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Register a new userThe Truth about Power Laws: Theory and Reality
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Consensus about the universality of the power law feature in complex networks is experiencing profound challenges. To shine fresh light on this controversy, we propose a generic theoretical framework in order to examine the power law property. First, we study a class of birth-and-death networks that is ubiquitous in the real world, and calculate its degree distributions. Our results show that the tails of its degree distributions exhibits a distinct power law feature, providing robust theoretical support for the ubiquity of the power law feature. Second, we suggest that in the real world two important factors, network size and node disappearance probability, point to the existence of the power law feature in the observed networks. As network size reduces, or as the probability of node disappearance increases, then the power law feature becomes increasingly difficult to observe. Finally, we suggest that an effective way of detecting the power law property is to observe the asymptotic (limiting) behaviour of the degree distribution within its effective intervals.
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In this paper, we study the structural properties of the complex bus network of Chennai. We formulate this extensive network structure by identifying each bus stop as a node, and a bus which stops at any two adjacent bus stops as an edge connecting the nodes. Rigorous statistical analysis of this data shows that the Chennai bus network displays small-world properties and a scale-free degree distribution with the power-law exponent, $gamma > 3$.
The power from wind and solar exhibits a nonlinear flickering variability, which typically occurs at time scales of a few seconds. We show that high-frequency monitoring of such renewable powers enables us to detect a transition, controlled by the field size, where the output power qualitatively changes its behaviour from a flickering type to a diffusive stochastic behaviour. We find that the intermittency and strong non-Gaussian behavior in cumulative power of the total field, even for a country-wide installation still survives for both renewable sources. To overcome the short time intermittency, we introduce a time-delayed feedback method for power output of wind farm and solar field that can change further the underlying stochastic process and suppress their strong non- gaussian fluctuations.
The preferential attachment (PA) process is a popular theory for explaining network power-law degree distributions. In PA, the probability that a new vertex adds an edge to an existing vertex depends on the connectivity of the target vertex. In real-world networks, however, each vertex may have asymmetric accessibility to information. Here we address this issue using a new network-generation mechanism that incorporates asymmetric accessibility to upstream and downstream information. We show that this asymmetric information accessibility directly affects the power-law exponent, producing a broad range of values that are consistent with observations. Our findings shed new light on the possible mechanisms in three important real-world networks: a citation network, a hyperlink network, and an online social network.
Power-grid systems constitute one of the most complex man-made spatially extended structures. These operate with strict operational bounds to ensure synchrony across the grid. This is particularly relevant for power-grid frequency, which operates strictly at $50,$Hz ($60,$Hz). Nevertheless, small fluctuations around the mean frequency are present at very short time scales $<2$ seconds and can exhibit highly complex spatio-temporal behaviour. Here we apply superstatistical data analysis techniques to measured frequency fluctuations in the Nordic Grid. We study the increment statistics and extract the relevant time scales and superstatistical distribution functions from the data. We show that different synchronous recordings of power-grid frequency have very distinct stochastic fluctuations with different types of superstatistics at different spatial locations, and with transitions from one superstatistics to another when the time lag of the increment statistics is changed.
Across many scientific domains, there is a common need to automatically extract a simplified view or coarse-graining of how a complex systems components interact. This general task is called community detection in networks and is analogous to searching for clusters in independent vector data. It is common to evaluate the performance of community detection algorithms by their ability to find so-called ground truth communities. This works well in synthetic networks with planted communities because such networks links are formed explicitly based on those known communities. However, there are no planted communities in real world networks. Instead, it is standard practice to treat some observed discrete-valued node attributes, or metadata, as ground truth. Here, we show that metadata are not the same as ground truth, and that treating them as such induces severe theoretical and practical problems. We prove that no algorithm can uniquely solve community detection, and we prove a general No Free Lunch theorem for community detection, which implies that there can be no algorithm that is optimal for all possible community detection tasks. However, community detection remains a powerful tool and node metadata still have value so a careful exploration of their relationship with network structure can yield insights of genuine worth. We illustrate this point by introducing two statistical techniques that can quantify the relationship between metadata and community structure for a broad class of models. We demonstrate these techniques using both synthetic and real-world networks, and for multiple types of metadata and community structure.
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