Reconstructing the topology of sparsely connected dynamical networks

Given a general physical network and measurements of node dynamics, methods are proposed for reconstructing the network topology. We focus on networks whose connections are sparse and where data are limited. Under these conditions, common in many biological networks, constrained optimization techniques based on the $L1$ vector norm are found to be superior for inference of the network connections.

Figure 1

(Color online) Connection diagram of system. The system is sparsely connected; each node has three incoming connections from other nodes.

Figure 2

(Color online) Isoclines of the $L1$ norm. The minimum-norm point on a linear space tends to have relatively few nonzero coordinates, making $L1$ minimization effective for computing sparse solutions.

Figure 3

Error in reconstructing coefficients versus the number of data points: (a) 10% sparsity and (b) 20% sparsity.

Figure 4

True positive rate for example system (3). The upper pair of traces use single trajectories of length 200 (400) points and apply the $L1$ optimization technique from the text. The mean TPR over 40 trajectories is shown. The lower pair of traces are the same, but using the ordinary least-squares approach.

Figure 5

True positive rate for example system (4). Settings are similar to those in Fig. 4. The circles denote the use of the $L1$ technique described in the text. The squares denote the use of the ordinary least-squares approach.