Influence Robustness of Nodes in Multiplex Networks against Attacks

TL;DR

This work addresses node-influence robustness in multiplex networks under targeted attacks by introducing MultiCoreRank, a core-decomposition–based centrality that propagates influence along the multiplex core lattice $G=(V,E,L)$. The method computes node influence through a BFS-style, level-wise aggregation over a $oldsymbol{k}$-core structure, capturing multi-layer interdependencies via a hierarchical core lattice. Empirical results across assortative, neutral, and disassortative multiplexes show that assortative networks exhibit greater resilience as the core structure remains more stable under attack, while disassortative networks fragment more quickly; correlations with traditional centralities validate the method’s relevance, especially its alignment with overlapping degree. The findings highlight the importance of inter-layer degree correlations for resilience and provide a practical tool for identifying influential nodes in complex multi-layer systems, with potential applications in designing robust infrastructure and social networks.

Abstract

Recent advances have focused mainly on the resilience of the monoplex network in attacks targeting random nodes or links, as well as the robustness of the network against cascading attacks. However, very little research has been done to investigate the robustness of nodes in multiplex networks against targeted attacks. In this paper, we first propose a new measure, MultiCoreRank, to calculate the global influence of nodes in a multiplex network. The measure models the influence propagation on the core lattice of a multiplex network after the core decomposition. Then, to study how the structural features can affect the influence robustness of nodes, we compare the dynamics of node influence on three types of multiplex networks: assortative, neutral, and disassortative, where the assortativity is measured by the correlation coefficient of the degrees of nodes across different layers. We found that assortative networks have higher resilience against attack than neutral and disassortative networks. The structure of disassortative networks tends to break down quicker under attack.

Figures (5)

• Figure 1: An example two-layer network, where solid lines signify edges belonging to the first layer, while dashed lines indicate edges associated with the second layer.
• Figure 2: The network after removing node $B$ and its edges in Figure \ref{['fig:example_network']}.
• Figure 3: The core lattice of the network in Figure \ref{['fig:example_network']}. The numbers in the core vectors are the minimum degrees of each layer in a core. (i.e. (2,1) consists of nodes with at least degree 2 on layer 1 and at least degree 1 on layer 2.) The level in which a core is in the lattice is given by the L-1 norm of its core vector. The core vector (0,3) is shown as an example of empty core while other empty cores are omitted.
• Figure 4: The number of cores remaining in the network after a percentage of nodes are removed. Two types of removal are performed 1) sorted attack based on MultiCoreRank and 2) uniformly random attack. (1) and (2) correspond to the two assortative networks, (3) and (4) correspond to the two neutral networks, and (5) and (6) correspond to the two disassortative networks.
• Figure 5: The change in percentage of cores as a percentage of nodes are removed in different types of networks. The top row shows the sorted attack results. The bottom row shows the random attack results. The exponential fit on the top row is an exponential function,$y=a\mathrm{e}^{-xb}$, fitted on the average y-value given by each dataset at each x-value.

Theorems & Definitions (2)

• definition thmcounterdefinition: $\boldsymbol{k}$-core percolation azimi2014k
• theorem thmcountertheorem: Core containment galimberti2020core