Connectivity distribution of spatial networks

Connectivity concepts in neuronal network modeling

Sustainable research on computational models of neuronal networks requires published models to be understandable, reproducible, and extendable. Missing details or ambiguities about mathematical concepts and assumptions, algorithmic implementations, or parameterizations hinder progress. Such flaws are unfortunately frequent and one reason is a lack of readily applicable standards and tools for model description. Our work aims to advance complete and concise descriptions of network connectivity but also to guide the implementation of connection routines in simulation software and neuromorphic hardware systems. We first review models made available by the computational neuroscience community in the repositories ModelDB and Open Source Brain, and investigate the corresponding connectivity structures and their descriptions in both manuscript and code. The review comprises the connectivity of networks with diverse levels of neuroanatomical detail and exposes how connectivity is abstracted in existing description languages and simulator interfaces. We find that a substantial proportion of the published descriptions of connectivity is ambiguous. Based on this review, we derive a set of connectivity concepts for deterministically and probabilistically connected networks and also address networks embedded in metric space. Beside these mathematical and textual guidelines, we propose a unified graphical notation for network diagrams to facilitate an intuitive understanding of network properties. Examples of representative network models demonstrate the practical use of the ideas. We hope that the proposed standardizations will contribute to unambiguous descriptions and reproducible implementations of neuronal network connectivity in computational neuroscience.

Levy geometric graphs

We present a family of graphs with remarkable properties. They are obtained by connecting the points of a random walk when their distance is smaller than a given scale. Their degree (number of neighbors) does not depend on the graph's size but only on the considered scale. It follows a gamma distribution and thus presents an exponential decay. Levy flights are particular random walks with some power-law increments of infinite variance. When building the geometric graphs from them, we show from dimensional arguments that the number of connected components (clusters) follows an inverse power of the scale. The distribution of the size of their components, properly normalized, is scale invariant, which reflects the self-similar nature of the underlying process. This allows to test if a graph (including nonspatial ones) could possibly result from an underlying Levy process. When the scale increases, these graphs never tend towards a single cluster, the giant component. In other words, while the autocorrelation of the process scales as a power of the distance, they never undergo a phase transition of percolation type. The Levy graphs may find applications in community detection and in the analysis of collective behaviors as in face-to-face interaction networks.

Temporal Graphs and Temporal Network Characteristics for Bio-Inspired Networks during Optimization

Temporal network analysis and time evolution of network characteristics are powerful tools in describing the changing topology of dynamic networks. This paper uses such approaches to better visualize and provide analytical measures for the changes in performance that we observed in Voronoi-type spatial coverage, particularly for the example of time-evolving networks with a changing number of wireless sensors being deployed. Specifically, our analysis focuses on the role different combinations of impenetrable obstacles and environmental noise play in connectivity and overall network structure. It is shown how the use of (i) temporal network graphs, and (ii) network centrality and regularity measures illustrate the differences between various options developed for the balancing act of energy and time efficiency in network coverage. Last, we compare the outcome of these measures with the less abstract classification variables, such as percent area covered and cumulative distance traveled.

Unique Neural Circuit Connectivity of Mouse Proximal, Middle, and Distal Colon Defines Regional Colonic Motor Patterns

BACKGROUND & AIMS

Colonic motor patterns have been described by a number of different groups, but the neural connectivity and ganglion architecture supporting patterned motor activity have not been elucidated. Our goals were to describe quantitatively, by region, the structural architecture of the mouse enteric nervous system and use functional calcium imaging, pharmacology, and electrical stimulation to show regional underpinnings of different motor patterns.

METHODS

Excised colon segments from mice expressing the calcium indicator GCaMP6f or GCaMP6s were used to examine spontaneous and evoked (pharmacologic or electrical) changes in GCaMP-mediated fluorescence and coupled with assessment of colonic motor activity, immunohistochemistry, and confocal imaging. Three-dimensional image reconstruction and statistical methods were used to describe quantitatively mouse colon myenteric ganglion structure, neural and vascular network patterning, and neural connectivity.

RESULTS

In intact colon, regionally specific myenteric ganglion size, architecture, and neural circuit connectivity patterns along with neurotransmitter-receptor expression underlie colonic motor patterns that define functional differences along the colon. Region-specific effects on spontaneous, evoked, and chemically induced neural activity contribute to regional motor patterns, as does intraganglionic functional connectivity. We provide direct evidence of neural circuit structural and functional regional differences that have only been inferred in previous investigations. We include regional comparisons between quantitative measures in mouse and human colon that represent an important advance in showing the usefulness and relevance of the mouse system for translation to the human colon.

CONCLUSIONS

There are several neural mechanisms dependent on myenteric ganglion architecture and functional connectivity that underlie neurogenic control of patterned motor function in the mouse colon.

Generic Emergence of Modularity in Spatial Networks

Landscape’s spatial structure has vast implications for the dynamics and distribution of species populations and ecological communities. However, the characterization of the structure of spatial networks has not received nearly as much attention as networks of species interactions counterparts. Recent experiments show the dynamical implications of modularity to buffer perturbations, and theory shows that several other processes might be impacted if spatial networks were modular, from disease transmission to gene flow. Yet the question is, are spatial networks actually modular? Even though some case studies have found modular structures, we lack a general answer to that question. Here, I show that modularity is a naturally emergent property of spatial networks. This finding is further reinforced by analyzing real patchy habitats. Furthermore, I show that there is no need for any other biological process other than dispersal in order to generate a significantly modular spatial network. Modularity is explained by the spatial heterogeneity in the density of habitat fragments. The fact that spatial networks are intrinsically modular might have direct consequences for population and evolutionary dynamics. Modules define the spatial limits of populations and the role each habitat fragment plays in ecological dynamics; they become the relevant scale at which a multitude of processes occur.

Inferring Cortical Connectivity from ECoG Signals Using Graph Signal Processing

A novel method to characterize connectivity between sites in the cerebral cortex of primates is proposed in this paper. Connectivity graphs for two macaque monkeys are inferred from Electrocorticographic (ECoG) activity recorded while the animals were alert. The locations of ECoG electrodes are considered as nodes of the graph, the coefficients of the auto-regressive (AR) representation of the signals measured at each node are considered as the signal on the graph and the connectivity strengths between the nodes are considered as the edges of the graph. Maximization of the graph smoothness defined from the Laplacian quadratic form is used to infer the connectivity map (adjacency matrix of the graph). The cortical evoked potential (CEP) map was obtained by stimulating different electrodes and recording the evoked potentials at the other electrodes. The maps obtained by the graph inference and the traditional method of spectral coherence are compared with the CEP map. The results show that the proposed method provides a description of cortical connectivity that is more similar to the stimulation-based measures than spectral coherence. The results are also tested by the surrogate map analysis in which the CEP map is randomly permuted and the distribution of the errors is obtained. It is shown that error between the two maps is comfortably outside the surrogate map error distribution. This indicates that the similarity between the map calculated by the graph inference and the CEP map is statistically significant.

Assortative mixing in spatially-extended networks

We focus on spatially-extended networks during their transition from short-range connectivities to a scale-free structure expressed by heavy-tailed degree-distribution. In particular, a model is introduced for the generation of such graphs, which combines spatial growth and preferential attachment. In this model the transition to heterogeneous structures is always accompanied by a change in the graph's degree-degree correlation properties: while high assortativity levels characterize the dominance of short distance couplings, long-range connectivity structures are associated with small amounts of disassortativity. Our results allow to infer that a disassortative mixing is essential for establishing long-range links. We discuss also how our findings are consistent with recent experimental studies of 2-dimensional neuronal cultures.

Quasirandom geometric networks from low-discrepancy sequences

We define quasirandom geometric networks using low-discrepancy sequences, such as Halton, Sobol, and Niederreiter. The networks are built in d dimensions by considering the d-tuples of digits generated by these sequences as the coordinates of the vertices of the networks in a d-dimensional I^{d} unit hypercube. Then, two vertices are connected by an edge if they are at a distance smaller than a connection radius. We investigate computationally 11 network-theoretic properties of two-dimensional quasirandom networks and compare them with analogous random geometric networks. We also study their degree distribution and their spectral density distributions. We conclude from this intensive computational study that in terms of the uniformity of the distribution of the vertices in the unit square, the quasirandom networks look more random than the random geometric networks. We include an analysis of potential strategies for generating higher-dimensional quasirandom networks, where it is know that some of the low-discrepancy sequences are highly correlated. In this respect, we conclude that up to dimension 20, the use of scrambling, skipping and leaping strategies generate quasirandom networks with the desired properties of uniformity. Finally, we consider a diffusive process taking place on the nodes and edges of the quasirandom and random geometric graphs. We show that the diffusion time is shorter in the quasirandom graphs as a consequence of their larger structural homogeneity. In the random geometric graphs the diffusion produces clusters of concentration that make the process more slow. Such clusters are a direct consequence of the heterogeneous and irregular distribution of the nodes in the unit square in which the generation of random geometric graphs is based on.

Ultra-small scale-free geometric networks

We consider a family of long-range percolation models (Gp)p>0on ℤdthat allow dependence between edges and have the following connectivity properties forp∈ (1/d, ∞): (i) the degree distribution of vertices inGphas a power-law distribution; (ii) the graph distance between pointsxandyis bounded by a multiple of logpdlogpd|x-y| with probability 1 -o(1); and (iii) an adversary can delete a relatively small number of nodes fromGp(ℤd∩ [0,n]d), resulting in two large, disconnected subgraphs.

Spatial network surrogates for disentangling complex system structure from spatial embedding of nodes

Networks with nodes embedded in a metric space have gained increasing interest in recent years. The effects of spatial embedding on the networks' structural characteristics, however, are rarely taken into account when studying their macroscopic properties. Here, we propose a hierarchy of null models to generate random surrogates from a given spatially embedded network that can preserve certain global and local statistics associated with the nodes' embedding in a metric space. Comparing the original network's and the resulting surrogates' global characteristics allows one to quantify to what extent these characteristics are already predetermined by the spatial embedding of the nodes and links. We apply our framework to various real-world spatial networks and show that the proposed models capture macroscopic properties of the networks under study much better than standard random network models that do not account for the nodes' spatial embedding. Depending on the actual performance of the proposed null models, the networks are categorized into different classes. Since many real-world complex networks are in fact spatial networks, the proposed approach is relevant for disentangling the underlying complex system structure from spatial embedding of nodes in many fields, ranging from social systems over infrastructure and neurophysiology to climatology.

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.

Spatially embedded growing small-world networks

Networks in nature are often formed within a spatial domain in a dynamical manner, gaining links and nodes as they develop over time. Motivated by the growth and development of neuronal networks, we propose a class of spatially-based growing network models and investigate the resulting statistical network properties as a function of the dimension and topology of the space in which the networks are embedded. In particular, we consider two models in which nodes are placed one by one in random locations in space, with each such placement followed by configuration relaxation toward uniform node density, and connection of the new node with spatially nearby nodes. We find that such growth processes naturally result in networks with small-world features, including a short characteristic path length and nonzero clustering. We find no qualitative differences in these properties for two different topologies, and we suggest that results for these properties may not depend strongly on the topology of the embedding space. The results do depend strongly on dimension, and higher-dimensional spaces result in shorter path lengths but less clustering.

Cascading failures in spatially-embedded random networks

Cascading failures constitute an important vulnerability of interconnected systems. Here we focus on the study of such failures on networks in which the connectivity of nodes is constrained by geographical distance. Specifically, we use random geometric graphs as representative examples of such spatial networks, and study the properties of cascading failures on them in the presence of distributed flow. The key finding of this study is that the process of cascading failures is non-self-averaging on spatial networks, and thus, aggregate inferences made from analyzing an ensemble of such networks lead to incorrect conclusions when applied to a single network, no matter how large the network is. We demonstrate that this lack of self-averaging disappears with the introduction of a small fraction of long-range links into the network. We simulate the well studied preemptive node removal strategy for cascade mitigation and show that it is largely ineffective in the case of spatial networks. We introduce an altruistic strategy designed to limit the loss of network nodes in the event of a cascade triggering failure and show that it performs better than the preemptive strategy. Finally, we consider a real-world spatial network viz. a European power transmission network and validate that our findings from the study of random geometric graphs are also borne out by simulations of cascading failures on the empirical network.

Analytical framework for recurrence network analysis of time series

Recurrence networks are a powerful nonlinear tool for time series analysis of complex dynamical systems. While there are already many successful applications ranging from medicine to paleoclimatology, a solid theoretical foundation of the method has still been missing so far. Here, we interpret an ɛ-recurrence network as a discrete subnetwork of a "continuous" graph with uncountably many vertices and edges corresponding to the system's attractor. This step allows us to show that various statistical measures commonly used in complex network analysis can be seen as discrete estimators of newly defined continuous measures of certain complex geometric properties of the attractor on the scale given by ɛ. In particular, we introduce local measures such as the ɛ-clustering coefficient, mesoscopic measures such as ɛ-motif density, path-based measures such as ɛ-betweennesses, and global measures such as ɛ-efficiency. This new analytical basis for the so far heuristically motivated network measures also provides an objective criterion for the choice of ɛ via a percolation threshold, and it shows that estimation can be improved by so-called node splitting invariant versions of the measures. We finally illustrate the framework for a number of archetypical chaotic attractors such as those of the Bernoulli and logistic maps, periodic and two-dimensional quasiperiodic motions, and for hyperballs and hypercubes by deriving analytical expressions for the novel measures and comparing them with data from numerical experiments. More generally, the theoretical framework put forward in this work describes random geometric graphs and other networks with spatial constraints, which appear frequently in disciplines ranging from biology to climate science.

Common and unique network dynamics in football games

The sport of football is played between two teams of eleven players each using a spherical ball. Each team strives to score by driving the ball into the opposing goal as the result of skillful interactions among players. Football can be regarded from the network perspective as a competitive relationship between two cooperative networks with a dynamic network topology and dynamic network node. Many complex large-scale networks have been shown to have topological properties in common, based on a small-world network and scale-free network models. However, the human dynamic movement pattern of this network has never been investigated in a real-world setting. Here, we show that the power law in degree distribution emerged in the passing behavior in the 2006 FIFA World Cup Final and an international “A” match in Japan, by describing players as vertices connected by links representing passes. The exponent values are similar to the typical values that occur in many real-world networks, which are in the range of , and are larger than that of a gene transcription network, . Furthermore, we reveal the stochastically switched dynamics of the hub player throughout the game as a unique feature in football games. It suggests that this feature could result not only in securing vulnerability against intentional attack, but also in a power law for self-organization. Our results suggest common and unique network dynamics of two competitive networks, compared with the large-scale networks that have previously been investigated in numerous works. Our findings may lead to improved resilience and survivability not only in biological networks, but also in communication networks.

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.

Understanding spatial connectivity of individuals with non-uniform population density

We construct a two-dimensional geometric graph connecting individuals placed in space within a given contact distance. The individuals are distributed using a measured country's density of population. We observe that while large clusters (group of individuals connected) emerge within some regions, they are trapped in detached urban areas owing to the low population density of the regions bordering them. To understand the emergence of a giant cluster that connects the entire population, we compare the empirical geometric graph with the one generated by placing the same number of individuals randomly in space. We find that, for small contact distances, the empirical distribution of population dominates the growth of connected components, but no critical percolation transition is observed in contrast to the graph generated by a random distribution of population. Our results show that contact distances from real-world situations as for WIFI and Bluetooth connections drop in a zone where a fully connected cluster is not observed, hinting that human mobility must play a crucial role in contact-based diseases and wireless viruses' large-scale spreading.

Stochastic model for scale-free networks with cutoffs

We propose and analyze a stochastic model which explains, analytically, the cutoff behavior of real scale-free networks previously modeled computationally by Amaral [Proc. Natl. Acad. Sci. U.S.A. 97, 11149 (2000)] and others. We present a mathematical model that can explain several existing computational scale-free network generation models. This yields a theoretical basis to understand cutoff behavior in complex networks, previously treated only with simulations using distinct models. Therefore, ours is an integrative approach that unifies the existing literature on cutoff behavior in scale-free networks. Furthermore, our mathematical model allows us to reach conclusions not hitherto possible with computational models: the ability to predict the equilibrium point of active vertices and to relate the growth of networks with the probability of aging. We also discuss how our model introduces a useful way to classify scale free behavior of complex networks.

WiFi networks and malware epidemiology

In densely populated urban areas WiFi routers form a tightly interconnected proximity network that can be exploited as a substrate for the spreading of malware able to launch massive fraudulent attacks. In this article, we consider several scenarios for the deployment of malware that spreads over the wireless channel of major urban areas in the US. We develop an epidemiological model that takes into consideration prevalent security flaws on these routers. The spread of such a contagion is simulated on real-world data for georeferenced wireless routers. We uncover a major weakness of WiFi networks in that most of the simulated scenarios show tens of thousands of routers infected in as little as 2 weeks, with the majority of the infections occurring in the first 24-48 h. We indicate possible containment and prevention measures and provide computational estimates for the rate of encrypted routers that would stop the spreading of the epidemics by placing the system below the percolation threshold.

Spatially embedded random networks

Many real-world networks analyzed in modern network theory have a natural spatial element; e.g., the Internet, social networks, neural networks, etc. Yet, aside from a comparatively small number of somewhat specialized and domain-specific studies, the spatial element is mostly ignored and, in particular, its relation to network structure disregarded. In this paper we introduce a model framework to analyze the mediation of network structure by spatial embedding; specifically, we model connectivity as dependent on the distance between network nodes. Our spatially embedded random networks construction is not primarily intended as an accurate model of any specific class of real-world networks, but rather to gain intuition for the effects of spatial embedding on network structure; nevertheless we are able to demonstrate, in a quite general setting, some constraints of spatial embedding on connectivity such as the effects of spatial symmetry, conditions for scale free degree distributions and the existence of small-world spatial networks. We also derive some standard structural statistics for spatially embedded networks and illustrate the application of our model framework with concrete examples.

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.

Enhancing robustness and immunization in geographical networks

We find that different geographical structures of networks lead to varied percolation thresholds, although these networks may have similar abstract topological structures. Thus, strategies for enhancing robustness and immunization of a geographical network are proposed. Using the generating function formalism, we obtain an explicit form of the percolation threshold qc for networks containing arbitrary order cycles. For three-cycles, the dependence of qc on the clustering coefficients is ascertained. The analysis substantiates the validity of the strategies with analytical evidence.

Geographical networks evolving with an optimal policy

In this article we propose a growing network model based on an optimal policy involving both topological and geographical measures. In this model, at each time step, a node, having randomly assigned coordinates in a 1x1 square, is added and connected to a previously existing node i, which minimizes the quantity ri2/kialpha, where ri is the geographical distance, ki the degree, and alpha a free parameter. The degree distribution obeys a power-law form when alpha=1, and an exponential form when alpha=0. When alpha is in the interval (0, 1), the network exhibits a stretched exponential distribution. We prove that the average topological distance increases in a logarithmic scale of the network size, indicating the existence of the small-world property. Furthermore, we obtain the geographical edge length distribution, the total geographical length of all edges, and the average geographical distance of the whole network. Interestingly, we found that the total edge length will sharply increase when alpha exceeds the critical value alphac=1, and the average geographical distance has an upper bound independent of the network size. All the results are obtained analytically with some reasonable approximations, which are well verified by simulations.

Ultra-small scale-free geometric networks

We consider a family of long-range percolation models (G p ) p>0 on ℤ d that allow dependence between edges and have the following connectivity properties for p ∈ (1/d, ∞): (i) the degree distribution of vertices in G p has a power-law distribution; (ii) the graph distance between points x and y is bounded by a multiple of log pd log pd | x - y | with probability 1 - o(1); and (iii) an adversary can delete a relatively small number of nodes from G p (ℤ d ∩ [0, n] d ), resulting in two large, disconnected subgraphs.

Geographical effects on the path length and the robustness in complex networks

The short paths between any two nodes and the robustness of connectivity are advanced properties of scale-free (SF) networks; however, they may be affected by geographical constraints in realistic situations. We consider geographical networks with the SF structure based on planar triangulation for online routings, and suggest scaling relations between the average distance or number of hops on the optimal paths and the network size. We also show that the tolerance to random failures and attacks on hubs is weakened in geographical networks, and that even then it is possible for the extremely vulnerable ones to be improved by adding with the local exchange of links.

Geographical effects on cascading breakdowns of scale-free networks

Cascading breakdowns of real networks have resulted in severe accidents in recent years. In this paper, we study the effects of geographical structure on the cascading phenomena of load-carrying scale-free networks. Our essential finding is that when networks are more geographically constrained, i.e., more locally interconnected, they tend to have larger cascading breakdowns. Explanations are provided in terms of the effects of cycles and the distributions of betweenness over degrees.

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 data sets 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 three- and four-node subgraphs, in both directed and nondirected geometric networks. We also analyze geometric networks with arbitrary degree sequences and models with a bias 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.

Self-similar disk packings as model spatial scale-free networks

The network of contacts in space-filling disk packings, such as the Apollonian packing, is examined. These networks provide an interesting example of spatial scale-free networks, where the topology reflects the broad distribution of disk areas. A wide variety of topological and spatial properties of these systems is characterized. Their potential as models for networks of connected minima on energy landscapes is discussed.

Epidemic spreading in a hierarchical social network

A model of epidemic spreading in a population with a hierarchical structure of interpersonal interactions is described and investigated numerically. The structure of interpersonal connections is based on a scale-free network. Spatial localization of individuals belonging to different social groups, and the mobility of a contemporary community, as well as the effectiveness of different interpersonal interactions, are taken into account. Typical relations characterizing the spreading process, like a range of epidemic and epidemic curves, are discussed. The influence of preventive vaccinations on the spreading process is investigated. The critical value of preventively vaccinated individuals that is sufficient for the suppression of an epidemic is calculated. Our results are compared with solutions of the master equation for the spreading process and good agreement of the character of this process is found.

Scale-free network on a vertical plane

A scale-free network is grown in the Euclidean space with a global directional bias. On a vertical plane, nodes are introduced at unit rate at randomly selected points and a node is allowed to be connected only to the subset of nodes which are below it using the attachment probability, pi(i)(t) approximately k(i)(t)l(alpha). Our numerical results indicate that the directed scale-free network for alpha=0 belongs to a different universality class compared to the isotropic scale-free network. For alpha<alpha(c), the degree distribution is stretched exponential in general which takes a pure exponential form in the limit of alpha-->- infinity. The link length distribution is calculated analytically for all values of alpha.