  In this paper we introduce and study the topology of clique complexes of
multigraphs without loops. These clique complexes generalize tournaplexes,
which were recently introduced by Govc, Levi, and Smith for the topological
study of brain functional networks. We study a general construction of
edge-inflated simplicial posets, which generalize clique complexes of
multigraphs. The poset fiber theorem of Bj\"{o}rner, Wachs, and Welker is
applied to obtain the homotopy wedge decomposition of an edge-inflated
simplicial poset. The homological corollary of this result allows to
parallelize the homology computations for edge inflated complexes, in
particular, for clique complexes of multigraphs and tournaplexes. We provide
functorial versions of some results to be used in computations of persistent
homology. Finally, we introduce a general notion of simplex inflations and
prove homotopy wedge decompositions for this class of spaces.
