Investigating instances where lexical selection fails can lead to deeper insights into the cognitive machinery and architecture supporting successful word retrieval and speech production. In this paper, we used a multiplex lexical network approach that combines semantic and phonological similarities among words to model the structure of the mental lexicon. Network measures at different levels of analysis (degree, network distance, and closeness centrality) were used to investigate the influence of network structure on picture naming accuracy and errors by people with Anomic, Broca's, Conduction, and Wernicke's aphasia. Our results reveal that word retrieval is influenced by the multiplex lexical network structure in at least two ways--(a) the accuracy of production and error type on incorrect productions were influenced by the degree and closeness centrality of the target word, and (b) error type also varied in terms of network distance between the target word and produced error word. Taken together, the analyses demonstrate that network science techniques, particularly the use of the multiplex lexical network to simultaneously represent semantic and phonological relationships among words, reveal how the structure of the mental lexicon influences language processes beyond traditionally examined psycholinguistic variables. We propose a framework for how the multiplex lexical network approach allows for understanding the influence of mental lexicon structure on word retrieval processes, with an eye toward a better understanding of the nature of clinical impairments, like aphasia.