Towards Optimal Robustness of Network Controllability: An Empirical Necessary Condition

Abstract

To better understand the correlation between network topological features and the robustness of network controllability in a general setting, this paper suggests a practical approach to searching for optimal network topologies with given numbers of nodes and edges. Since theoretical analysis is impossible at least in the present time, exhaustive search based on optimization techniques is employed, firstly for a group of small-sized networks that are realistically workable, where exhaustive means 1) all possible network structures with the given numbers of nodes and edges are computed and compared, and 2) all possible node-removal sequences are considered. An empirical necessary condition (ENC) is observed from the results of exhaustive search, which shrinks the search space to quickly find an optimal solution. ENC shows that the maximum and minimum in-and out-degrees of an optimal network structure should be almost identical, or within a very narrow range, i.e., the network should be extremely homogeneous. Edge rectification towards the satisfaction of the ENC is then designed and evaluated. Simulation results on large-sized synthetic and realworld networks verify the effectiveness of both the observed ENC and the edge rectification scheme. As more operations of edge rectification are performed, the network is getting closer to exactly satisfying the ENC, and consequently the robustness of the network controllability is enhanced towards optimum.

A Framework of Hierarchical Attacks to Network Controllability

“…For these networks, the attacker is unable or uneasy to find targets to attack in order to destruct the controllability. This finding is consistent with, and actually extends the applicability of, the previous findings: 1) dense and homogeneous networks have better controllability [5]; 2) extremely-homogeneous topology has the optimal controllability robustness [57]. Nevertheless, critical nodes and edges will expose themselves during the attack process, as the network becomes sparser.…”

“…Input : adjacency matrix A; feature F ; number of nodes N Output: index j of the node to be attacked Nine typical directed synthetic network models are adopted for simulation, namely the Erdös-Rényi random-graph (ER) network [51], Newman-Watts small-world (SW) network [52], generic scale-free (SF) network [37], [53], [54], q-snapback (QS) network [55], q-snapback network with redirected edges (QR) [56], random triangle (RT) network [24], and random rectangle (RR) network [24], extremely homogeneous (HO) network [57], and onion-like (OL) network [58].…”

“…Recall that the HO networks were empirically verified with optimal controllability robustness before [57]. Given an Nnode and M -edge configuration, the HO network satisfies M/N ≤ k in,out i ≤ M/N , i = 1, 2, .…”

Controllability robustness of heat exchanger network based on complex network theory

“…Pang et al 8 focused on the effect of interaction strength on the controllability robustness of edges, and used the edge attributes to assess and analyze the controllability robustness of complex systems with arbitrary structures. Lou et al 9 used an exhaustive search based on optimization technology to narrow the search space to quickly find the optimal solution with controllability robustness. However, the method used by the predecessors is computationally complex for large-scale heat exchanger networks and has certain limitations.…”

The network asymmetry caused by the degree correlation and its effect on the bimodality in control

“…There are intensive studies on the original framework of network controllability [13,14,15,16,17,18,19,20,21,22], with extensions to temporal or multilayer networks [7,23,24,25,26], cost of control [7,27,28,29,30,31,32], and biological and social networks [33,34,35,36,37,38,39,40,41,42,43]. Yet, except for a few studies [44,45,46,47], most of the previous works mainly consider "neutral" networks, whose degree distributions do not have correlations.…”