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This paper produces an efficient Semidefinite Programming (SDP) solution forcommunity detection that incorporates non-graph data, which in this context isknown as side information. SDP is an efficient solution for standard communitydetection on graphs. We formulate a semi-definite relaxation for the maximumlikelihood estimation of node labels, subject to observing both graph andnon-graph data. This formulation is distinct from the SDP solution of standardcommunity detection, but maintains its desirable properties. We calculate theexact recovery threshold for three types of non-graph information, which inthis paper are called side information: partially revealed labels, noisylabels, as well as multiple observations (features) per node with arbitrary butfinite cardinality. We find that SDP has the same exact recovery threshold inthe presence of side information as maximum likelihood with side information.Thus, the methods developed herein are computationally efficient as well asasymptotically accurate for the solution of community detection in the presenceof side information. Simulations show that the asymptotic results of this papercan also shed light on the performance of SDP for graphs of modest size.

Semidefinite Programming for Community Detection with Side Information