The traditional Power Grid has been designed in a hierarchical fashion, with Energy pushed from the large scale production facilities towards the end users. But with the increasing availability of micro and medium scale generating facilities, the situation is changing. Many end users can now produce energy and share it over the Power Grid. Naturally, end users need to have incentives to do so and might want to be able to act in an open decentralized energy market. In the present work, we offer a novel analysis of the Medium and Low Voltage Power Grids of the North Netherlands using statistical tools from the Complex Network Analysis field. We use a weighted model based on actual Grid data and propose a set of statistical measures to evaluate the adequacy of the current infrastructure for a decentralized energy market. Further, we use the insight gained by the analysis to propose parameters that tie the statistical topological measures to economic factors that might influence the attractiveness to the end users in participating in such a decentralized energy market, thus identifying what are the important topological parameters to work on to facilitate such open decentralized markets.
Casals @cite_15 analyse the whole European and try to extract non-topological reliability measures investigating the topological properties of the network. The analysed is the end composed of almost 2800 nodes that span across the European continent. The assumption is that node degree distribution follows an exponential decay for every single network composing the European network, each one having a characteristic parameter that is related to the of the specific Grid. Although this study is based on relevant samples, but half of the considered Grids are small both in size and order (below 100 nodes). The most interesting aspect is the use of new indicators to assess network reliability, but, as the same authors explicitly state, these metrics need more test and a deeper study. As remarked for other works, there is no mention about using weights to characterize the edges in the networks.
@cite_15
2135031396 2104812688 2067269323 2084983846
The topological properties of high-voltage electrical power transmission networks in several UE countries (the Italian 380 kV, the French 400 kV and the Spanish 400 kV networks) have been studied from available data. An assessment of the vulnerability of the networks has been given by measuring the level of damage introduced by a controlled removal of links. Topological studies could be useful to make vulnerability assessment and to design specific action to reduce topological weaknesses. Motivation: Analogous to biological sequence comparison, comparing cellular networks is an important problem that could provide insight into biological understanding and therapeutics. For technical reasons, comparing large networks is computationally infeasible, and thus heuristics, such as the degree distribution, clustering coefficient, diameter, and relative graphlet frequency distribution have been sought. It is easy to demonstrate that two networks are different by simply showing a short list of properties in which they differ. It is much harder to show that two networks are similar, as it requires demonstrating their similarity in all of their exponentially many properties. Clearly, it is computationally prohibitive to analyze all network properties, but the larger the number of constraints we impose in determining network similarity, the more likely it is that the networks will truly be similar. Results: We introduce a new systematic measure of a network's local structure that imposes a large number of similarity constraints on networks being compared. In particular, we generalize the degree distribution, which measures the number of nodes 'touching' k edges, into distributions measuring the number of nodes 'touching' k graphlets, where graphlets are small connected non-isomorphic subgraphs of a large network. Our new measure of network local structure consists of 73 graphlet degree distributions of graphlets with 2--5 nodes, but it is easily extendible to a greater number of constraints (i.e. graphlets), if necessary, and the extensions are limited only by the available CPU. Furthermore, we show a way to combine the 73 graphlet degree distributions into a network 'agreement' measure which is a number between 0 and 1, where 1 means that networks have identical distributions and 0 means that they are far apart. Based on this new network agreement measure, we show that almost all of the 14 eukaryotic PPI networks, including human, resulting from various high-throughput experimental techniques, as well as from curated databases, are better modeled by geometric random graphs than by Erdos--Reny, random scale-free, or Barabasi--Albert scale-free networks. Availability: Software executables are available upon request. Contact: natasha@ics.uci.edu We study the property of certain complex networks of being both sparse and highly connected, which is known as “good expansion” (GE). A network has GE properties if every subset S of nodes (up to 50 of the nodes) has a neighborhood that is larger than some “expansion factor” φ multiplied by the number of nodes in S. Using a graph spectral method we introduce here a new parameter measuring the good expansion character of a network. By means of this parameter we are able to classify 51 real-world complex networks — technological, biological, informational, biological and social — as GENs or non-GENs. Combining GE properties and node degree distribution (DD) we classify these complex networks in four different groups, which have different resilience to intentional attacks against their nodes. The simultaneous existence of GE properties and uniform degree distribution contribute significantly to the robustness in complex networks. These features appear solely in 14 of the 51 real-world networks studied here. At the other extreme we find that ∼40 of all networks are very vulnerable to targeted attacks. They lack GE properties, display skewed DD — exponential or power-law — and their topologies are changed more dramatically by targeted attacks directed at bottlenecks than by the removal of network hubs. This paper studies networks where all nodes are distributed on a unit square A=Δ [- [1 2], [1 2]]2 following a Poisson distribution with known density ρ and a pair of nodes separated by an Euclidean distance x are directly connected with probability grρ(x)=Δg(x rρ), independent of the event that any other pair of nodes are directly connected. Here, g:[0,∞)→ [0,1] satisfies the conditions of rotational invariance, nonincreasing monotonicity, integral boundedness, and g(x)=o(1 (x2log2x)) ; further, rρ=√ (logρ+b) (Cρ) where C=∫ℜ2g(||x||)dx and b is a constant. Denote the aforementioned network by G(Xρ,grρ,A). We show that as ρ→ ∞, 1) the distribution of the number of isolated nodes in G(Xρ,grρ,A) converges to a Poisson distribution with mean e-b ; 2) asymptotically almost surely (a.a.s.) there is no component in G(Xρ,grρ,A) of fixed and finite order k >; 1; c) a.a.s. the number of components with an unbounded order is one. Therefore, as ρ→ ∞, the network a.a.s. contains a unique unbounded component and isolated nodes only; a sufficient and necessary condition for G(Xρ,grρ,A) to be a.a.s. connected is that there is no isolated node in the network, which occurs when b→ ∞ as ρ→ ∞. These results expand recent results obtained for connectivity of random geometric graphs from the unit disk model and the fewer results from the log-normal model to the more general and more practical random connection model.