Faculty and Staff

Two classes of scaling behaviours, namely the super-linear scaling of links or activities, and the sub-linear scaling of area, diversity, or time elapsed with respect to size have been found to prevail in the growth of complex networked systems. Despite some pioneering modelling approaches proposed for specific systems, whether there exists some general mechanisms that account for the origins of such scaling behaviours in different contexts, especially in socioeconomic systems, remains an open question. We address this problem by introducing a geometric network model without free parameter, finding that both super-linear and sub-linear scaling behaviours can be simultaneously reproduced and that the scaling exponents are exclusively determined by the dimension of the Euclidean space in which the network is embedded. We implement some realistic extensions to the basic model to offer more accurate predictions for cities of various scaling behaviours and the Zipf distribution reported in the literature and observed in our empirical studies. All of the empirical results can be precisely recovered by our model with analytical predictions of all major properties. By virtue of these general findings concerning scaling behaviour, our models with simple mechanisms gain new insights into the evolution and development of complex networked systems.

The Dawn Framing Camera (FC) has imaged the northern hemisphere of the Asteroid (4) Vesta at high spatial resolution and coverage. This study represents the first investigation of the overall geology of the northern hemisphere (22–90°N, quadrangles Av-1, 2, 3, 4 and 5) using these unique Dawn mission observations. We have compiled a morphologic map and performed crater size–frequency distribution (CSFD) measurements to date the geologic units. The hemisphere is characterized by a heavily cratered surface with a few highly subdued basins up to ∼200 km in diameter. The most widespread unit is a plateau (cratered highland unit), similar to, although of lower elevation than the equatorial Vestalia Terra plateau. Large-scale troughs and ridges have regionally affected the surface. Between ∼180°E and ∼270°E, these tectonic features are well developed and related to the south pole Veneneia impact (Saturnalia Fossae trough unit), elsewhere on the hemisphere they are rare and subdued (Saturnalia Fossae cratered unit). In these pre-Rheasilvia units we observed an unexpectedly high frequency of impact craters up to ∼10 km in diameter, whose formation could in part be related to the Rheasilvia basin-forming event. The Rheasilvia impact has potentially affected the northern hemisphere also with S–N small-scale lineations, but without covering it with an ejecta blanket. Post-Rheasilvia impact craters are small (<60 km in diameter) and show a wide range of degradation states due to impact gardening and mass wasting processes. Where fresh, they display an ejecta blanket, bright rays and slope movements on walls. In places, crater rims have dark material ejecta and some crater floors are covered by ponded material interpreted as impact melt.

Oppia Quadrangle Av-10 (288–360°E, ±22°) is a junction of key geologic features that preserve a rough history of Asteroid (4) Vesta and serves as a case study of using geologic mapping to define a relative geologic timescale. Clear filter images, stereo-derived topography, slope maps, and multispectral color-ratio images from the Framing Camera on NASA’s Dawn spacecraft served as basemaps to create a geologic map and investigate the spatial and temporal relationships of the local stratigraphy. Geologic mapping reveals the oldest map unit within Av-10 is the cratered highlands terrain which possibly represents original crustal material on Vesta that was then excavated by one or more impacts to form the basin Feralia Planitia. Saturnalia Fossae and Divalia Fossae ridge and trough terrains intersect the wall of Feralia Planitia indicating that this impact basin is older than both the Veneneia and Rheasilvia impact structures, representing Pre-Veneneian crustal material. Two of the youngest geologic features in Av-10 are Lepida (∼45 km diameter) and Oppia (∼40 km diameter) impact craters that formed on the northern and southern wall of Feralia Planitia and each cross-cuts a trough terrain. The ejecta blanket of Oppia is mapped as ‘dark mantle’ material because it appears dark orange in the Framing Camera ‘Clementine-type’ color-ratio image and has a diffuse, gradational contact distributed to the south across the rim of Rheasilvia. Mapping of surface material that appears light orange in color in the Framing Camera ‘Clementine-type’ color-ratio image as ‘light mantle material’ supports previous interpretations of an impact ejecta origin. Some light mantle deposits are easily traced to nearby source craters, but other deposits may represent distal ejecta deposits (emplaced >5 crater radii away) in a microgravity environment.

We used Dawn spacecraft data to identify and delineate geological units and landforms in the Marcia quadrangle of Vesta as a means to assess the role of the large, relatively young impact craters Marcia (∼63 km diam.) and Calpurnia (∼53 km diam.) and their surrounding ejecta field on the local geology. We also investigated a local topographic high with a dark-rayed crater named Aricia Tholus, and the impact crater Octavia that is surrounded by a distinctive diffuse mantle. Crater counts and stratigraphic relations suggest that Marcia is the youngest large crater on Vesta, in which a putative impact melt on the crater floor ranges in age between ∼40 and 60 Ma (depending upon choice of chronology system), and Marcia’s ejecta blanket ranges in age between ∼120 and 390 Ma (depending upon choice of chronology system).

We interpret the geologic units in and around Marcia crater to mark a major vestan time-stratigraphic event, and that the Marcia Formation is one of the geologically youngest formations on Vesta. Marcia crater reveals pristine bright and dark material in its walls and smooth and pitted terrains on its floor. The smooth unit we interpret as evidence of flow of impact melts and (for the pitted terrain) release of volatiles during or after the impact process. The distinctive dark ejecta surrounding craters Marcia and Calpurnia is enriched in OH- or H-bearing phases and has a variable morphology, suggestive of a complex mixture of impact ejecta and impact melts including dark materials possibly derived from carbonaceous chondrite-rich material. Aricia Tholus, which was originally interpreted as a putative vestan volcanic edifice based on lower resolution observations, appears to be a fragment of an ancient impact basin rim topped by a dark-rayed impact crater. Octavia crater has a cratering model formation age of ∼280–990 Ma based on counts of its ejecta field (depending upon choice of chronology system), and its ejecta field is the second oldest unit in this quadrangle. The relatively young craters and their related ejecta materials in this quadrangle are in stark contrast to the surrounding heavily cratered units that are related to the billion years old or older Rheasilvia and Veneneia impact basins and Vesta’s ancient crust preserved on Vestalia Terra.

We studied high-resolution images of asteroid Vesta's surface (~70 and 20–25 m/pixel) obtained during the High- and Low-Altitude Mapping Orbits (HAMO, LAMO) of NASA's Dawn mission to assess the formation mechanisms responsible for a variety of lobate, flow-like features observed across the surface. We searched for evidence of volcanic flows, based on prior mathematical modeling and the well-known basaltic nature of Vesta's crust, but no unequivocal morphologic evidence of ancient volcanic activity has thus far been identified. Rather, we find that all lobate, flow-like features on Vesta appear to be related either to impact or erosional processes. Morphologically distinct lobate features occur in and around impact craters, and most of these are interpreted as impact ejecta flows, or possibly flows of impact melt. Estimates of melt production from numerical models and scaling laws suggests that large craters like Marcia (~60 km diameter) could have potentially produced impact melt volumes ranging from tens of millions of cubic meters to a few tens of cubic kilometers, which are relatively small volumes compared to similar-sized lunar craters, but which are consistent with putative impact melt features observed in Dawn images. There are also examples of lobate flows that trend downhill both inside and outside of crater rims and basin scarps, which are interpreted as the result of gravity-driven mass movements (slumps and landslides).

Given a complex geospatial network with nodes distributed in a two-dimensional region of physical space, can the locations of the nodes be determined and their connection patterns be uncovered based solely on data? We consider the realistic situation where time series/signals can be collected from a single location. A key challenge is that the signals collected are necessarily time delayed, due to the varying physical distances from the nodes to the data collection centre. To meet this challenge, we develop a compressive-sensing-based approach enabling reconstruction of the full topology of the underlying geospatial network and more importantly, accurate estimate of the time delays. A standard triangularization algorithm can then be employed to find the physical locations of the nodes in the network. We further demonstrate successful detection of a hidden node (or a hidden source or threat), from which no signal can be obtained, through accurate detection of all its neighbouring nodes. As a geospatial network has the feature that a node tends to connect with geophysically nearby nodes, the localized region that contains the hidden node can be identified.

Recent works revealed that the energy required to control a complex network depends on the number of driving signals and the energy distribution follows an algebraic scaling law. If one implements control using a small number of drivers, e.g. as determined by the structural controllability theory, there is a high probability that the energy will diverge. We develop a physical theory to explain the scaling behaviour through identification of the fundamental structural elements, the longest control chains (LCCs), that dominate the control energy. Based on the LCCs, we articulate a strategy to drastically reduce the control energy (e.g. in a large number of real-world networks). Owing to their structural nature, the LCCs may shed light on energy issues associated with control of nonlinear dynamical networks.

A challenging problem in network science is to control complex networks. In existing frameworks of structural or exact controllability, the ability to steer a complex network toward any desired state is measured by the minimum number of required driver nodes. However, if we implement actual control by imposing input signals on the minimum set of driver nodes, an unexpected phenomenon arises: due to computational or experimental error there is a great probability that convergence to the final state cannot be achieved. In fact, the associated control cost can become unbearably large, effectively preventing actual control from being realized physically. The difficulty is particularly severe when the network is deemed controllable with a small number of drivers. Here we develop a physical controllability framework based on the probability of achieving actual control. Using a recently identified fundamental chain structure underlying the control energy, we offer strategies to turn physically uncontrollable networks into physically controllable ones by imposing slightly augmented set of input signals on properly chosen nodes. Our findings indicate that, although full control can be theoretically guaranteed by the prevailing structural controllability theory, it is necessary to balance the number of driver nodes and control cost to achieve physical control.

Network reconstruction is a fundamental problem for understanding many complex systems with unknown interaction structures. In many complex systems, there are indirect interactions between two individuals without immediate connection but with common neighbors. Despite recent advances in network reconstruction, we continue to lack an approach for reconstructing complex networks with indirect interactions. Here we introduce a two-step strategy to resolve the reconstruction problem, where in the first step, we recover both direct and indirect interactions by employing the Lasso to solve a sparse signal reconstruction problem, and in the second step, we use matrix transformation and optimization to distinguish between direct and indirect interactions. The network structure corresponding to direct interactions can be fully uncovered. We exploit the public goods game occurring on complex networks as a paradigm for characterizing indirect interactions and test our reconstruction approach. We find that high reconstruction accuracy can be achieved for both homogeneous and heterogeneous networks, and a number of empirical networks in spite of insufficient data measurement contaminated by noise. Although a general framework for reconstructing complex networks with arbitrary types of indirect interactions is yet lacking, our approach opens new routes to separate direct and indirect interactions in a representative complex system.

Controlling complex networks has become a forefront research area in network science and engineering. Recent efforts have led to theoretical frameworks of controllability to fully control a network through steering a minimum set of driver nodes. However, in realistic situations not every node is accessible or can be externally driven, raising the fundamental issue of control efficacy: if driving signals are applied to an arbitrary subset of nodes, how many other nodes can be controlled? We develop a framework to determine the control efficacy for undirected networks of arbitrary topology. Mathematically, based on non-singular transformation, we prove a theorem to determine rigorously the control efficacy of the network and to identify the nodes that can be controlled for any given driver nodes. Physically, we develop the picture of diffusion that views the control process as a signal diffused from input signals to the set of controllable nodes. The combination of mathematical theory and physical reasoning allows us not only to determine the control efficacy for model complex networks and a large number of empirical networks, but also to uncover phenomena in network control, e.g., hub nodes in general possess lower control centrality than an average node in undirected networks.