This paper studies m firms competing on location over a finite subset of vertices of a graph, with the aim of attracting consumers located at some vertices of this graph. It is shown that firm i (i = 1, …, m) wants to establish its mi facilities at the mi-median of the graph whose vertices are weighted by consumers' purchasing power provided that consumers' tastes are sufficiently dispersed in probabilistic terms among the different firms. Examples based on some standard network configurations illustrate the property.