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During the 2020 general election there were unprecedented efforts to undermine and cast doubt on the integrity of American electoral institutions and the officials in charge of administering the election. In this paper, we first propose a networks-based method to uncover social media accounts of local election officials and construct a dataset of their social media behavior. Then, we employ Joint Sentiment Topic modelling (a topic modeling method that jointly uncovers topical and sentimental structure in text data) to detect online animosity directed at election official accounts. Finally, we use dimensionality reduction techniques to measure the dynamics of this behavior. Unlike previous survey-based approaches, our framework allows for real-time and dynamic analysis of salient online political behavior. We use these dynamic measures of animosity to validate theoretical predictions related to the loser's effect on the public's trust in election outcomes. Using our methodology, we measure attempts by political elites to encourage online rhetoric against local election officials, exacerbating the loser's effect. We observe in our real-time data that, in response, LEOs took to Twitter to reinforce trust in the electoral institution. Studying the dynamics of hostility against LEOs and their subsequent defenses is prudent for all elections moving forward, especially in a time of consistently increasing partisanship and eroding trust in political institutions--this motivated the need for a method to study these interactions. We have developed and validated such a method, along with a novel data collection method.