Physics

We present exact solutions for the size of the giant connected component (GCC) of graphs composed of higher-order homogeneous cycles, including weak cycles and cliques, following bond percolation. We use our theoretical result to find the location of the percolation threshold of the model, providing analytical solutions where possible. We expect the results derived here to be useful to a wide variety of applications including graph theory, epidemiology, percolation and lattice gas models as well as fragmentation theory. We also examine the Erd?s-Gallai theorem as a necessary condition on the graphicality of configuration model networks comprising higher-order clique sub-graphs.

Random walks have been proven to be useful for constructing various algorithms to gain information on networks. Algorithm node2vec employs biased random walks to realize embeddings of nodes into low-dimensional spaces, which can then be used for tasks such as multi-label classification and link prediction. The usefulness of node2vec in these applications is considered to be contingent upon properties of random walks that the node2vec algorithm uses. In the present study, we theoretically and numerically analyze random walks used by the node2vec. The node2vec random walk is a second-order Markov chain. We exploit the mapping of its transition rule to a transition probability matrix among directed edges to analyze the stationary probability, relaxation times, and coalescence time. In particular, we provide a multitude of evidence that node2vec random walk accelerates diffusion when its parameters are tuned such that walkers avoid both back-tracking and visiting a neighbor of the previously visited node, but not excessively.

We present analytical results for the structural evolution of random networks undergoing contraction processes via generic node deletion scenarios, namely, random deletion, preferential deletion and propagating deletion. Focusing on configuration model networks, which exhibit a given degree distribution P 0 (k) and no correlations, we show using a rigorous argument that upon contraction the degree distributions of these networks converge towards a Poisson distribution. To this end, we use the relative entropy S t =S[ P t (k)||π(k|⟨K ⟩ t )] of the degree distribution P t (k) of the contracting network at time t with respect to the corresponding Poisson distribution π(k|⟨K ⟩ t ) with the same mean degree ⟨K ⟩ t as a distance measure between P t (k) and Poisson. The relative entropy is suitable as a distance measure since it satisfies S t ≥0 for any degree distribution P t (k) , while equality is obtained only for P t (k)=π(k|⟨K ⟩ t ) . We derive an equation for the time derivative d S t /dt during network contraction and show that the relative entropy decreases monotonically to zero during the contraction process. We thus conclude that the degree distributions of contracting configuration model networks converge towards a Poisson distribution. Since the contracting networks remain uncorrelated, this means that their structures converge towards an Erd{\H o}s-Rényi (ER) graph structure, substantiating earlier results obtained using direct integration of the master equation and computer simulations [I. Tishby, O. Biham and E. Katzav, {\it Phys. Rev. E} {\bf 100}, 032314 (2019)]. We demonstrate the convergence for configuration model networks with degenerate degree distributions (random regular graphs), exponential degree distributions and power-law degree distributions (scale-free networks).

We formulate a generalized susceptible exposed infectious recovered (SEIR) model on a graph, describing the population dynamics of an open crowded place with an arbitrary topology. As a sample calculation, we discuss three simple cases, both analytically, and numerically, by means of a cellular automata simulation of the individual dynamics in the system. As a result, we provide the infection ratio in the system as a function of controllable parameters, which allows for quantifying how acting on the human behavior may effectively lower the disease spread throughout the system.

We design an optimal group boarding method using a stochastic cellular automata model for passenger movements, which is extended by a virus transmission approach. Furthermore, a new mathematical model is developed to determine an appropriate seat layout for groups. The proposed seating layout is based on the idea that group members are allowed to have close contact and that groups should have a distance among each other. The sum of individual transmission rates is taken as the objective function to derive scenarios with a low level transmission risk. After the determination of an appropriate seat layout, the cellular automata is used to derive and evaluate a corresponding boarding sequence aiming at both short boarding times and low risk of virus transmission. We find that the consideration of groups in a pandemic scenario will significantly contribute to a faster boarding (reduction of time by about 60%) and less transmission risk (reduced by 85%), which reaches the level of boarding times in pre-pandemic scenarios.

With rising shares of renewables and the need to properly assess trade-offs between transmission, storage and sectoral integration as balancing options, building a bridge between energy system models and detailed power flow studies becomes increasingly important, but is computationally challenging. W compare approximations for two nonlinear phenomena, power flow and transmission losses, in linear capacity expansion problems that co-optimise investments in generation, storage and transmission infrastructure. We evaluate different flow representations discussing differences in investment decisions, nodal prices, the deviation of optimised flows and losses from simulated AC power flows, and the computational performance. By using the open European power system model PyPSA-Eur we obtain detailed and reproducible results aiming at facilitating the selection of a suitable power flow model. Given the differences in complexity, the optimal choice depends on the application, the user's available computational resources, and the level of spatial detail considered. Although the commonly used transport model can already identify key features of a cost-efficient system while being computationally performant, deficiencies under high loading conditions arise due to the lack of a physical grid representation. Moreover, disregarding transmission losses overestimates optimal grid expansion by 20%. Adding a convex relaxation of quadratic losses with two or three tangents to the linearised power flow equations and accounting for changing line impedances as the network is reinforced suffices to represent power flows and losses adequately in design studies. We show that the obtained investment and dispatch decisions are then sufficiently physical to be used in more detailed nonlinear simulations of AC power flow in order to better assess their technical feasibility.

The dense social contact networks and high mobility in congested urban areas facilitate the rapid transmission of infectious diseases. Typical mechanistic epidemiological models are either based on uniform mixing with ad-hoc contact processes or need real-time or archived population mobility data to simulate the social networks. However, the rapid and global transmission of the novel coronavirus (SARS-CoV-2) has led to unprecedented lockdowns at global and regional scales, leaving the archived datasets to limited use. While it is often hypothesized that population density is a significant driver in disease propagation, the disparate disease trajectories and infection rates exhibited by the different cities with comparable densities require a high-resolution description of the disease and its drivers. In this study, we explore the impact of the creation of containment zones on travel patterns within the city. Further, we use a dynamical network-based infectious disease model to understand the key drivers of disease spread at sub-kilometer scales demonstrated in the city of Ahmedabad, India, which has been classified as a SARS-CoV-2 hotspot. We find that in addition to the contact network and population density, road connectivity patterns and ease of transit are strongly correlated with the rate of transmission of the disease. Given the limited access to real-time traffic data during lockdowns, we generate road connectivity networks using open-source imageries and travel patterns from open-source surveys and government reports. Within the proposed framework, we then analyze the relative merits of social distancing, enforced lockdowns, and enhanced testing and quarantining mitigating the disease spread.

With the expansion of renewable energies in Germany, imminent grid congestion events occur more often. One approach for avoiding curtailment of renewable energies is to cover excess feed-in by demand response. As curtailment is often a local phenomenon, in this work we determine the regional demand response potential for the 401 German administrative districts. The load regionalisation is based on weighting factors derived from population and employment statistics, locations of industrial facilities, etc. Using periodic and temperature-dependent load profiles and technology specific parameters, load shifting potentials were determined with a temporal resolution of 15 minutes. Our analysis yields that power-to-heat technologies provide the highest potentials, followed by residential appliances, commercial and industrial loads. For the considered 2030 scenario, power-to-gas and e-mobility also contribute a significant potential. The cumulated load increase potential of all technologies ranges from 5−470 MW per administrative district. The median value is 25 MW , which would suffice to avoid the curtailment of 8 classical wind turbines. Further, we calculated load shifting cost-potential curves for each district. Industrial processes and power-to-heat in district heating have the lowest load shifting investment cost, due to the largest installed capacities per facility. We distinguished between different size classes of the installed capacity of heat pumps, yielding 23% lower average investment cost for heat pump flexibilisation in the city of Berlin compared to a rural district. The variable costs of most considered load shifting technologies remain under the average compensation costs for curtailment of renewable energies of 110 \euro{}/MWh . As all results and the developed code are published under open source licenses, they can be integrated into energy system models.

This paper assesses the technical, financial, and market potential of the rooftop Solar Photovoltaic (PV) system on residential buildings in major cities namely Kathmandu valley, Pokhara, and Biratnagar of Nepal. Three sets of questionnaires were prepared each for residential households, PV suppliers, and solar project financing institutions. From the field survey, it is found that the average rooftop area available for PV installation is 14.5 sq.m, 12.45 sq.m, and 19 sq.m in Kathmandu, Pokhara, and Biratnagar cities respectively. Considering 557,027 residential buildings in Kathmandu; 77,523 in Pokhara and 33,075 in Biratnagar, total rooftop PV power potential in all three cities are found to be 970 MWp which could generate 1,310 GWh/year that comes out to be 35% of the electricity sold by Nepal Electricity Authority (NEA) in fiscal year 2014/15. Based on the 1.5 kWp PV system design per household and market price of 2016, the Levelized Cost of Electricity (LCOE) changes from NRs 8/kWh to NRs 20/kWh for basic lighting to a full load consisting of domestic electrical appliances. The technical barriers for the grid connection of rooftop solar in Nepal are not a major issue now as Nepal Electricity Authority has set clear guidelines for it.

This paper analyzes the impact of COVID-19 on the populations and equity markets of 92 countries. We compare country-by-country equity market dynamics to cumulative COVID-19 case and death counts and new case trajectories. First, we examine the multivariate time series of cumulative cases and deaths, particularly regarding their changing structure over time. We reveal similarities between the case and death time series, and key dates that the structure of the time series changed. Next, we classify new case time series, demonstrate five characteristic classes of trajectories, and quantify discrepancy between them with respect to the behavior of waves of the disease. Finally, we show there is no relationship between countries' equity market performance and their success in managing COVID-19. Each country's equity index has been unresponsive to the domestic or global state of the pandemic. Instead, these indices have been highly uniform, with most movement in March.