Resilience centrality in complex networks

Resilience describes a system's ability to adjust its activity to retain the basic functionality when errors or failures occur in components (nodes) of the network. Due to the complexity of a system's structure, different components in the system exhibit diversity in the ability to affect the resilience of the system, bringing us a great challenge to protect the system from collapse. A fundamental problem is therefore to propose a physically insightful centrality index, with which to quantify the resilience contribution of a node in any systems effectively. However, existing centrality indexes are not suitable for the problem because they only consider the network structure of the system and ignore the impact of underlying dynamic characteristics. To break the limits, we derive a new centrality index: resilience centrality from the 1D dynamic equation of systems, with which we can quantify the ability of nodes to affect the resilience of the system accurately. Resilience centrality unveils the long-sought relations between the ability of nodes in a system's resilience and network structure of the system: the capacity is mainly determined by the degree and weighted nearest-neighbor degree of the node, in which weighted nearest-neighbor degree plays a prominent role. Further, we demonstrate that weighted nearest-neighbor degree has a positive impact on resilience centrality, while the effect of the degree depends on a specific parameter, average weighted degree ${\beta }_{\mathrm{eff}}$, in the 1D dynamic equation. To test the performance of our approach, we construct four real networks from data, which corresponds to two complex systems with entirely different dynamic characteristics. The simulation results demonstrate the effectiveness of our resilience centrality, providing us theoretical insights into the protection of complex systems from collapse.

Figure 1

Schematic diagram of resilience centrality. (a) Topological structure of a five-node ecological network. (b) Resilience function of ecological network, satisfying the nonlinear dynamic Eq. (17). We remove nodes from the original network, respectively, to simulate the failure of each node and construct five new networks. Using Eqs. (2) and (3), we calculate $x_{\mathrm{eff}}$ and ${\beta }_{\mathrm{eff}}$ for each network. Failure of node 4 leads to the collapse of the system, indicating that node 4 has a stronger ability to affect the system's resilience than other nodes. (c) Network mapping by resilience centrality, in which size of a node corresponds to its resilience centrality. (d) Three other centrality indexes are introduced as baselines to evaluate the performance of resilience centrality. Distribution of resilience centrality conforms to numerical results in (b), indicating that resilience centrality captures the ability of nodes accurately.

Figure 2

Error analysis for the simplified resilience centrality in four real networks. (a)–(d) Distribution of relative errors for nodes in the four real networks: insect network, flower network, email network, and dolphin network. Simulation results show that relative errors of simplified resilience centrality for most nodes are less than 0.05, especially those in flower network. (e)–(h) Match degree between ranking of nodes based on real resilience centrality and ranking based on simplified resilience centrality for four real networks, in which ranking based on real resilience centrality is chosen as ground truth, that is, diagonal lines of each subfigure. TR represents the top-ranked nodes in the ranking vector based on real resilience centrality. CR represents number of correctly ranked nodes by simplified resilience centrality, that is, number of overlapping nodes both appearing in the set of top-ranked nodes based on real resilience centrality and in the set based on simplified resilience centrality. The four lines related to simplified resilience centrality for four real networks cover the diagonal gray line except for a tiny proportion nodes, indicating that the simplified form of resilience centrality has very high accuracy for ranking of nodes.

Figure 3

Variation of $x_{\mathrm{eff}}(\mathrm{\Delta }{x}_{\mathrm{eff}}$) describes change of the system's state: shifts from normal operation to collapse state, or vice versa, which is used as ground truth to test performance of four centrality indexes. (a), (e) Resilience centraliy vs. $\mathrm{\Delta }{x}_{\mathrm{eff}}$ of each node for insect network [34] and flower network [33], which correspond to mutualistic dynamic Eq. (17). (i), (m) Resilience centrality vs. $\mathrm{\Delta }{x}_{\mathrm{eff}}$ of each node for email network [40] and dolphin network [39], which correspond to SIS dynamic Eq. (18). Similar results are found for degree centrality (b), (f), (j), (n), for betweenness centrality (c), (g), (k), (o), and for closeness centrality (d), (h), (l), (p).

Figure 4

Performance of four centrality indexes in perspective of system collapse. (a)–(d) Match degree between ranking of nodes based on four centrality indexes and the real ranking based on numerical simulation for insect network, flower network, email network, and dolphin network. Simulation results show the good performance of resilience centrality. However, diversity of performance for degree centrality is observed in the four networks: in flower network, degree centrality achieves similar performance to resilience centrality and better than other two centrality indexes significantly; while performance of degree centrality worse than resilience centrality in the other networks. (e)–(h) ROC cuves for insect network, flower network, email network and dolphin network, in which top 30 percent of nodes in real ranking vector of nodes based on numerical simulation are selected as “positive.” (i)–(l) Similar results are found for ROC curves with top 50 percent of nodes as “positive.”

Figure 5

Relation between resilience centrality and topological features. (a)–(d) Nomalized distribution of degree (d) for insect network, flower network, email network, and dolphin network. The distribution of insect network and two social networks follow an approximately exponential distribution, or a short power-low distribution, while distribution of flower network follows an approximately log-normal distribution. (e)–(h) Similar results are found for weighted nearest-neighbor degree (D). (i)–(l) Similar results are also found for resilience centrality (R), indicating that only a few of nodes have the ability to affect resilience of corresponding system significantly. (m)–(p) Theoretical results of Eq. (16) for insect network, flower network, email network, and dolphin network, which are visualized by contour plots. Red crosses correspond to nodes in the four network. We notice that weighted nearest-neighbor degree is far higher than a degree for all four networks, indicating that weighted nearest-neighbor degree plays a prominent role in resilience centrality.