Evolving networks with disadvantaged long-range connections

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.

A study of cascading failures in real and synthetic power grid topologies

Using the direct current power flow model, we study cascading failures and their spatial and temporal properties in the U.S. Western Interconnection (USWI) power grid. We show that yield (the fraction of demand satisfied after the cascade) has a bimodal distribution typical of a first-order transition. The single line failure leads either to an insignificant power loss or to a cascade which causes a major blackout with yield less than 0.8. The former occurs with high probability if line tolerance α (the ratio of the maximal load a line can carry to its initial load) is greater than 2, while a major blackout occurs with high probability in a broad range of 1 < α < 2. We also show that major blackouts begin with a latent period (with duration proportional to α) during which few lines overload and yield remains high. The existence of the latent period suggests that intervention during early stages of a cascade can significantly reduce the risk of a major blackout. Finally, we introduce the preferential Degree And Distance Attachment model to generate random networks with similar degree, resistance, and flow distributions to the USWI. Moreover, we show that the Degree And Distance Attachment model behaves similarly to the USWI against failures.

A distribution model of functional connectome based on criticality and energy constraints

The analysis of the network structure of the functional connectivity data constructed from fMRI images provides basic information about functions and features of the brain activity. We focus on the two features which are considered as relevant to the brain activity, the criticality and the constraint regarding energy consumptions. Within a wide variety of complex systems, the critical state occurs associated with a phase transition between distinct phases, random one and order one. Although the hypothesis that human brain activity is also in a state of criticality is supported by some experimental results, it still remains controversial. One issue is that experimental distributions exhibit deviations from the power law predicted by the criticality. Based on the assumption that constraints on brain from the biological costs cause these deviations, we derive a distribution model. The evaluation using the information criteria indicates an advantage of this model in fitting to experimental data compared to other representative distribution models, the truncated power law and the power law. Our findings also suggest that the mechanism underlying this model is closely related to the cost effective behavior in human brain with maximizing the network efficiency for the given network cost.

Structural power and the evolution of collective fairness in social networks

From work contracts and group buying platforms to political coalitions and international climate and economical summits, often individuals assemble in groups that must collectively reach decisions that may favor each part unequally. Here we quantify to which extent our network ties promote the evolution of collective fairness in group interactions, modeled by means of Multiplayer Ultimatum Games (MUG). We show that a single topological feature of social networks—which we call structural power—has a profound impact on the tendency of individuals to take decisions that favor each part equally. Increased fair outcomes are attained whenever structural power is high, such that the networks that tie individuals allow them to meet the same partners in different groups, thus providing the opportunity to strongly influence each other. On the other hand, the absence of such close peer-influence relationships dismisses any positive effect created by the network. Interestingly, we show that increasing the structural power of a network leads to the appearance of well-defined modules—as found in human social networks that often exhibit community structure—providing an interaction environment that maximizes collective fairness.

Structural and functional properties of spatially embedded scale-free networks

Scale-free networks have been studied mostly as non-spatially embedded systems. However, in many realistic cases, they are spatially embedded and these constraints should be considered. Here, we study the structural and functional properties of a model of scale-free (SF) spatially embedded networks. In our model, both the degree and the length of links follow power law distributions as found in many real networks. We show that not all SF networks can be embedded in space and that the largest degree of a node in the network is usually smaller than in nonembedded SF networks. Moreover, the spatial constraints (each node has only few neighboring nodes) introduce degree-degree anticorrelations (disassortativity) since two high degree nodes cannot stay close in space. We also find significant effects of space embedding on the hopping distances (chemical distance) and the vulnerability of the networks.

Percolation of spatially constrained Erdős-Rényi networks with degree correlations

Motivated by experiments on activity in neuronal cultures [ J. Soriano, M. Rodríguez Martínez, T. Tlusty and E. Moses Proc. Natl. Acad. Sci. 105 13758 (2008)], we investigate the percolation transition and critical exponents of spatially embedded Erdős-Rényi networks with degree correlations. In our model networks, nodes are randomly distributed in a two-dimensional spatial domain, and the connection probability depends on Euclidian link length by a power law as well as on the degrees of linked nodes. Generally, spatial constraints lead to higher percolation thresholds in the sense that more links are needed to achieve global connectivity. However, degree correlations favor or do not favor percolation depending on the connectivity rules. We employ two construction methods to introduce degree correlations. In the first one, nodes stay homogeneously distributed and are connected via a distance- and degree-dependent probability. We observe that assortativity in the resulting network leads to a decrease of the percolation threshold. In the second construction methods, nodes are first spatially segregated depending on their degree and afterwards connected with a distance-dependent probability. In this segregated model, we find a threshold increase that accompanies the rising assortativity. Additionally, when the network is constructed in a disassortative way, we observe that this property has little effect on the percolation transition.

A latent parameter node-centric model for spatial networks

Spatial networks, in which nodes and edges are embedded in space, play a vital role in the study of complex systems. For example, many social networks attach geo-location information to each user, allowing the study of not only topological interactions between users, but spatial interactions as well. The defining property of spatial networks is that edge distances are associated with a cost, which may subtly influence the topology of the network. However, the cost function over distance is rarely known, thus developing a model of connections in spatial networks is a difficult task. In this paper, we introduce a novel model for capturing the interaction between spatial effects and network structure. Our approach represents a unique combination of ideas from latent variable statistical models and spatial network modeling. In contrast to previous work, we view the ability to form long/short-distance connections to be dependent on the individual nodes involved. For example, a node's specific surroundings (e.g. network structure and node density) may make it more likely to form a long distance link than other nodes with the same degree. To capture this information, we attach a latent variable to each node which represents a node's spatial reach. These variables are inferred from the network structure using a Markov Chain Monte Carlo algorithm. We experimentally evaluate our proposed model on 4 different types of real-world spatial networks (e.g. transportation, biological, infrastructure, and social). We apply our model to the task of link prediction and achieve up to a 35% improvement over previous approaches in terms of the area under the ROC curve. Additionally, we show that our model is particularly helpful for predicting links between nodes with low degrees. In these cases, we see much larger improvements over previous models.

Graph theoretical analysis of resting magnetoencephalographic functional connectivity networks

Complex networks have been observed to comprise small-world properties, believed to represent an optimal organization of local specialization and global integration of information processing at reduced wiring cost. Here, we applied magnitude squared coherence to resting magnetoencephalographic time series in reconstructed source space, acquired from controls and patients with schizophrenia, and generated frequency-dependent adjacency matrices modeling functional connectivity between virtual channels. After configuring undirected binary and weighted graphs, we found that all human networks demonstrated highly localized clustering and short characteristic path lengths. The most conservatively thresholded networks showed efficient wiring, with topographical distance between connected vertices amounting to one-third as observed in surrogate randomized topologies. Nodal degrees of the human networks conformed to a heavy-tailed exponentially truncated power-law, compatible with the existence of hubs, which included theta and alpha bilateral cerebellar tonsil, beta and gamma bilateral posterior cingulate, and bilateral thalamus across all frequencies. We conclude that all networks showed small-worldness, minimal physical connection distance, and skewed degree distributions characteristic of physically-embedded networks, and that these calculations derived from graph theoretical mathematics did not quantifiably distinguish between subject populations, independent of bandwidth. However, post-hoc measurements of edge computations at the scale of the individual vertex revealed trends of reduced gamma connectivity across the posterior medial parietal cortex in patients, an observation consistent with our prior resting activation study that found significant reduction of synthetic aperture magnetometry gamma power across similar regions. The basis of these small differences remains unclear.

Features and heterogeneities in growing network models

Many complex networks from the World Wide Web to biological networks grow taking into account the heterogeneous features of the nodes. The feature of a node might be a discrete quantity such as a classification of a URL document such as personal page, thematic website, news, blog, search engine, social network, etc., or the classification of a gene in a functional module. Moreover the feature of a node can be a continuous variable such as the position of a node in the embedding space. In order to account for these properties, in this paper we provide a generalization of growing network models with preferential attachment that includes the effect of heterogeneous features of the nodes. The main effect of heterogeneity is the emergence of an "effective fitness" for each class of nodes, determining the rate at which nodes acquire new links. The degree distribution exhibits a multiscaling behavior analogous to the the fitness model. This property is robust with respect to variations in the model, as long as links are assigned through effective preferential attachment. Beyond the degree distribution, in this paper we give a full characterization of the other relevant properties of the model. We evaluate the clustering coefficient and show that it disappears for large network size, a property shared with the Barabási-Albert model. Negative degree correlations are also present in this class of models, along with nontrivial mixing patterns among features. We therefore conclude that both small clustering coefficients and disassortative mixing are outcomes of the preferential attachment mechanism in general growing networks.

Preferential attachment in growing spatial networks

We obtain the degree distribution for a class of growing network models on flat and curved spaces. These models evolve by preferential attachment weighted by a function of the distance between nodes. The degree distribution of these models is similar to that of the fitness model of Bianconi and Barabási, with a fitness distribution dependent on the metric and the density of nodes. We show that curvature singularities in these spaces can give rise to asymptotic Bose-Einstein condensation, but transient condensation can be observed also in smooth hyperbolic spaces with strong curvature. We provide numerical results for spaces of constant curvature (sphere, flat, and hyperbolic space) and we discuss the conditions for the breakdown of this approach and the critical points of the transition to distance-dominated attachment. Finally, we discuss the distribution of link lengths.

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.

Geographical networks stochastically constructed by a self-similar tiling according to population

In real communication and transportation networks, the geographical positions of nodes are very important for the efficiency and the tolerance of connectivity. Considering spatially inhomogeneous positions of nodes according to a population, we introduce a multiscale quartered (MSQ) network that is stochastically constructed by recursive subdivision of polygonal faces as a self-similar tiling. It has several advantages: the robustness of connectivity, the bounded short path lengths, and the shortest distance routing algorithm in a distributive manner. Furthermore, we show that the MSQ network is more efficient with shorter link lengths and more suitable with lower load for avoiding traffic congestion than other geographical networks which have various topologies ranging from river to scale-free networks. These results will be useful for providing an insight into the future design of ad hoc network infrastructures.

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.

Small-world brain networks

Many complex networks have a small-world topology characterized by dense local clustering or cliquishness of connections between neighboring nodes yet a short path length between any (distant) pair of nodes due to the existence of relatively few long-range connections. This is an attractive model for the organization of brain anatomical and functional networks because a small-world topology can support both segregated/specialized and distributed/integrated information processing. Moreover, small-world networks are economical, tending to minimize wiring costs while supporting high dynamical complexity. The authors introduce some of the key mathematical concepts in graph theory required for small-world analysis and review how these methods have been applied to quantification of cortical connectivity matrices derived from anatomical tract-tracing studies in the macaque monkey and the cat. The evolution of small-world networks is discussed in terms of a selection pressure to deliver cost-effective information-processing systems. The authors illustrate how these techniques and concepts are increasingly being applied to the analysis of human brain functional networks derived from electroencephalography/magnetoencephalography and fMRI experiments. Finally, the authors consider the relevance of small-world models for understanding the emergence of complex behaviors and the resilience of brain systems to pathological attack by disease or aberrant development. They conclude that small-world models provide a powerful and versatile approach to understanding the structure and function of human brain systems.

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.

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

Many real networks are equipped with short diameters, high clustering, and power-law degree distributions. With preferential attachment and network growth, the model by Barabási and Albert simultaneously reproduces these properties, and geographical versions of growing networks have also been analyzed. However, nongrowing networks with intrinsic vertex weights often explain these features more plausibly, since not all networks are really growing. We propose a geographical nongrowing network model with vertex weights. Edges are assumed to form when a pair of vertices are spatially close and/or have large summed weights. Our model generalizes a variety of models as well as the original nongeographical counterpart, such as the unit disk graph, the Boolean model, and the gravity model, which appear in the contexts of percolation, wire communication, mechanical and solid physics, sociology, economy, and marketing. In appropriate configurations, our model produces small-world networks with power-law degree distributions. We also discuss the relation between geography, power laws in networks, and power laws in general quantities serving as vertex weights.

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.

Lattice scale-free networks with weighted linking

Recently, models of the scale-free (SF) networks on lattices were investigated, which consider the influence of the embedded space on the networks. Since a lot of real networks exist on the 2D global surface, it is helpful to discuss these models. In this paper, based on the lattice SF networks model, a linking weight is added, thus an additional parameter which can control the clustering coefficient is introduced. Depending on the linking weight, the properties of the model change smoothly from the lattice SF model to the SF random graphs.

Urban economy as a scale-free network

We present empirical evidence that land values are scale free and introduce a network model that reproduces the observations. The network approach to urban modeling is based on the assumption that the market dynamics that generates land values can be represented as a growing scale-free network. Our results suggest that the network properties of trade between specialized activities cause land values, and likely also other observables such as population, to be power-law distributed. In addition to being an attractive avenue for further analytical inquiry, the network representation is also applicable to empirical data and is thereby attractive for predictive modeling.

Clustering properties of a generalized critical Euclidean network

Many real-world networks exhibit a scale-free feature, have a small diameter, and a high clustering tendency. We study the properties of a growing network, which has all these features, in which an incoming node is connected to its ith predecessor of degree k(i) with a link of length l using a probability proportional to k(beta)(i)l(alpha). For alpha>-0.5, the network is scale-free at beta=1 with the degree distribution P(k) proportional to k(-gamma) and gamma=3.0 as in the Barabási-Albert model (alpha=0,beta=1). We find a phase boundary in the alpha-beta plane along which the network is scale-free. Interestingly, we find a scale-free behavior even for beta>1 for alpha<-0.5, where the existence of a different universality class is indicated from the behavior of the degree distribution and the clustering coefficients. The network has a small diameter in the entire scale-free region. The clustering coefficients emulate the behavior of most real networks for increasing negative values of alpha on the phase boundary.