  Community detection or clustering is a crucial task for understanding the
structure of complex systems. In some networks, nodes are permitted to be
linked by either "positive" or "negative" edges; such networks are called
signed networks. Discovering communities in signed networks is more challenging
than that in unsigned networks. In this study, we innovatively develop a
non-backtracking matrix of signed networks, theoretically derive a
detectability threshold for this matrix, and demonstrate the feasibility of
using the matrix for community detection. We further improve the developed
matrix by considering the balanced paths in the network (referred to as a
balanced non-backtracking matrix). Simulation results demonstrate that the
algorithm based on the balanced nonbacktracking matrix significantly
outperforms those based on the adjacency matrix and the signed non-backtracking
matrix. The proposed (improved) matrix shows great potential for detecting
communities with or without overlap.
