A Model for the Emergence of Cooperation, Interdependence, And Structure in Evolving Networks

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A Model for the Emergence of Cooperation, Interdependence, and Structure in EvolvingNetworksAuthor(s): Sanjay Jain and Sandeep KrishnaReviewed work(s):Source: Proceedings of the National Academy of Sciences of the United States of America,Vol. 98, No. 2 (Jan. 16, 2001), pp. 543-547Published by: National Academy of Sciences

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A m o d e l o r t h e emergen einterdependencen d s t r u t uv o l v i n g networks

Sanjay Jain*t*? and Sandeep Krishna*?l*Centre for Theoretical Studies and?Physics Department, Indian Institute of ScienceSanta Fe, NM 87501; and *Jawaharlal Nehru Centre for Advanced Scientific ResearclCommunicated by Murray Gell-Mann, Santa Fe Institute, Santa Fe, NM, November 1Evolutionproduces complex and structurednetworks of interact- using components in chemical, biological, and social systems. We codescribe a simple mathematical model for the evolution of anidealized chemicalsystem to study how a network of cooperativemolecularspecies arises and evolves to become more complex andstructured.The network is modeled by a directed weighted graphwhose positive and negative links represent catalytic and in-hibitory interactions among the molecular species, and whichevolves as the least populated species (typically those that goextinct) are replaced by new ones. A small autocatalytic set,appearing by chance, provides the seed for the spontaneousgrowth of connectivity and cooperation in the graph. A highlystructuredchemical organization arises inevitably as the autocat- Ttalytic set enlarges and percolates through the network in a shortanalyticallydetermined timescale. This self organization does notrequirethe presence of self-replicating species. The network also prexhibits catastrophes over long timescales triggered by the chanceelimination of keystone species, followed by recoveries. foPC

tructured networks of interacting components are a hallmark Yiof several complex systems, for example, the chemical net- trhwork of molecular species in cells (1), the web of interdependentbiological species in ecosystems (2, 3), and social and economic ornetworks of interacting agents in societies (4-7). The structure reof these networks is a product of evolution, shaped partlyby the coenvironment and physical constraints and partly by the popula- eqtion (or other) dynamics in the system. For example, imagine a hpond on the prebiotic earth containing a set of interacting hmolecular species with some concentrations. The interactions thamong the species in the pond affect how the populations evolve thiwith time. If a population goes to zero, or if new molecularspecies enter the pond from the environment (through storms, unfloods, or tides), the effective chemical network existing in the se(pond changes. We discuss a mathematical model that attempts Poto incorporate this interplay between a network, populations, re;and the environment in a simple and idealized fashion. The prmodel [including an earlier version (8, 9)] was inspired by the inideas and results in refs. 10-18. Related but different models are chstudied in refs. 19-21. GriThe Model folThe system consists of s species labeled by the index i = 1,2,... ,s. prThe network of interactions between species is specified by the chs x s real matrix C- {i}. The network can be visualized as a =directed graph whose nodes represent the species. A nonzero ij asis represented by a directed weighted link from node j to nodei. If cij > 0, then the corresponding link is a cooperative link:species j catalyzes the production of species i. If cij < 0, it is a Abtdestructive link: the presence ofj causes a depletion of i (22). ?TOjai

ThePopulation Dynamics.The model contains another dynamical Thivariable x (x,.... Xs),where Xistands for the relative popu- ?17lation of the ith species (0 ? xi ? 1, Ei=t xi = 1)). The time Artevolution of x depends on the interaction coefficients C, as is Art

f cooperationr i n

, Bangalore 560 012, India; tSanta Fe Institute, 1399 Hyde Park Road,i, Bangalore 560 064, India6, 2000 (received for review April 28, 2000)aal in population models. The specific evolution rule wensider is

xi = fi if xi>O or fi20,=0 if xi = O and fi


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species to itself can only be inhibitive, i.e., autocatalytic orself-replicating individual species are not allowed. The variablex is initialized by choosing eachxi randomlybetween 0 and 1 andthen rescaling all xi uniformly such that ==1 xi = 1. Theevolution of the network proceeds in three steps:(i) Keeping the network fixed, the populations are evolvedaccording to Eq. 1 for a time T, which is large enough for x toget reasonably close to its attractor. We denote Xi xi(T).(ii) The set of nodes i with the least value of Xi is determined.We call this the set of least fit nodes, identifying the relativepopulation of a species in the attractor (or, more specifically, atT) with its fitness in the environment defined by the graph.One of the least fit nodes is chosen randomly (say io) andremoved from the system along with all its links, leaving a graphof s - 1 species.(iii) A new node is added to the graph so that it again has snodes. The links of the added node (ciio ndciq, or i = 1, ... , s) areassignedrandomlyaccording to the same rule as for the nodes inthe initialgraph.The newspeciesisgivenasmall relativepopulationxio= xo, and the other populationsare rescaled to keep Es= xi = 1.This process, from step i onwards,is iterated many times.The rules for the evolution of the network C are intended tocapture two key features of natural evolution, namely selectionand novelty. The species that has the least population in theattractorconfiguration is the one most likely to be eliminated ina large fluctuation in a possible hostile environment. Often, theleast value ofXi is zero. Thus the model implements selection byeliminating from the network a species that has become extinctor has the least chance of survival (18). [Relaxing in various waysthe assumption (18) that only the least populated species isremoved does not change the qualitative picture presented here;details on the robustness of the model to various deformationswill be presented elsewhere.] Novelty is introduced in thenetwork in the form of a new species. This species has on averagethe same connectivity as the initial set of species, but its actualconnections with the existing set are drawn randomly. E.g., if astorm brings into a prebiotic pond a new molecular species fromthe environment, the new species might be statistically similar tothe one being eliminated, but its actual catalytic and inhibitoryinteractions with the surviving species can be quite different. FigAnother common feature of natural evolution is that populations Nuitypically evolve on a fast timescale compared with the network. witThis is captured in the model by having the xi relax to their timattractor before the network is updated. The idealization of a nulfixed total number of species s is one that we hope to relax in intfuture work. moThe model described above differs from the one studied inrefs. 8 and 9 in that it allows negative links and varying link sastrengths, and that the population dynamics, given by Eq. 1 is no anclonger linear. The earlier model had only fixed point attractors; po|here limit cycles are also observed. Because C now has negativeentries, the formalism of nonnegative matrices no longer applies. ofResults diEmergenceof Cooperationand Interdependence.Fig. 1 shows a ithsample run. The same qualitative behavior is seen in each of itsseveral hundred runs performed for p values ranging from qu0.00002 to 0.01 and for s = 100, 150, 200. That the ratio of Sp(number of cooperative to destructive links at first remains (+constant at unity (statistically) and then increases by more than Yan order of magnitude is evidence of the emergence of coop- noeration. Fig. 1 also shows how a measure of the mutual inter- idependence of the species changes with time. This measure, sp(interdependency, denoted d, is defined as d (1/s)S=1 di, cowhere di is the dependency of the ith node. di-= klckihk, intwhere h: is 1 if there exists a directed path from k to i and 0 anotherwise. di is the sum of (the absolute value of) the strengths of

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a100908070 Entireraph ecomesQ60 I an ACS at n=3643

50o40 ACSappears

40

40 at n=1904 1

100 1000 2000 3000 4000 5000 6000 7000 8000 9000n

b140

120-

100

80 r

60 f

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0 1000 2000 3000 4000 5000 6000 7000 8000 9000n1. A run with parameter values s = 100, p = 0.005, and xo = 10-4. (a)nber of populated species, sl, in the attractor of Eq.1 (i.e., number of nodeshXi > 0) after the nth addition of a new species (i.e., after n graph updatee steps). (b) The number, /+, of positive links (ci > 0) in the graph (blue); thenber, /-, of negative links (green); and interdependency, d, of the specieshe network (red). The curves have three distinct regions. Initiallys1 issmall;st of the species have zero relative populations. I+ and /- also do not vary:h from their initial (random graph) value (_ps2/2 = 25) and remain

,roximately equal. d hovers about its initial low value. In the second region+ and a show a sharp increase, and /_ decreases. In the third region si, /+i d level off (but with fluctuations), and almost all species have nonzerorulations in contrast to the initial period.

all links that eventually feed into i along some directed path.lescribes not just the character of the neighborhood of thespecies but also the long-range connections that affectdynamics. The increase in d by an order of magnitude is aantitative measure of the increase of interdependence of:cies in the network. The increase in the total density of links+ I_)/s is another aspect of the increase of complexity of thetem. Note that in the model selection rewards only perfor-nce as measured in terms of relative population; the rules dot select for higher cooperativityper se. Because a new speciesequally likely to have positive or negative links with other:cies, the introduction of novelty is also not biased in favor of)perativity. That this behavior is not a consequence of anyrinsic bias in the model that favors the increase of cooperationi interdependence is evidenced by the flat initial region of allthe curves.

Jainand Krishna

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reasons. If the new node forms an ACS of its own with nodesoutside the dominant ACS, and the new ACS has a higherpopulation growth rate (as determined by Eq. 1) than the oldACS, it drives the species of the latter to extinction and becomesthe new dominant ACS. Alternatively, the new node could be adestructiveparasite: it receives one or more positive links fromand gives one or more negative links to the dominant ACS. Thenpart or whole of the ACS may join the set of least-fit nodes.Structuresthat diminish the size of the dominant ACS or destroyit appear rarely. For example, in Fig. 1, destructive parasitesappeared 6 times at n = 3,388, 3,478, 3,576, 3,579, 3,592, and3,613. In each case, sl decreased by 1.Emergenceof Structure.At n = n2, the whole graph becomes anACS; the entire system can collectively self replicate despite theexplicit absence of individual self replicators. Such a fullyautocatalytic set is a very nonrandom structure. Consider agraph of s nodes and let the probability of a positive link existingbetween any pair of nodes be p*. Such a graph has on averagem * =p*(s - 1) incoming or outgoing positive links per node. Forthe entire graph to be an ACS, each node must have at least oneincoming positive link, i.e., each row of the matrix C must containat least one positive element. Hence the probability, P, for the ofentire random graph to be an ACS is (fc

p/P = probability that every row has at least one positive entry tP/= [probability that a row has at least one positive entry]s re= [1 - (probability that every entry of a row is < 0)]s= - (1 -p*)Sl]S ex= [1 - (1 - m*/(s - 1))s-1]s. ofFor large s and m* - 0(1), fevCOP (1 - e-m)s = e-, [3] pi

where a is positive and 0(1). At n = n2, we find in all our runs pethat l+(n2) = l* is greater than s but of orders, i.e., m* - 0(1). ex]Thus dynamical evolution in the model via the ACS mechanismconverts a random organization into a highly structured one that Cais exponentially unlikely to appear by chance. In the displayed chrun at n = n2, the graph had 117 positive links. The probability tinthat a random graph with s = 100 nodes and m* = 1.17 would ofbe an ACS is given by Eq. 3 to be _10-16. sigSuch a structurewould take an exponentiallylong time to arise theby pure chance. The reason it arises inevitably n a short timescale imin the present model is the following: a small ACS can appear by pochance quitereadilyand,once appeared, t grows exponentially ast caiacross the graphbythe mechanism outlined earlier. The dynamical s1appearanceof such a structuremaybe regardedas the emergence larof organizationalorder. The appearance of exponentiallyun- A(likely structures n the prebioticcontext has been a puzzle.That in thethe present model such structures nevitablyform in a short time pImay be relevantfor the origin of life problem. mavThe Self-OrganizationTimescale n a PrebioticScenario. We now orl

speculate on a possible application to prebiotic chemical evolu-o

tion. Imagine the molecular species to be small peptide chains bawith weak catalytic activity in a prebiotic pond alluded to earlier.The pond periodically receives an influx of new molecularspecies being randomlygenerated elsewhere in the environment ev(through tides, storms, or floods. Between these influxes of Di,novelty, the pond behaves as a well stirred reactor where thepopulations of existent molecular species evolve according to (a Wrealistic version of) Eq. 1 and reach their attractorconfiguration. naUnder the assumption that the present model captures what inthappens in such a pond, the growth timescale (Eq. 2) for a highly plCstructured almost fully autocatalytic chemical organization in bythe pond is Tg= 2/p in units of the graph update time step. In strthis scenario, the latter time unit corresponds to the periodicity foi

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00

90 |

20

o 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Fin xl l

Fig. 3. The run of Fig. la displayed over a much longer timescale.

the influx of new molecular species, hence it ranges from 1 dayr tides) to 1 year (for floods). Further, in the present model2is the probability that a random small peptide will catalyzeproduction of another (26), and this has been estimated in12 as being in the range 10-5-10-1?. With p/2 10-8, forimple, the timescale for a highly structured chemical organi-ion to grow in the pond would be estimated to be of the order106-108years. It is believed that life originated on Earth in av hundred million years after the oceans condensed. Theseisiderations suggest that it might be worthwhile to empiricallyidown the catalytic probability p (introduced in ref. 26) forptides, catalytic RNA, lipids, etc., on the one hand, and)lore chemically more realistic models on the other.:astrophes ndRecoveries nNetworkDynamics.After n = n2,theiracter of the network evolution changes again. For the firstie, the least-fit node will be one of the ACS members. Mostthe time elimination of the node does not affect the ACSnificantly, and s, fluctuates between s and s - 1. Sometimesleast-fit node could be a keystone species, which plays anportant organizational role in the network despite its lowpulation. When such a node is eliminated, many other nodesi get disconnected from the ACS, resulting in large dips innd d and subsequently large fluctuations in 1+and 1_.Thesege extinction events can be seen in Fig. 3. Occasionally, the;S can even be destroyed completely. The system recovers ontimescale Tgafter large extinctions if the ACS is not com-tely destroyed; if it is, and the next few updates obliterate themory of previous structures in the graph, then again a time on-rage Ta elapses before an ACS arises, and the self-;anization process begins anew. It may be of interest (espe-Ily n ecology, economics, and finance) that network dynamics;ed on a fitness selection and the incremental introductionnovelty, as discussed here, can by itself cause catastrophic:ntswithout the presence of large external perturbations.cussion? have described an evolutionary model in which the dy-nics of species' populations (fast variables) and the graph oferactions among them (slow variables) are mutually cou-d. The network dynamics displays self organization seededthe chance but inevitable appearance of a small cooperativeucture, namely an ACS. In a dynamics that penalizes specieslow population performance, the collective cooperativity

Jainand Krishna



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of the ACS members makes the set relatively robust against cirdisruption.New speciesthathappento latch on cooperatively beto this structurepreferentially urvive, furtherenlargingtheACS in the process. Eventually the graph acquires a highly wenonrandom tructure.We have discussed he time evolution of noquantitative measures of cooperation, interdependence, and astructure of the network, which capture various aspects of the ancomplexity of the system. dyIt is noteworthy that collectively replicating ACSs arise even tecthough individual species are not self replicating. Thus thepresent mechanism is different from the hypercycle (27), where Wa template is needed to produce copies of existing species. Unlike hethe hypercycle, the ACS is not disrupted by parasites and short shi

1. Lodish, H., Baltimore, D., Berk, A., Zipursky,S. L., Matsudaira, P. & Darnell,J. E. (1995) Molecular Cell Biology (Scientific American Books, New York).2. Cohen, J. E., Briand, F. & Newman, C. M. (1990) CommunityFood Webs:Data 15.and Theory(Springer, New York).3. Pimm, S. L. (1991) The Balance of Nature?EcologicalIssues in the Conservationof Species and Communities (Univ. of Chicago Press, Chicago).4. Wellman, B. & Berkovitz, S. D., eds. (1988) Social Structures:A Network 17.Approach (Cambridge Univ. Press, Cambridge, U.K.). 18.5. Axelrod, R. (1997) The Complexityof Cooperation (Princeton Univ. Press, 19.Princeton). 20.6. Watts, D. J. & Strogatz, S. H. (1998) Nature (London) 393, 440-442.

7. Barabasi, A.-L. & Albert, R. (1999) Science 286, 509-512. 21.8. Jain, S. & Krishna, S. (1998) Phys. Rev. Lett. 81, 5684-5687.9. Jain, S. & Krishna, S. (1999) Comput. Phys. Commun. 121-122, 116-121. 2210. Dyson, F. (1985) Origins of Life (Cambridge Univ. Press, Cambridge, U.K.).11. Kauffman, S. A. (1986) J. Theor. Biol. 119, 1-24. 24.12. Kauffman,S. A (1993)TheOigins of Order OxfordUniv.Press,Oxford,U.K). 25.13. Farmer, J. D., Kauffman, S. A. & Packard, N. H. (1986) Physica D22, 50-67. 26.14. Bagley, R. J. & Farmer, J. D. (1991) in Artificial Life II, eds. Langton, C. G., 27.

Jain and Krishna

cuits andgrowsin complexity,as evidenced in all ourruns. It candisrupted,however, when it loses a keystone species.It is also worth mentioning one departure from ref. 12, in thatfind that a fullyautocatalytic ystem orpercolatingACS)ist needed a priori for self organization. In the present model,;mallACS, once formed, typically expands (see also ref. 15)d eventually percolates the whole network dynamically. Thisnamicalprocessmightbe relevant or economictakeoff and:hnological growth in societies.L hank J. D. Farmer, W. C. Saslaw, and anonymous referees forpful comments on the manuscript. S.J. acknowledges the Associate-p of the Abdus Salam International Centre for Theoretical Physics.Taylor,C., Farmer, . D. & Rasmussen, . (Addison-Wesley, edwoodCity,CA),pp.93-140.Bagley,R. J., Farmer, . D. & Fontana,W. (1991) in ArtificialLifeII, eds.Langton,C.G.,Taylor,C., Farmer, . D. & Rasmussen, . (Addison-Wesley,RedwoodCity,CA),pp. 141-158.Fontana,W.(1991) nArtificial ife I,eds.Langton,C.G.,Taylor,C., Farmer,J. D. & Rasmussen, . (Addison-Wesley, edwoodCity, CA),pp. 159-209.Fontana,W. & Buss,L. (1994)Bull.Math.Biol.56, 1-64.Bak,P. & Sneppen,K. (1993)Phys.Rev.Lett.71, 4083-4086.Happel,R. & Stadler,P. F. (1998)J. Theor.Biol. 195,329-338.Yasutomi,A. & Tokita,K. (1998) Preprint,http://paradox.harvard.edu/-tokita/list.html.Segr6,D., Ben-Eli,D. & Lancet,D. (2000)Proc.Natl.Acad. Sci. USA97,4112-4117.Odum,E. P. (1953)Fundamentalsf Ecology Saunders, hiladelphia).Ashmore,P. G. (1963)Catalysisnd Inhibitionf Chemical eactionsButter-worth,London).Eigen,M. (1971)Naturwissenschaften8, 465-523.Rossler,0. E. (1971)Z. Naturforschung6b,741-746.Kauffman, . A. (1971)J. Cybernetics, 71-96.Eigen,M. & Schuster,P. (1979)TheHypercycleSpringer,Berlin).

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A Model for the Emergence of Cooperation, Interdependence, And Structure in Evolving Networks

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