  Strong inter-dependence in complex systems can manifest as partially
bipartite networks characterized by interactions occurring primarily between
distinct groups of nodes (identified as modules). In this paper, we show that
the anti-modular character of such networks, e.g., those defined by the
adjacent occurrence of alphabetic characters in corpora of natural language
texts, can result in striking structural properties which place them outside
the well-known regular/small-world/random network paradigm. Using an ensemble
of model networks whose modularity can be tuned, we demonstrate that strong
module size heterogeneity in anti-modular random networks imparts them with
higher communication efficiency and lower clustering than their randomized
counterparts, making them infra small-world. Passage to anti-modularity is
associated with characteristic changes in spectral properties of the network,
including a delocalization transition exhibited by the principal eigenvector
(PEV) of the normalized Laplacian. This is accompanied by the emergence of
prominent bimodality in the distribution of PEV components, which can function
as a signature for identifying anti-modular organization in empirical networks.
