Fig 1.
Comparing nodes in different graphs by closeness centrality and Ω.
The black nodes in the two graphs all have closeness 3/5, which ranks them as the most important node in panel A but of intermediate importance in panel B. The value 3/5 is thus insufficient for ranking the nodes importance. Closeness manages to rank the nodes within each graph correctly with respect to Ω for the infection rate β = 1/2 (except it does not split the blue nodes of the graph in panel A), but ranks the white nodes of panel B too high in both graphs together.
Fig 2.
The unfolding of the SIR configuration tree.
(left) For the triangle graph with one initial infected node, the outbreak is a tree of configurations. The subtrees whose root nodes are labeled (a) unfold symmetrically (only one is shown); the same for (b). (right) For a path in the tree, the transition probabilities are shown.
Fig 3.
The discrete probability distribution for the number of edges M across all nonisomorphic, connected, undirected graphs of 8 ≤ N ≤ 10. The shaded areas mark values outside of the bounds of M (N − 1 ≤ M ≤ N(N − 1)/2).
Table 1.
In matrix notation: x is the vector of node centralities, A is the graph adjacency matrix, λ1 is the largest eigenvalue of A, D is the degree matrix (the diagonal matrix of node degrees), 1 is the vector of ones, and I is the identity matrix (the diagonal matrix of ones). Other notation: dij is the number of edges on the shortest path between nodes i and j, σjk is the number of shortest paths between nodes j and k, σjk|i is the number of shortest path between nodes j and k which pass through i.
Fig 4.
How the expected outbreak size Ω varies with a spectral centrality measure and the node degree across all small graphs.
Panels A, B and C show Ω for any node starting an outbreak, in any graph of size N = 10, against the eigenvector centrality of that node. Panel A shows data for β = 1/16; B for β = 1 and C for β = 16. The color denotes the degree centrality.
Fig 5.
How the expected outbreak size Ω varies with a centrality measure and the edge density across all small graphs.
As Fig 4, except that the color here denotes the network density.
Fig 6.
Example graphs with large Ω diversity.
Panel A shows the graph with largest range of Ω values for β = 1/16; B for β = 1 and C for β = 16.
Fig 7.
How predictive is a single node centrality?.
The coefficient of determination R2 when Ω(β) is estimated over graphs of N nodes. The centralities appear in decreasing order of the minimum R2 across β values at N = 10: 0.69 for closeness, 0.65 for degree, 0.54 for coreness, but only 0.12 for betweenness.
Fig 8.
The most predictive pairs of node centralities.
The coefficient of determination R2 when Ω(β) is estimated over graphs of N nodes. The centrality pairs appear in decreasing order of the minimum R2 across β values at N = 10: from 0.91 for degree and PageRank, to 0.88 for all three other combinations.
Fig 9.
The predictability of one node metric and density (the number of edges in the graph of a node).
The coefficient of determination R2 when Ω(β) is estimated over graphs of N nodes similar to Fig 8. The panels show the top four centralities in terms of the minimum R2 value over all parameter combinations: R2 = 0.92 for both PageRank and Katz, 0.88 for eigenvector centrality and 0.78 for degree.
Fig 10.
Combinations between PageRank or Katz centrality and other measures.
The leftmost markers represent single-feature predictions; the rest are combinations with other measures.
Fig 11.
The most predictive triplets of node centralities (including the number of edges).
The coefficient of determination R2 when Ω(β) is estimated over graphs of N nodes. The centrality triplets appear in decreasing order of the largest minimum R2 across β values at N = 10: these four combinations reach R2 values between 0.96 and 0.95 (and many other triplets, not shown here, also score above R2 = 0.90).
Fig 12.
Size scaling of the best predictability for one, two and three features.
The three panels represent less (A), medium (B) and more (C) contagious diseases.
Fig 13.
Prediction maps for the combination of degree and PageRank at different transmission rates.
In each of these subpanels for nine infection rates β, the degree centrality is given on the x-axis and PageRank on the y-axis. The diagonal grid shows regions where no real graph exists. N = 8 for all panels.
Fig 14.
Prediction maps for the combination of Katz centrality and density at different transmission rates.
This figure corresponds to Fig 13 but is for Katz centrality and density instead of degree and PageRank.
Fig 15.
Prediction maps for the combination of PageRank and degree centrality for different graph sizes.
This figure corresponds to Fig 13 but the transmission rate is fixed to β = 1 and we vary the system size. To be able to compare different systems sizes, we plot Ω/N rather than Ω.