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RESEARCH ARTICLE

Universality of the emergent scaling in finite

random binary percolation networks

Chongpu Zhai

1☯

, Dorian Hanaor

1,2☯

, Yixiang Gan

1☯

*

1The School of Civil Engineering, The University of Sydney, Sydney, New South Wales, Australia, 2Institute

for Materials Science and Technology, Technische Universita¨t Berlin, Berlin, Germany

☯These authors contributed equally to this work.

*yixang.gan@sydney.edu.au

Abstract

In this paper we apply lattice models of finite binary percolation networks to examine the

effects of network configuration on macroscopic network responses. We consider both

square and rectangular lattice structures in which bonds between nodes are randomly

assigned to be either resistors or capacitors. Results show that for given network geome-

tries, the overall normalised frequency-dependent electrical conductivities for different

capacitor proportions are found to converge at a characteristic frequency. Networks with

sufficiently large size tend to share the same convergence point uninfluenced by the bound-

ary and electrode conditions, can be then regarded as homogeneous media. For these net-

works, the span of the emergent scaling region is found to be primarily determined by the

smaller network dimension (width or length). This study identifies the applicability of power-

law scaling in random two phase systems of different topological configurations. This under-

standing has implications in the design and testing of disordered systems in diverse

applications.

Introduction

The bulk behavior of complex systems comprising disordered multi-phase components is of

importance in diverse applications including supercapacitors and batteries [1–4], dielectric

material characterization [5–10], the mechanics of structures [11–13], fracturing process [14],

thermal analysis [15] and soil probing [16]. In such systems, various parameters govern the

electrical, thermal, chemical, and/or mechanical properties of components of a system across

multiple scales from molecular up to macroscopic length-scale. Experimental and computa-

tional research efforts are increasingly conducted in order to gain insights into the manner in

which these properties combine across scales to determine overall system performance.

In particular, the AC conductivity of systems that can be schematically represented as mix-

tures of electrical components has been the subject of numerous investigations that have

shown power-law scaling with frequency arising through different relaxation mechanisms [11,

17–19]. Above a critical frequency this scaling of AC conductivity is described by Jonscher’s

power law[20] and has been experimentally observed across diverse conductor-dielectric

PLOS ONE | DOI:10.1371/journal.pone.0172298 February 16, 2017 1 / 11

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OPEN ACCESS

Citation: Zhai C, Hanaor D, Gan Y (2017)

Universality of the emergent scaling in finite

random binary percolation networks. PLoS ONE 12

(2): e0172298. doi:10.1371/journal.pone.0172298

Editor: Dante R. Chialvo, Consejo Nacional de

Investigaciones Cientificas y Tecnicas, ARGENTINA

Received: November 22, 2016

Accepted: February 2, 2017

Published: February 16, 2017

Copyright: ©2017 Zhai et al. This is an open

access article distributed under the terms of the

Creative Commons Attribution License, which

permits unrestricted use, distribution, and

reproduction in any medium, provided the original

author and source are credited.

Data Availability Statement: All relevant data are

included within the paper.

Funding: Financial support for this research is from

the Australian Research Council through grants

DE130101639 is greatly appreciated. The funder

had no role in study design, data collection and

analysis, decision to publish, or preparation of the

manuscript.

Competing interests: The authors have declared

that no competing interests exist.

composites and porous materials [5,7,18,21,22] with this scaling being termed the “Universal

Dielectric Response” [17,18,21,23]. This emergent property does not arise directly from any

particular physical or chemical properties of the involved components, but rather is a conse-

quence of the way components combine [11,19,21,24,25]. Such dielectric mixtures have

been effectively approximated as a random network of resistors and capacitors [18,19,24]

with representative conductors exhibiting a constant conductance 1/Rand dielectric compo-

nents exhibiting a variable complex admittance iωC, which is directily proportional to an

angular frequency ω, as illustrated in Fig 1. Useful asymptotic formula for the emergent net-

work admittance including both the effects of component proportions and the network size

can be obtained based on the spectral method [21,26] and the averaging approach [11,21,27].

However, establishing a more rigorous estimation necessitates numerical analysis.

From previous numerical studies [11,17–19,21,23,25,28], the typical obtained conductiv-

ity-frequency spectrum of a square lattice resistor-capacitor (RC) network can be divided into

three regions of angular frequency, ω, governed by the proportion of capacitors, p

c

, and net-

work size, N(a) an emergent region for intermediate frequencies; (b) two percolation regions;

(c) transitions between the two above-mentioned [21]. Symmetry is found between the overall

responses at high and low frequencies, which can be correlated to percolation behavior [26,29,

30]. For low and high frequencies, current tends to percolate predominantly through resistors

and capacitors respectively, as these will exhibit relatively lower impedance, as presented in Fig

1. For intermediate frequencies, where the values of admittance for the resistors and the

Fig 1. A lattice network containing W×L = 5×5 randomly distributed resistors and capacitors. The network width Wand

length Lindicate the numbers of horizontal and vertical elements in a single chain, respectively. The values of B

0

= 2, B

1

= 4, and

B

2

= 6 present the number of nodes connected directly to the electrodes. Percolation paths formed by resistors and capacitors are

shown in thick blue and orange lines, corresponding respectively to the dominating modes at low and high frequencies, for the B

0

configuration.

doi:10.1371/journal.pone.0172298.g001

Emergent scaling in binary percolation networks

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capacitors are close, we observe the power-law emergent behavior whereby conductivity is

proportional to ω

α

, with αp[11,21,27].

Motivated by the applicability of these networks for representing real-world disordered sys-

tems, in this paper we reexamine the universality of the emergent power-law scaling observed

in previous work by further considering the significance of the network aspect ratio and

boundary conditions on the convergence point and the span of the emergent region, the two

key network characteristics of universal scaling behavior.

Methods

In this paper, we extend the square lattice RC networks to rectangular ones with N=W×Lele-

ments distributed between two bus-bars, one of which is grounded and the other raised to a

potential, |V

0

|e

iωt

. A system of complex number linear equations is set up by applying Kirchh-

off’s current law (for complex currents) on each individual node of the RC network. For the

node k, we get

IkðtÞ ¼ Sn

jjIj;kjeiðotþφj;kÞ¼Sn

jðjVj;kjeiðotþφj;kÞ jVkjeiðotþφkÞÞ=Zj;k¼0;ð1Þ

where I

k

is the sum of the currents (negative or possitive) flowing towards the node k, from

connected components. The impedance of a component connected to node k,Z

j,k

, is random-

ized to be either Ror 1/iωC. The voltage potential of the connected node j,jVj;kjeiðotþφj;kÞ, with

respect to that of the node k, is represented as complex-valued function of time, t, with φ

j,k

being the relative phase. The value of nis determined by the location in the network, equaling

the number of connected components. More specifically, n= 4 for ones located away from the

boundaries, n= 2 for the lattice corners, n= 3 for the nodes on the boundaries excluding cor-

ners. The two electrodes are also regarded as nodes with n=B. The electrode dimension, B, is

defined as the number of nodes connected directly to the electrodes, e.g., B=W+ 1 when all

the elements along the boundary side are connected to the electrode. Each single node is repre-

sented by a corresponding linear algebraic equation, resulting in (W+ 1) ×(L+ 1) equations

for all the nodes. Two additional equations can be obtained from the electrodes. By solving

these equations, the potential of each node and the current going through each bonds in the

network can be calculated. Thus, for a given applied potential difference between electrodes,

|V

0

|e

iωt

, the macroscopic admittance can be given by Y¼ jI0jeiðotþφIÞ=jV0jeiot, where jI0jeiðotþφIÞ

is the obtained overall current flowing into the ground. Additionally, Frank-Lobb techniques

can be employed to reduce the network size, thus improving the computational efficiency [31].

Here, the network aspect ratio and the electrode dimension are considered as variables, in

order to investigate the influence of network configuration and boundary conditions on mac-

roscopic responses. The network size is determined by its width, W, and length, L, which rep-

resent the numbers of components in a single chain along the horizontal and vertical

directions, respectively. In the network circuit, two electrodes of identical dimension are con-

nected to elements located symmetrically in the center of the two vertical boundaries. The fre-

quency-dependent macroscopic responses obtained from different configurations, in terms of

network length, L, and width, W, are normalized through

~

Y¼jYjRL

Wþ1;e

o¼oRC;ð2Þ

where Rand Care the resistance and capacitance values of resistors and capacitors in the con-

sidered network, respectively. This normalization process is applied in order to include the sig-

nificance of all the elements in the overall network behavior, represented by the equivalent

admittance, by considering rules for simple series and parallel combinations of components.

Emergent scaling in binary percolation networks

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The span of the emergent region, S, is defined as the horizontal distance between the intersec-

tions of the power-law function, y¼opc, with top and bottom percolation admittances, e

Y1

and e

Y2(averaged from multiple simulations), for low and high frequencies, respectively, as

shown in Fig 2.

The network behaviors shown in Fig 2 in the frequency domain are primarily governed by

percolation effects [32], which are closely linked to the frequency-dependent conductivity of

each single bond in the network. In the studied rectangular RC networks with two types of

bonds (i.e., resistor and capacitor, the admittance ratio of capacitor elements with respect to

resistors is iωRC) have been considered to describe the responses of random binary networks.

The observed universal scaling behavior in Fig 2 can be also found in other networks

Fig 2. Normalised admittance module as a function of frequency. Numerical results obtained from three groups of differently

configured networks (denoted as W×L_B) are presented, with capacitor proportions, p

c

, varying from 0% (corresponding curves are

shown in black) to, 25% (red), 50% (blue), 75% (green), and to 100% (brown). The phase responses are depicted in the inset. For

each network configuration with a given capacitor proportion, five simulations have been realized.

doi:10.1371/journal.pone.0172298.g002

Emergent scaling in binary percolation networks

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containing two types of elements, indexed by aand b, exhibiting differences which can be

described in the form of S

a

=ωS

b

, e.g., mechanical stiffness, thermal conductivity, and chemi-

cal reaction rate, etc. [11,14,25].

Results and analysis

The AC electrical responses of three groups of different sized networks (expressed in the form

of W×L_B, representing network width×length_electrode dimension: 20 ×20_21, 100 ×

20_101, and 20 ×100_21) with varying p

c

are plotted in Fig 2. The three types of regions

observed in the conductance spectroscopy of square networks can be also observed here in

rectangular ones. For a given network geometry, the obtained admittance spectroscopies for

various p

c

, are found to intersect at a convergence point with the characteristic frequency of

ω= 1/RC where resistors and capacitors contribute equally to the overall conduction. This

point also appears to be the center of the emergent region. The normalized characteristic

admittance (the values of ~

Yat the characteristic frequency) at the convergence point is close to

1. This indicates that the network at the characteristic frequency perform effectively as a

mono-element network, which has a phase angle reaching the extremum value, as is shown in

Fig 2. It is further evidenced that the normalized emergent regions of the three groups of net-

works coincide with each other presenting universal features. The normalized admittance in

this common emergent region appears to be uninfluenced by the network aspect ratio (length/

width) or the electrode dimension. However, differences can be found at percolation regions

along with the corresponding transition regions. Variation of the width or length can poten-

tially change the percolation thresholds which will determine the responses, following a resis-

tive-percolated (plateaued) or capacitive-percolated (upwards or downwards) trend,

corresponding to the low and high frequency ranges, respectively.

Statistical analysis of the normalised characteristic admittance for different-sized square

networks (from 5 ×5 to 600 ×600) with B=W+ 1 was conducted and the standard deviation

(STD) of the normalized characteristic admittance is presented in Fig 3. It is found that all val-

ues of normalized characteristic admittance obtained with various p

c

(averaging over ten simu-

lations) are in the range of (0.95, 1.05) for networks with more than 10 ×10 elements. The

variance tends to diminish as the network size increases. This can be explained by considering

boundary effects that relatively smaller networks have higher percentage of boundary elements

(connected to four other elements rather than six in the bulk region). Responses of larger net-

works perform with little influence from the boundary. For a given sized square network a

larger variation is found for cases of p= 1/2, as such conditions lead to an equal likelihood of

resistive-percolated and capacitive-percolated network responses at low and high frequencies.

Consequently, there are four possible qualitatively different types of response for any realiza-

tion of the system.[21] Different available responses potentially introduce dispersion and

uncertainty of the network behavior in both percolation and emergent regions, as can be seen

in Fig 3.

We consider 2D rectangular networks with various L/W and B/(W+ 1) ratios, in order to

study the effects of network size and electrode dimension on frequency dependent responses.

The variations of normalized characteristic admittance for rectangular networks (not shown)

are comparable to those of square networks. Here, the convergence-divergence behavior is

tested with three groups of rectangular networks, which have the fixed width, W, of 20, 50, or

100 elements, respectively. The results obtained from the three groups coincide with each

other, and typical results for L/W = 1.5, 0.8, 0.2 are shown in the Fig 4A. The contour of nor-

malized characteristic admittance shown in Fig 4B presents a clear trend approaching 1, as the

network length and the electrode dimension increase. For an RC network with a given size, a

Emergent scaling in binary percolation networks

PLOS ONE | DOI:10.1371/journal.pone.0172298 February 16, 2017 5 / 11

smaller value of electrode dimension tends to constrict the current to fewer paths at zones near

the electrodes, thus, effectively reducing the network length. However, this influence will

diminish as the network lengthens. Networks with large enough length tend to share the same

intersection point uninfluenced by the boundary condition. In this case, these networks can be

defined as homogeneous systems, with the responses in emergent region unaffected by the net-

work configuration and the electrode dimension. A smaller length/width ratio or electrode

dimension will usually lead to normalized characteristic admittance values smaller than 1.

Fig 3. Mapping of the standard deviation of normalised characteristic admittance values.For varying capacitor proportions

from 0.1 to 1.0, different-sized square networks (from 5 ×5 to 600 ×600) were considered. For a given network size and capacitor

proportion, ten RC networks were generated and used in the simulations to obtain the averaged normalised characteristic

admittance, represented by the black dot. The STD of these points are used for mapping with the colour indicating the STD values, as

detailed in the legend.

doi:10.1371/journal.pone.0172298.g003

Emergent scaling in binary percolation networks

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Fig 4. Influences of network aspect ratio and electrode dimension on the values of characteristic

admittance. (A) The dependence of characteristic admittance on the B/(W+ 1) ratio, for cases of different

widths, W, of 20, 50, and 100 but same L/W, including 0.2, 0.8, and 1.5 (corresponding to ▲,■, and ●,

respectively). (B) Mapping of characteristic admittance values obtained from different-sized networks with

varying L/Wand B/(W+ 1) ratios. The square network is marked by the black dot.

doi:10.1371/journal.pone.0172298.g004

Emergent scaling in binary percolation networks

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For cases of electrodes of various geometries on the boundary or embedded in RC net-

works, the influence of the electrodes is mainly induced by the elements connected directly to

the electrodes. As the network size increases, boundary elements and electrode-affected ele-

ments will have a decreasing percentage. Therefore, the electrode size effect along with the

boundary effect will be unobservable for a sufficiently large network. The trend can be found

in Fig 4B that the characteristic admittance values for an increasingly large network locate well

upwards and to the right from the red zone, asymptotically approaching 1. This universal char-

acteristic can be also extended for an infinite network, with the normalized intersection value

to be 1.

The power law dependence of the electrical responses on frequency is of a universal nature

for a wide range of complex RC networks. To further interpret this emergent behavior, we

investigate the spans of the power-law emergent region. The convergence point tends to be the

geometric center of all the emergent regions for various p

c

. By considering the region center

reported here combined with the span, S, the emergent scaling behavior region can be well

described.

It has been observed [21] that, for a square network with p

c

= 0.5, the span of the power-law

emergent region increases without bound as the network size increases. In this paper, using

the results from square networks as references, we compare the spans of emergent regions

obtained for various sized rectangular networks with different electrode dimensions, as p

c

var-

ies. We found that when the normalized characteristic admittance value is sufficiently close to

1, it is the smaller dimension, S

min

= min(W,L) that determines the span of emergent region,

while the electrode dimension has little influence on the length. Different sized homogeneous

networks with the same s

min

tend to effectively present identical responses for the emergent

region with a given p

c

. However, discrepancies can be found for responses at low and high fre-

quencies dominated by percolation behavior. This likely to be the case also for an intersection

value far away from 1, but with lower accuracy due to the instability of the responses on

account of the boundary effects. Only the results with p

c

from 0 to 0.5 are discussed and pre-

sented in Fig 5, with the consideration of symmetricity of network responses.

The results shown here for varied network and electrode dimensions shed light on the

behavior of infinitely-large binary percolation-type network and large networks with irregular

boundaries (e.g., in the shape of spline curves) and electrodes (e.g., with the geometry of circu-

lar zones, embedded in the network, or unequal-sized electrodes). As long as the network size

is significantly larger than the electrode and boundary dimension, the presenting universal fea-

tures will not be influenced by the boundary and electrode conditions. This enables the net-

work responses to reach a robust and reliable status at the emergent region with the span being

determined by the effective network size in the order of De2(D

e

is the distance between positive

electrode and ground, indicating the shortest current path) and p

c

(dominating the slope of

universal power law). Additionally, evident trend presented in Fig 5 supports that infinitely

large emergent scaling regions can be observed for various capacitor proportions.

Conclusion

We studied the influences of network geometry and electrode dimension on the electrical

responses of rectangular RC networks. The universal scaling behavior can be fully character-

ized using the center and the length of the emergent region, i.e., the convergence point, and

the span, respectively. For both square and rectangular networks, a convergence point is

observed at the characteristic frequency, ω= 1/RC, which usually appears to be the center of

the emergent region. At this characteristic frequency, the normalized admittance value |Y|RL/

(W+ 1) approaches 1 as the length-to-width ratio and electrode dimension increase. For a

Emergent scaling in binary percolation networks

PLOS ONE | DOI:10.1371/journal.pone.0172298 February 16, 2017 8 / 11

defined homogeneous network, the span of the emergent range is primarily determined by the

shorter dimension of width and length. These observations provide a unified description for

the emergent scaling properties of network responses for random two-phase systems with

varying topological configurations. The comprehensive understanding of this emergent scaling

can guide the design and testing of disordered systems in terms of determining testing condi-

tions (e.g., the shape, size, location, and spacing of the fixtures), boundary conditions, and sys-

tem dimensions.

Author Contributions

Conceptualization: CZ DH YG.

Data curation: CZ.

Fig 5. Dependence of the emergent region span on network size and electrode dimension. The results for square networks are

shown by solid lines with error bars obtained across ten simulations for each point. Results of rectangular networks (10 ×100_11,

100 ×10_101 and 10 ×1000_5) are presented by dashed lines. The inset compares the spans of different-sized square networks with

p

c

= 0.5 with those of rectangular networks with various network configurations.

doi:10.1371/journal.pone.0172298.g005

Emergent scaling in binary percolation networks

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Formal analysis: CZ DH YG.

Funding acquisition: YG.

Investigation: CZ DH YG.

Methodology: CZ DH YG.

Project administration: YG.

Resources: YG.

Software: CZ.

Supervision: DH YG.

Validation: CZ YG.

Visualization: CZ YG.

Writing – original draft: CZ DH YG.

Writing – review & editing: CZ DH YG.

References

1. Bertei A, Choi H-W, Pharoah J, Nicolella C. Percolating behavior of sintered random packings of

spheres. Powder technology. 2012; 231:44–53.

2. Vo

¨lker B, McMeeking RM. Impact of particle size ratio and volume fraction on effective material parame-

ters and performance in solid oxide fuel cell electrodes. Journal of Power Sources. 2012; 215:199–215.

3. Zhai C, Hanaor D, Proust G, Brassart L, Gan Y. Interfacial electro-mechanical behaviour at rough sur-

faces. Extreme Mechanics Letters. 2016; 9:422–429.

4. Sabatier J, Aoun M, Oustaloup A, Gre

´goire G, Ragot F, Roy P. Fractional system identification for lead

acid battery state of charge estimation. Signal processing. 2006; 86(10):2645–57.

5. Papathanassiou A, Sakellis I, Grammatikakis J. Universal frequency-dependent ac conductivity of con-

ducting polymer networks. Applied Physics Letters. 2007; 91(12):122911.

6. Dyre JC, Maass P, Roling B, Sidebottom DL. Fundamental questions relating to ion conduction in disor-

dered solids. Reports on Progress in Physics. 2009; 72(4):046501.

7. Bakkali H, Dominguez M, Batlle X, Labarta A. Universality of the electrical transport in granular metals.

Scientific Reports. 2016; 6.

8. Li W, Schwartz RW. ac conductivity relaxation processes in CaCu3Ti4O12 ceramics: Grain boundary

and domain boundary effects. Applied physics letters. 2006; 89(24):242906.

9. Unuma T, Umemoto A, Kishida H. Anisotropic terahertz complex conductivities in oriented polythio-

phene films. Applied Physics Letters. 2013; 103(21):213305.

10. Nawroj AI, Swensen JP, Dollar AM. Electrically Conductive Bulk Composites through a Contact-Con-

nected Aggregate. PloS one. 2013; 8(12):e82260. doi: 10.1371/journal.pone.0082260 PMID: 24349239

11. Murphy K, Hunt G, Almond DP. Evidence of emergent scaling in mechanical systems. Philosophical

Magazine. 2006; 86(21–22):3325–38.

12. Picu R. Mechanics of random fiber networks—a review. Soft Matter. 2011; 7(15):6768–85.

13. Tighe BP. Dynamic critical response in damped random spring networks. Physical review letters. 2012;

109(16):168303. doi: 10.1103/PhysRevLett.109.168303 PMID: 23215140

14. Moreira A, Oliveira C, Hansen A, Arau

´jo N, Herrmann H, Andrade J Jr. Fracturing highly disordered

materials. Physical review letters. 2012; 109(25):255701. doi: 10.1103/PhysRevLett.109.255701

PMID: 23368480

15. Pollock H, Hammiche A. Micro-thermal analysis: techniques and applications. Journal of Physics D:

Applied Physics. 2001; 34(9):R23.

16. Deschamps R, Siddiqui S, Drnevich V. Time domain reflectometry development for use in geotechnical

engineering. 2000.

17. Almond D, Vainas B. The dielectric properties of random R-C networks as an explanation of theuniver-

sal’power law dielectric response of solids. Journal of Physics: Condensed Matter. 1999; 11(46):9081.

Emergent scaling in binary percolation networks

PLOS ONE | DOI:10.1371/journal.pone.0172298 February 16, 2017 10 / 11

18. Almond DP, Bowen C. Anomalous power law dispersions in ac conductivity and permittivity shown to

be characteristics of microstructural electrical networks. Physical review letters. 2004; 92(15):157601.

doi: 10.1103/PhysRevLett.92.157601 PMID: 15169318

19. Mccullen NJ, Almond DP, Budd CJ, Hunt GW. The robustness of the emergent scaling property of

random RC network models of complex materials. Journal of Physics D: Applied Physics. 2009; 42

(6):064001.

20. Jonscher AK. The universal dielectric response. nature. 1977; 267:673–9.

21. Almond D, Budd C, Freitag M, Hunt G, McCullen N, Smith N. The origin of power-law emergent scaling

in large binary networks. Physica A: Statistical Mechanics and its Applications. 2013; 392(4):1004–27.

22. Galvão RKH, Hadjiloucas S, Kienitz KH, Paiva HM, Afonso RJM. Fractional order modeling of large

three-dimensional RC networks. Circuits and Systems I: Regular Papers, IEEE Transactions on. 2013;

60(3):624–37.

23. Bouamrane R, Almond DP. The ‘emergent scaling’phenomenon and the dielectric properties of random

resistor–capacitor networks. Journal of Physics: Condensed Matter. 2003; 15(24):4089.

24. Creyssels M, Falcon E, Castaing B. Scaling of ac electrical conductivity of powders under compression.

Physical Review B. 2008; 77(7):075135.

25. Almond DP, Budd CJ, McCullen NJ. Emergent behaviour in large electrical networks. Approximation

Algorithms for Complex Systems: Springer; 2011. p. 3–26.

26. Jonckheere T, Luck J. Dielectric resonances of binary random networks. Journal of Physics A: Mathe-

matical and General. 1998; 31(16):3687.

27. Milton GW. Bounds on the complex dielectric constant of a composite material. Applied Physics Letters.

1980; 37:300–2.

28. Almond DP, Bowen C, Rees D. Composite dielectrics and conductors: simulation, characterization and

design. Journal of Physics D: Applied Physics. 2006; 39(7):1295.

29. Clerc J, Giraud G, Laugier J, Luck J. The electrical conductivity of binary disordered systems, percola-

tion clusters, fractals and related models. Advances in Physics. 1990; 39(3):191–309.

30. Bertei A, Nicolella C. A comparative study and an extended theory of percolation for random packings

of rigid spheres. Powder technology. 2011; 213(1):100–8.

31. Frank D, Lobb C. Highly efficient algorithm for percolative transport studies in two dimensions. Physical

Review B. 1988; 37(1):302.

32. Sakellis I, Papathanassiou A, Grammatikakis J. Scaling and universality of ac conductivity and dielectric

response in disordered materials under pressure. Applied Physics Letters. 2010; 97(4):042904.

Emergent scaling in binary percolation networks

PLOS ONE | DOI:10.1371/journal.pone.0172298 February 16, 2017 11 / 11