The socle series of a Leavitt path algebra

We investigate the ascending Loewy socle series of Leavitt path algebras $L_K(E)$ for an arbitrary graph $E$ and field $K$. We classify those graphs $E$ for which $L_K(E)=S_{\lambda}$ for some element $S_{\lambda}$ of the Loewy socle series. We then show that for any ordinal $\lambda$ there exists a graph $E$ so that the Loewy length of $L_K(E)$ is $\lambda$. Moreover, $\lambda \leq \omega$ (the first infinite ordinal) if $E$ is a row-finite graph.