The rise of social media platforms has fundamentally altered the public discourse by providing easy to use and ubiquitous forums for the exchange of ideas and opinions. Elected officials often use such platforms for communication with the broader public to disseminate information and engage with their constituencies and other public officials. In this work, we investigate whether Twitter conversations between legislators reveal their real-world position and influence by analyzing multiple Twitter networks that feature different types of link relations between the Members of Parliament (MPs) in the United Kingdom, and an identical dataset for politicians within Ireland. We develop and apply a matrix factorization technique that allows the analyst to emphasize nodes with contextual local network structures by specifying network statistics that guide the factorization solution. Leveraging only link relation data, we find that important politicians in Twitter networks are associated with real-world leadership positions, and that rankings from the proposed method are correlated with the number of future media headlines.