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Register a new userRandom Spatial Networks: Small Worlds without Clustering, Traveling Waves, and Hop-and-Spread Disease Dynamics
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Random network models play a prominent role in modeling, analyzing and understanding complex phenomena on real-life networks. However, a key property of networks is often neglected: many real-world networks exhibit spatial structure, the tendency of a node to select neighbors with a probability depending on physical distance. Here, we introduce a class of random spatial networks (RSNs) which generalizes many existing random network models but adds spatial structure. In these networks, nodes are placed randomly in space and joined in edges with a probability depending on their distance and their individual expected degrees, in a manner that crucially remains analytically tractable. We use this network class to propose a new generalization of small-world networks, where the average shortest path lengths in the graph are small, as in classical Watts-Strogatz small-world networks, but with close spatial proximity of nodes that are neighbors in the network playing the role of large clustering. Small-world effects are demonstrated on these spatial small-world networks without clustering. We are able to derive partial integro-differential equations governing susceptible-infectious-recovered disease spreading through an RSN, and we demonstrate the existence of traveling wave solutions. If the distance kernel governing edge placement decays slower than exponential, the population-scale dynamics are dominated by long-range hops followed by local spread of traveling waves. This provides a theoretical modeling framework for recent observations of how epidemics like Ebola evolve in modern connected societies, with long-range connections seeding new focal points from which the epidemic locally spreads in a wavelike manner.
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Networks with underlying metric spaces attract increasing research attention in network science, statistical physics, applied mathematics, computer science, sociology, and other fields. This attention is further amplified by the current surge of activity in graph embedding. In the vast realm of spatial network models, only a few reproduce even the most basic properties of real-world networks. Here, we focus on three such properties---sparsity, small worldness, and clustering---and identify the general subclass of spatial homogeneous and heterogeneous network models that are sparse small worlds and that have nonzero clustering in the thermodynamic limit. We rely on the maximum entropy approach where network links correspond to noninteracting fermions whose energy dependence on spatial distances determines network small worldness and clustering.
Small-worlds represent efficient communication networks that obey two distinguishing characteristics: a high clustering coefficient together with a small characteristic path length. This paper focuses on an interesting paradox, that removing links in a network can increase the overall clustering coefficient. Reckful Roaming, as introduced in this paper, is a 2-localized algorithm that takes advantage of this paradox in order to selectively remove superfluous links, this way optimizing the clustering coefficient while still retaining a sufficiently small characteristic path length.
We establish a relationship between the Small-World behavior found in complex networks and a family of Random Walks trajectories using, as a linking bridge, a maze iconography. Simple methods to generate mazes using Random Walks are discussed along with related issues and it is explained how to interpret mazes as graphs and loops as shortcuts. Small-World behavior was found to be non-logarithmic but power-law in this model, we discuss the reason for this peculiar scaling
Many diseases display heterogeneity in clinical features and their progression, indicative of the existence of disease subtypes. Extracting patterns of disease variable progression for subtypes has tremendous application in medicine, for example, in early prognosis and personalized medical therapy. This work present a novel, data-driven, network-based Trajectory Clustering (TC) algorithm for identifying Parkinsons subtypes based on disease trajectory. Modeling patient-variable interactions as a bipartite network, TC first extracts communities of co-expressing disease variables at different stages of progression. Then, it identifies Parkinsons subtypes by clustering similar patient trajectories that are characterized by severity of disease variables through a multi-layer network. Determination of trajectory similarity accounts for direct overlaps between trajectories as well as second-order similarities, i.e., common overlap with a third set of trajectories. This work clusters trajectories across two types of layers: (a) temporal, and (b) ranges of independent outcome variable (representative of disease severity), both of which yield four distinct subtypes. The former subtypes exhibit differences in progression of disease domains (Cognitive, Mental Health etc.), whereas the latter subtypes exhibit different degrees of progression, i.e., some remain mild, whereas others show significant deterioration after 5 years. The TC approach is validated through statistical analyses and consistency of the identified subtypes with medical literature. This generalizable and robust method can easily be extended to other progressive multi-variate disease datasets, and can effectively assist in targeted subtype-specific treatment in the field of personalized medicine.
Parkinsons disease (PD) is a common neurodegenerative disease with a high degree of heterogeneity in its clinical features, rate of progression, and change of variables over time. In this work, we present a novel data-driven, network-based Trajectory Profile Clustering (TPC) algorithm for 1) identification of PD subtypes and 2) early prediction of disease progression in individual patients. Our subtype identification is based not only on PD variables, but also on their complex patterns of progression, providing a useful tool for the analysis of large heterogenous, longitudinal data. Specifically, we cluster patients based on the similarity of their trajectories through a time series of bipartite networks connecting patients to demographic, clinical, and genetic variables. We apply this approach to demographic and clinical data from the Parkinsons Progression Markers Initiative (PPMI) dataset and identify 3 patient clusters, consistent with 3 distinct PD subtypes, each with a characteristic variable progression profile. Additionally, TPC predicts an individual patients subtype and future disease trajectory, based on baseline assessments. Application of our approach resulted in 74% accurate subtype prediction in year 5 in a test/validation cohort. Furthermore, we show that genetic variability can be integrated seamlessly in our TPC approach. In summary, using PD as a model for chronic progressive diseases, we show that TPC leverages high-dimensional longitudinal datasets for subtype identification and early prediction of individual disease subtype. We anticipate this approach will be broadly applicable to multidimensional longitudinal datasets in diverse chronic diseases.
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