Estimating topological properties of weighted networks from limited information

A problem typically encountered when studying complex systems is the limitedness of the information available on their topology, which hinders our understanding of their structure and of the dynamical processes taking place on them. A paramount example is provided by financial networks, whose data are privacy protected: Banks publicly disclose only their aggregate exposure towards other banks, keeping individual exposures towards each single bank secret. Yet, the estimation of systemic risk strongly depends on the detailed structure of the interbank network. The resulting challenge is that of using aggregate information to statistically reconstruct a network and correctly predict its higher-order properties. Standard approaches either generate unrealistically dense networks, or fail to reproduce the observed topology by assigning homogeneous link weights. Here, we develop a reconstruction method, based on statistical mechanics concepts, that makes use of the empirical link density in a highly nontrivial way. Technically, our approach consists in the preliminary estimation of node degrees from empirical node strengths and link density, followed by a maximum-entropy inference based on a combination of empirical strengths and estimated degrees. Our method is successfully tested on the international trade network and the interbank money market, and represents a valuable tool for gaining insights on privacy-protected or partially accessible systems

Resolving structural variability in network models and the brain

Large-scale white matter pathways crisscrossing the cortex create a complex pattern of connectivity that underlies human cognitive function. Generative mechanisms for this architecture have been difficult to identify in part because little is known about mechanistic drivers of structured networks. Here we contrast network properties derived from diffusion spectrum imaging data of the human brain with 13 synthetic network models chosen to probe the roles of physical network embedding and temporal network growth. We characterize both the empirical and synthetic networks using familiar diagnostics presented in statistical form, as scatter plots and distributions, to reveal the full range of variability of each measure across scales in the network. We focus on the degree distribution, degree assortativity, hierarchy, topological Rentian scaling, and topological fractal scaling---in addition to several summary statistics, including the mean clustering coefficient, shortest path length, and network diameter. The models are investigated in a progressive, branching sequence, aimed at capturing different elements thought to be important in the brain, and range from simple random and regular networks, to models that incorporate specific growth rules and constraints. We find that synthetic models that constrain the network nodes to be embedded in anatomical brain regions tend to produce distributions that are similar to those extracted from the brain. We also find that network models hardcoded to display one network property do not in general also display a second, suggesting that multiple neurobiological mechanisms might be at play in the development of human brain network architecture. Together, the network models that we develop and employ provide a potentially useful starting point for the statistical inference of brain network structure from neuroimaging data.Comment: 24 pages, 11 figures, 1 table, supplementary material

Measuring the dimension of partially embedded networks

Scaling phenomena have been intensively studied during the past decade in the context of complex networks. As part of these works, recently novel methods have appeared to measure the dimension of abstract and spatially embedded networks. In this paper we propose a new dimension measurement method for networks, which does not require global knowledge on the embedding of the nodes, instead it exploits link-wise information (link lengths, link delays or other physical quantities). Our method can be regarded as a generalization of the spectral dimension, that grasps the network's large-scale structure through local observations made by a random walker while traversing the links. We apply the presented method to synthetic and real-world networks, including road maps, the Internet infrastructure and the Gowalla geosocial network. We analyze the theoretically and empirically designated case when the length distribution of the links has the form P(r) ~ 1/r. We show that while previous dimension concepts are not applicable in this case, the new dimension measure still exhibits scaling with two distinct scaling regimes. Our observations suggest that the link length distribution is not sufficient in itself to entirely control the dimensionality of complex networks, and we show that the proposed measure provides information that complements other known measures

Locating influential nodes via dynamics-sensitive centrality

With great theoretical and practical significance, locating influential nodes of complex networks is a promising issues. In this paper, we propose a dynamics-sensitive (DS) centrality that integrates topological features and dynamical properties. The DS centrality can be directly applied in locating influential spreaders. According to the empirical results on four real networks for both susceptible-infected-recovered (SIR) and susceptible-infected (SI) spreading models, the DS centrality is much more accurate than degree, $k$-shell index and eigenvector centrality.Comment: 6 pages, 1 table and 2 figure

Cascading Power Outages Propagate Locally in an Influence Graph that is not the Actual Grid Topology

In a cascading power transmission outage, component outages propagate non-locally, after one component outages, the next failure may be very distant, both topologically and geographically. As a result, simple models of topological contagion do not accurately represent the propagation of cascades in power systems. However, cascading power outages do follow patterns, some of which are useful in understanding and reducing blackout risk. This paper describes a method by which the data from many cascading failure simulations can be transformed into a graph-based model of influences that provides actionable information about the many ways that cascades propagate in a particular system. The resulting "influence graph" model is Markovian, in that component outage probabilities depend only on the outages that occurred in the prior generation. To validate the model we compare the distribution of cascade sizes resulting from $n-2$ contingencies in a $2896$ branch test case to cascade sizes in the influence graph. The two distributions are remarkably similar. In addition, we derive an equation with which one can quickly identify modifications to the proposed system that will substantially reduce cascade propagation. With this equation one can quickly identify critical components that can be improved to substantially reduce the risk of large cascading blackouts.Comment: Accepted for publication at the IEEE Transactions on Power System

Null Models of Economic Networks: The Case of the World Trade Web

In all empirical-network studies, the observed properties of economic networks are informative only if compared with a well-defined null model that can quantitatively predict the behavior of such properties in constrained graphs. However, predictions of the available null-model methods can be derived analytically only under assumptions (e.g., sparseness of the network) that are unrealistic for most economic networks like the World Trade Web (WTW). In this paper we study the evolution of the WTW using a recently-proposed family of null network models. The method allows to analytically obtain the expected value of any network statistic across the ensemble of networks that preserve on average some local properties, and are otherwise fully random. We compare expected and observed properties of the WTW in the period 1950-2000, when either the expected number of trade partners or total country trade is kept fixed and equal to observed quantities. We show that, in the binary WTW, node-degree sequences are sufficient to explain higher-order network properties such as disassortativity and clustering-degree correlation, especially in the last part of the sample. Conversely, in the weighted WTW, the observed sequence of total country imports and exports are not sufficient to predict higher-order patterns of the WTW. We discuss some important implications of these findings for international-trade models.Comment: 39 pages, 46 figures, 2 table

Using Incomplete Information for Complete Weight Annotation of Road Networks -- Extended Version

We are witnessing increasing interests in the effective use of road networks. For example, to enable effective vehicle routing, weighted-graph models of transportation networks are used, where the weight of an edge captures some cost associated with traversing the edge, e.g., greenhouse gas (GHG) emissions or travel time. It is a precondition to using a graph model for routing that all edges have weights. Weights that capture travel times and GHG emissions can be extracted from GPS trajectory data collected from the network. However, GPS trajectory data typically lack the coverage needed to assign weights to all edges. This paper formulates and addresses the problem of annotating all edges in a road network with travel cost based weights from a set of trips in the network that cover only a small fraction of the edges, each with an associated ground-truth travel cost. A general framework is proposed to solve the problem. Specifically, the problem is modeled as a regression problem and solved by minimizing a judiciously designed objective function that takes into account the topology of the road network. In particular, the use of weighted PageRank values of edges is explored for assigning appropriate weights to all edges, and the property of directional adjacency of edges is also taken into account to assign weights. Empirical studies with weights capturing travel time and GHG emissions on two road networks (Skagen, Denmark, and North Jutland, Denmark) offer insight into the design properties of the proposed techniques and offer evidence that the techniques are effective.Comment: This is an extended version of "Using Incomplete Information for Complete Weight Annotation of Road Networks," which is accepted for publication in IEEE TKD