Fig 1.
Two examples of multilayer networks.
(a) has two layers: A and B. (b) has three layers A, B and C. Node A1 and node B1 are two representatives of entity ‘1’, they are connected by a cross-layer edge (dash line).
Fig 2.
An example of one propagation of multiple influences (colors) in a multilayer network G.
(1) is a double-layer network G. The uppercase and lowercase of one letter represents one entity, uppercase and lowercase nodes are the representatives of the entity in two layers respectively. Red node ‘A’, blue node ‘a’ and blue node ‘B’ are three seeds of two influences respectively. (2) is an outcome of G under MMIC, i.e. a sample ω. Pink and pale blue nodes are the nodes who have received red and blue activations, respectively. Because entity ‘Aa’ has one representative ‘A’ as a red seed, and one representative ‘a’ as a blue seed, entity ‘Aa’ decides to each color with a probability of 1/2. Entity ‘Bb’ only owns blue seed as its representative, so ‘Bb’ ultimately decides to blue. Entity ‘Cc’ only receives red activations, so ‘Cc’ ultimately decides to red. Entity Dd receives red activation once and blue activation once. Therefore, entity Dd decides to each color with a probability of 1/2.
Fig 3.
An example of calculating influence spread using Monte Carlo method.
The 10000 samples are the outcomes of 10000 propagations of G under MMIC model respectively. The red node is the seed A. The pink nodes are the active nodes in each sample. The dark nodes are the inactive nodes. The red edges are the live edges. σi(A) is the number of active entities in sample i. The influence spread of node A is the average of all σi(A)(where i = 1, 2, …, 10000).
Table 1.
Notations.
Fig 4.
An example of random RRE of entity Bb in a multilayer network.
(1) is a bilayer network G, the pink nodes ‘B’ and ‘b’ are representatives of entity Bb. After a reverse propagation (along the opposite direction of edges) from ‘B’ and ‘b’, we have a sample ω, i.e., (2). The active nodes (B,b,c,C,A) which constitute a random RRE of entity Bb.
Fig 5.
The procedure of RRE method (Algorithm 3).
Table 2.
Network info.
Fig 6.
The influence spread of three methods and the running time of RRE method.
The number of iterations to calculate influence spread is 10000. For (a) and (c), abscissa is the number of seeds; ordinates represent the influence spread; red cross, blue dot and green star line represent the methods of random, degree and RRE respectively. For (b) and (d), abscissa is the number of seeds, ordinates represent the running time (s) of RRE method.
Fig 7.
The procedure of Fair1 (Algorithm 4) when budgets are 1:2:3 and for i = 1, 2, …, 6.
Fig 8.
The procedure of Fair2 (Algorithm 5) when budgets are 1:2:3 and for i = 1, 2, …, 6.
Fig 9.
The procedure of solutions of Fair seed allocation problem.
Fig 10.
The results of Fair1 and Fair2 on multilayer network Wiki-Vote0.
Abscissa is the number of seeds; ordinates represent the value of f; red cross, blue dot and green star line represent Random, Fair1 and Fair2 respectively. Subfigure (a)-(f) are of different proportion of budgets.
Fig 11.
The results of Fair1 and Fair2 on multilayer network Bitcoin-alpha0 abscissa is the number of seeds; ordinates represent the value of f; red cross, blue dot and green star line represent Random, Fair1 and Fair2 respectively.
Subfigure (a)-(f) are of different proportion of budgets.