How a complex network is connected crucially impacts its dynamics and function. Percolation, the transition to extensive connectedness on gradual addition of links, was long believed to be continuous, but recent numerical evidence of ‘explosive percolation’ suggests that it might also be discontinuous if links compete for addition. Here we analyse the microscopic mechanisms underlying discontinuous percolation processes and reveal a strong impact of single-link additions. We show that in generic competitive percolation processes, including those showing explosive percolation, single links do not induce a discontinuous gap in the largest cluster size in the thermodynamic limit. Nevertheless, our results highlight that for large finite systems single links may still induce substantial gaps, because gap sizes scale weakly algebraically with system size. Several essentially macroscopic clusters coexist immediately before the transition, announcing discontinuous percolation. These results explain how single links may drastically change macroscopic connectivity in networks where links add competitively.
Percolation, the transition to large-scale connectedness of networks on gradual addition of links, occurs during growth and evolutionary processes in a large variety of natural, technological and social systems1,2,3,4,5,6. Percolation arises in atomic and molecular solids in physics as well as in social, biological and artificial networks2,7,8,9,10. In the more complex of these systems, adding links often is a competitive process. For instance, a human host carrying a virus may travel at any given time to one but not to another geographic location and therefore infect other people only at one of the places11,12. Across all percolating systems, once the number of added links exceeds a certain critical value, extensively large connected components (clusters) emerge that dominate the system.
Given the breadth of experimental, numerical and empirical studies, as well as several theoretical results and analytic arguments13,14,15,16, percolation was commonly believed to exhibit a continuous transition where the relative size of the largest cluster increases continuously from zero in the thermodynamic limit once the number of links crosses a certain threshold. Therefore, recent work by Achlioptas, D’Souza and Spencer17 came as a surprise because it suggested a new class of random percolating systems that exhibit ‘explosive percolation’18. Close to some threshold value, the system they considered shows a steep increase of the largest-cluster size on increasing the number of links; moreover, numerical scaling analysis of finite-size systems suggests a discontinuous percolation transition. This study initiated several follow-up works (for example refs 10, 19, 20, 21, 22, 23 24, 25, 26) confirming the original results for a number of system modifications. These in particular support that competition in the addition of links is crucial; the key mechanisms underlying discontinuous percolation, however, are still not well understood. Although some works have considered single-link addition processes22,27 the impact of individual link additions on competitive percolation transitions is unknown.
Gaining one or a few links may have drastic consequences for a network’s growth and its overall dynamics, depending on whether or not such individual links qualitatively alter the global connectivity of a network. For instance, spontaneous activity in developing neural circuits may become persistent after establishing some extra synaptic connections28,29. Similarly, during beginning pandemics the specific travel patterns of a single infected person may substantially change the number of infecteds on a timescale of months11.
Here we identify how microscopic single-link additions impact competitive processes. We find that, in generic percolation processes, single links do not induce macroscopic gaps in the largest cluster size as the system size . Nevertheless, the gap sizes decay weakly algebraically as N−β with often small β such that gaps are essentially macroscopic, that is, substantially large even for systems of macroscopic size N≈1023. Such gaps, induced by single links, occur at the point of percolation transitions, are a key signature of discontinuous percolation and are announced by several coexisting, essentially macroscopic clusters.
How discontinuous are the transitions?
Consider a family of competitive percolation processes where potentially new links compete with others for addition (Fig. 1). Starting with an empty graph of a large number N of isolated nodes (no links), links sequentially add in competition with others. For edge competition, for each single-link addition, m potential links are randomly selected. The link for which the sum of the sizes of the clusters containing its two end-nodes is smallest wins the competition and adds. Intra-cluster links are possible; these can only broaden the transition compared with disallowing them. For m=1, this process is non-competitive and identical to random Erdös–Rényi percolation14, whereas for m=2 it specializes to the process introduced before17. For all m≥2, this kind of competition promotes that during gradual addition of links smaller clusters tend to be connected (to form larger ones) before larger clusters grow. With increasing m, the process becomes more strongly competitive, because more potentially new links actually compete. If m is maximal, all potential links in the network compete for addition and we have global competition.
Taking the sum of cluster sizes in edge competitive processes seems somewhat arbitrary because, for example, taking the product30, or, for that matter, any convex function of the two cluster sizes, has similar competitive effects; compare ref. 17. We thus consider also clique competition, which does not suffer from this ambiguity. For clique competition, we randomly draw a fixed number k of nodes and connect those two of them contained in the two smallest clusters. Here k=2 describes non-competitive random percolation and for all k≥3 competition has the same principal effect on changes in cluster sizes as edge competition; compare refs 23, 26 for the special case k=3. We remark that for maximal possible k we again have global competition.
For large finite systems, single realizations of genuinely competitive processes (m≥2 or k≥3; compare Fig. 2) exhibit macroscopic changes in the size C1 of the largest cluster C1. In fact, numerical scaling studies (Supplementary Fig. S1) confirm that the transition regime in the plane spanned by ℓ=L/N and c1=C1/N shows an change of c1 in a region of width Δℓ that scales as N−γ,γ>0, for large N (compare also ref. 17). These results may suggest that in the limit of infinite systems there is a discontinuous gap in the curve characterizing competitive percolation in the ℓ−c1 plane.
Further investigating the microscopic dynamics of the transition, however, seeds doubt about any such gap. If the largest gap is macroscopic (extensive), we call such transitions strongly discontinuous, otherwise weakly discontinuous (see Supplementary Information for an exact definition). For weakly discontinuous transitions, the curve in the ℓ–c1 plane does in fact not exhibit any such macroscopic gap for a single-link addition in the thermodynamic limit.
Evaluating the largest jump size from extensive numerical simulations of systems up to size 222 already suggests (Fig. 3) that it scales algebraically as independent of whether the process is non-competitive, minimally competitive (k=3,m=2) or exhibits even stronger forms of competition (k≥4,m≥3). As we find that β>0 for all such processes, we have and thus the transitions are all weakly discontinuous. The only exception seems to be global competition, where we find β indistinguishable from zero and for all N, indicating a strongly discontinuous transition, equation (1). The set of all numerical analyses therefore suggests that competitive percolation transitions are generically weakly discontinuous, and single links do not induce a gap in c1 in the thermodynamic limit . Nevertheless, as the gap sizes scale weakly algebraically with system size (2) with often small β such gaps may still be essentially macroscopic, that is, substantially large even for macroscopic systems of large finite size N.
Finite-size scaling and coexisting large clusters
Further extensive numerical scaling analysis reveals that the gaps in the generic competitive percolation processes we consider indeed occur coincident with the point where the largest cluster size is discontinuous (Supplementary Fig. S2). Moreover, immediately before the transition, not only the largest gap size, but also the second-largest cluster, the third-largest cluster and so on, seem essentially macroscopic (Fig. 5). In particular, the size of the maximum second-largest cluster generically is exactly equal to the maximum gap size, see Supplementary Information for a derivation. Thus for small β the largest cluster is essentially non-unique, in contrast to standard continuous percolation transitions. This is consistent with refs 23, 26, where cluster sizes and their ranks were studied qualitatively. Finally, analytical arguments (see Supplementary Information) also demonstrate that the percolation strength, defined as the difference in largest-cluster size immediately after and immediately before the gap20,25, is exactly equal to the size of the second-largest cluster before the transition, which in turn scales with the same exponent β as the gap size (2). Further recent work also considers continuity properties of the transition regime34 on the coarse scale but does not study the impact of single links. As shown above, single-link additions induce several new distinctive features of percolation transitions and thus serve as key mechanisms controlling competitive percolation processes.
Interestingly, the so-called k-cores of the evolving graph, serving as the key example of the drastic impact of single links in traditional percolation theory35,36, exhibit dynamics very similar to that for Erdös–Rényi percolation, even for extreme processes with . The k-core of a graph is the largest subgraph with minimum degree at least k. As numerical simulations indicate (Supplementary Fig. S6), the size of the 2-core increases continuously from zero whereas k-cores for all k≥3 exhibit a discontinuous jump induced by single-link additions. These results hold for both Erdös–Rényi and competitive percolation processes. Even for extreme processes with the 2-core is still continuous, but with the location of the transition moved to larger values compared with the point of percolation. The dynamics of k-cores is thus very similar for competitive and standard, non-competitive percolation processes, in stark contrast to the dynamics of the largest-cluster size. This is true even though, as shown above, the latter is also strongly influenced by single-link additions.
These results explain how increasing the level of competition strengthens the impact of single-link additions, thus controlling the type of transition. In particular, the exponent β tells us to what extent single-link additions change macroscopic connectivity. For generic competitive processes β is smaller than for non-competitive ones (see Fig. 3), but our numerical and analytic results indicate that they are still distinct from zero. Only processes with global competition or other extreme forms of competition yield β=0 and thus a discontinuous gap induced by single-link addition. Other, more generic processes typically exhibit β>0 and thus a weakly discontinuous transition.
It is important to note that percolation processes with only moderate competition may already yield very small positive exponents and thus essentially macroscopic gaps (see Fig. 2). Here we use ‘essentially macroscopic’ to mean that the addition of single links in systems of physically large size induces gaps that are of relevant size (substantial fraction of system size) and that the gap sizes increase with stronger competition (for example increasing k), yielding a decreasing exponent β0 as kN. As a consequence, even processes actually exhibiting weakly discontinuous transitions may show large gaps in systems of physically relevant size (compare with Fig. 3). For instance, if β=0.02, a system of macroscopic, but finite size N=1023 exhibits a gap of , although formally as . For many real processes with already moderate forms of competition, we expect exponents β close to zero, and thus conjecture that single links may have a strong impact on how such a network becomes connected.
In summary, our results demonstrate how, in competitive percolation, keeping the growth rate of the largest cluster small strengthens the impact of single-link additions that merge smaller clusters. Growing (i) and overtaking (ii) markedly distinguish the microscopic dynamics in systems exhibiting competitive percolation. The more largest-cluster growth is suppressed, the more relevant the discontinuous gap becomes in large systems of given finite size. Single-link additions may then induce an essentially macroscopic gap even for weakly discontinuous transitions if competition is sufficiently strong.
Our main results do not rely on the specific random network percolation process or the specific forms of competition considered and thus suggest that the phenomena described are largely independent of the underlying system. In particular, we expect that the addition of single links can also have a strong impact on lattices (compare refs 21, 37), where so far only macroscopic observables have been studied and the impact of single links remains unknown.
Interestingly, a protein homology network has recently been identified10 showing macroscopic features akin to explosive percolation. Individual links may also induce abrupt changes in several other growing networked systems, possibly with severe consequences for the systems’ dynamics and function (compare refs 35, 36 38). For instance, growing one or a few extra synaptic connections in a neuronal circuit may strongly alter the global connectivity and thus the overall activity of the circuit28,29,39; specific infected individuals travelling to one but not another location may drastically change the patterns of infectious diseases11; and the macroscopic properties of complex systems exhibiting competitive aggregation dynamics of physical or biological units may exhibit abrupt phase transitions induced by a small set of specific individual bonds newly established; compare, for example, refs 40, 41. Our study thus not only provides recipes (by looking for certain competitive cluster formation) to identify real systems that could exhibit a (weakly) discontinuous percolation transition, but also shows that and how single-link additions in such systems may induce essentially macroscopic gaps, and in turn a collective, very abrupt change of structure and dynamics.
The current study answers how single-link dynamics underlies competitive percolation in general, but does not tell us how single-link additions are actually generated and controlled in any given real system. Future work must bridge this gap and establish how competitive percolation, and in particular the creation of essentially macroscopic jumps due to single-link additions, is influenced by predefined structure, for example for percolation processes on lattices and in geometrical or topological confinement occurring in nature10,19,20,21,23.
We thank N. Goldenfeld and I. Kanter for discussions. Supported by the Federal Ministry of Education and Research (BMBF) Germany under grant number 01GQ1005B (A.L. and M.T.) and by a grant of the Max Planck Society to M.T.
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Scientific Reports (2015)