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On Consistent Vertex Nomination Schemes

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Figure 1: A visual representation of the classical Vertex Nomination framework: Given acommunity of interest in a network (here the red community) and some examplesof vertices that are/are not part of the community of interest (colored red andgreen, respectively), rank the remaining vertices in the network into a nominationlist, with those vertices from the community of interest concentrating at the topof the nomination list.
Figure 2: A visual representation of the generalized Vertex Nomination framework: Given avertex of interest v∗ (colored red) in a graph G1 = (V1, E1), find the correspondingvertex of interest u∗ (if it exists) in a second graph G2 = (V2, E2), ranking thevertices of G2 into a nomination list so that u∗ ideally appears at the top of thenomination list.
Figure 3: An illustration of the “label-independence” property of VN schemes. If the bluevertex in o1(g2) (resp., o2(g2)) is o1(u) (resp., o2(u)) for u ∈ V2, then we requirethe ranks of I(o1(u); o1(g2)) (outlined in red in the network o1(g2) and coloredred
Figure 4: An example of the consistency criterion, Equation (2), in action., v∗, {4,Φ5 5}) = rΦ(g1, g2, o2, v∗Φ(g1, g2, o2, v∗∗∗3panel (a) shows a well-defined nomination scheme while the right panel (b)shows an ill-defined scheme

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