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A derangement of a graph G = (V, E) is an injective function f :Not all graphs admit a derangement and previous results have characterized graphs with derangements using neighborhood conditions for subsets of V . We establish an alternative criterion for the existence of derangements on a graph. We analyze strict Nash equilibria of the biologically motivated Territorial Raider Game, a multi-player competition for resources in a spatially structured population based on animal raiding and defending behavior. We find that a graph G admits a derangement if and only if there is a strict Nash equilibrium of the Territorial Raider game on G.

Many organisms maintain collective territories and compete on behalf of the fitness of the overall group. Inspired by this concept, the territorial raider model is a graph-based resource competition in which populations have fixed home locations and a limited range of sites accessible for raiding. In our present extension of the model, groups control "colonies" or "armies" which can be divided across multiple locations. We present Nash equilibria for games played on both regular graphs and regular bipartite graphs, and we also examine differences that emerge when populations are composed of discrete units (pack scale) or when they are continuously divisible (colony scale). Reliance upon defense over aggressive raiding is greater here than in the original model where populations had to totally commit to a singular action. This defensive posture increases with the advantage of the local population and also varies with the degree of the graph's connectivity. When discrete units are employed, multiple strategies emerge.

While many studies have explored the use of social media and behavioral changes of individuals, few examined the utility of using social media for suicide detection and prevention. The study by Jashinsky et al. identified specific language patterns associated with a set of twelve suicide risk factors. The authors extended these methods to assess the significance of the language used on Twitter for suicide detection. This article quantifies the use of Twitter to express suicide related language, and its potential to detect users at high risk of suicide. The authors searched Twitter for tweets indicative of 12 suicide risk factors. This paper divided Twitter users into two groups: “high risk” and “at risk” based on two of the risk factors (“self-harm” and “prior suicide attempts”) and examined language patterns by computing co-occurrences of terms in tweets which helped identify relationships between suicide risk factors in both groups.