological properties change markedly when the network is expanded from a thin slice to a three-dimensional cube. Typical small-world properties were found to persist as network size was increased. The simulated networks were very similar to two-dimensional and three-dimensional lattice networks. According to the used metrics, these lattice networks also have small-world properties, indicating that lattice likeness is sufficient to become classified as a small-world network. Our results explain why FRC networks have small-world properties and provide a framework for simulating realistic FRC networks.

Generalized Erdős numbers for network analysis

The identification of relationships in complex networks is critical in a variety of scientific contexts. This includes the identification of globally central nodes and analysing the importance of pairwise relationships between nodes. In this paper, we consider the concept of topological proximity (or 'closeness') between nodes in a weighted network using the generalized Erdős numbers (GENs). This measure satisfies a number of desirable properties for networks with nodes that share a finite resource. These include: (i) real-valuedness, (ii) non-locality and (iii) asymmetry. We show that they can be used to define a personalized measure of the importance of nodes in a network with a natural interpretation that leads to new methods to measure centrality. We show that the square of the leading eigenvector of an importance matrix defined using the GENs is strongly correlated with well-known measures such as PageRank, and define a personalized measure of centrality that is also well correlated with other existing measures. The utility of this measure of topological proximity is demonstrated by showing the asymmetries in both the dynamics of random walks and the mean infection time in epidemic spreading are better predicted by the topological definition of closeness provided by the GENs than they are by other measures.

Factorization threshold models for scale-free networks generation

Background

Several models for producing scale-free networks have been suggested; most of them are based on the preferential attachment approach. In this article, we suggest a new approach for generating scale-free networks with an alternative source of the power-law degree distribution.

Methods

The model derives from matrix factorization methods and geographical threshold models that were recently proven to show good results in generating scale-free networks. We associate each node with a vector having latent features distributed over a unit sphere and with a weight variable sampled from a Pareto distribution. We join two nodes by an edge if they are spatially close and/or have large weights.

Results and conclusion

The network produced by this approach is scale free and has a power-law degree distribution with an exponent of 2. In addition, we propose an extension of the model that allows us to generate directed networks with tunable power-law exponents.

CD30 cell graphs of Hodgkin lymphoma are not scale-free--an image analysis approach

MOTIVATION

Hodgkin lymphoma (HL) is a type of B-cell lymphoma. To diagnose the subtypes, biopsies are taken and immunostained. The slides are scanned to produce high-resolution digital whole slide images (WSI). Pathologists manually inspect the spatial distribution of cells, but little is known on the statistical properties of cell distributions in WSIs. Such properties would give valuable information for the construction of theoretical models that describe the invasion of malignant cells in the lymph node and the intercellular interactions.

RESULTS

In this work, we define and discuss HL cell graphs. We identify CD30(+) cells in HL WSIs, bringing together the fields of digital imaging and network analysis. We define special graphs based on the positions of the immunostained cells. We present an automatic analysis of complete WSIs to determine significant morphological and immunohistochemical features of HL cells and their spatial distribution in the lymph node tissue under three different medical conditions: lymphadenitis (LA) and two types of HL. We analyze the vertex degree distributions of CD30 cell graphs and compare them to a null model. CD30 cell graphs show higher vertex degrees than expected by a random unit disk graph, suggesting clustering of the cells. We found that a gamma distribution is suitable to model the vertex degree distributions of CD30 cell graphs, meaning that they are not scale-free. Moreover, we compare the graphs for LA and two subtypes of HL. LA and classical HL showed different vertex degree distributions. The vertex degree distributions of the two HL subtypes NScHL and mixed cellularity HL (MXcHL) were similar.

AVAILABILITY AND IMPLEMENTATION

The CellProfiler pipeline used for cell detection is available at https://sourceforge.net/projects/cellgraphs/.

CONTACT

ina.koch@bioinformatik.uni-frankfurt.de

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Random geometric graph description of connectedness percolation in rod systems

The problem of continuum percolation in dispersions of rods is reformulated in terms of weighted random geometric graphs. Nodes (or sites or vertices) in the graph represent spatial locations occupied by the centers of the rods. The probability that an edge (or link) connects any randomly selected pair of nodes depends upon the rod volume fraction as well as the distribution over their sizes and shapes, and also upon quantities that characterize their state of dispersion (such as the orientational distribution function). We employ the observation that contributions from closed loops of connected rods are negligible in the limit of large aspect ratios to obtain percolation thresholds that are fully equivalent to those calculated within the second-virial approximation of the connectedness Ornstein-Zernike equation. Our formulation can account for effects due to interactions between the rods, and many-body features can be partially addressed by suitable choices for the edge probabilities.

Superlinear and sublinear urban scaling in geographical networks modeling cities

Using a geographical scale-free network to describe relations between people in a city, we explain both superlinear and sublinear allometric scaling of urban indicators that quantify activities or performances of the city. The urban indicator Y(N) of a city with the population size N is analytically calculated by summing up all individual activities produced by person-to-person relationships. Our results show that the urban indicator scales superlinearly with the population, namely, Y(N)∝N(β) with β>1, if Y(N) represents a creative productivity and the indicator scales sublinearly (β<1) if Y(N) is related to the degree of infrastructure development. These results coincide with allometric scaling observed in real-world urban indicators. We also show how the scaling exponent β depends on the strength of the geographical constraint in the network formation.

Intergroup networks as random threshold graphs

Similar-minded people tend to form social groups. Due to pluralistic homophily as well as a sort of heterophily, people also participate in a wide variety of groups. Thus, these groups generally overlap with each other; an overlap between two groups can be characterized by the number of common members. These common members can play a crucial role in the transmission of information between the groups. As a step towards understanding the information dissemination, we perceive the system as a pruned intergroup network and show that it maps to a very basic graph theoretic concept known as a threshold graph. We analyze several structural properties of this network such as degree distribution, largest component size, edge density, and local clustering coefficient. We compare the theoretical predictions with the results obtained from several online social networks (LiveJournal, Flickr, YouTube) and find a good match.

Identifying highly connected counties compensates for resource limitations when evaluating national spread of an invasive pathogen

Surveying invasive species can be highly resource intensive, yet near-real-time evaluations of invasion progress are important resources for management planning. In the case of the soybean rust invasion of the United States, a linked monitoring, prediction, and communication network saved U.S. soybean growers approximately $200 M/yr. Modeling of future movement of the pathogen (Phakopsora pachyrhizi) was based on data about current disease locations from an extensive network of sentinel plots. We developed a dynamic network model for U.S. soybean rust epidemics, with counties as nodes and link weights a function of host hectarage and wind speed and direction. We used the network model to compare four strategies for selecting an optimal subset of sentinel plots, listed here in order of increasing performance: random selection, zonal selection (based on more heavily weighting regions nearer the south, where the pathogen overwinters), frequency-based selection (based on how frequently the county had been infected in the past), and frequency-based selection weighted by the node strength of the sentinel plot in the network model. When dynamic network properties such as node strength are characterized for invasive species, this information can be used to reduce the resources necessary to survey and predict invasion progress.

Scale-free networks embedded in fractal space

The impact of an inhomogeneous arrangement of nodes in space on a network organization cannot be neglected in most real-world scale-free networks. Here we propose a model for a geographical network with nodes embedded in a fractal space in which we can tune the network heterogeneity by varying the strength of the spatial embedding. When the nodes in such networks have power-law distributed intrinsic weights, the networks are scale-free with the degree distribution exponent decreasing with increasing fractal dimension if the spatial embedding is strong enough, while the weakly embedded networks are still scale-free but the degree exponent is equal to γ = 2 regardless of the fractal dimension. We show that this phenomenon is related to the transition from a noncompact to compact phase of the network and that this transition accompanies a drastic change of the network efficiency. We test our analytically derived predictions on the real-world example of networks describing the soil porous architecture.

Efficient physical embedding of topologically complex information processing networks in brains and computer circuits

Nervous systems are information processing networks that evolved by natural selection, whereas very large scale integrated (VLSI) computer circuits have evolved by commercially driven technology development. Here we follow historic intuition that all physical information processing systems will share key organizational properties, such as modularity, that generally confer adaptivity of function. It has long been observed that modular VLSI circuits demonstrate an isometric scaling relationship between the number of processing elements and the number of connections, known as Rent's rule, which is related to the dimensionality of the circuit's interconnect topology and its logical capacity. We show that human brain structural networks, and the nervous system of the nematode C. elegans, also obey Rent's rule, and exhibit some degree of hierarchical modularity. We further show that the estimated Rent exponent of human brain networks, derived from MRI data, can explain the allometric scaling relations between gray and white matter volumes across a wide range of mammalian species, again suggesting that these principles of nervous system design are highly conserved. For each of these fractal modular networks, the dimensionality of the interconnect topology was greater than the 2 or 3 Euclidean dimensions of the space in which it was embedded. This relatively high complexity entailed extra cost in physical wiring: although all networks were economically or cost-efficiently wired they did not strictly minimize wiring costs. Artificial and biological information processing systems both may evolve to optimize a trade-off between physical cost and topological complexity, resulting in the emergence of homologous principles of economical, fractal and modular design across many different kinds of nervous and computational networks.

Brains are often compared to computers but, apart from the trivial fact that both process information using a complex physical pattern of connections, it has been unclear whether this is more than just a metaphor. In our work, we rigorously uncover novel quantitative organizational principles that underlie the network organization of the human brain, high performance computer circuits, and the nervous system of the nematode C. elegans. We show through a topological and physical analysis of connectivity data that each of these systems is cost-efficiently embedded in physical space; they are organized as economical modular networks, paying a modest premium in wiring cost for the functional advantages of high dimensional topology. We also show that the fractal properties of human brain network connectivity can be used to explain allometric scaling relations between grey and white matter volumes in the brains of a wide range of differently sized mammals—from mouse opossum to sea lion—further suggesting that these principles of nervous system design are highly conserved across species. We propose that market-driven human invention and natural selection have negotiated trade-offs between cost and complexity in design of information processing networks and convergently come to similar conclusions.

Threshold Graph Limits and Random Threshold Graphs

We study the limit theory of large threshold graphs and apply this to a variety of models for random threshold graphs. The results give a nice set of examples for the emerging theory of graph limits.

Neural Network Model for Word Sense Disambiguation Using Up/Down State and Morphoelectrotonic Transform

We propose a neural network model for word sense disambiguation using up/down state neurons and morphoelectrotonic transform. Relations between stimulus words and associated words are implemented in this neural network using an associative ontology. This new neural coding model disambiguates word senses in input sentences by using neural network firing dynamics. Whether to put a new link between two neurons is decided using the cooccurrence frequency between two words corresponding to the neurons and the attenuation rate of morphoelectrotonic potential between the two neurons. The distance of the new link is obtained with a learning mechanism by calculating morphoelectrotonic transform from the morphoelectrotonic potential of the two neurons. By analyzing learning behavior using average shortest path lengths and clustering coefficients, we show that this model has a small-world structure.

Understanding network concepts in modules

Background

Network concepts are increasingly used in biology and genetics. For example, the clustering coefficient has been used to understand network architecture; the connectivity (also known as degree) has been used to screen for cancer targets; and the topological overlap matrix has been used to define modules and to annotate genes. Dozens of potentially useful network concepts are known from graph theory.

Results

Here we study network concepts in special types of networks, which we refer to as approximately factorizable networks. In these networks, the pairwise connection strength (adjacency) between 2 network nodes can be factored into node specific contributions, named node 'conformity'. The node conformity turns out to be highly related to the connectivity. To provide a formalism for relating network concepts to each other, we define three types of network concepts: fundamental-, conformity-based-, and approximate conformity-based concepts. Fundamental concepts include the standard definitions of connectivity, density, centralization, heterogeneity, clustering coefficient, and topological overlap. The approximate conformity-based analogs of fundamental network concepts have several theoretical advantages. First, they allow one to derive simple relationships between seemingly disparate networks concepts. For example, we derive simple relationships between the clustering coefficient, the heterogeneity, the density, the centralization, and the topological overlap. The second advantage of approximate conformity-based network concepts is that they allow one to show that fundamental network concepts can be approximated by simple functions of the connectivity in module networks.

Conclusion

Using protein-protein interaction, gene co-expression, and simulated data, we show that a) many networks comprised of module nodes are approximately factorizable and b) in these types of networks, simple relationships exist between seemingly disparate network concepts. Our results are implemented in freely available R software code, which can be downloaded from the following webpage: http://www.genetics.ucla.edu/labs/horvath/ModuleConformity/ModuleNetworks

Growing networks under geographical constraints

Inspired by the structure of technological weblike systems, we discuss network evolution mechanisms which give rise to topological properties found in real spatial networks. Thus, we suggest that the peculiar structure of transport and distribution networks is fundamentally determined by two factors. These are the dependence of the spatial interaction range of vertices on the vertex attractiveness (or importance within the network) and on the inhomogeneous distribution of vertices in space. We propose and analyze numerically a simple model based on these generating mechanisms which seems, for instance, to be able to reproduce known structural features of the Internet.

Extraction and analysis of traffic and topologies of transportation networks

The knowledge of real-life traffic patterns is crucial for a good understanding and analysis of transportation systems. These data are quite rare. In this paper we propose an algorithm for extracting both the real physical topology and the network of traffic flows from timetables of public mass transportation systems. We apply this algorithm to timetables of three large transportation networks. This enables us to make a systematic comparison between three different approaches to construct a graph representation of a transportation network; the resulting graphs are fundamentally different. We also find that the real-life traffic pattern is very heterogenous, in both space and traffic flow intensities, which makes it very difficult to approximate the node load with a number of topological estimators.

Crossovers in scale-free networks on geographical space

Complex networks are characterized by several topological properties: degree distribution, clustering coefficient, average shortest path length, etc. Using a simple model to generate scale-free networks embedded on geographical space, we analyze the relationship between topological properties of the network and attributes (fitness and location) of the vertices in the network. We find there are two crossovers for varying the scaling exponent of the fitness distribution.