Influence maximization in complex networks through optimal percolation

Journal name:
Nature
Volume:
524,
Pages:
65–68
Date published:
DOI:
doi:10.1038/nature14604
Received
Accepted
Published online

The whole frame of interconnections in complex networks hinges on a specific set of structural nodes, much smaller than the total size, which, if activated, would cause the spread of information to the whole network1, or, if immunized, would prevent the diffusion of a large scale epidemic2, 3. Localizing this optimal, that is, minimal, set of structural nodes, called influencers, is one of the most important problems in network science4, 5. Despite the vast use of heuristic strategies to identify influential spreaders6, 7, 8, 9, 10, 11, 12, 13, 14, the problem remains unsolved. Here we map the problem onto optimal percolation in random networks to identify the minimal set of influencers, which arises by minimizing the energy of a many-body system, where the form of the interactions is fixed by the non-backtracking matrix15 of the network. Big data analyses reveal that the set of optimal influencers is much smaller than the one predicted by previous heuristic centralities. Remarkably, a large number of previously neglected weakly connected nodes emerges among the optimal influencers. These are topologically tagged as low-degree nodes surrounded by hierarchical coronas of hubs, and are uncovered only through the optimal collective interplay of all the influencers in the network. The present theoretical framework may hold a larger degree of universality, being applicable to other hard optimization problems exhibiting a continuous transition from a known phase16.

At a glance

Figures

  1. The non-backtracking (NB) matrix and weak nodes.
    Figure 1: The non-backtracking (NB) matrix and weak nodes.

    a, The largest eigenvalue λ of exemplified on a simple network. The optimal strategy for immunization and spreading minimizes λ by removing the minimum number of nodes (optimal influencers) that destroys all the loops. Left panel, the action of the matrix is on the directed edges of the network. The entry encodes the occupancy (n3 = 1) or vacancy (n3 = 0) of node 3. In this particular case, the largest eigenvalue is λ = 1. Centre panel, non-optimal removal of a leaf, n4 = 0, which does not decrease λ. Right panel, optimal removal of a loop, n3 = 0, which decreases λ to zero. b, A NB walk is a random walk that is not allowed to return back along the edge that it just traversed. We show a NB open walk ( = 3), a NB closed walk with a tail ( = 4), and a NB closed walk with no tails ( = 5). The NB walks are the building blocks of the diagrammatic expansion to calculate λ. c, Representation of the global minimum over n of the largest eigenvalue λ of versus q. When qqc, the minimum is at λ = 0. Then, G = 0 is stable (still, non-optimal configurations exist with λ > 1 for which G > 0). When q < qc, the minimum of the largest eigenvalue is always λ > 1, the solution G = 0 is unstable, and then G > 0. At the optimal percolation transition, the minimum is at n* with λ(n*, qc) = 1. For q = 0, we find λ = κ − 1 (κ = left fencek2right fence/left fencekright fence, where k is the node degree) which is the largest eigenvalue of for random networks25 with all nodes present (ni = 1). When λ = 1, the giant component is reduced to a tree plus one single loop (unicyclic graph), which is suddenly destroyed at the transition qc to become a tree, causing the abrupt fall of λ to zero. d, Ball(i, ) of radius around node i is the set of nodes at distance from i, and ∂Ball is the set of nodes on the boundary. The shortest path from i to j is shown in red. e, Example of a weak node: a node with a small number of connections surrounded by hierarchical coronas of hubs at different levels.

  2. Exact optimal solution and performance of CI in synthetic networks.
    Figure 2: Exact optimal solution and performance of CI in synthetic networks.

    a, G(q) in an ER network (N = 2 × 105, left fencekright fence = 3.5, error bars are s.e.m. over 20 realizations). We show the true optimal solution found with EO (‘×’ symbol), and also using CI, HDA, PR, HD, CC, EC and k-core methods. The other methods are not scalable and perform worse than HDA and are treated in Supplementary Information sections VI and VII (Extended Data Figs 8, 9). CI is close to the optimal obtained with EO in Supplementary Information section IV. Note that EO can estimate the extrapolated optimal value of qc, but it cannot provide the optimal configuration for large systems. Inset, qc (obtained at the peak of the second-largest cluster) for the three best methods versus left fencekright fence. b, G(q) for a SF network with degree exponent γ = 3, maximum degree kmax = 103, minimum degree kmin = 2 and N = 2 × 105 (error bars are s.e.m. over 20 realizations). Inset, qc versus γ. The continuous blue line is the HD analytical result computed in Supplementary Information section IIG (Extended Data Fig. 1b). c, Example of SF network with γ = 3 after the removal of 15% of nodes, using the three methods HD, HDA and CI. CI produces a much reduced giant component G (red nodes).

  3. Performance of CI in large-scale real social networks.
    Figure 3: Performance of CI in large-scale real social networks.

    a, Giant component G(q) of Twitter users19 (N = 469,013) computed using CI, HDA, PR, HD and k-core strategies (other heuristics have prohibitive running times for this system size). b, Percentage of fake influencers or false positives (PFI, equation (120) in Supplementary Information) in Twitter as a function of q, defined as the percentage of non-optimal influencers identified by the HD algorithm in comparison with CI. Below , PFI reaches as much as ~40%, indicating the failure of HD in optimally finding the top influencers. Indeed, to obtain G = 0, HD has to remove a much larger number of fake influencers, which at reaches PFI 48%. c, An example of the many weak nodes found in Twitter. These crucial influencers were missed by all heuristic strategies. d, G(q) for a social network of 1.4 × 107 mobile phone users in Mexico representing an example of big data to test the scalability and performance of the algorithm in real networks. CI immunizes this social network using half a million fewer people than the best heuristic strategy (HDA), saving ~35% of the vaccine stockpile.

  4. High-degree (HD) threshold.
    Extended Data Fig. 1: High-degree (HD) threshold.

    a, HD influence threshold qc as a function of the degree distribution exponent γ of scale-free networks in the ensemble with kmax = mN1/(γ−1) and N right arrow ∞. The curves refer to different values of the minimum degree m: 1 (red), 2 (blue), 3 (black). The fragility of SF networks (small qc) is notable for m = 1 (the case calculated in ref. 10). In this case (m = 1), the network contains many leaves, and reduces to a star at γ = 2, which is trivially destroyed by removing the only single hub, explaining the general fragility in this case. Furthermore, in this same case, the network becomes a collection of dimers with k = 1 when γ right arrow ∞, which is still trivially fragile. This also explains why qc right arrow 0 for γ ≥ 4. Therefore, the fragility in the case m = 1 has its roots in these two limiting trivial cases. Removing the leaves (m = 2) results in a 2-core, which is already more robust. For the 3-core m = 3, qc 0.4–0.5 provides a quite robust network, and has the expected asymptotic limit to a non-zero qc of a random regular graph with k = 3 as γ right arrow ∞, qc right arrow (k − 2)/(k − 1) = 0.5. Thus, SF networks become robust in these more realistic cases, and the search for other attack strategies becomes even more important. b, HD influence threshold qc as a function of the degree distribution exponent of scale-free networks with minimum degree m = 2 in the ensemble where kmax is fixed and does not scale with N. The curves refer to different values of the cut-off kmax: 102 (red), 103 (green), 105 (blue), 108 (magenta), and kmax = ∞ (black), and show that for a typical kmax degree of 103, for instance in social networks, the network is fairly robust with qc 0.2 for all γ. The curve with m = 2 and kmax = 103 is replotted in the inset of Fig. 2b.

  5. Replica Symmetry (RS) estimation of the maximum eigenvalue.
    Extended Data Fig. 2: Replica Symmetry (RS) estimation of the maximum eigenvalue.

    Main panel, the eigenvalue , equation (92) in Supplementary Information for the two-body interaction = 1, obtained by minimizing the energy function with the RS cavity method. The curve was computed on an ER graph of N = 10,000 nodes and average degree left fencekright fence = 3.5 and then averaged over 40 realizations of the network (error bars are s.e.m.). Inset, comparison between the RS cavity method and EO (extremal optimization) for an ER graph of left fencekright fence = 3.5 and N = 128. The curves are averaged over 200 realizations (error bars are s.e.m.).

  6. EO estimation of the maximum eigenvalue.
    Extended Data Fig. 3: EO estimation of the maximum eigenvalue.

    Eigenvalue λ(q) obtained by minimizing the energy function (n) with τEO (τ-extremal optimization), plotted as a function of the fraction of removed nodes q. The panels are for different orders of the interactions. The curves in each panel refer to different sizes of ER networks with average connectivity left fencekright fence = 3.5. Each curve is an average over 200 instances (error bars are s.e.m.). The value qc where λ(qc) = 1 is the threshold for a particular N and many-body interaction.

  7. Estimation of optimal threshold  with EO.
    Extended Data Fig. 4: Estimation of optimal threshold with EO.

    a, Critical threshold qc as a function of the system size N, obtained with EO from Extended Data Fig. 3, of ER networks with left fencekright fence = 3.5 and varying size. The curves refer to different orders of the many-body interactions. The data show a linear behaviour as a function of N−2/3, typical of spin glasses, for each many-body interaction ρ. The extrapolated value is obtained at the y intercept. b, Thermodynamic critical threshold as a function of the order of the interactions ρ from a. The data scale linearly with 1/ρ. From the y intercept of the linear fit we obtain the thermodynamic limit of the infinite-body optimal value .

  8. Comparison of the CI algorithm for different radii [ell] of the Ball([ell]).
    Extended Data Fig. 5: Comparison of the CI algorithm for different radii of the Ball().

    We use = 1, 2, 3, 4, 5, on a ER graph with average degree left fencekright fence = 3.5 and N = 105 (the average is taken over 20 realizations of the network, error bars are s.e.m.). For = 3 the performance is already practically indistinguishable from = 4, 5. The stability analysis we developed to minimize qc is strictly valid only when G = 0, since the largest eigenvalue of the modified NB matrix controls the stability of the solution G = 0, and not the stability of the solution G > 0. In the region where G > 0 we use a simple and fast procedure to minimize G explained in Supplementary Information section VA. This explains why there is a small dependence on having a slightly larger G for larger , when G > 0 in the region q 0.15.

  9. Illustration of the algorithm used to minimize G(q) for q < qc.
    Extended Data Fig. 6: Illustration of the algorithm used to minimize G(q) for q < qc.

    Starting from the completely fragmented network at q = qc, the Nqc influencers are reinserted with their original degree and connected to their original neighbours with the following criterion: each node is assigned and index c(i) given by the number of clusters it would join if it were reinserted in the network. For example, the red node has c(red) = 2, while the blue one has c(blue) = 3. The node with the smallest c(i) is reinserted in the network: in this case the red node. Then the c(i)s are recalculated and the new node with the smallest c(i) is found and reinserted. These steps are repeated until all the removed nodes are reinserted in the network.

  10. Test of the decimation fraction.
    Extended Data Fig. 7: Test of the decimation fraction.

    Giant component G as a function of the fraction of removed nodes q using CI, for an ER network of N = 105 nodes and average degree left fencekright fence = 3.5. The profiles of the curves are drawn for different percentages of nodes fixed at each step of the decimation algorithm.

  11. Comparison of the performance of CI, BC and EGP in destroying G.
    Extended Data Fig. 8: Comparison of the performance of CI, BC and EGP in destroying G.

    We also include HD, HDA, EC, CC, k-core and PR. We use a scale-free (SF) network with degree exponent γ = 2.5, average degree left fencekright fence = 4.68, and N = 104. We use the same parameters as in ref. 11.

  12. Comparison with BP for a network immunization.
    Extended Data Fig. 9: Comparison with BP for a network immunization.

    a, Fraction of infected nodes f as a function of the fraction of immunized nodes q in the susceptible-infected-removed (SIR) model from the BP solution. We use an ER random graph of N = 200 nodes and average degree left fencekright fence = 3.5. The fraction of initially infected nodes is p = 0.1 and the inverse temperature β = 3.0. The profiles are drawn for different values of the transmission probability w: 0.4 (red curve), 0.5 (green), 0.6 (blue), 0.7 (magenta). Also shown are the results of the fixed density BP algorithm (open circles). b, Chemical potential μ as a function of the immunized nodes q from BP. We use an ER random graph of N = 200 nodes and average degree left fencekright fence = 3.5. The fraction of the initially infected nodes is p = 0.1 and the inverse temperature β = 3.0. The profiles are drawn for different values of the transmission probability w: 0.4 (red curve), 0.5 (green), 0.6 (blue), 0.7 (magenta). Also shown are the results of the fixed density BP algorithm (open circles) for the region where the chemical potential is non-convex. c, Comparison between the giant components obtained with CI, HDA, HD and BP. We use an ER network of N = 103 and left fencekright fence = 3.5. We also show the solution of CI from Fig. 2a for N = 105. We find in order of performance: CI, HDA, BP and HD. (The average is taken over 20 realizations of the network, error bars are s.e.m.) d, Comparison between the giant components obtained with CI, HDA, HD and BPD. We use a SF network with degree exponent γ = 3.0, minimum degree kmin = 2, and N = 104 nodes.

  13. Fraction of infected nodes f(q) as a function of the fraction of immunized nodes q in SIR from BP.
    Extended Data Fig. 10: Fraction of infected nodes f(q) as a function of the fraction of immunized nodes q in SIR from BP.

    We use the following parameters: initial fraction of infected people p = 0.1, and transmission probability w = 0.5. We use an ER network of N = 103 nodes and left fencekright fence = 3.5. We compare CI, HDA and BP. All strategies give similar performance, owing to the large value of the initial infection p, which washes out the optimization performed by any sensible strategy, in agreement with the results shown in figure 12a of ref. 14.

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Author information

Affiliations

  1. Levich Institute and Physics Department, City College of New York, New York, New York 10031, USA

    • Flaviano Morone &
    • Hernán A. Makse

Contributions

Both authors contributed equally to the work presented in this paper.

Competing financial interests

The authors declare no competing financial interests.

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Extended data figures and tables

Extended Data Figures

  1. Extended Data Figure 1: High-degree (HD) threshold. (171 KB)

    a, HD influence threshold qc as a function of the degree distribution exponent γ of scale-free networks in the ensemble with kmax = mN1/(γ−1) and N right arrow ∞. The curves refer to different values of the minimum degree m: 1 (red), 2 (blue), 3 (black). The fragility of SF networks (small qc) is notable for m = 1 (the case calculated in ref. 10). In this case (m = 1), the network contains many leaves, and reduces to a star at γ = 2, which is trivially destroyed by removing the only single hub, explaining the general fragility in this case. Furthermore, in this same case, the network becomes a collection of dimers with k = 1 when γ right arrow ∞, which is still trivially fragile. This also explains why qc right arrow 0 for γ ≥ 4. Therefore, the fragility in the case m = 1 has its roots in these two limiting trivial cases. Removing the leaves (m = 2) results in a 2-core, which is already more robust. For the 3-core m = 3, qc 0.4–0.5 provides a quite robust network, and has the expected asymptotic limit to a non-zero qc of a random regular graph with k = 3 as γ right arrow ∞, qc right arrow (k − 2)/(k − 1) = 0.5. Thus, SF networks become robust in these more realistic cases, and the search for other attack strategies becomes even more important. b, HD influence threshold qc as a function of the degree distribution exponent of scale-free networks with minimum degree m = 2 in the ensemble where kmax is fixed and does not scale with N. The curves refer to different values of the cut-off kmax: 102 (red), 103 (green), 105 (blue), 108 (magenta), and kmax = ∞ (black), and show that for a typical kmax degree of 103, for instance in social networks, the network is fairly robust with qc 0.2 for all γ. The curve with m = 2 and kmax = 103 is replotted in the inset of Fig. 2b.

  2. Extended Data Figure 2: Replica Symmetry (RS) estimation of the maximum eigenvalue. (119 KB)

    Main panel, the eigenvalue , equation (92) in Supplementary Information for the two-body interaction = 1, obtained by minimizing the energy function with the RS cavity method. The curve was computed on an ER graph of N = 10,000 nodes and average degree left fencekright fence = 3.5 and then averaged over 40 realizations of the network (error bars are s.e.m.). Inset, comparison between the RS cavity method and EO (extremal optimization) for an ER graph of left fencekright fence = 3.5 and N = 128. The curves are averaged over 200 realizations (error bars are s.e.m.).

  3. Extended Data Figure 3: EO estimation of the maximum eigenvalue. (157 KB)

    Eigenvalue λ(q) obtained by minimizing the energy function (n) with τEO (τ-extremal optimization), plotted as a function of the fraction of removed nodes q. The panels are for different orders of the interactions. The curves in each panel refer to different sizes of ER networks with average connectivity left fencekright fence = 3.5. Each curve is an average over 200 instances (error bars are s.e.m.). The value qc where λ(qc) = 1 is the threshold for a particular N and many-body interaction.

  4. Extended Data Figure 4: Estimation of optimal threshold with EO. (197 KB)

    a, Critical threshold qc as a function of the system size N, obtained with EO from Extended Data Fig. 3, of ER networks with left fencekright fence = 3.5 and varying size. The curves refer to different orders of the many-body interactions. The data show a linear behaviour as a function of N−2/3, typical of spin glasses, for each many-body interaction ρ. The extrapolated value is obtained at the y intercept. b, Thermodynamic critical threshold as a function of the order of the interactions ρ from a. The data scale linearly with 1/ρ. From the y intercept of the linear fit we obtain the thermodynamic limit of the infinite-body optimal value .

  5. Extended Data Figure 5: Comparison of the CI algorithm for different radii of the Ball(). (120 KB)

    We use = 1, 2, 3, 4, 5, on a ER graph with average degree left fencekright fence = 3.5 and N = 105 (the average is taken over 20 realizations of the network, error bars are s.e.m.). For = 3 the performance is already practically indistinguishable from = 4, 5. The stability analysis we developed to minimize qc is strictly valid only when G = 0, since the largest eigenvalue of the modified NB matrix controls the stability of the solution G = 0, and not the stability of the solution G > 0. In the region where G > 0 we use a simple and fast procedure to minimize G explained in Supplementary Information section VA. This explains why there is a small dependence on having a slightly larger G for larger , when G > 0 in the region q 0.15.

  6. Extended Data Figure 6: Illustration of the algorithm used to minimize G(q) for q < qc. (138 KB)

    Starting from the completely fragmented network at q = qc, the Nqc influencers are reinserted with their original degree and connected to their original neighbours with the following criterion: each node is assigned and index c(i) given by the number of clusters it would join if it were reinserted in the network. For example, the red node has c(red) = 2, while the blue one has c(blue) = 3. The node with the smallest c(i) is reinserted in the network: in this case the red node. Then the c(i)s are recalculated and the new node with the smallest c(i) is found and reinserted. These steps are repeated until all the removed nodes are reinserted in the network.

  7. Extended Data Figure 7: Test of the decimation fraction. (99 KB)

    Giant component G as a function of the fraction of removed nodes q using CI, for an ER network of N = 105 nodes and average degree left fencekright fence = 3.5. The profiles of the curves are drawn for different percentages of nodes fixed at each step of the decimation algorithm.

  8. Extended Data Figure 8: Comparison of the performance of CI, BC and EGP in destroying G. (190 KB)

    We also include HD, HDA, EC, CC, k-core and PR. We use a scale-free (SF) network with degree exponent γ = 2.5, average degree left fencekright fence = 4.68, and N = 104. We use the same parameters as in ref. 11.

  9. Extended Data Figure 9: Comparison with BP for a network immunization. (238 KB)

    a, Fraction of infected nodes f as a function of the fraction of immunized nodes q in the susceptible-infected-removed (SIR) model from the BP solution. We use an ER random graph of N = 200 nodes and average degree left fencekright fence = 3.5. The fraction of initially infected nodes is p = 0.1 and the inverse temperature β = 3.0. The profiles are drawn for different values of the transmission probability w: 0.4 (red curve), 0.5 (green), 0.6 (blue), 0.7 (magenta). Also shown are the results of the fixed density BP algorithm (open circles). b, Chemical potential μ as a function of the immunized nodes q from BP. We use an ER random graph of N = 200 nodes and average degree left fencekright fence = 3.5. The fraction of the initially infected nodes is p = 0.1 and the inverse temperature β = 3.0. The profiles are drawn for different values of the transmission probability w: 0.4 (red curve), 0.5 (green), 0.6 (blue), 0.7 (magenta). Also shown are the results of the fixed density BP algorithm (open circles) for the region where the chemical potential is non-convex. c, Comparison between the giant components obtained with CI, HDA, HD and BP. We use an ER network of N = 103 and left fencekright fence = 3.5. We also show the solution of CI from Fig. 2a for N = 105. We find in order of performance: CI, HDA, BP and HD. (The average is taken over 20 realizations of the network, error bars are s.e.m.) d, Comparison between the giant components obtained with CI, HDA, HD and BPD. We use a SF network with degree exponent γ = 3.0, minimum degree kmin = 2, and N = 104 nodes.

  10. Extended Data Figure 10: Fraction of infected nodes f(q) as a function of the fraction of immunized nodes q in SIR from BP. (98 KB)

    We use the following parameters: initial fraction of infected people p = 0.1, and transmission probability w = 0.5. We use an ER network of N = 103 nodes and left fencekright fence = 3.5. We compare CI, HDA and BP. All strategies give similar performance, owing to the large value of the initial infection p, which washes out the optimization performed by any sensible strategy, in agreement with the results shown in figure 12a of ref. 14.

Supplementary information

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