Research ArticleA Novel Antifragility Measure Based on Satisfaction and ItsApplication to Random and Biological Boolean Networks

Omar K Pineda 12 Hyobin Kim 2 and Carlos Gershenson 234

1Posgrado en Ciencia e Ingenierıa de la Computacion Universidad Nacional Autonoma de Mexico 04510 CDMX Mexico2Centro de Ciencias de la Complejidad Universidad Nacional Autonoma de Mexico 04510 CDMX Mexico3Instituto de Investigaciones en Matematicas Aplicadas y en Sistemas Universidad Nacional Autonoma de Mexico04510 CDMX Mexico4ITMO University St Petersburg 199034 Russia

Correspondence should be addressed to Carlos Gershenson cggunammx

Received 17 December 2018 Accepted 23 April 2019 Published 28 May 2019

Academic Editor Massimiliano Zanin

Copyright copy 2019 Omar K Pineda et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Antifragility is a property that enhances the capability of a system in response to external perturbations Although the concept hasbeen applied in many areas a practical measure of antifragility has not been developed yet Here we propose a simply calculablemeasure of antifragility based on the change of ldquosatisfactionrdquo before and after adding perturbations and apply it to randomBooleannetworks (RBNs) Using the measure we found that ordered RBNs are the most antifragile Also we demonstrated that sevenbiological systems are antifragile Ourmeasure and results can be used in various applications of Boolean networks (BNs) includingcreating antifragile engineering systems identifying the genetic mechanism of antifragile biological systems and developing newtreatment strategies for various diseases

1 Introduction

Antifragility suggested by Taleb is defined as a property toenhance the capability of a system in response to externalstressors [1] It is beyond resilience or robustness Whilethe resilientrobust systems resist stress and stay the sameantifragile systems not only withstand stress but also benefitfrom it The immune system is a representative exampleof antifragile systems When exposed to diverse germs atan early age our immune system strengthens and thusovercomes new diseases in the future

The concept of antifragility has been actively appliedin numerous areas such as risk analysis [2 3] physics [4]molecular biology [5 6] transportation planning [7 8]engineering [9ndash11] and aerospace and computer science [12ndash15] However a practical measure of antifragility has notbeen developed yet Here we propose a novel measure forantifragility based on the change of complexity We userandomBoolean networks (RBNs) as a case study to illustrateour measure We quantify the complexity by assessing theextent of how much the node states of a RBN are maintained

and changed during state transitions We perturb the net-work flipping the node states with the structure of thenetwork fixed Calculating the variation of the complexity inthe network before and after adding the perturbations wemeasure antifragility

BNs have a wide range of applications from biochemicalsystems [16ndash20] to economic systems [21] from socialnetworks [22 23] to robots [24] Our antifragility measurecan be utilized in various applications of BNs For instanceone could create antifragile engineered systems or identify thegenetic mechanisms of antifragile biological systems

The rest of our article is structured as follows In the sec-tion of ldquoMeasurement of Antifragility in RBNsrdquo we describeRBNs complexity of RBNs perturbations to RBNs and howto assess antifragility in RBNs In the section ldquoExperimentsrdquomethods and parameter setting for simulations are explainedIn the section of ldquoResults and Discussionrdquo the results of theantifragility of RBNs and several biological BNs are presentedand analyzed The section of ldquoConclusionsrdquo summarizes andcloses the article

HindawiComplexityVolume 2019 Article ID 3728621 10 pageshttpsdoiorg10115520193728621

2 Complexity

With perturbations0

5

10

15

20

25

30

35

40

0 10

Node0 10

Node

Tim

e0

5

10

15

20

25

30

35

40

Tim

e

Without perturbations

(a)

0 10

Node

0

5

10

15

20

25

30

35

40

Tim

e

0 10

Node

0

5

10

15

20

25

30

35

40

Tim

e

With perturbationsWithout perturbations

(b)

Figure 1 Schematic diagrams showing state transitions of (a) critical and (b) chaotic RBNs with119873 = 20119883 = 2 and119874 = 1 The left side is thenetwork without perturbations and the right one is the network with perturbations with the same initial states Each square represents thestate of a node (white = 0 black = 1) The state transitions were calculated from the initial states at the top to states at the bottom during 119879= 40 (a) 119870 = 2 (critical) ∮ = minus00958 (complexity is increased by perturbations antifragile) (b) 119870 = 3 (chaotic) ∮ = 00388 (complexity isdecreased by perturbations fragile)

2 Measurement of Antifragility in RBNs

21 Random Boolean Networks RBNs were proposed asmodels of gene regulatory networks by Kauffman [26 27] ARBN consists of N nodes representing genes Each node cantake either 0 (off inhibited) or 1 (on activated) as its stateThe node state is determined by the states of input nodesand Boolean functions assigned to each node Every node hasK input nodes (or input links) Self-inputs are allowed Thelinks are wired randomly and the Boolean functions are alsorandomly assigned Once the links and the Boolean functionsset up they remain fixed

In Figures 1(a) and 1(b) the left plots show how randomlychosen initial states are updated over time The plots aresimulated until 119879 = 40 A state space refers to the set of allthe possible configurations (2N) and all the transitions amongthem Being deterministic classic RBNs have one and onlyone successor for each state In the state space repeated statesare attractors which can be fixed points or limit cycles Theother states that lead to the attractors are basin of attractionof the attractors

Depending on the structure of the state space thereare three dynamical regimes in RBNs ordered chaotic andcritical The first two are phases while the critical regime liesat the phase transition Ordered dynamics are characterizedby the change of few node states which is related to highstability Chaotic dynamics are characterized by the changeof most node states which is associated with high variabilityCritical dynamics balance the stability of the ordered regimewith the variability of the chaotic regime [28 29] Thedynamical regimes can be varied by 119870 For RBNs withinternal homogeneity (ie the probability that a gene isactivated [30]) 119901 = 05119870 = 1 is ordered119870 = 2 is critical and

119870 gt 2 is chaotic on average [31] Other properties of RBNscan be used to regulate dynamical regimes [32]

22 Complexity of RBNs It is well known that complexadaptive systems are equipped with stability and flexibilitysimultaneously Here complexity signifies a balance betweenregularity and change which allows systems to adapt robustly[27 33 34] From an information viewpoint the regularityensures that useful information survives while the changeenables the systems to explore new possibilities essential foradaptability Living organisms or computer systems need notonly stability to survive or to maintain information but alsoflexibility to evolve and adapt to their environment Followingthis concept of complexity we developed a quantitativemeasure [28] Using our previous approach we can measurethe complexity of RBNs In this study complexity is presentedas quantities computed according to our measure

The complexity is calculated based on Shannonrsquos informa-tion entropy Its equation is as follows

119864119894 = minus (1199010 log2 1199010 + 1199011 log2 1199011) (1)

119862 = 4 times 119864 times (1 minus 119864) (2)

where119864119894 is the ldquoemergencerdquo of node 119894119901119895 is the probability thatthe state of the node is 119895 (119895 = 0 1) among the states of node 119894updated at each time step until simulation time119879119862 (0 le 119862 le1) is the complexity of the network and 119864 (0 le 119864 le 1) is theaverage of the emergence values for all the nodes Specifically1199010 (1199011) is calculated by counting the number of 0s (1s) innode 119894 until simulation time 119879 For example in the left plotof Figure 1(a) 1199010 and 1199011 of the last node are 240 and 3840respectively Because RBNs are deterministic systems once

Complexity 3

E

C

1 minus E

02 04 06 08 1000

p0(p1)

00

02

04

06

08

10

Figure 2 Relationship between change 119864 regularity 1 minus 119864 andcomplexity 119862 in RBNs [25]

initial states are determined state transitions from them toattractors are also determined Thus 119864 and 119862 are dependenton initial states

When 119864 is calculated 1199010 and 1199011 in (1) should not beconfusedwith119901whichwasmentioned as internal homogene-ity in previous section 1199010 and 1199011 are values measured fromstate transitions Meanwhile 119901 is a parameter used to createBoolean functions assigned to each node in a RBN In theBoolean functions each value is determined with probability119901 of being one or probability 1 minus 119901 of being zero119864 1minus119864 and119862 are time-dependent because they focus on

the dynamics of node states119864 indicates howmuch new statesare produced over time (ie change) As the complement of119864 1 minus 119864 represents how much existing states are maintained(ie regularity) 119862 means how successfully both of them aremet Numerically 119862 reaches maximum when the emergence119864 is 05 (119864 = 05 997888rarr 119862 = 1) It is when the expression of anyone of the two states is highly probable ie 1199010 or 1199011 cong 089for each node [25 29]Meanwhile119862 becomes 0when the twostates are evenly distributed (1199010 = 1199011 = 05 119864 = 1) or only onestate has maximum probability (1199010 or 1199011 = 1 119864 = 0)

Figure 2 illustrates a mathematical relation betweenchange 119864 regularity 1minus119864 and complexity119862 in RBNs [25] Asseen in the figure high complexity is achievedwhen119864= 1minus119864whichmeans an optimal balance between keeping and chang-ing the states of the network For perturbed RBNs Figure 1(a)shows that the antifragile network maintains original statesoverall and simultaneously explores new states by means ofperturbations Figure 1(b) represents that most of the statesin the fragile network change with perturbations whichindicates that the network does not maintain information ina noisy environment

23 Network Perturbations We express network perturba-tions due to external stressors as the change of node statesin a RBN We flip the states of 119883 nodes randomly chosenwhere the perturbations are added with frequency 119874 duringsimulation run time 119879 In other words the perturbations areadded whenever the time step 119905 is divisible by 119874 (119905 mod 119874= 0) For example 119883 = 2 119874 = 3 and 119879 = 99 mean that the

states of two nodes randomly chosen in each configurationare flipped every three time steps until the simulation runtime becomes 99 By comparing the state transitions of theoriginal network and its perturbed network we can observehow the perturbations propagate over time (Figure 1)

In our study the degree of perturbations is defined asfollows

119909 = 119883 times (119879119874)119873 times 119879 (3)

where 0 le 119909 le 124 Antifragility of RBNs We define (anti)fragility ∮ as

∮ = minus120590 times 119909 (4)

where 120590 is the difference of ldquosatisfactionrdquo before and afterperturbations while 119909 is the degree of perturbations Toprevent the influence of node size of a network we calibratethe values of ∮ by multiplying 119909 The satisfaction 120590 is thedegree to which the goals of an agent have been achieved [35]In the context of RBNs each node of the network can be seenas an agent We can arbitrarily define their goal as reachinga balance between change and regularity which is achievedwhen the nodes have high complexity values Thus in RBNsthe satisfaction is measured with complexity Depending onhow the satisfaction changes before and after perturbationsthe RBN is classified fragile robust or antifragile

The satisfaction can be measured differently dependingon the particular systems eg performance value and fitnessIf the satisfaction is decreased with perturbations then thesystem is fragile If the satisfaction does not change before andafter adding perturbations then the system is robust If thesatisfaction increases with perturbations then the system isantifragile Notice that 120590 and 119909 should be normalized tothe interval [-1 1] [0 1] respectively

The perturbations119909 for RBNs were defined in the previ-ous section We can define the ldquosatisfactionrdquo of a RBN basedon its complexity Since high complexity offers a balancebetween robustness and adaptability we can arbitrarily preferRBNs with high complexity Using the complexity measurepresented previously 120590 is calculated by the following equa-tion

120590 = 119862 minus 1198620 (5)

where 1198620 is complexity of a network before adding pertur-bations and 119862 is complexity of the network after addingperturbationsThe same initial states are used at 119905=0 Becausethe value of complexity is between 0 and 1 -1 le 120590 le 1

Negative values of ∮ mean that the RBN is antifragileand positive values mean that the RBN is fragile Values closeto zero indicate that the RBN is robust As shown in (4) ∮has the opposite sign of 120590 Hence the negative values of∮ indicate that 119862 is larger than 1198620 (ie the complexity ofa system is improved by external perturbations) while thepositive values represent that 1198620 is greater than 119862 (ie thecomplexity is lowered by the perturbations) The value of 0

4 Complexity

Table 1 Parameter settings for experiments

Figure 119873 119879 119883 119874 of different networks of initial states3(a) 100 200 1100 1 50 103(b) 100 200 40 150 50 104(a) 100 2000 0 0 1000 14(b) 100 200 1100 1 50 104(c) 100 200 1100 1 50 105 119873 200 1119873 1 1 10006 100 200 1100 130 50 107 119873 200 1119873 120 1 5000

refers to the fact that complexity does not change before andafter perturbations which indicates that the RBN is robustFigures 1(a) and 1(b) show the values of∮ calculated from theexamples of critical and chaotic RBNs

antifragile robust fragile

-1 0 1

In a RBN the value of ∮ can be different depending oninitial states because 120590 is determined by the states of nodesThus using multiple initial states we calculate average ∮ fora RBN and represent it as a system property

3 Experiments

We performed two sets of experiments one for RBNs and theother for biological BNs

First to measure antifragility of RBNs we generatedordered critical and chaotic RBNs composed of 100 nodes(119870 = 1 (ordered) 2 (critical) 3 4 5 (chaotic)) with internalhomogeneity 119901 = 05 [31] For each RBN we randomlychose 10 different initial states and then examined their statetransitions until simulation time119879= 200 respectively For thesame RBN taking the same initial states varying perturbednode size 119883 and perturbation frequency 119874 we obtainedthe state transitions of the perturbed RBN until 119879 = 200By comparing complexity before and after perturbations wecalculated mean of antifragility for the 10 initial states Themeasured values shown in the plots are average calculatedfrom 50 different RBNs per 119870

Secondly to measure antifragility of biological BNs weused the following seven biological network models

(i) CD4+T cell differentiation and plasticity [36] (119873= 18)It is a model representing how CD4+ T cells orches-trate immune responses depending on environmentalsignals and immunological challenges

(ii) Mammalian cell-cycle [37] (119873 = 20) It is a modelexplaining the mechanism of action of the cell cyclecheckpoints in mammalian cells

(iii) Cardiac development [38] (119873 = 15) It is a modelreferring to how the first heart field (FHF) and secondheart field (SHF) are formed by differential expressionof transcription and signaling factors during cardiacdevelopmental processes

(iv) Metabolic interactions in the gut microbiome [39](119873 = 12) It is a model describing interactive host-microbiota metabolic processes

(v) Death receptor signaling [40] (119873 = 28) It is a modelrelated to the activation of death receptors (TNFR andFas) that determine either survival or cell death

(vi) Arabidopsis thaliana cell-cycle [41] (119873 = 14) It is amodel explaining the mechanism of plant cell-cycleand cell differentiation in A thaliana

(vii) Tumor cell invasion and migration [42] (119873 = 32) It isa model representing the mechanism and interplaysbetween pathways that are involved in the process ofmetastasis

For each network we randomly chose 1000 different initialstates and then investigated their state transitions until119879 = 200 Changing 119883 and 119874 we computed antifragilitySpecifications of parameters for the simulation follow Table 1Our simulator for antifragility was implemented in Python(the source code is available at httpsgithubcomOkarim1RBNgit)

4 Results and Discussion

41 Antifragility in RBNs Figure 3 shows average ∮ ofordered (119870 = 1) critical (119870 = 2) and chaotic (119870 = 3 4 5)RBNs depending on perturbed node size119883 and perturbationfrequency 119874 The ordered and critical RBNs had negativevalues (antifragility) in certain ranges of 119883 and 119874 while thechaotic RBNs all had zero or positive values in the givenranges This means that the ordered and critical RBNs can beantifragile if they have the ldquorightrdquo amount of perturbationsHowever chaotic RBNs are just robust or fragile againstperturbations

As shown in Figure 3(a) the values of the ordered andcritical RBNs were lower than zero and got smaller andsmaller as 119883 increased which indicates that their dynamicschange more and more to antifragile However the valuesincreased beyond certain119883 values and even the critical RBNschanged from antifragile to fragile (X gt 20) From this wefound that neither too large nor too small but a moderatelevel of perturbations can induce greater antifragility Thesedynamics are similar to the slower-is-faster effect where amoderate level of speed can lead to better traffic flow ratherthan that of the highest speed of individuals [43]

Complexity 3

E

C

1 minus E

02 04 06 08 1000

p0(p1)

00

02

04

06

08

10

Figure 2 Relationship between change 119864 regularity 1 minus 119864 andcomplexity 119862 in RBNs [25]

initial states are determined state transitions from them toattractors are also determined Thus 119864 and 119862 are dependenton initial states

When 119864 is calculated 1199010 and 1199011 in (1) should not beconfusedwith119901whichwasmentioned as internal homogene-ity in previous section 1199010 and 1199011 are values measured fromstate transitions Meanwhile 119901 is a parameter used to createBoolean functions assigned to each node in a RBN In theBoolean functions each value is determined with probability119901 of being one or probability 1 minus 119901 of being zero119864 1minus119864 and119862 are time-dependent because they focus on

the dynamics of node states119864 indicates howmuch new statesare produced over time (ie change) As the complement of119864 1 minus 119864 represents how much existing states are maintained(ie regularity) 119862 means how successfully both of them aremet Numerically 119862 reaches maximum when the emergence119864 is 05 (119864 = 05 997888rarr 119862 = 1) It is when the expression of anyone of the two states is highly probable ie 1199010 or 1199011 cong 089for each node [25 29]Meanwhile119862 becomes 0when the twostates are evenly distributed (1199010 = 1199011 = 05 119864 = 1) or only onestate has maximum probability (1199010 or 1199011 = 1 119864 = 0)

Figure 2 illustrates a mathematical relation betweenchange 119864 regularity 1minus119864 and complexity119862 in RBNs [25] Asseen in the figure high complexity is achievedwhen119864= 1minus119864whichmeans an optimal balance between keeping and chang-ing the states of the network For perturbed RBNs Figure 1(a)shows that the antifragile network maintains original statesoverall and simultaneously explores new states by means ofperturbations Figure 1(b) represents that most of the statesin the fragile network change with perturbations whichindicates that the network does not maintain information ina noisy environment

23 Network Perturbations We express network perturba-tions due to external stressors as the change of node statesin a RBN We flip the states of 119883 nodes randomly chosenwhere the perturbations are added with frequency 119874 duringsimulation run time 119879 In other words the perturbations areadded whenever the time step 119905 is divisible by 119874 (119905 mod 119874= 0) For example 119883 = 2 119874 = 3 and 119879 = 99 mean that the

states of two nodes randomly chosen in each configurationare flipped every three time steps until the simulation runtime becomes 99 By comparing the state transitions of theoriginal network and its perturbed network we can observehow the perturbations propagate over time (Figure 1)

In our study the degree of perturbations is defined asfollows

119909 = 119883 times (119879119874)119873 times 119879 (3)

where 0 le 119909 le 124 Antifragility of RBNs We define (anti)fragility ∮ as

∮ = minus120590 times 119909 (4)

where 120590 is the difference of ldquosatisfactionrdquo before and afterperturbations while 119909 is the degree of perturbations Toprevent the influence of node size of a network we calibratethe values of ∮ by multiplying 119909 The satisfaction 120590 is thedegree to which the goals of an agent have been achieved [35]In the context of RBNs each node of the network can be seenas an agent We can arbitrarily define their goal as reachinga balance between change and regularity which is achievedwhen the nodes have high complexity values Thus in RBNsthe satisfaction is measured with complexity Depending onhow the satisfaction changes before and after perturbationsthe RBN is classified fragile robust or antifragile

The satisfaction can be measured differently dependingon the particular systems eg performance value and fitnessIf the satisfaction is decreased with perturbations then thesystem is fragile If the satisfaction does not change before andafter adding perturbations then the system is robust If thesatisfaction increases with perturbations then the system isantifragile Notice that 120590 and 119909 should be normalized tothe interval [-1 1] [0 1] respectively

The perturbations119909 for RBNs were defined in the previ-ous section We can define the ldquosatisfactionrdquo of a RBN basedon its complexity Since high complexity offers a balancebetween robustness and adaptability we can arbitrarily preferRBNs with high complexity Using the complexity measurepresented previously 120590 is calculated by the following equa-tion

120590 = 119862 minus 1198620 (5)

where 1198620 is complexity of a network before adding pertur-bations and 119862 is complexity of the network after addingperturbationsThe same initial states are used at 119905=0 Becausethe value of complexity is between 0 and 1 -1 le 120590 le 1

Negative values of ∮ mean that the RBN is antifragileand positive values mean that the RBN is fragile Values closeto zero indicate that the RBN is robust As shown in (4) ∮has the opposite sign of 120590 Hence the negative values of∮ indicate that 119862 is larger than 1198620 (ie the complexity ofa system is improved by external perturbations) while thepositive values represent that 1198620 is greater than 119862 (ie thecomplexity is lowered by the perturbations) The value of 0

4 Complexity

Table 1 Parameter settings for experiments

Figure 119873 119879 119883 119874 of different networks of initial states3(a) 100 200 1100 1 50 103(b) 100 200 40 150 50 104(a) 100 2000 0 0 1000 14(b) 100 200 1100 1 50 104(c) 100 200 1100 1 50 105 119873 200 1119873 1 1 10006 100 200 1100 130 50 107 119873 200 1119873 120 1 5000

refers to the fact that complexity does not change before andafter perturbations which indicates that the RBN is robustFigures 1(a) and 1(b) show the values of∮ calculated from theexamples of critical and chaotic RBNs

antifragile robust fragile

-1 0 1

In a RBN the value of ∮ can be different depending oninitial states because 120590 is determined by the states of nodesThus using multiple initial states we calculate average ∮ fora RBN and represent it as a system property

3 Experiments

We performed two sets of experiments one for RBNs and theother for biological BNs

First to measure antifragility of RBNs we generatedordered critical and chaotic RBNs composed of 100 nodes(119870 = 1 (ordered) 2 (critical) 3 4 5 (chaotic)) with internalhomogeneity 119901 = 05 [31] For each RBN we randomlychose 10 different initial states and then examined their statetransitions until simulation time119879= 200 respectively For thesame RBN taking the same initial states varying perturbednode size 119883 and perturbation frequency 119874 we obtainedthe state transitions of the perturbed RBN until 119879 = 200By comparing complexity before and after perturbations wecalculated mean of antifragility for the 10 initial states Themeasured values shown in the plots are average calculatedfrom 50 different RBNs per 119870

Secondly to measure antifragility of biological BNs weused the following seven biological network models

(i) CD4+T cell differentiation and plasticity [36] (119873= 18)It is a model representing how CD4+ T cells orches-trate immune responses depending on environmentalsignals and immunological challenges

(ii) Mammalian cell-cycle [37] (119873 = 20) It is a modelexplaining the mechanism of action of the cell cyclecheckpoints in mammalian cells

(iii) Cardiac development [38] (119873 = 15) It is a modelreferring to how the first heart field (FHF) and secondheart field (SHF) are formed by differential expressionof transcription and signaling factors during cardiacdevelopmental processes

(iv) Metabolic interactions in the gut microbiome [39](119873 = 12) It is a model describing interactive host-microbiota metabolic processes

(v) Death receptor signaling [40] (119873 = 28) It is a modelrelated to the activation of death receptors (TNFR andFas) that determine either survival or cell death

(vi) Arabidopsis thaliana cell-cycle [41] (119873 = 14) It is amodel explaining the mechanism of plant cell-cycleand cell differentiation in A thaliana

(vii) Tumor cell invasion and migration [42] (119873 = 32) It isa model representing the mechanism and interplaysbetween pathways that are involved in the process ofmetastasis

For each network we randomly chose 1000 different initialstates and then investigated their state transitions until119879 = 200 Changing 119883 and 119874 we computed antifragilitySpecifications of parameters for the simulation follow Table 1Our simulator for antifragility was implemented in Python(the source code is available at httpsgithubcomOkarim1RBNgit)

4 Results and Discussion

41 Antifragility in RBNs Figure 3 shows average ∮ ofordered (119870 = 1) critical (119870 = 2) and chaotic (119870 = 3 4 5)RBNs depending on perturbed node size119883 and perturbationfrequency 119874 The ordered and critical RBNs had negativevalues (antifragility) in certain ranges of 119883 and 119874 while thechaotic RBNs all had zero or positive values in the givenranges This means that the ordered and critical RBNs can beantifragile if they have the ldquorightrdquo amount of perturbationsHowever chaotic RBNs are just robust or fragile againstperturbations

As shown in Figure 3(a) the values of the ordered andcritical RBNs were lower than zero and got smaller andsmaller as 119883 increased which indicates that their dynamicschange more and more to antifragile However the valuesincreased beyond certain119883 values and even the critical RBNschanged from antifragile to fragile (X gt 20) From this wefound that neither too large nor too small but a moderatelevel of perturbations can induce greater antifragility Thesedynamics are similar to the slower-is-faster effect where amoderate level of speed can lead to better traffic flow ratherthan that of the highest speed of individuals [43]

4 Complexity

Table 1 Parameter settings for experiments

Figure 119873 119879 119883 119874 of different networks of initial states3(a) 100 200 1100 1 50 103(b) 100 200 40 150 50 104(a) 100 2000 0 0 1000 14(b) 100 200 1100 1 50 104(c) 100 200 1100 1 50 105 119873 200 1119873 1 1 10006 100 200 1100 130 50 107 119873 200 1119873 120 1 5000

refers to the fact that complexity does not change before andafter perturbations which indicates that the RBN is robustFigures 1(a) and 1(b) show the values of∮ calculated from theexamples of critical and chaotic RBNs

antifragile robust fragile

-1 0 1

In a RBN the value of ∮ can be different depending oninitial states because 120590 is determined by the states of nodesThus using multiple initial states we calculate average ∮ fora RBN and represent it as a system property

3 Experiments

We performed two sets of experiments one for RBNs and theother for biological BNs

First to measure antifragility of RBNs we generatedordered critical and chaotic RBNs composed of 100 nodes(119870 = 1 (ordered) 2 (critical) 3 4 5 (chaotic)) with internalhomogeneity 119901 = 05 [31] For each RBN we randomlychose 10 different initial states and then examined their statetransitions until simulation time119879= 200 respectively For thesame RBN taking the same initial states varying perturbednode size 119883 and perturbation frequency 119874 we obtainedthe state transitions of the perturbed RBN until 119879 = 200By comparing complexity before and after perturbations wecalculated mean of antifragility for the 10 initial states Themeasured values shown in the plots are average calculatedfrom 50 different RBNs per 119870

Secondly to measure antifragility of biological BNs weused the following seven biological network models

(i) CD4+T cell differentiation and plasticity [36] (119873= 18)It is a model representing how CD4+ T cells orches-trate immune responses depending on environmentalsignals and immunological challenges

(ii) Mammalian cell-cycle [37] (119873 = 20) It is a modelexplaining the mechanism of action of the cell cyclecheckpoints in mammalian cells

(iii) Cardiac development [38] (119873 = 15) It is a modelreferring to how the first heart field (FHF) and secondheart field (SHF) are formed by differential expressionof transcription and signaling factors during cardiacdevelopmental processes

(iv) Metabolic interactions in the gut microbiome [39](119873 = 12) It is a model describing interactive host-microbiota metabolic processes

(v) Death receptor signaling [40] (119873 = 28) It is a modelrelated to the activation of death receptors (TNFR andFas) that determine either survival or cell death

(vi) Arabidopsis thaliana cell-cycle [41] (119873 = 14) It is amodel explaining the mechanism of plant cell-cycleand cell differentiation in A thaliana

(vii) Tumor cell invasion and migration [42] (119873 = 32) It isa model representing the mechanism and interplaysbetween pathways that are involved in the process ofmetastasis

For each network we randomly chose 1000 different initialstates and then investigated their state transitions until119879 = 200 Changing 119883 and 119874 we computed antifragilitySpecifications of parameters for the simulation follow Table 1Our simulator for antifragility was implemented in Python(the source code is available at httpsgithubcomOkarim1RBNgit)

4 Results and Discussion

41 Antifragility in RBNs Figure 3 shows average ∮ ofordered (119870 = 1) critical (119870 = 2) and chaotic (119870 = 3 4 5)RBNs depending on perturbed node size119883 and perturbationfrequency 119874 The ordered and critical RBNs had negativevalues (antifragility) in certain ranges of 119883 and 119874 while thechaotic RBNs all had zero or positive values in the givenranges This means that the ordered and critical RBNs can beantifragile if they have the ldquorightrdquo amount of perturbationsHowever chaotic RBNs are just robust or fragile againstperturbations

As shown in Figure 3(a) the values of the ordered andcritical RBNs were lower than zero and got smaller andsmaller as 119883 increased which indicates that their dynamicschange more and more to antifragile However the valuesincreased beyond certain119883 values and even the critical RBNschanged from antifragile to fragile (X gt 20) From this wefound that neither too large nor too small but a moderatelevel of perturbations can induce greater antifragility Thesedynamics are similar to the slower-is-faster effect where amoderate level of speed can lead to better traffic flow ratherthan that of the highest speed of individuals [43]

Complexity 5

Average Fragility

06

04

02

00

minus02

Frag

ility

0 20 40 60 80 100

X

K= 1K= 2K= 3

K= 4K= 5

(a)

Average Fragility

K= 1K= 2K= 3

K= 4K= 5

0 10 20 30 40 50

O

minus02

minus01

00

Frag

ility

01

02

(b)

Figure 3 Average ∮ of ordered critical and chaotic RBNs depending on 119883 and 119874 The error bars represent the standard error ofmeasurements for 50 different networks at 10 different initial states ran by 200 steps (a)119873 = 100 and 119874 = 1 (b)119873 = 100 and119883 = 40

Average Complexity

Initi

al C

ompl

exity

10

08

06

04

02

00

10 2 3 4 5

K

(a)

0 20 40 60 80 100

XK= 1K= 2K= 3

K= 4K= 5

Average Complexity

00

02

04

06

08

10

Fina

l Com

plex

ity

(b)

0 20 40 60 80 100

XK= 1K= 2K= 3

K= 4K= 5

Difference in Complexity

minus06minus04minus020002Δ

04060810

(c)

Figure 4 Initial and final complexity for 119870 = 1 2 3 4 5 with 119873 = 100 The error bars represent the standard error of measurements for50 different networks at 10 different initial states run by 200 steps (a) Complexity before adding perturbations (b) Complexity after addingperturbations (c) Difference of complexity before and after perturbations

Meanwhile in Figure 3(b) antifragility of the orderedand critical RBNs decreased overall as 119874 grew (ie theperiod of adding perturbations became longer and longer)Furthermore all the RBNs were robust in the case of thatthe perturbations were not added frequently although theperturbed nodes were 40 (119883 = 40) From these results wefound that the more frequently perturbations are added themore antifragile a system is particularly for the orderedRBNs Moreover how often perturbations are added hasa greater effect on antifragility than how many nodes areperturbed Thus it is essential that moderate perturbationsare added frequently in order to obtain maximal antifragility

Based on Figure 3 we are able to see that the orderedRBNs are the most antifragile Figure 4 clearly accounts forthe reason In Figure 4(a) the complexity before addingperturbations was the lowest at 119870 = 1 However as shown

in Figure 4(b) the complexity after adding perturbationsincreased most greatly and the value was also the largestexcept for the early range of 119883 at 119870 = 1 Therefore thedifference was the largest at119870 = 1 (Figure 4(c)) which led theordered RBNs to be most antifragile

Our result for complexity before perturbations is the sameas previous studies showing that critical RBNs have the mostappropriate balance between regularity and change [25 2844] In Figure 4(a) for low119870 the complexity was low whichrepresents that the ordered RBNs have high robustness andfew changesThat is there is few or no information emergingFor high 119870 the complexity was also low which reflects thatthe chaotic RBNs have high variability and many changesAlmost all the nodes carry novel emergent information Formedium connectivities (2 lt 119870 lt 3) there was a balancebetween regularity and change leading to high complexity

6 Complexity

O=1

CD4+ T CellMammalianCardiacMetabolic

DeathArabidopsisTumour Cell

minus02

minus01

00

01

02

Frag

ility

03

04

05

0 10 15 20 255 30

X

Figure 5 ∮ of biological Boolean networks The error bars represent the standard error of measurements for 1000 different initial states runby 200 steps

Probability at K=1

10

20

30

O

20 40 60 80 100X

00 02 04 06 08 10(a)

Probability at K=2

10

20

30

O

20 40 60 80 100X

00 02 04 06 08 10(b)

20 40 60 80 100X

00 02 04 06 08 10

Probability at K=3

10

20

30

O

(c)

20 40 60 80 100X

00 02 04 06 08 10

Probability at K=4

10

20

30

O

(d)

Figure 6 Probability of generating antifragile networks depending on 119883 and 119874 for ordered critical and chaotic RBNs with 119873 = 100 119879 =200 119901 = 05 50 different networks were used 10 different initial states were randomly chosen for each network (a) 119870 = 1 (b) 119870 = 2 (c) 119870 =3 (d) 119870 = 4

This is consistent with the dynamics of critical RBNs wherecriticality is found theoretically at 119870 = 2 (when N997888rarr infin)and for finite systems at 2 lt 119870 lt 3 due to a finite-size effect[25]

However the result is changed by adding perturbationsIn Figure 4(b) the ordered RBNs had the biggest complexityexcluding the early range of119883 which means that the ordered

RBNs show the optimal balance between regularity andchange in the presence of noise This illustrates that systemscan exhibit different properties in accordance with the pres-ence of external stressors Such phenomenon was recentlyobserved in a neural network as well [45] where neuralsystems showed different dynamical behaviors depending onthe presenceabsence of external inputs

Complexity 7

1

5

10

15

20

O

5 10 15

X

10

08

06

04

02

00

Probability for CD4+ T Cell

(a)

Probability for Mammalian Cell Cycle

20

15

10

5

1

O5 15 2010

X

00

02

04

06

08

10

(b)

5 10 15

X

Probability for Cardiac development

20

15

10

5

1

O

00

02

04

06

08

10

(c)Probability for Metabolic Interactions in the Gut Microbiome

20

15

10

5

1

O

00

02

04

06

08

10

5 10

X(d)

5 10 15 20 25

X

10

08

06

04

02

00

Probability for Death Receptor Signaling

20

15

10

5

1

O

(e)

10

08

06

04

02

00

Probability for Arabidopsis thaliana Cell Cycle

20

15

10

5

1

O5 10

X(f)

Probability for Tumour Cell Invasion and Migration

00

02

04

06

08

10

20

15

10

5

1

O

5 25 3015 2010

X(g)

Figure 7 Probability of generating antifragile networks depending on 119883 and 119874 for different biological Boolean networks with 119879 = 2005000 different initial states were used for each network (a) CD4+ T cell differentiation and plasticity (b) Mammalian cell-cycle (c) Cardiacdevelopment (d) Metabolic interactions in the gut microbiome (e) Death receptor signaling (f) A thaliana cell-cycle (g) Tumor cell invasionand migration

42 Antifragility in Biological BNs Boolean networks havebeen extensively used as models of genetic or cellular regula-tion in the fields of computational and systems biology [36ndash42] because they can capture interesting features of biolog-ical systems despite their simplicity Using seven biologicalBoolean network models we measured the values of ∮ ofbiological systems

We first consider a volatile environment where perturba-tions are added every time step (119874 = 1) Figure 5 shows thatfor this high level of noise the network of A thaliana cell-cycle is fragile the networks of death receptor signaling and

tumor cell invasion andmigration are robust in a certain rangeof 119883 and fragile in the rest of the range and the networksof CD4+ T cell differentiation and plasticitymammalian cell-cycle cardiac development and metabolic interactions in thegut microbiome are antifragile against perturbations Whencomparing with Figure 3(a) we found that antifragility of thebiological networks except for A thaliana cell-cycle is similarto that of ordered or critical RBNs

To obtain more generalized dynamics we investigatedthe probability of generating antifragile networks in a diverserange of 119883 and 119874 Figure 6 is a heat map showing the

8 Complexity

probability for RBNs As shown in the figure the ordered andcritical RBNs can produce antifragile networks However iftoo large perturbations are added in a volatile environment(ie119874 = 1) both of them do not exhibit antifragile dynamicsIn the case of the chaotic RBNs they cannot produceantifragile networks in any range of119883 and 119874

Figure 7 is a heat map for the seven BNs They allshow antifragile dynamics like the ordered or critical RBNsAmong the heat maps the most interesting networks areA thaliana cell-cycle and CD4+ T cell differentiation andplasticity We found that A thaliana cell-cycle repeatedlyproduces antifragile networks at regular intervals dependingon the values of119874 Based onmany studies demonstrating thatliving organisms are ordered or critical [46ndash49] we can inferthat A thaliana might have been evolved in environmentswhere particular dimensions of perturbations are addedmorefrequently than other biological systems We also foundthat CD4+ T cell differentiation and plasticity are the mostantifragile of the ones studied probably because it has themost variable environment It indicates that our antifragilitymeasure successfully captures the property of the immunesystem mentioned as a representative example of antifragilesystems

5 Conclusions

In this study we proposed a new measure of (anti)fragilityand applied it to RBNs Considering an environment givento a system as a noise source we observed how systemproperties can be varied depending on the degree of per-turbations We found that ordered and critical RBNs showantifragile dynamics and especially ordered RBNs are mostantifragile against the perturbations Also biological systemsshow antifragile dynamics

In addition to the findings we gained a meaningfulinsight to environments as external stressors The highcomplexity with an optimal balance between regularity andchange was acquired when moderate perturbations wereadded very frequently It means that ldquooptimalityrdquo depends onthe precise variability of the environment How can systemsbe antifragile or robust for varying levels of noise Whichmechanisms can be used to adjust the internal variabil-ity depending on the external variability These questionsdemand further studies but possible answers are alreadybeing explored based on the results presented here

Based on the findings and insight by adjusting the sizeand frequency of perturbations we can control system prop-erties from fragile through robust to antifragile dynamics Itmay help to understand dynamical behaviors of biologicalsystems depending on environmental conditions and developnew treatment strategies for various diseases including canceror AIDS eg how can we decrease the antifragility of cancercells or pathogens This should reduce their adaptability andpotentially improve treatments

Data Availability

Our simulator and data are available at httpsgithubcomOkarim1RBNgit

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

Omar K Pineda and Hyobin Kim contributed equally to thiswork

Acknowledgments

We are grateful to Dario Alatorre Ewan Colman LuisAngel Escobar Jose Luis Mateos Dante Perez and FernandaSanchez-Puig for useful comments and discussions I wouldlike to acknowledge Dr Gershenson for his guidance duringthe MSc course of Adaptive Computation The final work ofthe course served as a foundation for this paper [Omar KPineda]This researchwas partially supported by CONACYTand DGAPA UNAM

References

[1] N Taleb Antifragile Things That Gain from Disorder RandomHouse New York NY USA 2012

[2] J Derbyshire andGWright ldquoPreparing for the future Develop-ment of an rsquoantifragilersquomethodology that complements scenarioplanning by omitting causationrdquo Technological Forecasting ampSocial Change vol 82 no 1 pp 215ndash225 2014

[3] T Aven ldquoThe concept of antifragility and its implications for thepractice of risk analysisrdquo Risk Analysis vol 35 no 3 pp 476ndash483 2015

[4] A Naji M Ghodrat H Komaie-Moghaddam and R Pod-gornik ldquoAsymmetric Coulomb fluids at randomly chargeddielectric interfaces Anti-fragility overcharging and chargeinversionrdquoThe Journal of Chemical Physics vol 141 no 17 articleno 174704 2014

[5] M Grube L Muggia and C Gostincar ldquoNiches and adapta-tions of polyextremotolerant black fungirdquo in Polyextremophilesvol 27 pp 551ndash566 Springer 2013

[6] A Danchin P M Binder and S Noria ldquoAntifragility andtinkering in biology (and in business) flexibility provides anefficient epigenetic way to manage riskrdquo Gene vol 2 no 4 pp998ndash1016 2011

[7] J S Levin S P Brodfuehrer and W M Kroshl ldquoDetectingantifragile decisions and models Lessons from a conceptualanalysis model of service life extension of aging vehiclesrdquoin Proceedings of the 8th Annual IEEE International SystemsConference SysCon 2014 pp 285ndash292 Canada April 2014

[8] R Isted ldquoThe use of antifragility heuristics in transport plan-ningrdquo in Proceedings of Australian Institute of Traffic Planningand Management (AITPM) National Conference vol 3 2014

[9] K H Jones ldquoEngineering antifragile systems A change indesign philosophyrdquo Procedia Computer Science vol 32 pp 870ndash875 2014

[10] M Lichtman M T Vondal T C Clancy and J H ReedldquoAntifragile communicationsrdquo IEEE Systems Journal vol 12 no1 pp 659ndash670 2018

[11] E Verhulst ldquoApplying systems and safety engineering principlesfor antifragilityrdquo Procedia Computer Science vol 32 pp 842ndash849 2014

6 Complexity

O=1

CD4+ T CellMammalianCardiacMetabolic

DeathArabidopsisTumour Cell

minus02

minus01

00

01

02

Frag

ility

03

04

05

0 10 15 20 255 30

X

Figure 5 ∮ of biological Boolean networks The error bars represent the standard error of measurements for 1000 different initial states runby 200 steps

Probability at K=1

10

20

30

O

20 40 60 80 100X

00 02 04 06 08 10(a)

Probability at K=2

10

20

30

O

20 40 60 80 100X

00 02 04 06 08 10(b)

20 40 60 80 100X

00 02 04 06 08 10

Probability at K=3

10

20

30

O

(c)

20 40 60 80 100X

00 02 04 06 08 10

Probability at K=4

10

20

30

O

(d)

Figure 6 Probability of generating antifragile networks depending on 119883 and 119874 for ordered critical and chaotic RBNs with 119873 = 100 119879 =200 119901 = 05 50 different networks were used 10 different initial states were randomly chosen for each network (a) 119870 = 1 (b) 119870 = 2 (c) 119870 =3 (d) 119870 = 4

This is consistent with the dynamics of critical RBNs wherecriticality is found theoretically at 119870 = 2 (when N997888rarr infin)and for finite systems at 2 lt 119870 lt 3 due to a finite-size effect[25]

However the result is changed by adding perturbationsIn Figure 4(b) the ordered RBNs had the biggest complexityexcluding the early range of119883 which means that the ordered

RBNs show the optimal balance between regularity andchange in the presence of noise This illustrates that systemscan exhibit different properties in accordance with the pres-ence of external stressors Such phenomenon was recentlyobserved in a neural network as well [45] where neuralsystems showed different dynamical behaviors depending onthe presenceabsence of external inputs

Complexity 7

1

5

10

15

20

O

5 10 15

X

10

08

06

04

02

00

Probability for CD4+ T Cell

(a)

Probability for Mammalian Cell Cycle

20

15

10

5

1

O5 15 2010

X

00

02

04

06

08

10

(b)

5 10 15

X

Probability for Cardiac development

20

15

10

5

1

O

00

02

04

06

08

10

(c)Probability for Metabolic Interactions in the Gut Microbiome

20

15

10

5

1

O

00

02

04

06

08

10

5 10

X(d)

5 10 15 20 25

X

10

08

06

04

02

00

Probability for Death Receptor Signaling

20

15

10

5

1

O

(e)

10

08

06

04

02

00

Probability for Arabidopsis thaliana Cell Cycle

20

15

10

5

1

O5 10

X(f)

Probability for Tumour Cell Invasion and Migration

00

02

04

06

08

10

20

15

10

5

1

O

5 25 3015 2010

X(g)

Figure 7 Probability of generating antifragile networks depending on 119883 and 119874 for different biological Boolean networks with 119879 = 2005000 different initial states were used for each network (a) CD4+ T cell differentiation and plasticity (b) Mammalian cell-cycle (c) Cardiacdevelopment (d) Metabolic interactions in the gut microbiome (e) Death receptor signaling (f) A thaliana cell-cycle (g) Tumor cell invasionand migration

42 Antifragility in Biological BNs Boolean networks havebeen extensively used as models of genetic or cellular regula-tion in the fields of computational and systems biology [36ndash42] because they can capture interesting features of biolog-ical systems despite their simplicity Using seven biologicalBoolean network models we measured the values of ∮ ofbiological systems

We first consider a volatile environment where perturba-tions are added every time step (119874 = 1) Figure 5 shows thatfor this high level of noise the network of A thaliana cell-cycle is fragile the networks of death receptor signaling and

tumor cell invasion andmigration are robust in a certain rangeof 119883 and fragile in the rest of the range and the networksof CD4+ T cell differentiation and plasticitymammalian cell-cycle cardiac development and metabolic interactions in thegut microbiome are antifragile against perturbations Whencomparing with Figure 3(a) we found that antifragility of thebiological networks except for A thaliana cell-cycle is similarto that of ordered or critical RBNs

To obtain more generalized dynamics we investigatedthe probability of generating antifragile networks in a diverserange of 119883 and 119874 Figure 6 is a heat map showing the

8 Complexity

probability for RBNs As shown in the figure the ordered andcritical RBNs can produce antifragile networks However iftoo large perturbations are added in a volatile environment(ie119874 = 1) both of them do not exhibit antifragile dynamicsIn the case of the chaotic RBNs they cannot produceantifragile networks in any range of119883 and 119874

Figure 7 is a heat map for the seven BNs They allshow antifragile dynamics like the ordered or critical RBNsAmong the heat maps the most interesting networks areA thaliana cell-cycle and CD4+ T cell differentiation andplasticity We found that A thaliana cell-cycle repeatedlyproduces antifragile networks at regular intervals dependingon the values of119874 Based onmany studies demonstrating thatliving organisms are ordered or critical [46ndash49] we can inferthat A thaliana might have been evolved in environmentswhere particular dimensions of perturbations are addedmorefrequently than other biological systems We also foundthat CD4+ T cell differentiation and plasticity are the mostantifragile of the ones studied probably because it has themost variable environment It indicates that our antifragilitymeasure successfully captures the property of the immunesystem mentioned as a representative example of antifragilesystems

5 Conclusions

In this study we proposed a new measure of (anti)fragilityand applied it to RBNs Considering an environment givento a system as a noise source we observed how systemproperties can be varied depending on the degree of per-turbations We found that ordered and critical RBNs showantifragile dynamics and especially ordered RBNs are mostantifragile against the perturbations Also biological systemsshow antifragile dynamics

In addition to the findings we gained a meaningfulinsight to environments as external stressors The highcomplexity with an optimal balance between regularity andchange was acquired when moderate perturbations wereadded very frequently It means that ldquooptimalityrdquo depends onthe precise variability of the environment How can systemsbe antifragile or robust for varying levels of noise Whichmechanisms can be used to adjust the internal variabil-ity depending on the external variability These questionsdemand further studies but possible answers are alreadybeing explored based on the results presented here

Based on the findings and insight by adjusting the sizeand frequency of perturbations we can control system prop-erties from fragile through robust to antifragile dynamics Itmay help to understand dynamical behaviors of biologicalsystems depending on environmental conditions and developnew treatment strategies for various diseases including canceror AIDS eg how can we decrease the antifragility of cancercells or pathogens This should reduce their adaptability andpotentially improve treatments

Data Availability

Our simulator and data are available at httpsgithubcomOkarim1RBNgit

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

Omar K Pineda and Hyobin Kim contributed equally to thiswork

Acknowledgments

We are grateful to Dario Alatorre Ewan Colman LuisAngel Escobar Jose Luis Mateos Dante Perez and FernandaSanchez-Puig for useful comments and discussions I wouldlike to acknowledge Dr Gershenson for his guidance duringthe MSc course of Adaptive Computation The final work ofthe course served as a foundation for this paper [Omar KPineda]This researchwas partially supported by CONACYTand DGAPA UNAM

References

[1] N Taleb Antifragile Things That Gain from Disorder RandomHouse New York NY USA 2012

[2] J Derbyshire andGWright ldquoPreparing for the future Develop-ment of an rsquoantifragilersquomethodology that complements scenarioplanning by omitting causationrdquo Technological Forecasting ampSocial Change vol 82 no 1 pp 215ndash225 2014

[3] T Aven ldquoThe concept of antifragility and its implications for thepractice of risk analysisrdquo Risk Analysis vol 35 no 3 pp 476ndash483 2015

[4] A Naji M Ghodrat H Komaie-Moghaddam and R Pod-gornik ldquoAsymmetric Coulomb fluids at randomly chargeddielectric interfaces Anti-fragility overcharging and chargeinversionrdquoThe Journal of Chemical Physics vol 141 no 17 articleno 174704 2014

[5] M Grube L Muggia and C Gostincar ldquoNiches and adapta-tions of polyextremotolerant black fungirdquo in Polyextremophilesvol 27 pp 551ndash566 Springer 2013

[6] A Danchin P M Binder and S Noria ldquoAntifragility andtinkering in biology (and in business) flexibility provides anefficient epigenetic way to manage riskrdquo Gene vol 2 no 4 pp998ndash1016 2011

[7] J S Levin S P Brodfuehrer and W M Kroshl ldquoDetectingantifragile decisions and models Lessons from a conceptualanalysis model of service life extension of aging vehiclesrdquoin Proceedings of the 8th Annual IEEE International SystemsConference SysCon 2014 pp 285ndash292 Canada April 2014

[8] R Isted ldquoThe use of antifragility heuristics in transport plan-ningrdquo in Proceedings of Australian Institute of Traffic Planningand Management (AITPM) National Conference vol 3 2014

[9] K H Jones ldquoEngineering antifragile systems A change indesign philosophyrdquo Procedia Computer Science vol 32 pp 870ndash875 2014

[10] M Lichtman M T Vondal T C Clancy and J H ReedldquoAntifragile communicationsrdquo IEEE Systems Journal vol 12 no1 pp 659ndash670 2018

[11] E Verhulst ldquoApplying systems and safety engineering principlesfor antifragilityrdquo Procedia Computer Science vol 32 pp 842ndash849 2014

Complexity 7

1

5

10

15

20

O

5 10 15

X

10

08

06

04

02

00

Probability for CD4+ T Cell

(a)

Probability for Mammalian Cell Cycle

20

15

10

5

1

O5 15 2010

X

00

02

04

06

08

10

(b)

5 10 15

X

Probability for Cardiac development

20

15

10

5

1

O

00

02

04

06

08

10

(c)Probability for Metabolic Interactions in the Gut Microbiome

20

15

10

5

1

O

00

02

04

06

08

10

5 10

X(d)

5 10 15 20 25

X

10

08

06

04

02

00

Probability for Death Receptor Signaling

20

15

10

5

1

O

(e)

10

08

06

04

02

00

Probability for Arabidopsis thaliana Cell Cycle

20

15

10

5

1

O5 10

X(f)

Probability for Tumour Cell Invasion and Migration

00

02

04

06

08

10

20

15

10

5

1

O

5 25 3015 2010

X(g)

Figure 7 Probability of generating antifragile networks depending on 119883 and 119874 for different biological Boolean networks with 119879 = 2005000 different initial states were used for each network (a) CD4+ T cell differentiation and plasticity (b) Mammalian cell-cycle (c) Cardiacdevelopment (d) Metabolic interactions in the gut microbiome (e) Death receptor signaling (f) A thaliana cell-cycle (g) Tumor cell invasionand migration

42 Antifragility in Biological BNs Boolean networks havebeen extensively used as models of genetic or cellular regula-tion in the fields of computational and systems biology [36ndash42] because they can capture interesting features of biolog-ical systems despite their simplicity Using seven biologicalBoolean network models we measured the values of ∮ ofbiological systems

We first consider a volatile environment where perturba-tions are added every time step (119874 = 1) Figure 5 shows thatfor this high level of noise the network of A thaliana cell-cycle is fragile the networks of death receptor signaling and

tumor cell invasion andmigration are robust in a certain rangeof 119883 and fragile in the rest of the range and the networksof CD4+ T cell differentiation and plasticitymammalian cell-cycle cardiac development and metabolic interactions in thegut microbiome are antifragile against perturbations Whencomparing with Figure 3(a) we found that antifragility of thebiological networks except for A thaliana cell-cycle is similarto that of ordered or critical RBNs

To obtain more generalized dynamics we investigatedthe probability of generating antifragile networks in a diverserange of 119883 and 119874 Figure 6 is a heat map showing the

8 Complexity

probability for RBNs As shown in the figure the ordered andcritical RBNs can produce antifragile networks However iftoo large perturbations are added in a volatile environment(ie119874 = 1) both of them do not exhibit antifragile dynamicsIn the case of the chaotic RBNs they cannot produceantifragile networks in any range of119883 and 119874

Figure 7 is a heat map for the seven BNs They allshow antifragile dynamics like the ordered or critical RBNsAmong the heat maps the most interesting networks areA thaliana cell-cycle and CD4+ T cell differentiation andplasticity We found that A thaliana cell-cycle repeatedlyproduces antifragile networks at regular intervals dependingon the values of119874 Based onmany studies demonstrating thatliving organisms are ordered or critical [46ndash49] we can inferthat A thaliana might have been evolved in environmentswhere particular dimensions of perturbations are addedmorefrequently than other biological systems We also foundthat CD4+ T cell differentiation and plasticity are the mostantifragile of the ones studied probably because it has themost variable environment It indicates that our antifragilitymeasure successfully captures the property of the immunesystem mentioned as a representative example of antifragilesystems

5 Conclusions

In this study we proposed a new measure of (anti)fragilityand applied it to RBNs Considering an environment givento a system as a noise source we observed how systemproperties can be varied depending on the degree of per-turbations We found that ordered and critical RBNs showantifragile dynamics and especially ordered RBNs are mostantifragile against the perturbations Also biological systemsshow antifragile dynamics

In addition to the findings we gained a meaningfulinsight to environments as external stressors The highcomplexity with an optimal balance between regularity andchange was acquired when moderate perturbations wereadded very frequently It means that ldquooptimalityrdquo depends onthe precise variability of the environment How can systemsbe antifragile or robust for varying levels of noise Whichmechanisms can be used to adjust the internal variabil-ity depending on the external variability These questionsdemand further studies but possible answers are alreadybeing explored based on the results presented here

Based on the findings and insight by adjusting the sizeand frequency of perturbations we can control system prop-erties from fragile through robust to antifragile dynamics Itmay help to understand dynamical behaviors of biologicalsystems depending on environmental conditions and developnew treatment strategies for various diseases including canceror AIDS eg how can we decrease the antifragility of cancercells or pathogens This should reduce their adaptability andpotentially improve treatments

Data Availability

Our simulator and data are available at httpsgithubcomOkarim1RBNgit

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

Omar K Pineda and Hyobin Kim contributed equally to thiswork

Acknowledgments

We are grateful to Dario Alatorre Ewan Colman LuisAngel Escobar Jose Luis Mateos Dante Perez and FernandaSanchez-Puig for useful comments and discussions I wouldlike to acknowledge Dr Gershenson for his guidance duringthe MSc course of Adaptive Computation The final work ofthe course served as a foundation for this paper [Omar KPineda]This researchwas partially supported by CONACYTand DGAPA UNAM

References

[1] N Taleb Antifragile Things That Gain from Disorder RandomHouse New York NY USA 2012

[2] J Derbyshire andGWright ldquoPreparing for the future Develop-ment of an rsquoantifragilersquomethodology that complements scenarioplanning by omitting causationrdquo Technological Forecasting ampSocial Change vol 82 no 1 pp 215ndash225 2014

[3] T Aven ldquoThe concept of antifragility and its implications for thepractice of risk analysisrdquo Risk Analysis vol 35 no 3 pp 476ndash483 2015

[4] A Naji M Ghodrat H Komaie-Moghaddam and R Pod-gornik ldquoAsymmetric Coulomb fluids at randomly chargeddielectric interfaces Anti-fragility overcharging and chargeinversionrdquoThe Journal of Chemical Physics vol 141 no 17 articleno 174704 2014

[5] M Grube L Muggia and C Gostincar ldquoNiches and adapta-tions of polyextremotolerant black fungirdquo in Polyextremophilesvol 27 pp 551ndash566 Springer 2013

[6] A Danchin P M Binder and S Noria ldquoAntifragility andtinkering in biology (and in business) flexibility provides anefficient epigenetic way to manage riskrdquo Gene vol 2 no 4 pp998ndash1016 2011

[7] J S Levin S P Brodfuehrer and W M Kroshl ldquoDetectingantifragile decisions and models Lessons from a conceptualanalysis model of service life extension of aging vehiclesrdquoin Proceedings of the 8th Annual IEEE International SystemsConference SysCon 2014 pp 285ndash292 Canada April 2014

[8] R Isted ldquoThe use of antifragility heuristics in transport plan-ningrdquo in Proceedings of Australian Institute of Traffic Planningand Management (AITPM) National Conference vol 3 2014

[9] K H Jones ldquoEngineering antifragile systems A change indesign philosophyrdquo Procedia Computer Science vol 32 pp 870ndash875 2014

[10] M Lichtman M T Vondal T C Clancy and J H ReedldquoAntifragile communicationsrdquo IEEE Systems Journal vol 12 no1 pp 659ndash670 2018

[11] E Verhulst ldquoApplying systems and safety engineering principlesfor antifragilityrdquo Procedia Computer Science vol 32 pp 842ndash849 2014

8 Complexity

probability for RBNs As shown in the figure the ordered andcritical RBNs can produce antifragile networks However iftoo large perturbations are added in a volatile environment(ie119874 = 1) both of them do not exhibit antifragile dynamicsIn the case of the chaotic RBNs they cannot produceantifragile networks in any range of119883 and 119874

Figure 7 is a heat map for the seven BNs They allshow antifragile dynamics like the ordered or critical RBNsAmong the heat maps the most interesting networks areA thaliana cell-cycle and CD4+ T cell differentiation andplasticity We found that A thaliana cell-cycle repeatedlyproduces antifragile networks at regular intervals dependingon the values of119874 Based onmany studies demonstrating thatliving organisms are ordered or critical [46ndash49] we can inferthat A thaliana might have been evolved in environmentswhere particular dimensions of perturbations are addedmorefrequently than other biological systems We also foundthat CD4+ T cell differentiation and plasticity are the mostantifragile of the ones studied probably because it has themost variable environment It indicates that our antifragilitymeasure successfully captures the property of the immunesystem mentioned as a representative example of antifragilesystems

5 Conclusions

In this study we proposed a new measure of (anti)fragilityand applied it to RBNs Considering an environment givento a system as a noise source we observed how systemproperties can be varied depending on the degree of per-turbations We found that ordered and critical RBNs showantifragile dynamics and especially ordered RBNs are mostantifragile against the perturbations Also biological systemsshow antifragile dynamics

In addition to the findings we gained a meaningfulinsight to environments as external stressors The highcomplexity with an optimal balance between regularity andchange was acquired when moderate perturbations wereadded very frequently It means that ldquooptimalityrdquo depends onthe precise variability of the environment How can systemsbe antifragile or robust for varying levels of noise Whichmechanisms can be used to adjust the internal variabil-ity depending on the external variability These questionsdemand further studies but possible answers are alreadybeing explored based on the results presented here

Based on the findings and insight by adjusting the sizeand frequency of perturbations we can control system prop-erties from fragile through robust to antifragile dynamics Itmay help to understand dynamical behaviors of biologicalsystems depending on environmental conditions and developnew treatment strategies for various diseases including canceror AIDS eg how can we decrease the antifragility of cancercells or pathogens This should reduce their adaptability andpotentially improve treatments

Data Availability

Our simulator and data are available at httpsgithubcomOkarim1RBNgit

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Authorsrsquo Contributions

Omar K Pineda and Hyobin Kim contributed equally to thiswork

Acknowledgments

We are grateful to Dario Alatorre Ewan Colman LuisAngel Escobar Jose Luis Mateos Dante Perez and FernandaSanchez-Puig for useful comments and discussions I wouldlike to acknowledge Dr Gershenson for his guidance duringthe MSc course of Adaptive Computation The final work ofthe course served as a foundation for this paper [Omar KPineda]This researchwas partially supported by CONACYTand DGAPA UNAM

References

[1] N Taleb Antifragile Things That Gain from Disorder RandomHouse New York NY USA 2012

[2] J Derbyshire andGWright ldquoPreparing for the future Develop-ment of an rsquoantifragilersquomethodology that complements scenarioplanning by omitting causationrdquo Technological Forecasting ampSocial Change vol 82 no 1 pp 215ndash225 2014

[3] T Aven ldquoThe concept of antifragility and its implications for thepractice of risk analysisrdquo Risk Analysis vol 35 no 3 pp 476ndash483 2015

[4] A Naji M Ghodrat H Komaie-Moghaddam and R Pod-gornik ldquoAsymmetric Coulomb fluids at randomly chargeddielectric interfaces Anti-fragility overcharging and chargeinversionrdquoThe Journal of Chemical Physics vol 141 no 17 articleno 174704 2014

[5] M Grube L Muggia and C Gostincar ldquoNiches and adapta-tions of polyextremotolerant black fungirdquo in Polyextremophilesvol 27 pp 551ndash566 Springer 2013

[6] A Danchin P M Binder and S Noria ldquoAntifragility andtinkering in biology (and in business) flexibility provides anefficient epigenetic way to manage riskrdquo Gene vol 2 no 4 pp998ndash1016 2011

[7] J S Levin S P Brodfuehrer and W M Kroshl ldquoDetectingantifragile decisions and models Lessons from a conceptualanalysis model of service life extension of aging vehiclesrdquoin Proceedings of the 8th Annual IEEE International SystemsConference SysCon 2014 pp 285ndash292 Canada April 2014

[8] R Isted ldquoThe use of antifragility heuristics in transport plan-ningrdquo in Proceedings of Australian Institute of Traffic Planningand Management (AITPM) National Conference vol 3 2014

[9] K H Jones ldquoEngineering antifragile systems A change indesign philosophyrdquo Procedia Computer Science vol 32 pp 870ndash875 2014

[10] M Lichtman M T Vondal T C Clancy and J H ReedldquoAntifragile communicationsrdquo IEEE Systems Journal vol 12 no1 pp 659ndash670 2018

[11] E Verhulst ldquoApplying systems and safety engineering principlesfor antifragilityrdquo Procedia Computer Science vol 32 pp 842ndash849 2014

Complexity 9

[12] C A Ramirez and M Itoh ldquoAn initial approach towardsthe implementation of human error identification servicesfor antifragile systemsrdquo in Proceedings of the SICE AnnualConference (SICE) pp 2031ndash2036 September 2014

[13] A Abid M T Khemakhem S Marzouk et al ldquoTowardantifragile cloud computing infrastructuresrdquo Procedia Com-puter Science vol 32 pp 850ndash855 2014

[14] M Monperrus ldquoPrinciples of antifragile softwarerdquo in Proceed-ings of Companion to the first International Conference on theArt Science and Engineering of Programming ACM vol 32 pp1ndash4 Brussels Belgium April 2017

[15] L Guang E Nigussie J Plosila and H Tenhunen ldquoPositioningantifragility for clouds on public infrastructuresrdquo ProcediaComputer Science vol 32 pp 856ndash861 2014

[16] R Serra M Villani A Barbieri S A Kauffman and AColacci ldquoOn the dynamics of randomBoolean networks subjectto noise attractors ergodic sets and cell typesrdquo Journal ofTheoretical Biology vol 265 no 2 pp 185ndash193 2010

[17] A L Bauer T L Jackson Y Jiang and T Rohlf ldquoReceptorcross-talk in angiogenesis mapping environmental cues tocell phenotype using a stochastic Boolean signaling networkmodelrdquo Journal of Theoretical Biology vol 264 no 3 pp 838ndash846 2010

[18] M K Morris J Saez-Rodriguez P K Sorger and D ALauffenburger ldquoLogic-based models for the analysis of cellsignaling networksrdquo Biochemistry vol 49 no 15 pp 3216ndash32242010

[19] R Albert and H G Othmer ldquoThe topology of the regulatoryinteractions predicts the expression pattern of the segmentpolarity genes in Drosophila melanogasterrdquo Journal of Theoret-ical Biology vol 223 no 1 pp 1ndash18 2003

[20] F Li T Long Y Lu Q Ouyang and C Tang ldquoThe yeast cell-cycle network is robustly designedrdquo Proceedings of the NationalAcadamy of Sciences of the United States of America vol 101 no14 pp 4781ndash4786 2004

[21] M Paczuski K E Bassler and A Corral ldquoSelf-organizednetworks of competing boolean agentsrdquo Physical Review Lettersvol 84 no 14 pp 3185ndash3188 2000

[22] M E J Newman ldquoSpread of epidemic disease on networksrdquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 66 no 1 Article ID 016128 11 pages 2002

[23] P Ramo S A Kauffman J Kesseli and O Yli-Harja ldquoMeasuresfor information propagation in Boolean networksrdquo Physica DNonlinear Phenomena vol 227 no 1 pp 100ndash104 2007

[24] A Roli M Manfroni C Pinciroli and M Birattari ldquoOnthe design of Boolean network robotsrdquo in Proceedings ofthe European Conference on the Applications of EvolutionaryComputation pp 43ndash52 Springer 2011

[25] N Fernandez CMaldonado andC Gershenson ldquoInformationmeasures of complexity emergence self-organization home-ostasis and autopoiesisrdquo inGuided Self-Organization InceptionM Prokopenko Ed vol 9 pp 19ndash51 Springer 2014

[26] S A Kauffman ldquoMetabolic stability and epigenesis in randomlyconstructed genetic netsrdquo Journal of Theoretical Biology vol 22no 3 pp 437ndash467 1969

[27] S A Kauffman The Origins of Order Self-Organization andSelection in Evolution Oxford University Press 1993

[28] C Gershenson ldquoIntroduction to random Boolean networksrdquoin Proceedings of the Ninth International Conference on theSimulation and Synthesis of Living Systems (ALife IX) pp 160ndash173 2004

[29] C Gershenson and N Fernandez ldquoComplexity and informa-tion Measuring emergence self-organization and homeostasisat multiple scalesrdquo Complexity vol 18 no 2 pp 29ndash44 2012

[30] C C Walker and W R Ashby ldquoOn temporal characteristics ofbehavior in certain complex systemsrdquo Kybernetik vol 3 no 2pp 100ndash108 1966

[31] B Derrida and Y Pomeau ldquoRandom networks of automata Asimple annealed approximationrdquo EPL (Europhysics Letters) vol1 no 2 pp 45ndash49 1986

[32] C Gershenson ldquoGuiding the self-organization of randomBoolean networksrdquoTheory in Biosciences vol 131 no 3 pp 181ndash191 2012

[33] C G Langton ldquoComputation at the edge of chaos phasetransitions and emergent computationrdquo Physica D NonlinearPhenomena vol 42 no 1ndash3 pp 12ndash37 1990

[34] R Lopez-Ruiz H L Mancini and X Calbet ldquoA statisticalmeasure of complexityrdquo Physics Letters A vol 209 no 5-6 pp321ndash326 1995

[35] C Gershenson ldquoThe sigma profile A formal tool to studyorganization and its evolution at multiple scalesrdquo Complexityvol 16 no 5 pp 37ndash44 2011

[36] M E Martinez-Sanchez L Mendoza C Villarreal and E RAlvarez-Buylla ldquoAMinimal regulatory network of extrinsic andintrinsic factors recovers observed patterns of CD4+ T celldifferentiation and plasticityrdquo PLoS Computational Biology vol11 no 6 article no 1004324 2015

[37] O Sahin H Frohlich C Lobke et al ldquoModeling ERBBreceptor-regulated G1S transition to find novel targets for denovo trastuzumab resistancerdquo BMC Systems Biology vol 3 no1 2009

[38] F Herrmann A Groszlig D Zhou H A Kestler and M KuhlldquoA Boolean model of the cardiac gene regulatory networkdetermining first and second heart field identityrdquo PLoS ONEvol 7 no 10 article no 46798 2012

[39] S N Steinway M B Biggs T P Loughran J A Papin and RAlbert ldquoInference of network dynamics and metabolic interac-tions in the gut microbiomerdquo PLoS Computational Biology vol11 no 6 article no 1004338 2015

[40] L Calzone L Tournier S Fourquet et al ldquoMathematicalmodelling of cell-fate decision in response to death receptorengagementrdquo PLoS Computational Biology vol 6 no 3 articleno 1000702 2010

[41] E Ortiz-Gutierrez K Garcıa-Cruz E Azpeitia et al ldquoAdynamic gene regulatory networkmodel that recovers the cyclicbehavior of arabidopsis thaliana cell cyclerdquo PLoS ComputationalBiology vol 11 no 9 article no e1004486 2015

[42] D P Cohen L Martignetti S Robine et al ldquoMathematicalmodelling ofmolecular pathways enabling tumour cell invasionand migrationrdquo PLoS Computational Biology vol 11 no 11article no e1004571 2015

[43] C Gershenson and D Helbing ldquoWhen slower is fasterrdquo Com-plexity vol 21 no 2 pp 9ndash15 2015

[44] S A Kauffman Investigations Oxford University Press 2000[45] J Schuecker S Goedeke and M Helias ldquoOptimal sequence

memory in driven random networksrdquo Physical Review X vol8 no 4 Article ID 041029 2018

[46] I Shmulevich S A Kauffman andMAldana ldquoEukaryotic cellsare dynamically ordered or critical but not chaoticrdquo Proceedingsof the National Acadamy of Sciences of the United States ofAmerica vol 102 no 38 pp 13439ndash13444 2005

10 Complexity

[47] E Balleza E R Alvarez-Buylla A Chaos S Kauffman IShmulevich and M Aldana ldquoCritical dynamics in geneticregulatory networks Examples from four kingdomsrdquo PLoSONE vol 3 no 6 article no 2456 2008

[48] M Villani L La Rocca S A Kauffman and R Serra ldquoDynam-ical criticality in gene regulatory networksrdquo Complexity vol2018 Article ID 5980636 14 pages 2018

[49] B C Daniels H Kim D Moore et al ldquoCriticality dis-tinguishes the ensemble of biological regulatory networksrdquoPhysical Review Letters vol 121 no 13 article no 138102 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Complexity

[47] E Balleza E R Alvarez-Buylla A Chaos S Kauffman IShmulevich and M Aldana ldquoCritical dynamics in geneticregulatory networks Examples from four kingdomsrdquo PLoSONE vol 3 no 6 article no 2456 2008

[48] M Villani L La Rocca S A Kauffman and R Serra ldquoDynam-ical criticality in gene regulatory networksrdquo Complexity vol2018 Article ID 5980636 14 pages 2018

[49] B C Daniels H Kim D Moore et al ldquoCriticality dis-tinguishes the ensemble of biological regulatory networksrdquoPhysical Review Letters vol 121 no 13 article no 138102 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom