# Percolation on a maximally disassortative network

###### Abstract

We propose a maximally disassortative (MD) network model which realizes a maximally negative degree-degree correlation, and study its percolation transition to discuss the effect of a strong degree-degree correlation on the percolation critical behaviors. Using the generating function method for bipartite networks, we analytically derive the percolation threshold and a critical exponent. For the MD scale-free networks, whose degree distribution is , we show that the criticalities on the MD networks and corresponding uncorrelated networks are the same for but are different for . A strong degree-degree correlation significantly affects the percolation critical behaviors in heavy-tailed scale-free networks. Our analytical results for the critical exponents are numerically confirmed by a finite-size scaling argument.

## I Introduction

Numerous complex systems are abstracted as networks consisting of simplified elements (nodes) and their connections (edges) e.g., the Internet, World Wide Web, and prey/predator relations in ecosystems Caldarelli_text (). It is well known that most real-world networks are scale-free, such that the degree distribution obeys the power-law function: , where is called the degree exponent. The robustness of networks to failures and attacks has been frequently discussed by considering the network percolation problem. Some studies concerning percolation models on networks have shown that scale-free networks are extremely robust to the random removal of nodes and edges; however, they are fragile to the targeted removal of the high-degree nodes Newman_text ().

Percolation on networks has been studied in order to also clarify the relation between the critical phenomena and underlying network structures. In a seminal work concerning the criticality of the percolation transitions on networks, Cohen et al. Cohen02 () analytically derived the relation between the percolation critical exponents and degree exponent, , for (degree-)uncorrelated scale-free networks. Specifically, an unconventional universality class emerges for , and a mean-field class is observed for . In addition, the fractal dimension of a percolating cluster at a criticality Cohen03 () and the upper critical dimension for uncorrelated scale-free networks Wu07 () have been established.

In real-world networks, a degree-degree correlation, which is the correlation of the degrees of the nodes directly connected by an edge, would arise Newman02 (); Newman03 (). In a network with a positive (negative) degree-degree correlation, similar (dissimilar) degree nodes tend to connect to each other. A degree-degree correlated structure can affect critical phenomena on a network. Goltsev et al. Goltsev08 () treated percolation on degree-degree correlated networks analytically using the eigenvectors and associated eigenvalues of a branching matrix defined by the conditional probability, , according to which a random neighbor of a degree- node has degree . They showed the necessary and sufficient conditions that the critical behavior of a percolation on a degree-degree correlated network is the same as that on uncorrelated networks with an identical degree distribution. When a network does not satisfy any of their conditions, its critical behavior does not coincide with that of the corresponding uncorrelated networks. Two strongly correlated networks analyzed in Goltsev08 () actually violate one of their conditions, and so, exhibit an atypical universality class depending on the details of the network structure. Further studies are needed to attain a better understanding of the criticality of correlated networks; however, there is little research on the investigation of percolation transition on strongly correlated networks owing to the lack of other solvable models.

In this paper, we propose a solvable model in which the networks realize a maximally negative degree-degree correlation. Hereinafter, we refer the networks as maximally disassortative (MD) networks. Applying the generating function method for bipartite networks to the percolation on the MD networks, we analytically derive the percolation threshold and critical exponent, , related to the relative size of the giant component. For the MD scale-free networks with , an unconventional criticality is observed: the critical exponent, , acquires a value that is different from that of the uncorrelated networks. Contrastingly, the MD scale-free networks with belong to the same universality class as that of the uncorrelated ones. Our analytical estimations are confirmed by a finite-size scaling analysis near the zero percolation threshold Radicchi15 ().

The remainder of this paper is organized as follows. In Sec. II, we introduce the MD networks described as bipartite networks. Starting with recalling the generating function approach for percolation on bipartite networks in Sec. III, we analyze the criticality of the percolation transition on the MD networks in Sec. IV. In Sec. V, the result presented in Sec. IV is validated by Monte-Carlo simulations. Section VI is devoted to the conclusion and discussions.

## Ii Maximally disassortative network

Let us construct our MD networks as bipartite networks in which each node belongs to either of the two groups, A or B, and each edge connects a group A node and group B node (Fig. 1). First, the number, , of nodes in group A is given, and the number of stubs, i.e., degree of each node in group A is assigned by a predetermined degree distribution, . In this study, we consider a power-law degree distribution for group A, i.e.,

(1) |

for large . Next, to realize a maximally negative degree-degree correlation, we designate a minimum degree, , of the network, prepare nodes in group B, and let all the group B nodes have degree . Specifically, the degree distribution, , for group B is given by

(2) |

where is the Kronecker delta. For a network to be bipartite, the total number of degrees in group A should be equal to that in group B. Then, the number, , of nodes in group B is determined from the relation, , or equivalently,

(3) |

where is the average degree of group A. An edge is formed by randomly selecting a stub from each of groups A and B and joining the stubs. This process is repeated until no stub exists to realize. Then, an MD network with the degree distribution,

(4) |

is realized. Here, . Because any group A node is connected to only the group B nodes having the minimum degree, , the degree-degree correlation of the entire network is totally negative, i.e., disassortative. This type of network realizes an MD structure, in that any edge swapping increases the Spearman’s rank correlation coefficient for the network Fujiki17 ().

## Iii Percolation on bipartite networks

We briefly recall the generating function method for percolation on bipartite networks with arbitrary and Newman02-2 () prior to the analysis of the MD networks. We start with the generating functions for the degree distributions of groups A and B, i.e.,

(5) | ||||

(6) |

Similarly, the generating functions, and , for the so-called excess degree distributions of groups A and B are given by

(7) | ||||

(8) |

respectively. Here, and are the first moments of and , respectively.

Let us consider the site percolation process on a bipartite network. Each node is occupied with probability and unoccupied with probability . Let [] be the generating function for the probability of reaching a branch of a finite size by an edge outgoing from a node in group B (A). Under the assumption that a given network is locally tree-like, and satisfy the following equations:

(9) | ||||

(10) |

Considering that and are the probabilities to reach a finite branch by an edge outgoing from a group B node and group A node, respectively, and in the percolating phase, i.e., . Thus, substituting for Eqs. (9) and (10), we have

(11) | ||||

(12) |

To derive the percolation threshold for the bipartite network, we introduce the generating function, , for the probability of a node belonging to a cluster of a finite size:

(13) |

where [] is the generating function for the probability of a node in group A (B) belonging to a cluster of a finite size as

(14) | ||||

(15) |

Thus, the size of the giant component, , is given by

(16) |

and the average size of the finite clusters, , is given as . In the non-percolating phase () where , this average cluster size reduces to

(17) |

where and are

(18) |

and

(19) |

respectively. The percolation threshold is given as the point at which in Eq. (17) diverges. Equations (17)–(19) lead to the result that diverges at given by

(20) |

where . The percolation threshold (20) corresponds to that for a bond percolation on bipartite networks, which has been previously obtained by several approaches Newman02-2 (); Allard09 (); Hooyberghs10 (); Bianconi17 ().

## Iv Criticality of percolation on MD network

To discuss the effect of the MD structures on the criticality of a scale-free network, we concentrate on the MD networks having () and . Applying Eq. (20) to the MD networks, we obtain the percolation threshold as

(21) |

Because Eq. (8) reduces to in the present case, we have

(22) |

from Eqs. (11) and (12). At , where is a positive infinitesimal value, is slightly smaller than unity, i.e., . Here, is the order parameter and a positive infinitesimal value. Hence, Eq. (22) at is

(23) |

The summation in Eq. (23) determines the critical behavior of the percolation on the MD networks. For with in which , the summation in Eq. (23) has an asymptotic form,

(24) |

where is a constant. Substituting Eq. (24) into Eq. (23), we obtain

(25) |

Because for , the leading term of the right-hand side in Eq. (24) is , we approximate by in Eq. (25) and obtain

(26) |

Consequently, the order parameter, , is related to the difference, , as

(27) |

which implies that the critical exponent, , related to the relative size, , of the giant component, , is for the MD networks. This value corresponds to that for the uncorrelated scale-free networks, , having the same degree exponent Cohen02 (); Goltsev08 (). For , we easily find the mean-field result:

(28) |

i.e., .

When , the percolation threshold is , and the summation in Eq. (23) becomes

(29) |

Substituting Eq. (29) and into Eq. (23), we derive

(30) |

Thus, near , the order parameter behaves as

(31) |

with for . This indicates that for , is different from the critical exponent for the uncorrelated networks, Cohen02 (). It should be also mentioned that for , the giant component size (16) behaves as

(32) |

To summarize, the relation, , holds for , whereas holds for , as displayed in Table 1.

## V Numerical check

In this section, we confirm the validity of our theoretical result for as discussed in the previous section. Along with the finite-size scaling argument in Stauffer (), we assume that for a network with a finite percolation threshold, , the relative size, , of the giant component near the percolation threshold behaves as

(33) |

where is a scaling function and the critical exponent, , is related to the correlation length, , as . The dependence of the pseudo percolation threshold, , is given as

(34) |

where is and is a constant Stauffer (). Substituting Eq. (34) into Eq. (33), we have the dependence of at the pseudo percolation threshold as

(35) |

where . When , in which , and the giant component size behaves as Eq. (32), we expect a finite-size scaling of as

(36) |

and

(37) |

instead of Eqs. (33) and (35). Fitting the numerical data by Eq. (37) for [Eq. (35) for ] with the help of Eq. (34) yields the critical exponents, and Radicchi15 (). Thus, we numerically obtain for the MD networks with a given value of to validate the relation, [], theoretically expected for the MD networks with ().

The MD networks employed for our numerical confirmations are generated as follows. First, according to the Dorogovtsev–Mendes–Samukhin (DMS) model Dorogo00 (), a degree sequence obeying with nodes and , which represents group A comment1 (), is generated. Second, nodes with are prepared as group B. Finally, to construct a network, we repeatedly choose a stub at random from each of the groups and join the stubs until all stubs are used up. The entire network has nodes and the average degree of . For comparisons, uncorrelated networks with an identical degree sequence are realized by randomizing the MD networks under preserving the degree of each node. The site percolation process is performed numerous times on the MD networks and uncorrelated ones. Using the Newman–Ziff algorithm Newman01 (), we obtain the average size of the largest component generated by the site percolation process on the networks.

First, we examine the relative size, , of the largest component. Figure 2 shows (the solid line) for the MD networks with degree exponent and (the dashed line) for the corresponding uncorrelated networks. The percolation threshold, , for the MD networks is larger than that for the uncorrelated ones, as shown in Table 1. On the other hand, both the MD networks and uncorrelated ones have the zero percolation threshold, , for (the inset of Fig. 2).

Next, by numerically evaluating the critical exponents, and , for the uncorrelated scale-free networks, we confirm the validity of the present scalings (34) and (37). Figure 3(a) shows the dependence of the pseudo percolation threshold, , for the uncorrelated scale-free networks with . The red filled circles represent the simulation result. Here, the pseudo percolation threshold, , for the uncorrelated networks is estimated by substituting the numerically obtained degree distributions for the Molly–Reed criterion, , where Newman01 (); Cohen00 (). According to [3], we theoretically obtain as when the maximum degree behaves as (Dorogo02, , see the inset of Fig.3(a)). The dashed line is drawn by using the theoretical value, for . The red filled circles lie on the dashed line, confirming that Eq. (34) for the uncorrelated networks is correct. Figure 3(b) shows the dependence of the largest component size over the pseudo percolation threshold, . The red filled circles representing the simulation results lie on the dashed line drawn using Eq. (37) with the theoretical values of and . Thus the finite-size scalings for , Eqs. (34) and (37), succeed in capturing the criticality of the percolation on the uncorrelated networks, as was reported in Radicchi15 ().

Finally, we consider the criticality on the MD networks. The black filled squares in Fig. 3(a) represent the simulation results for the MD scale-free networks with . The slope of the black filled squares is estimated as and is different from that for the uncorrelated networks (the red filled circles), which indicates that the criticality of the percolation on the MD networks differs from that on the uncorrelated ones. Here, the solid line is a guide to the eye with a slope of . Substituting and for , we have for . In Fig. 3(b), we depict a solid line with the slope, . The line is parallel to the black filled squares, which represent the simulation results for the percolation on the MD networks, thereby supporting the theoretical result for . In the inset of Fig. 3(b), we observe a correspondence with the critical exponent, , for the MD networks and uncorrelated networks for the degree exponent, . The results exhibit the validity of our theoretical arguments in Sec. IV.

## Vi Conclusion & Discussion

We have studied the site percolation on a maximally disassortative (MD) network, which has a maximally negative degree-degree correlation and can be regarded as a bipartite network. Based on the generating function method for bipartite networks, we have clarified the percolation threshold and criticality of the MD networks. We have found that the criticality of the site percolation on the MD networks is different from that on the uncorrelated networks when the network is heavy-tailed so that with . For , the criticality of the percolation on the MD networks corresponds to that on the uncorrelated networks. This conclusion has been numerically confirmed by a finite-size scaling analysis.

It should be mentioned how the bond percolation behaves in the present network; the site-bond percolation universality in the uncorrelated random networks breaks in terms of the scaling for the giant component size, , if the percolation threshold is zero Radicchi15 (). We can obtain the exponent, , for the bond percolation on the MD networks similar to the description in Secs III and IV or using the analysis of a biased bond percolation Hooyberghs10-2 (). In Hooyberghs10-2 (), Hooyberghs et al. investigated a biased bond percolation on networks. In the biased bond percolation, an edge is removed with a probability proportional to the power of the degree product of its two ends. Considering the limit of the -state Potts model, they developed an analytical treatment for the biased bond percolation on uncorrelated networks. Their analytical treatment was generalized to bipartite networks whose groups obey their respective power-law degree distributions Hooyberghs10 (). Considering the ordinary bond percolation on the MD networks as a particular case of the biased bond percolation on bipartite networks, we find that for the MD networks, exponent of the bond percolation is same as that of the site percolation ( in Table 1) irrespective of . Thus, this implies that the site-bond percolation universality coincides in terms of the order parameter, . With respect to the singularity of the giant component size, , the bond percolation on the MD networks shows when (), which is different from the case of the site percolation [Eq. (32)]. In the latter case, the site-bond percolation universality breaks in terms of the giant component size, , as was reported in Radicchi15 (). It can be confirmed numerically by means of the finite-size scaling argument for Monte-Carlo data (not shown).

In this study, we have focused on networks having only a nearest neighbor degree correlation. In general, however, real-world networks have a long-range degree correlation, which cannot be captured by any nearest neighbor degree correlation Fujiki17 (); Rybski10 (); Fujiki18 (); Mayo15 (); Fujiki19 (). Little is known about what long-range correlated structures induce. Some numerical studies Noh07 (); Menche11 (); Valdez11 () on correlated networks with the tunable degree-degree correlation have suggested that an unusual type of phase transition originates from something beyond the nearest neighbor degree correlation. Further studies to understand how correlated structures beyond the nearest neighbor degree correlations affect the critical phenomena of the networks should be conducted.

###### Acknowledgements.

S.M. and T.H. acknowledge the financial support from JSPS (Japan) KAKENHI Grant Number JP18KT0059. S.M. was supported by a Grant-in-Aid for Early-Career Scientists (No. 18K13473) and Grant-in-Aid for JSPS Research Fellow (No. 18J00527) from the Japan Society for the Promotion of Science (JSPS) for performing this work. T.H. acknowledges the financial support from JSPS (Japan) KAKENHI Grant Numbers JP16H03939 and JP19K03648.## References

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