Why temporal networks are more controllable: Link weight variation offers superiority

The control of temporal networks is of paramount importance to complex systems in diverse fields. Recent studies showed that temporal networks are more controllable than their static counterparts, in terms of control time, cost, and trajectory length. However, the underlying mechanism of this intriguing phenomenon remains elusive, partly due to the fact that multiple properties of a temporal network simultaneously change over time. Here, we explore a general model of temporal networks, and prove that the weight variation of a link is equivalent to attaching a virtual driver node to that link. Consequently, the random variation of link weights can significantly increase the dimension of controllable space and remarkably reduce control cost, which unveils the fundamental mechanism for the advantages of temporal networks in controllability. The finding of this mechanism leads to a graphic criterion that allows us to further discover that, degree-heterogeneous networks are more advantageous for enhancing controllability by link weight variation, and the favorable positions of weight variation are the incoming links of the nodes with a high outdegree and a low indegree. Our results are validated in both synthetic and empirical network data, together deepening the understanding of network temporality and shedding light on the long-standing problem of establishing graphic criteria for the controllability of general dynamic systems.

Figure 1

Controllability of temporal networks induced by link weight variation. (a), (b) A simple temporal network with two snapshots whose topologies are the same. The only difference between the two snapshots is that the weight of a link (red, from node 2 to node 3) is perturbed from $w_{32}$ to $w_{32}+\delta$, where $\delta$ is a random time-invariant number. If $\delta \ne 0$, the controllable dimension of the temporal network is $\text{dim}\left({\mathrm{\Omega }}_{\text{T}}\right)=3$. (c) The effective static network, without weight variation but with an additional virtual driver node (pointed by the red dashed arrow), is equivalent to the temporal network (b) with regard to controllability. If $\delta =0$, the network in (a) and (b) is degenerated to a static counterpart in (d) and (e) whose controllable dimension is $\text{dim}\left({\mathrm{\Omega }}_{\text{S}}\right)=2$. (f) If $w_{32}+\delta =0$, i.e., the link weight is altered from nonzero to zero, the temporal network becomes topology varying (g), (h) but its controllable dimension is still $\text{dim}\left({\mathrm{\Omega }}_{\text{T}}\right)=3$ (i). (j) A temporal network of two snapshots with complex topologies. The weight of the link (red) from node $p$ to node $q$ is altered, and $D$ denotes the shortest path length from the real driver node $v$ to node $p$. (k) The effective static network with an additional virtual driver node, whose controllability is equal to that of the temporal network in (j). (l), (m) Two specific examples demonstrate that link weight variation alone enhances network controllability [i.e., $\text{dim}\left({\mathrm{\Omega }}_{\text{T}}\right)>\text{dim}\left({\mathrm{\Omega }}_{\text{S}}\right)$].

Figure 2

Controllability enhancement vs degree heterogeneity. (a) $\mathrm{P}({\mathrm{\Omega }}_{\text{T}}>{\mathrm{\Omega }}_{\text{S}})$ represents the probability of that the network controllability is enhanced by link weight variation alone, where $M$ is the number of snapshots, each snapshot has 1000 nodes, and the node degrees follow a power-law distribution with exponent $\gamma$. A smaller $\gamma$ indicates higher heterogeneity. (b) The nodes are divided into three groups of equal size according to their indegrees $k_{\text{in}}$ or outdegrees $k_{\text{out}}$. The bars show the average increase of controllable dimension caused by perturbing the weights of links that point to nodes with low, medium, or high $k_{\text{in}}$ or $k_{\text{out}}$, where $\gamma =2.5$. The significance $p$ values in (b) are obtained through statistical $t$-tests, and the symbol of four asterisks indicates a $p$ value $<{10}^{-4}$.

Figure 3

Dependence of reduced control cost on distance $D$ from the driver node to the perturbed link for (a) chains, (b) circles, (d) random networks, and (e) six empirical networks (see Sec. III in SM [40]). (c) Reduced control cost vs the amount of weight perturbation for different links in a random network. The number of snapshots $M=2$, the duration of each snapshot $\tau ={10}^{-3}$, and the dashed lines represent best linear fits in semilog plots.

Figure 4

Control cost of real-world temporal networks, in comparison to that induced by link weight variation and also to aggregated static counterparts. The six empirical data sets are described in SM Sec. III [40]. In each network, $20\%$ nodes are randomly selected as driver nodes and the snapshot duration is $\tau ={10}^{-2}$.