THEINTERNET,mapped on the opposite page, is a scale-free network in thatsome siteS (starbursts and detail above) have a seemingly unlimited

number of connections to other sites. This map, made on February 6, 2003,traces the shortest routes from atest WebsinHo about 100,000 others,

using like colors for similar Webaddresses.

a -

Scientistshaverecentlydiscoveredthat variouscomplexsystemshaveantlnderlyihg~..'~tJ;i~e~tu"eg~Ye'l"rne(;lb9.$haredorganili ngprincipies.

Thisinsighthas important impli~ationsfor a hostofapplications,fromdrugdevelopmentto Internet security

BYALBERT-U\SZLOBARABASIANDERICBONABEAU

50 SCIENTIFIC AMERICAN MAY 2003

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The brain is a network of nerve cells con-

nected by axons, and cells themselves arenetworks of molecules connected by bio-chemical reactions. Societies, too, are net-

works of people linked by friendships,familial relationships and professionalties. On a larger scale, food webs and eco-systems can be represented as networksof species. And networks pervade tech-nology: the Internet, power grids andtransportation systems are but a few ex"amples. Even the language we are usingto convey these thoughts to you is a net-work, made up of words connected bysyntactic relationships,

Yet despite the importance and per-vasiveness of networks, scientists havehad little understanding of their structureand properties. How do the interactionsof several malfunctioning llodes in a com-plex genetic network resl).lt in cancer?How does diffusion occur so rapidly.incertain social and communications sys-tems, leading to epidemics ()fdiseases andcomputer viruses? How do ~ome net-works continue to function even after the

vast majority of their nodes have failed?Recent research has begun to ansWer

such questions. Over the past feW years,investigators from a variety of fields havediscovered that many .networks-froIIlthe World Wide Web to a cell's metabol-

ic system to actors in Hollywood-aredominated by a relatively sm.all numberof nodes that ani c:oll1l~ctedto many oth"er sites. Networks containing suchim-portant nodes, or hubs, tend to be whatwe call "scale-free," in the sense thatsome hubs have a seemingly unlimited

number of links and no node is typical ofthe others. These networks also behave in

certain predictable ways; for example,they are remarkably resistant to acciden-tal failures but extremely vulnerable tocoordinated attacks.

Such discoveries have dramaticallychanged what we thought we knew aboutthe complex interconnected world aroundus. Unexplained by previous network the~ories, hubs offer convincing proof thatvarious complex systems have a strict ar-chitecture, ruled by fundamental laws-laws that appear to apply equally to cells,col11puters, languages and society. Fur-thermore, these organizing principles havesignificant implications for developingbetter drugs, defending the Internet fromhackers, and halting the spread of deadlyepidemics, among other applicati()ns.

Networks without ScaleFOR MORE THAN 40 YEARS, sciencetreated all complex net\'\(orks as beingcompletely randoIIl. This paradigIIl has itsroots in the work of two H1ir1garianmath~ematicians, the inimitable Paul Erdos andhis close collaborator Alfred Renyi. In1959, aiming to describe networks seen incommunications and the lif~ sciences,Erdos and Renyi suggested that suc:h sys-tems could be effectively l110deledby con-necting their nodes withrandornlyplacedlinks. The simplicitYof their approach andthe elegance of so~e of their related theo-rems revitalized graph theory, leading tothe emergence of a field in mathematicsthat focuses on random networks.

An important prediction of random-

S2 SCIENTIFIC AMERICAN

network theory is that, despite the ran-dom placement of links, the resulting sys-tem will be deeply democratic: mostnodes will have approximately the samenumber of links. Indeed, in a random net-work the nodes follow a Poisson distrib-

ution with a bell shape, and it is extreme-ly rare to find nodes that have significant-ly more or fewer links than the average.Random networks are also called expo-nential, because the probability that anode is connected to k other sites de-

creases exponentially for large k.So in 1998, when we, together with

Hawoong Jeong and Reka Albert of theUniversity of Notre Dame, embarked ona project to map the World Wide Web,we expected to find a random network.Here's why: people follow their uniqueinterests when deciding what sites to linktheir Web documents to, and given the di-versity of everyone's .interests and thetremendous number of pages they canchoose from, the resulting pattern of con-nections should appear fairly random.

The measurements, however, defiedthat expectation. Software designed forthis project hopped from one Web pageto another and collected all the links it

could. Although this virtu:alrobot reachedonly a tiny fraction of the entire Web, themap it assembled revealed somethingquite surprising: a few highly connectedpages are essentially holding the WorldWide Web together. More than 80 per-cent ()fthe pages on the map had feWerthan four links, but a small minority, lessthan 0.01 percent of all nodes, had morethan 1,000. (A subsequent Web surveywould uncover one document that had

been referenced by more than two millionother pages!)

Counting how many Web pages haveex'1stly k links showed that the distribu-tion followed a so-called power law: theprobability that any node was connectedto k other nodes was proportional to Ykn.The value of n for incoming links was ap-proximately 2, so, for instance, any nodewas roughly four times as likely to havejust half the number of incoming links asanother node. Power laws are quite dif-ferent from the bell-shaped distributionsthat characterize random networks.

MAY 2003

RANDOMVERSUSSCALE-FREENETWORKS I

RANDOMNETWORKS,which resemble the U.S.highway system

(simplified in left map), consist of nodes with randomly placedconnections. In such systems, a plot of the distribution of node

linkages will follow a bell-shaped curve (left graph), with most

nodes having approximately the same number of links.In contrast, scale-free networks, which resemble,the U.S.

airline system (simplified in right map). contain hubs [red)-

RandomNetwork

nodes with a very high number of links. In such networks, the

distribution of node linkages follows a power law [center graph)

in that most nodes have just a few connections and some have

a tremendous number of links. In that sense, the system has no

"scale." The defining characteristic of such networks is that the

distribution of links, if plotted on a double-logarithmic scale

[right graph), results in a straight line.

Scale-FreeNetwork

Bell Curve ~istribution of NodeLinkages

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Specifically, a power law does not have apeak, as a bell curve does, but is instead de-scribed by a continuously decreasing func-tion. When plotted on a double-logarith-mic scale, a power law is a straight line[see illustration above]. In contrast to thedemocratic distribution of links seen in

random networks, power laws describesystems in which a few hubs, such as Ya-hoo and Google, dominate.

Hubs are simply forbidden in randomnetworks. When we began to map theWeb, we expected the nodes to follow abell-shaped distribution, as do people'sheights. Instead we discovered certainnodes that defied explanation, almost asif we had stumbled on a significant num-ber of people who were 100 feet tall, thus

prompting us to coin the term" scale-free."

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PowerLawDistribution of NodeLinkages

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Scale-Free Networks AboundOVER THE PAST several years, re-searchers have uncovered scale-free struc"

tures in a stunning range of systems.When we studied the World Wide Web,we looked at the virtual network of Web

pages connected to one another by hy-perlinks. In contrast, .ty1ichalisFaloutsosof the University of California at River-side, Petros Falotitsos of the University ofToronto arid Christos Faloutsos of Car-

negie MelloQ Uq~versity .analyzed tbephysical structure of the Internet. Thesethree computer-scientist brothers investi-gated the routers connected by optical orother communications lines and found

that the topology of that network, too, isscale-free.

Researchers have also discovered that

some social networks are scale-free. A col-laboration between scientists from Boston

University and Stockholm University, forinstance, has shown that a netWork ofsexual relationships among people inSweden followed a poWer law: althoughmost individuals had only a few sexualpartners during their lifetime, a few (thehubs) had hundreds. A recent study ledby Stefan Bornholdt of the University ofKiel in Germany concluded that the net-work of people connected bye-mail islikewise scahfree. Sidney Redner ofBoston University demonstrated that thenetwork of scientific papers, connectedby citations, follows a power law as well.And Mark Newman of the University ofMichigan at Ann Arbor examined col-laborations among scientists in several

SCIENTIFIC AMERICAN 53

disciplines,indudingphysicians and com-puter scientists, and found that those net-works were also scale-free, corroboratinga study we conducted focusing on math-ematicians and neurologists. (Interesting-ly, one of the largest hubs in the mathe-matics community is Erdos himself, whowrote more than 1,400 papers with nofewer than 500 co-authors.)

Scale-free networks can occur in busi-ness. Walter W. Powell of Stanford Uni-

versity, Douglas R. White of the Univer-sity of California at Irvine, Kenneth W.Koput of the University of Arizona, andJason-Owen Smith 6f the University ofMichigan studied the formation of al-liance networks in the U.S. biotechnolo:

gy industry and discovered definite hubs~for instance, companies such as Genzyme,Chiron and Genentech had a dispropor-tionately large number of partnershipswith other firms. Researchers in Italy tooka deeper look at that network. Using datacollected by the University of Siena's Phar-maceuticallndustry Database, which nowprovides information for around 20,100R&D agreements among more than 7,200organizations, they found that the hubsdetected by Powell and his colleagues wereactually part of a scale-free network.

Even the network of actors in Holly-wood-,.,popularized by the game Six De-grees of Kevin Bacon, in which playerstry to connect actors to Bacon via themovies in which they have appeared to-gether-is scale-free. A quantitative analy-

sis of that network showed that it, too,is dominated by hubs. Specifically, al-though most actors have only a few linksto others, a handful of actors, includingRod Steiger and Donald Pleasence, havethousands of connections. (Incidentally,on a list of most connected actors, Bacon

ranked just 876th.)On a more serious note, scale-free

networks are present in the biologicalrealm. With Zoltan Oltvai, a cell biologistfrom Northwestern University, we founda scale-free structure in the cellular meta-

bolic networks of 43 different organismsfrom all three domains of life, includingArchaeoglobus fulgidus{an archaeb<lc-terium), Escherichia coli (a eubacterium)and Caenorhabditis elegans (a eukary-ote). In such networks, cells burn food bysplitting complex molecules to release en-ergy. Each node is a particular molecule,and each link is a biochemical reaction.

We found that most molecules participatein just one or two reactions, but a few (thehubs), such as water and adenosine tri-phosphate, playa role in most of them.

We discovered that the protein-inter-action network of cells is scale-free as well.

In such a network, two proteins are "con-nected" if they are known to interact witheach other. Whenwe investigated Baker'syeast, one of the simplest eukaryotic (nu-cleus-containing) cells, with thousands ofproteins, we discovered a scale-free topol-ogy: although most proteins interact withonly one or two others, a few are able to

54 SCIENTIFIC AMERICAN

attach themselves physically to a hugenumber. We found a similar result in the

protein-interaction network of an organ-ism that is very different from yeast, a sim-ple bacterium called Helicobacter pylori.

Indeed, the more that scientists stud-ied networks, the more they uncoveredscale-free structures. These findings raisedan important question: How can systemsas fundamentally different as the cell andthe Internet have the same architecture

and obey the same laws? Not only arethese various networks scale-free, theyalso share an intriguing property: for rea-sons not yet known, the value of n in thekn term of thepower law tends to fall be-tween 2 and 3.

The Rich Get RicherPERHAPS A Mo..RE BASIC question iswhy randpm-network theory fails to ex-plain the existence of hubs. A closer ex-amination of the work of Erdos and Ren-

yi reveals two reasons.In developing their model, Erdos and

Renyi assumed that they had the full in-ventory of nodes before they placed thelinks. In contrast, the number of docu-ments. on the Web is anything but con-stant. In 1990 the Web had only one page.Now it has more than three billion. Most

networks have expanded similarly. Hol-lywood had only a handful of actors in1890, but as new people joined the trade,the network grew to include more thanhalf a million, with the rookies connect-ing to veteran actors. The Internet hadonly a few routers about three decadesago, but it gradually grew to have mil-lions, with the new routers always linkingto those that were already part of the net-work. Thanks to the growing nature ofreal networks, older nodes had greateropportunities to acquire links.

Furthermore, all nodes are not equal.When deciding where to link their Webpage, people can choose from a few billionlocations. Yet most of us are familiar with

only a tiny fraction of the full Web, andthat subset tends to include the more con-

nected sites because they are easier to find.By simply linking to those nodes, peopleexercise and reinforce a bias toward them.

This process of "preferential attachment"occurs elsewhere. In Hollywood the more

MAY2003

BIRTHOFASCALE-FREENETWORKA SCALE-FREE NETWORKgrows incrementally from two to 11 nodes in this example. When deciding where to establish a link, a new node

(green) prefers to attach to an existing node (red) that already has many other connections. These two basic mechanisms-growth

and preferential attachment-will eventually lead to the system's being dominated by hubs, nodes having an enormous number of links.

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connected actors are more likely to bechosen for new roles. On the Internet the

more connected routers, which typicallyhave greater bandwidth, are more desir-able for new users. In the U.S. biotech in-

dustry, well-established companies such asGenzyme tend to attract more alliances,which further increases their desirabilityfor future partnerships. Likewise, themost cited articles in the scientific litera-ture stimulate even more researchers to

read and cite them, a phenomenon thatnoted sociologist Robert K. Mertoncalled the Matthew effect, after a passagein the New Testament: "For unto everyone that hath shall be given, and he shallhave abundance."

These two mechanisms-growth andpreferential attachment-help to explainthe existence of hubs: as new nodes ap-pear, they tend to connect to the moreconnected sites, and these popular loca-tions thus acquire more links over timethan their less connected neighbors. Andthis "rich get richer" process will gener-ally favor the early nodes, which are morelikely to eventually become hubs.

Along with Reka Albert, we have usedcomputer simulations and calculations to

show that a growing network with pref-erential attachment will indeed becomescale-free, with its distribution of nodes

following a power law. Although this the-oretical model is simplistic and needs tobe adapted to specific situations, it does

appear to confirm our explanation for

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why scale-free networks are so ubiquitousin the real world.

Growth and preferential attachmentcan even help explicate the presence ofscale-free networks in biological systems.Andreas Wagner of the University ofNew Mexico and David A. Fell of Oxford

Brookes University in England havefound, for instance, that the most-con-nected molecules in the E. coli metabolic

network tend to have an early evolution-ary history: some are believed to be rem-nants of the so-called RNA world (theevolutionary step before the eInergence ofDNA), and others are coiJ;].poneptsohremost ancient metabolic pathways.

Interestingly, the mechanism of pref-erential attachment tends to be linear. In

other words, a new node is twice as Hke-ly to link to an existing node that hastwice as many connections as its neigh-bor. Redner and his colleagues at BostonUniversity and elsewhere have investigat-ed different types of preferential attach-ment and have learned that if the mecha-

nism is faster than linear (for example, anew node is four times as likely to link to

~ ~

~~an existing node that has twice as manyconnections), one hub will tend to runaway with the lion's share of connections.In such "winner take all" scenarios, thenetwork eventually assumes a star topol-ogy with a central hub.

AnAchilles' HeelAS HUMANITY BECOMES increasing-ly dependent on power grids and com-munications webs, a much-voiced con-cern arises: Exactly how reliable are thesetypes of networks? The good news is thatcomplex systems can be amazingly re-silient against accidental failures. In fact,although hundreds of routers routinelymalfunction on the Internet at any mo-ment, the network rarely suffers majordisruptions. A similar degree of robust-ness characterizes. living systems: peoplerarely notice the consequences of thou-sands of errors in their cells, ranging frommutations to misfolded proteins. What isthe origin of this robustness?

Intuition tells us that the breakdownofa substantial number of nodes will re-

sult in a network's inevitable fragmenta-

ALBERT'L4SZL.6BARABASI. and ERIC BONABEAU study the behavior and characteristics of

myriad complex systems, ranging from the Internet to insect colonies. Barabasi is Emil T.

Hofman. Professor of Physics at the. University of Notre Dame, where he directs research

on complex networks. He is author of Linked: The New Science of Networks. Bonabeau is

chief scientist at Icosystem, a consulting firm based in Cambridge, Mass., that applies the

tools of complexity science to the discovery of business opportunities. He is co-author of

SwarmIntelligence: From Natural to ArtificialSystems. Thisis Bonabeau's second article

for Scientific American.

SCIENTIFIC AMERICAN 55

tion. This is certainly trUe for random net-works: if a critical fraction of nodes is re-

moved, these systems break into tiny,noncommunicating islands. Yet simula-tions of scale-free networks tell a different

story: as many as 80 percent of random-ly selected Internet routers can fail and theremaining ones will still form a compactcluster in which there will still be a pathbetween any two nodes. It is equally dif-ficult to disrupt a cell's protein-interactionnetwork: our measurements indicate that

even after a high level of random muta-tions are introduced, the unaffected pro-teins will continue to work together.

In general, scale-free networks dis-play an amazing robustness against ac-cidental failures, a property that is root-ed in their inhomogeneous topology. Therandom removal of nodes will take out

mainly the small ones because they aremuch more plentiful than hubs. And theelimination of small nodes will not dis-

rupt the network topology significantly,because they contain few links comparedwith the hubs, which connect to nearlyeverything. But a reliance on hubs has aserious drawback: vulnerability to attacks.

In a series of simulations, we found

that the removal of just a few key hubsfrom the Internet splintered the systeminto tiny groups of hopelessly isolatedrouters. Similarly, knockout experimentsin yeast have shown that the removal ofthe more highly connected proteins has asignificantly greater chance of killing theorganism than does the deletion of othernodes. These hubs are crucial-if muta-

tions make them dysfunctional, the cellwill most likely die.

A reliance on hubs can be advanta-

geous or not, depending on the system.Certainly, resistance to random break-down is good news for both the Internetand the caLIn addition, the cell's relianceon hubs provides pharmaceutical re-searchers with new strategies for selectingdrug targets, potentially leading to curesthat would kill only harmful cells or bac-teria by selectively targeting their hubs,while leaving healthy tissue unaffected.But the ability of a small group of well-in-formed hackers to crash the entire com-

munications infrastructure by targetingits hubs is a major reason for concern.

The Achilles' heel of scale-free net-

works raises a compelling question: Howmany hubs are essential? Recent research

56 SCIENTIFIC AMERICAN

suggests that, generally speaking, the si-multaneous elimination of as few as 5 to

15 percent of all hubs can crash a system.For the Internet, our experiments implythat a highly coordinated attack-first re-moving the largest hub, then the nextlargest, and so on-could cause signifi-cant disruptions after the elimination ofjust several hubs. Therefore, protectingthe hubs is perhaps the most effectiveway to avoid large-scale disruptionscaused by malicious cyber-attacks. Butmuch more work is required to deter-mine just how fragile specific networksare. For instance, could the failure of sev-eral hubs like Genzyme and Genentechlead to the collapse of the entire U.S. bio-tech industry?

Scale-Free EpidemicsKNOWLEDGE ABOUT scale-free net-

works has implications for understandingthe spread of computer viruses, diseasesand fads. Diffusion theories, intensivelystudied for decades by both epidemi-ologists and marketing experts, predict acritical threshold for the propagation ofa contagion throughout a population.Any virus, disease or fad that is less in-fectious than that well-defined threshold

will inevitably die out, whereas thoseabove the threshold will multiply expo-nentially, eventually penetrating the en-tire system.

Recently, though, Romualdo Pastor-Satorras of the Polytechnic University ofCatalonia in Barcelona and Alessandro

Vespigniani of the International Centerfor Theoretical Physics in Trieste, Italy,reached a disturbing conclusion. Theyfound that in a scale-free network the

threshold is zero. That is, all viruses, eventhose that are weakly contagious, willspread and persist in the system. This re-sult explains why Love Bug, the mostdamaging computer virus thus far (it shutdown the British Parliament in 2000),was still one of the most pervasive virus-es a year after its supposed eradication.

Because hubs are connected to manyother nodes, at least one hub will tend tobe infected by any corrupted node. Andonce a hub has been infected, it will passthe virus to numerous other sites, eventu-

ally compromising other hubs, which will

MAY2003

HOWROBUSTARERANDOMANDSCALE-FREENETWORKS? I

THEACCIDENTALFAILUREof a number of nodes in a random

network (top panels) can fracture the system into non-communicating islands. In contrast, scale-free networks are

RandomNetwork,Accidental Node Failure

Before

Scale-Free Network, Accidental Node Failure

Before

Scale.FreeNetwork, Attackon Hubs

Before

more robust in the face of such failures (middle panels).

But they are highly vulnerable to a coordinated attack againsttheir hubs (bottom panels).

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After

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then spread the virus throughout the en-

tire system.

The fact that biological viruses spread

in social networks, which in many cases

appear to be scale-free, suggests that sci-entists should take a second look at the

volumes of research written on the inter-

play of network topology and epidemics.

Specifically, in a scale-free network, the

traditionalpublichealth approach of ran-dom immunization could easily fail be-

cause it would very likely neglect a num-

ber ofthe hubs. In fact, nearly everyonewould have to be treated to ensure thatthe hubs were not missed. A vaccination

for measles, for instance, must reach 90percent of the population to be effective.

Instead of random immunizations,

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.0

thoggh, what if doctors targeted the hubs,or the most connected individuals? Re-search in scale-free networks indicates

that this alternative approach could beef-fective even if the immunizations reached

only a small fraction of the overall popu-lation, provided that the fraction con-tained the hubs.

But identifyingthe hubs in a social

SCIENTIFIC AMERICAN 57

network is much more difficult than de-

tecting them in other types of systems.Nevertheless, Reuven Cohen and ShlomoHavlin of Bar-Han University in Israel, to-gether with Daniel ben-Avraham ofClarkson University, have proposed aclever solution: immunize a small fraction

of the random acquaintances of arbitrar- .ily selected individuals, a procedure thatselects hubs with a high probability be-cause they are linked to many people.That approach, though, leads to a num-ber of ethical dilemmas. For instance,even if the hubs could be identified,should they have priority for immuniza-

tions and cures? Such issues notwith-

standing, targeting hubs could be themost pragmatic solution for the futuredistribution of AIDS or smallpox vaccinesin countries and regions that do not havethe resources to treat everyone.

In many business contexts, peoplewant to start, not stop, epidemics. Viralmarketing campaigns, for instance, oftenspecifically try to target hubs to speed theadoption of a product. Obviously, such astrategy is not new. Back in the 1950s, astudy funded by pharmaceutical giantPfizer discovered the important role thathubs play in how quickly a community of

It's a SmallWor(d,AfterAll

doctors begins using a new drug. Indeed,marketers have intuitively known forsome time that certain customers outshine

others in spreading promotional buzzabout products and fads. But recent workin scale-free networks provides the scien-tific framework and mathematical tools to

probe that phenomenon more rigorously.

FromTheoryto PracticeALTHOUGH SCALE-FREEnetworks

are pervasive, numerous prominent ex-ceptions exist. For example, the highwaysystem and power grid in the u.s. are notscale-free. Neither are most networks

one another.. - -.

The smalt~world property does not necessarily indicate the

presence of a.ny magic organizing principle. Even a large network

witurely random connections will be a small world. Consider

t,n u ~i av

in als know~anotherl.000,then a million, -:be just two handshakes away from you, a billion will be just three

away, and the earth's entire population will be well withfi:l four,

Given that fact, the notion that any two.strangers in the world are,

d average of . egr,IV tihntveal e c

Our simple calculation assumes that the people you know are

all strangers to one another. In reality, there is much overlap.

Indeed, society is fragmented into clusters of individuals-havingsimilar c' uc 'ncome.or interests). aJ~ature

ihathas!t~ --followingthe seminal workin the 1970-sof MarkGranovetter,then a graduate student at Harvard,Clustering\s also a generalproperty of many othe,rtypes of networks, In 1998 DuncanWattsand Steven Strqgat en both at CornellUniversity,

significant 2liJsteri !vari£tyil'6fsy~~ems:'fr'iimtlgridto the he'IIralnetwork of the Caenorhabditisefegans Worm.

Atfirst glance, isolated clusters of highly interconnectednodes appear to run counter t01he topology of scale-freenetworks, inwhich a number of hubs raoiate throughout thesyst~m,lin verytRing~!Re~ently,'H~weverrwe.havl".. ,i' . M

that the two operties are compatible:.a network can be bothhighly clustered and scale-free when small, tightly interlinkedclusters of nodes are connected into larger, less cohesive groups(teft), Thistype of hierarchy appears to exist in a number of

systems, fr~~Jhe WorldWide)Yeb(io,yV/)jchc sare:,groupings ofWebpages devotei:!to the same t ) to a cell(in whichclusters are teams of molecules responsible fora specific function). -A.-L.B. and e.B:

010

pieasking them to forwardthe correspondence to acquaintanceswho might be able to shepherd it closerto a target recipient: astockbroker in Boston. Totrack"each of the different paths,

11.,t1i1 ask e p ant\.~o mai w ey p d th erto,somec ,

the letters that eventually arrived at the final destination hadpassed through an ave-rageof six individuals-the basis of thepopular notion of "sixdegrees of separ'!tion" between everyone.

Althou am rk rdly:£onclu-'-cletter~ ney de th y stofkbrokerecen~ly learned that other networks exhibit this "small world"

property, We have found, for instance, that a path of just three

reactions will connect almost any pair of chemicals in a cell. And

HIERARCHICALC!-~STE cludpag the FrankLio w).,

be I to otherclusters'(green) sin Wright;Tamous homes orPennsylvania's attractions. Those sites, in turn, could be connected to

clusters (red) on famous architects or architecture in general.

58 SCIENTIFICAMERICAN

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MAY 2003

MAPOFINTERACTINGPROTEINSin yeast highlights tl'\!!discoverytha.~high1ylinked, or hub, proteinstend to be crucial for a cell's survival. Red denotes essential proteir"js(their removal will cause the cell

to die). Orange represents proteins of some importance [their removal will slow cell growth). Green

and yellow represent proteins of less.er or unknoWn significance, respectively.

'":z~'":z0

3<::I:~0

g~0u

seen in materials science. In a crystallat-tice, for instance, atoms have the samenumber of links to their neighbors. Withother networks, the data are inconclusive.The relatively small size of food webs,whiclI §how predator-prey relationships,has prevented scielltists from reaching aclear conclusion regarding that network'stype. And the absenc~ oflarge-scale con-nectivitymaps of the brain has kept re-searchers from knowing the nature ofthat important network as well.

Determining whetl~er a network isscale-free is important in understandingthe system's behavior, but other signifi-cantparameters merit attention, too.One such characteristic is the diameter,or path length, of a network: the largestnumber'gf hops required to get from onenode to another by following the shortestroute possible [seebox on opposite page].

Finally, knowledge of a network's gen-eral topology is just part of the story in un-derstanding the overall characteristics and

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b~havior of such systems. There might besteep~osts, for instance, with the additionof each link to a given node that couldprevent certain networks (such as the U.S.highway system) from becoming scale-free. In food Ghains, some prey are easierto catch than others, and that fact has aprofound effe.cton the overall ecosystem.With social networks, ties among house-hold members are much stronger thanconnections to c~sual acquaintances, sodiseases (and information) are more like-ly to spread through such linkages. Fortransportation, transmission and commu-nications systems (such as the Internet),

congestion along specific links is a majorconsideration: too much traffic on a par-ticular link can cause it to break down,

leading to the potential failure of otherlinks that must then handle the spillover.And the nodes themselves might not behomogeneous-certain Web pages havemore interesting content, for instance-which could greatly alter the preferential-attachment mechanism.

Because of these and other factors, sci-entists have only begun to uncover the be-havior of scale-free systems. Immunizinghubs, for instance, might not be sufficientto stop the spread of a disease; a more ef-fective solution might be found by con-sidering not just the number of connec-tions a person has but also the frequencyand duration of contact for those links.

In essence, we have studied complexnetworks first by ignoring the details oftheir individual links and nodes. By dis-tancing ourselves from those particulars,we have been able to better glimpse someof the organiziIlg principles behind theseseemingly incomprehensible systems. Atthe very least, knowledge from this en-deavor has led to the rethinking of manybasic assumptions. In the past, for exam-ple, researchers modeled the Internet as arandom network to test how a new rout-

ing protocol might affect system conges-tion. But we now know that the Internet is

a scale-free system with behavior that isdramatically different from a random net-work's. Consequently, investigators suchas John W. Byers and his colleagues atBoston University are revamping the com-puter models they have been using to sim-ulate the Internet. Similarly, knowledge ofthe properties of scale-free networks willbe valuable in a number of other fields, es-pecially as we move beyond network to-pologies to probe the intricate and oftensubtle dynamics taking place within thosecomplex systems. Ii!I1

MORE: TO E:XPLORE: . IAll the World's a Net, David Cohen in New Scientist, Vol. 174, No. 2338, pages 24-29; April 13, 2002.

Statistical Mechanics IIfComplex Networks. Reka Albert and Albert-Laszlo Barabasi in Reviews

of Modern Physics, Vol.74, pages 47-97; January 2002.

Linked: The New Science of Networks. Albert-Laszlo Barabasi. Perseus Publishing, 2002.

Evolution of Networks: From Biological Nets to the Internet and WWW.J.F.F. Mendes and Sergei N.Dorogo~sev. OxfOrdUniversity Press, 2003.

Find lillks tQ paper.!>on sca.le-free networks at www.nd.edu/-networks

SCIENTIFIC AMERICAN 59