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We consider convex risk measures in a spatial setting, where the outcome of a financial position depends on the states at different nodes of a network.  In analogy to the theory of Gibbs measures in Statistical Mechanics, we discuss the local specification of a global risk measure in terms of conditional local risk measures for the single nodes of the network, given their environment.  Under a condition of local law invariance, we show that a consistent local specification must be of entropic form.  Even in that case, a global risk measure may not be uniquely determined by the local specification, and this can be seen as a source of “systemic risk”, in analogy to the appearance of phase transitions in the theory of Gibbs measures

Spatial risk measures and their local specification: The locally law-invariant case