Abstract
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 singlelink 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.
Main
Percolation, the transition to largescale connectedness of networks on gradual addition of links, occurs during growth and evolutionary processes in a large variety of natural, technological and social systems^{1,2,3,4,5,6}. Percolation arises in atomic and molecular solids in physics as well as in social, biological and artificial networks^{2,7,8,9,10}. 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 places^{11,12}. 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 arguments^{13,14,15,16}, 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 Spencer^{17} 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 largestcluster size on increasing the number of links; moreover, numerical scaling analysis of finitesize systems suggests a discontinuous percolation transition. This study initiated several followup 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 singlelink addition processes^{22,27} 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 connections^{28,29}. Similarly, during beginning pandemics the specific travel patterns of a single infected person may substantially change the number of infecteds on a timescale of months^{11}.
Here we identify how microscopic singlelink 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≈10^{23}. 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 singlelink addition, m potential links are randomly selected. The link for which the sum of the sizes of the clusters containing its two endnodes is smallest wins the competition and adds. Intracluster links are possible; these can only broaden the transition compared with disallowing them. For m=1, this process is noncompetitive and identical to random Erdös–Rényi percolation^{14}, whereas for m=2 it specializes to the process introduced before^{17}. 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 product^{30}, 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 noncompetitive 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 C_{1} of the largest cluster C_{1}. In fact, numerical scaling studies (Supplementary Fig. S1) confirm that the transition regime in the plane spanned by ℓ=L/N and c_{1}=C_{1}/N shows an change of c_{1} 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−c_{1} 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–c_{1} plane does in fact not exhibit any such macroscopic gap for a singlelink addition in the thermodynamic limit.
Evaluating the largest jump size from extensive numerical simulations of systems up to size 2^{22} already suggests (Fig. 3) that it scales algebraically as independent of whether the process is noncompetitive, 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 c_{1} 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.
The impact of single links
So how can single links actually impact the dynamics of the transition? For the extreme case of global competition, exact analytical arguments reveal the occurrence of macroscopic jumps and give key insights about the nature of transitions in competitive percolation processes, that similarly hold for weakly discontinuous transitions (see below). We label all existing clusters by C_{i} and their sizes by C_{i}=C_{i} where the index i labels their size rank such that where denotes the total number of existing clusters. For global competition each newly added link joins the two smallest clusters in the entire system such that . For simplicity of presentation, we choose the system size N to be a power of 2. This ensures that up to L_{1}=N/2 new links only connect 1clusters (isolated nodes) to result in new 2clusters (two nodes with a single connecting link) such that the maximum cluster size remains C_{1}=2 for all L≤L_{1}. The subsequent N/4 links each connect 2clusters to 4clusters, keeping C_{1}=4 until L_{2}=3N/4. In general, new links added between L_{n−1} and L_{n} connect 2^{n−1}clusters to 2^{n}clusters, keeping C_{1}=2^{n} where L_{n}=((2^{n}−1)/2^{n})N, for all n≤log_{2}(N). In the final step, at L=N−1, the remaining two N/2clusters join and induce the largest gap analytically confirming the numerical findings (Figs 2 and 3). As a consequence, global competition (involving information about the entire system’s state for local link addition) implies a genuine gap of size 1/2 in the main order parameter c_{1}.
For weaker forms of competition, closely related linkadding mechanisms control the cluster joining dynamics. Inspecting the impact of singlelink additions on cluster joining dynamics in more detail, we identify three distinct mechanisms that may contribute towards increasing the size C_{1} of the current largest cluster in more general competitive processes. (i) Direct largestcluster growth: the largest cluster itself connects to a smaller cluster of size C_{i}<C_{1} and grows, C_{1}+C_{i}C_{1}, to stay the largest cluster. (ii) Overtaking: two smaller clusters of size C_{i},C_{j}<C_{1} join into one that is larger than the current largest cluster, C_{i}+C_{j}C_{1}, and the originally largest cluster becomes second largest, C_{1}C_{2}. (iii) Doubling: if there are several clusters of maximal size C_{1}=C_{2}=…=C_{ν} for some ν≥2, two of these join, C_{i}+C_{j}C_{1} for some i,j∈{1,…,ν}, creating a new largest cluster of twice the size of the original one.
For each singlelink addition, we denote the probability for direct cluster growth (i) by . We say that if the probability of direct growth of the largest cluster (i) is zero up to the point where only two clusters are left in the system and direct growth is the only remaining way the largest cluster could grow at all (see Supplementary Information for a more formal definition).
As we show in the following, an arbitrary percolation process with necessarily exhibits a genuine gap and thus a strongly discontinuous transition, that is stays positive in the limit of infinitely large system sizes N. As growth (i) is prohibited, the largestcluster size changes either by overtaking (ii) or by doubling (iii). During any such percolation process adding a link never more than doubles C_{1}. As a consequence, there is a certain L′ such that C_{1}(L′) is larger than N/3 but not larger than 2N/3. When C_{1} will be overtaken (or doubles) one more time at some L=L′+ΔL, the cluster previously largest becomes the second largest, C_{1}C_{2} (or disappears in the case of doubling). Thus it is guaranteed that during percolation two clusters of sizes C_{1}≥N/3 and C_{2}≥N/3 are generated, which necessarily join at some time L>L′. Therefore, in any such competitive process, prohibited growth implies that the largest gap is macroscopic, for all N. Hence, all competitive percolation processes with show strongly discontinuous transitions with a strong impact of singlelink additions. As we show in the Supplementary Information, such a gap necessarily occurs at or beyond ℓ_{c}=1; thus, for extremal competition with the percolation point, where the largest cluster becomes macroscopic, does not necessarily coincide with the point where the largest gap occurs.
Single links induce gaps in large finite systems
Nevertheless, many weakly discontinuous transitions still exhibit essentially macroscopic gaps for large finite systems. We conjecture that competitive percolation processes in nature (or engineering or the social world), in particular spatially extended systems with limitedrange interactions will naturally enable the largest cluster to grow, (as do all competitive percolation processes for nonglobal clique and edge competition), and they generically exhibit weakly discontinuous (if not continuous) percolation transitions^{31,32}. In specific limiting models analytic meanfield considerations yield thus confirming (2). For instance, in a model variant where the largest cluster joins with the smallest available with probability p, and otherwise the two smallest clusters join with probability 1−p, we analytically find that (see Supplementary Information for a detailed derivation) for 0≤p≪1 scales roughly linearly with p.
Notably, if largestcluster growth does not occur, p=0, we have β=0 and in the thermodynamic limit, consistent with equation (4). More importantly, these results show that even when the largest cluster grows the slightest, that is, for arbitrarily small p>0, the percolation transition is weakly discontinuous, because β>0 as soon as . Direct numerical simulations agree well with our analytical prediction (6); see Fig. 4. The finding that β>0 as soon as p>0 is consistent with the above general result (compare with equation (2) and equation (5)) that for arbitrarily small probability of cluster growth the percolation transition is already weakly discontinuous, often with small positive exponents β and thus essentially macroscopic gaps in large finite systems (see numerical example below). More generally, the results above suggest that any process with nonmaximal competition (including nonmaximal edge competition (m=2) showing ‘explosive percolation’^{17,19,20,33}) generically shows weakly discontinuous transitions.
Finitesize 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 secondlargest cluster, the thirdlargest cluster and so on, seem essentially macroscopic (Fig. 5). In particular, the size of the maximum secondlargest 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 nonunique, 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 largestcluster size immediately after and immediately before the gap^{20,25}, is exactly equal to the size of the secondlargest 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 regime^{34} on the coarse scale but does not study the impact of single links. As shown above, singlelink additions induce several new distinctive features of percolation transitions and thus serve as key mechanisms controlling competitive percolation processes.
Interestingly, the socalled kcores of the evolving graph, serving as the key example of the drastic impact of single links in traditional percolation theory^{35,36}, exhibit dynamics very similar to that for Erdös–Rényi percolation, even for extreme processes with . The kcore of a graph is the largest subgraph with minimum degree at least k. As numerical simulations indicate (Supplementary Fig. S6), the size of the 2core increases continuously from zero whereas kcores for all k≥3 exhibit a discontinuous jump induced by singlelink additions. These results hold for both Erdös–Rényi and competitive percolation processes. Even for extreme processes with the 2core is still continuous, but with the location of the transition moved to larger values compared with the point of percolation. The dynamics of kcores is thus very similar for competitive and standard, noncompetitive percolation processes, in stark contrast to the dynamics of the largestcluster size. This is true even though, as shown above, the latter is also strongly influenced by singlelink additions.
Discussion
These results explain how increasing the level of competition strengthens the impact of singlelink additions, thus controlling the type of transition. In particular, the exponent β tells us to what extent singlelink additions change macroscopic connectivity. For generic competitive processes β is smaller than for noncompetitive 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 singlelink 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=10^{23} 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 singlelink additions that merge smaller clusters. Growing (i) and overtaking (ii) markedly distinguish the microscopic dynamics in systems exhibiting competitive percolation. The more largestcluster growth is suppressed, the more relevant the discontinuous gap becomes in large systems of given finite size. Singlelink 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 identified^{10} 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 circuit^{28,29,39}; specific infected individuals travelling to one but not another location may drastically change the patterns of infectious diseases^{11}; 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 singlelink 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 singlelink dynamics underlies competitive percolation in general, but does not tell us how singlelink 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 singlelink additions, is influenced by predefined structure, for example for percolation processes on lattices and in geometrical or topological confinement occurring in nature^{10,19,20,21,23}.
References
 1.
Strogatz, S. H. Exploring complex networks. Nature 410, 268–276 (2001).
 2.
Newman, M. E. J., Watts, D. J. & Strogatz, S. H. Random graph models of social networks. Proc. Natl Acad. Sci. 99, 2566–2572 (2002).
 3.
Song, C., Havlin, S. & Makse, H. A. Origins of fractality in the growth of complex networks. Nature Phys. 2, 275–281 (2006).
 4.
Parshani, R., Bulyrev, S. V. & Havlin, S. Interdependent networks: Reducing the coupling strength leads to a change from a first to second order percolation transition. Phys. Rev. Lett. 105, 048701 (2010).
 5.
Parshani, R., Bulyrev, S. V., Stanley, H. E. & Havlin, S. Catastrophic cascade of failures in interdependent networks. Nature 464, 1025–1028 (2010).
 6.
BenAvraham, D. & Havlin, S. Diffusion and Reactions in Fractals and Disordered Systems (Cambridge Univ. Press, 2001).
 7.
Solomon, S. G., Weisbucha, L. d. A., Janc, N. & Stauffer, D. Social percolation model. Physica A 277, 239–247 (2000).
 8.
Goldstone, R. L. & Janssena, M. A. Computational models of collective behavior. Trends Cognitive Sci. 9, 424–430 (2005).
 9.
Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275–1335 (2008).
 10.
Rozenfeld, H. D., Gallos, L. K. & Makse, H. A. Explosive percolation in the human protein homology networks. Eur. Phys. J. E 75, 305–310 (2010).
 11.
Hufnagel, L., Brockmann, D. & Geisel, T. Forecast and control of epidemics in a globalized world. Proc. Natl Acad. Sci. USA 101, 15124–15129 (2004).
 12.
D’Souza, R. M. Complex networks: Structure comes to random graphs. Nature Phys. 5, 627–628 (2009).
 13.
Bunde, A. & Havlin, S. Fractals and Disordered Systems (Springer, 1996).
 14.
Bollobás, B. & Riordan, O. Percolation (Cambridge Univ. Press, 2006).
 15.
Stauffer, D. & Aharony, A. Introduction to Percolation Theory (Taylor and Francis, 1993).
 16.
Grimmett, G. Percolation (Springer, 1999).
 17.
Achlioptas, D., D’Souza, R. M. & Spencer, J. Explosive percolation in random networks. Science 323, 1453–1455 (2009).
 18.
Bohmann, T. Emergence of connectivity in networks. Science 323, 1438–1439 (2009).
 19.
Cho, Y. S., Kim, J. S., Park, J., Kahng, B. & Kim, D. Percolation transitions in scalefree networks under the Achlioptas process. Phys. Rev. Lett. 103, 135702 (2009).
 20.
Radicchi, F. & Fortunato, S. Explosive percolation in scalefree networks. Phys. Rev. Lett. 103, 168701 (2009).
 21.
Ziff, R. M. Explosive growth in biased dynamic percolation on twodimensional regular lattice networks. Phys. Rev. Lett. 103, 045701 (2009).
 22.
Cho, Y. S., Kahng, B. & Kim, D. Cluster aggregation model for discontinuous percolation transitions. Phys. Rev. E 81, 030103 (2010).
 23.
Friedman, E. J. & Landsberg, A. S. Construction and analysis of random networks with explosive percolation. Phys. Rev. Lett. 103, 255701 (2009).
 24.
Moreira, A. A., Oliveira, E. A., Reis, S. D. S., Herrmann, H. J. & Andrade, J. S. Hamiltonian approach for explosive percolation. Phys. Rev. E 81, 040101 (2010).
 25.
Radicchi, F. & Fortunato, S. Explosive percolation: A numerical analysis. Phys. Rev. E 81, 036110 (2010).
 26.
D’Souza, R. M. & Mitzenmacher, M. Local cluster aggregation models of explosive percolation. Phys. Rev. Lett. 104, 195702 (2010).
 27.
Manna, S. S. & Chatterjee, A. A new route to explosive percolation. Physica A 390, 177–182 (2011).
 28.
Breskin, I., Soriano, J., Moses, E. & Tlusty, T. Percolation in living neural networks. Phys. Rev. Lett. 97, 188102 (2006).
 29.
Soriano, J., Martinez, M. R., Tlusty, T. & Moses, E. Development of input connections in neural cultures. Proc. Natl Acad. Sci. USA 105, 13758–13763 (2008).
 30.
Beveridge, A., Bohman, T., Frieze, A. & Pikhurko, O. Product rule wins a competitive game. Proc. Am. Math. Soc. 135, 3061–3071 (2007).
 31.
Binder, K. Theory of firstorder phase transitions. Rep. Prog. Phys. 50, 783–859 (1987).
 32.
Goldenfeld, N. D. Lectures on Phase Transitions and the Renormalisation Group (AddisonWesley, 1992).
 33.
Ziff, R. M. Scaling behavior of explosive percolation on the square lattice. Phys. Rev. E 82, 051105 (2010).
 34.
da Costa, R. A., Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. Explosive percolation transition is actually continuous. Preprint at http://arxiv.org/abs/1009.2534v2 (2010).
 35.
Spencer, J. & Wormald, N. Birth control for giants. Combinatorica 27, 587–628 (2008).
 36.
Spencer, J. The giant component: The golden anniversary. Not. Am. Math. Soc. 57, 720–724 (2010).
 37.
Araújo, N. A. M. & Herrmann, H. J. Explosive percolation via control of the largest cluster. Phys. Rev. Lett. 105, 035701 (2010).
 38.
Timme, M. Does dynamics reflect topology in directed networks? Europhys. Lett. 76, 367–373 (2006).
 39.
Memmesheimer, R. M. & Timme, M. Designing complex networks. Physica D 224, 182–201 (2006).
 40.
Urbanc, B. et al. Aggregation and disaggregation of senile plaques in Alzheimer disease. Proc. Natl Acad. Sci. USA 94, 7612–7616 (1996).
 41.
Dinsmore, A. D., Prasad, V., Wong, I. Y. & Weitz, D. A. Microscopic structure and elasticity of weakly aggregated colloidal gels. Phys. Rev. Lett. 96, 185502 (2006).
Acknowledgements
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.
Author information
Affiliations
Max Planck Institute for Dynamics and SelfOrganization (MPI DS) Göttingen, Bunsenstr. 10, 37073 Göttingen, Germany
 Jan Nagler
 , Anna Levina
 & Marc Timme
Institute for Nonlinear Dynamics, Faculty of Physics, University of Göttingen, Bunsenstr. 10, 37073 Göttingen, Germany
 Jan Nagler
 & Marc Timme
Bernstein Center for Computational Neuroscience (BCCN) Göttingen, Bunsenstr. 10, 37073 Göttingen, Germany
 Anna Levina
 & Marc Timme
Authors
Search for Jan Nagler in:
Search for Anna Levina in:
Search for Marc Timme in:
Contributions
All authors conceived and designed the research, contributed analysis tools and analysed the data. J.N. carried out the numerical experiments. All authors worked out the theory and wrote the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Jan Nagler.
Supplementary information
PDF files
 1.
Supplementary Information
Supplementary Information
Rights and permissions
To obtain permission to reuse content from this article visit RightsLink.
About this article
Further reading

Failure and recovery in dynamical networks
Scientific Reports (2017)

Center of mass in complex networks
Scientific Reports (2017)

Percolation transition of cooperative mutational effects in colorectal tumorigenesis
Nature Communications (2017)

Inhomogeneous Site Percolation on an Irregular Bethe Lattice with Random Site Distribution
Journal of Statistical Physics (2017)

Explosive synchronization as a process of explosive percolation in dynamical phase space
Scientific Reports (2015)