Fig 1.
Graph representation of the Western US power grid (A), one possible module representation (B), and the internal structure of nodes and edges inside two selected modules, plus edges connecting nodes between the two modules (C).
Table 1.
Topological data for the ten real networks studied.
size of the networks (N), number of edges (E), mean degree (〈k〉), modularity (Q), relative size of the largest community (), fraction of edges linking distinct communities (Einter), and the overall efficiency gain of the MBA method (η, see Eq (3) for definition). For the four parameters related with community detection we display the values corresponding to the most efficient case among ten seeds of infomap (I) and ten seeds of Louvain (L). These data is presented for node and edge attacks.
Fig 2.
Comparison between the effect of betweenness-based attack, degree-based attack, longest path attack, random attack, and module-based attack for the Western US power grid network.
(A) Size of the biggest connected component in terms of the initial size, σ, as function of fraction of removed nodes, ρ. (B) Network and modular representations of US power grid. (C) Snapshots of the node-representation of the US power grid when 1%, 2% and 3% of nodes are removed using CBA and MBA methods.
Fig 3.
Size of the biggest connected component in terms of the initial size, σ, as function of fraction of removed nodes, ρ.
Vertex Module-based-attack (black squares), betweenness-based attack (red circles). (A) Western US power grid. (B) Euro Road. (C) Open flights. (D) US airports. (E) Facebook. (F) Twitter. (G) Google Plus. (H) Yeast protein. (I) H pylori. (J) C elegans. The intersection of the dashed blue lines corresponds to the point (σe, ρe) of maximum damage on the network using MBA. Network data details are given in Table 1.
Fig 4.
Size of the biggest connected component in terms of the initial size, σ, as function of fraction of removed edges, ρ.
Edges Module-based-attack (black squares), betweenness-based attack (red circles). (A) Western US power grid. (B) Euro road. (C) Open flights. (D) US airports. (E) Facebook. (F) Twitter. (G) Google Plus. (H) Yeast protein. (I) H pylori. (J) C elegans. The intersection of the dashed blue lines corresponds to the point (σe, ρe) of maximum damage on the network using MBA. Network data details are given in Table 1.
Fig 5.
Efficiency gain of vertex MBA, compared to vertex CBA (γ = σnull/σ), as a function of the fraction of removed nodes, ρ.
The network code is Facebook (FB), Twitter (TW), Google Plus (G+), US power grid (PG), Euro road (ER), Open flights (OF), US airports (UA), Yeast protein (YP), H pylori (HP), and C elegans (CE). Infrastructural networks are colored red, biological are colored green, and social are colored blue.
Fig 6.
Overall efficiency gain (η) of the MBA method relative to the CBA method as function of modularity, Q, for nodes and edges removal.
The vertical axis is in logarithmic scale and the horizontal axis is linear. The networks attacked are Facebook (FB), Twitter (TW), Google Plus (G+), US power grid (PG), Euro road (ER), Open flights (OF), US airports (UA), Yeast protein (YP), H pylori (HP), and C elegans (CE).