Signed graph embedding: when everybody can sit closer to friends than enemies

Signed graphs are graphs with signed edges. They are commonly used to represent positive and negative relationships in social networks. While balance theory and clusterizable graphs deal with signed graphs to represent social interactions, recent empirical studies have proved that they fail to reflect some current practices in real social networks. In this paper we address the issue of drawing signed graphs and capturing such social interactions. We relax the previous assumptions to define a drawing as a model in which every vertex has to be placed closer to its neighbors connected via a positive edge than its neighbors connected via a negative edge in the resulting space. Based on this definition, we address the problem of deciding whether a given signed graph has a drawing in a given $\ell$-dimensional Euclidean space. We present forbidden patterns for signed graphs that admit the introduced definition of drawing in the Euclidean plane and line. We then focus on the $1$-dimensional case, where we provide a polynomial time algorithm that decides if a given complete signed graph has a drawing, and constructs it when applicable

Surface water flood warnings in England: overview, Assessment and recommendations based on survey responses and workshops

Following extensive surface water flooding (SWF) in England in summer 2007, progress has been made in improving the management and prediction of this type of flooding. A rainfall threshold-based extreme rainfall alert (ERA) service was launched in 2009 and superseded in 2011 by the surface water flood risk assessment (SWFRA). Through survey responses from local authorities (LAs) and the outcome of workshops with a range of flood professionals, this paper examines the understanding, benefits, limitations and ways to improve the current SWF warning service. The current SWFRA alerts are perceived as useful by district and county LAs, although their understanding of them is limited. The majority of LAs take action upon receipt of SWFRA alerts, and their reactiveness to alerts appears to have increased over the years and as SWFRA superseded ERA. This is a positive development towards increased resilience to SWF. The main drawback of the current service is its broad spatial resolution. Alternatives for providing localised SWF forecast and warnings were analysed, and a two-tier national-local approach, with pre-simulated scenario-based local SWF forecasting and warning systems, was deemed most appropriate by flood professionals given current monetary, human and technological resources