WI – State-of-the-Art

Scale-Free Networks The Impact of Fat Tailed Distribution on Diffusion and Communication Processes

works [ErRe59]. ER-networks are random like the degree distribution, the clustering in the sense that the links between the coefficient and the average length will The Authors nodes of a network are created with the be presented to enable a better understand- help of a random variable. Recently, fol- ing of the two network models “ER” and Oliver Hein lowing the introduction of what were “scale-free”. Diffusion processes in net- called “scale-free” networks by Barabasi works depend very much on network to- Michael Schwind and Albert [BaAl99], the field has received pology. Both network models will be ex- Wolfgang Ko¨nig growing attention from scientific research. amined in terms of their different diffusion Scale-free networks seem to match real behavior. Finally, the potential impact for Dipl.-Inform. Oliver Hein world applications much better than ER- real-life applications will be shown by the Dipl.-Wirtsch.-Ing. Michael Schwind network models [Bara03]. The term scale- introduction of a new application of scale- Prof. Dr. Wolfgang Ko¨nig free refers to the distribution principle of free networks: The simulation of the impli- Johann Wolfgang Goethe-Universita¨t how many links there are per node. cations of for the price Institut fu¨r Wirtschaftsinformatik Mertonstr. 17 This article is intended to provide an in- building process in a security trader net- 60325 Frankfurt am Main troduction into the metrics of modern net- work that relies on a communication net- {ohein, schwind, koenig}@is-frankfurt.de work analysis. The basic network measures work.

Executive Summary

1 Introduction The study of network topologies provides interesting insights into the way the principles on which the construction of connected systems are based influence diffusion dynamics and communication processes in many socio-technical systems. The widespread presence of networked systems in technical applications as well as & Empirical research has shown that there are principles for the construction of social net- in the business world makes network re- works and their technical derivatives, like e-mail networks, the , publication co- search useful for future applications in in- authoring, or business collaboration. formation systems (IS) research. Research & Such real world networks attach new members over time and the mode of attachment into networks means not just analyzing the prefers existing members that are already well connected. This principle is called “prefer- topology of networks, but also examining ential attachment” and leads to the emergence of “scale-free” networks. the dynamics of the processes that take & Scale-free networks seem to be a better fit for the description of real world networks than place in them. Research into diffusion pro- the random networks used so far. Their behavior in terms of diffusion and communication cesses in networks has been shown to pro- processes is fundamentally different from that of random networks. vide especially fruitful insights into several & To illustrate the value of scale-free networks for applications in information systems re- kinds of IS domains such as e.g. informa- search, examples will be given to illustrate their usefulness for real world network model- tion technology (IT) robustness in system ing. A communication network of security traders will show what impact network topol- failure situations, denial-of-services at- ogy has on the dynamics of complex socio-technical systems. tacks, and computer virus spreading. For decades network research was domi- Stichworte: Random Networks, Scale-Free Networks, Communication Networks, Socio- nated by random networks, also called Er- Technical Networks, Diffusion do¨ s and Renyi networks or just ER-net-

WIRTSCHAFTSINFORMATIK 48 (2006) 4, S. 267–275 268 Oliver Hein, Michael Schwind, Wolfgang Ko¨nig

bold edges) with four friends. Out of six possible friendship relationships (dashed lines) between the four friends (grey cir- cles), only two friendships exist (bold edges). Using the ratio between existing and possible relations, a may be computed. If a node has z nearest neighbors, a maximum of zðz 1Þ=2 edges is possible between them. Watts and Stro- gatz defined the clustering coefficient for node v as the ratio of the number i of exist- ing nodes to the possible number of edges between the direct neighbors of node v [WaSt98]: 2i Figure 1 The Basic Concept of a Network Degree Distribution Cv ¼ : ð1Þ zðz 1Þ As for the degree distribution, the cluster- ing coefficient plays an important role The article is intended to emphasize the 2.1 Basics of Network Analyses when analyzing networks in terms of im- importance of choosing the right network portant properties like diffusion, which model for the right purpose. The funda- In an undirected network graph, the degree will be discussed in subsequent paragraphs. mental differences between ER- and scale- of a node is defined as the number of edges The path length between two nodes of a free networks will become apparent in the it possesses. Figure 1a shows an example of network is defined as the number of edges sample application presented. a node with degree 4. If all the nodes of a between them. The minimal path length is network that share the same degree are the shortest path between two nodes. The counted and the results are sorted by in- is the average of all the 2 Network Models creasing degree a function like that in Fig- minimal path lengths between all pairs of ure 1c is most likely to occur. There are no nodes in a network. nodes with degree zero, because these This paragraph presents ER- and scale-free nodes would not be connected to the net- networks as well as their main characteris- 2.2 ER-Networks work. The network in figure 1c exhibits a tics and the processes by which they are majority of nodes with degree one to three Since the seminal paper of Erdo¨ s and Renyi created. ER-networks, because of their di- and rather fewer nodes with a degree great- in 1959, the random domi- minishing importance for modeling real er than seven. Chart 1b is idealized; the de- nated scientific thinking [Bara03]. Real world networks, are only briefly intro- gree distribution of networks does not world networks had been thought to be duced to show their important differences have to be continuous. too complex to understand and therefore from scale-free networks. The basics of If one node of the network in figure 1c held to be random. In the absence of other network analysis are discussed first to en- were randomly chosen, the probability of well-understood network models, random able a better understanding. obtaining a node with only one to three networks were widely used when modeling edges is much higher than that of obtaining networks. a node with a degree higher than seven. The process of creating an ER-network Therefore it is possible to define a prob- depends on probability p. For a network ability distribution function Pðj; v; NÞ that with n nodes each possible pair of distinct returns the probability that node v has j nodes are connected with an edge with edges within network N [DoMe03]. The probability p. concept of the degree distribution of a net- An ER-network has the property that work has important consequences for the the majority of nodes have a degree that is properties of a network and will be fre- close to the average degree of the overall quently used in the paragraphs that follow. network and that there is not much devia- It will be seen that deviation from the nor- tion from the average below and above it. mal distribution will lead to new results in It has been shown that the distribution of terms of diffusion within networks. links follows a (fig- Clustering within networks is another ure 3). Knowing the degree distribution, important factor when analyzing networks. the average path length, and the clustering It is interesting to know how well nodes coefficient of ER-networks, it is feasible to are interconnected within a specified area analyze their different behavior compared of the network. A similar task is the ques- to scale-free networks. Figure 2 The Bold Node is Connected tion how many of a person’s friends also ER-networks are still used in some types with Bold Edges to its Four Direct know each other. Figure 2 presents an ex- of models. The following chapters will Neighbors ample of a person (bold node with four show that ER-networks may not be applic-

WIRTSCHAFTSINFORMATIK 48 (2006) 4, S. 267–275 Scale-Free Networks 269 able for every purpose, because they lack Since the important observation of Al- some of the properties of other classes of bert et al. [AlJB99], scientists have empiri- networks. cally analyzed many real world networks with the scale-free property. The physical structure of the internet (router level, do- 2.3 Scale-Free Networks main level, web links), social networks like e-mail networks, the structure of software When Albert et al. [AlJB99] started to map modules, and many more examples show the internet in 1999, they did not know surprising scale-free structures (table 1). that they were about to influence network g is the degree exponent and ranges be- research in a sustained way. Because of the tween 1.8 and 2.5. The factor is derived by diverse interests of every internet user and counting the number of edges per node the gigantic number of web pages, the lin- and plotting the results by increasing de- kages between web pages were thought to gree within a chart with log-log-scale. The Figure 3 The Poisson Degree Distribu- be randomly linked as a random network. slope of the resulting straight line is g (fig- tion of the Number of Edges of ER-Net- The results of their study have disagreed ure 4 right). What is the reason for the si- works with the Characteristic Bell with this expectation in a surprising way. milar g in table 1 and how do the different Shape around the Average Number of Only a few pages have the majority of networks compare? First of all they are all Edges per Node [DoMe03] links, whereas most pages are only very evolving networks, which means that they sparsely connected. More than 80% of all develop over time. ER-networks in con- pages visited have 4 links or less, only trast are created with the number of nodes 0.01% of the pages are linked to more than already fixed. Second, they are created by with two new edges. The new edges are 1,000 other pages (some up to two mil- , which means that connected to existing nodes with a prob- lion). newly introduced nodes connect with a ability that corresponds to the number of If the nodes with one, two, three, etc. higher probability to existing nodes with edges of the existing nodes. The “richest” connections are counted, and the numbers many links. If the evolution of networks is nodes will receive most of the newly added are plotted into a chart as in the middle of the appropriate explanation for the distinct edges. figure 4, the distribution of edges becomes scale-free property, then the growth path The generalized creation process of a visible. There are many nodes with only a of networks deserves particular attention. scale-free-network is defined by Barabasi few links and a few nodes with a large Starting with a small number of nodes, like and Albert [BaBo03]: Start with a small number of links. If the distribution of the internet in 1990, new nodes are added number of nodes m0. During each step, in- edges is plotted in a logarithmic chart, as in over time. Older nodes therefore have a sert a new node with connections to figure 4 right, the power-law nature of the greater probability of acquiring new links. m m0 existing nodes. The appears in a straight curve Furthermore, already well-connected Pv of node v being connected to a new with a slope of g from equation 2. The nodes are much easier to find and will in node depends on its degree jv. The higher difference from ER-networks now be- consequence acquire more new links. its degree, the greater the probability that it comes evident (figure 4 left). The probabil- Coupled with the expanding nature of will receive new connections (“the rich get ity distribution function PðjÞ of the degree many networks this might explain the oc- richer”): j of scale-free networks is described by: currence of hubs, which hold a much high- P Pv ¼ jv jk : ð3Þ PðjÞjg ð2Þ er number of links than the average node. Figure 6 presents an idea of how scale-free k with j > 0 and g > 0, with g called the networks may evolve over time for the first The process of preferential attachment, as scale-free exponent. The term scale-free is a 8 steps. At each step, a new node is added defined by Barabasi and Albert, is linear.A mathematical expression which stands for power-law distributions as in equation 2. The power-law distributions belong to the class of leptokurtic or “fat-tailed” distribu- tions. They deviate from the Poisson distri- bution in two ways: they have higher peaks and “fat” tails. Figure 5 shows a power-law distribution in relation to a Poisson distri- bution. The peak on the left side of the chart is higher and the tail on the right side is thicker, therefore called “fat tail”. Com- pared to the Poisson distribution, the power-law distribution shows the greater probability of nodes with a number of links close to the average of all links (see the peak with 2 links in the example of fig- Figure 4 Poisson Distribution of Links for ER-Networks (left) and Scale-free Networks ure 5), and a greater probability of nodes (middle, right in log-log scale with the characteristic straight slope of a ) with many links (4 or more links in the tail [BaBo03] in figure 5).

WIRTSCHAFTSINFORMATIK 48 (2006) 4, S. 267–275 270 Oliver Hein, Michael Schwind, Wolfgang Ko¨nig

ter understanding of how scale-free net- works may arise and be analyzed. It may 0,40 help to enable the reader to use the right network model for the right application. The scale-free exponent g makes the exam- 0,35 ples from table 1 comparable. They all are similar in the sense that the probability dis- 0,30 tribution of their links is almost the same. Higher Peak However, only almost, because e-mail net- 0,25 works with a g of 1.81 and software with a g of 2.5 show the greatest difference among 0,20 the examples in table 1. How can this dif- ference be explained? The scale-free expo- probability p 0,15 Fat Tail nent g within PðjÞjg determines the probability PðjÞ of the occurrence of nodes 0,10 with degree j within the network. With a rising degree j the probability of the exis- 0,05 tence of nodes with degree j within the net- work decreases. In other words: the steeper 0,00 the slope, the more concentrated the ana- 2345678910 lyzed network is. It means that a higher g leads to a lower probability of nodes with degree j many links. Carried forward to the exam- Poisson Distribution Power-law Distribution ple of e-mail and software networks, e-mail networks with a g of 1.81 have a higher Figure 5 Poisson Distribution and Power-Law Distribution probability of nodes with many links, than is the case for software networks with a g of 2.5. Therefore, it may be concluded that e-mail networks possess a higher quantity of “super-nodes” with many links com- pared to software networks. Let us apply these findings to the five networks shown in table 1: The structure of the Internet can be studied on different levels (table 1, line 1–3). The physical level is composed of the servers, switches and routers. If a router is taken as a node and its connections to other routers as links, a arises and can be studied for its network properties. A scale-free ex- ponent g of 2.1 (table 1, line 2) for an ob- served router network of 228.298 routers shows the existence of some major router hubs and many small routers within their network periphery. The linkage between web pages – WWW – is another Internet Figure 6 The Emergence of a Scale-Free Network as a Result of the Preferential At- structure that forms a large network (ta- tachment of Newly Added Edges and Nodes to Already Existing Nodes [BaBo03] ble 1, line 3). For the WWW it is possible to analyze the in-degree, how many links lead to a web page, and the out-degree, new node is twice as likely to link to an ex- sulting scale-free exponent g is, as in the how many links lead out of the web page to isting node that has twice as many links as examples in table 1, between 1.8 and 2.5, other web pages. If the web pages are taken its neighbor. If the probability Pv is in- depending on how many links m a new as nodes and the links as edges between creased even more the network evolves to a node will link to existing nodes. The high them another large network is created. As star topology with one central hub correlation of empirically examined real in the example with the routers, the distri- [BaBo03]. world networks with networks generated bution of links between the web pages of To return to the examples of table 1, it by the Barabasi-Albert-algorithm indicates the WWW seems to have the scale-free may now be possible to explain why these that preferential attachment does indeed property. networks are similar in terms of network seem to be a valid explanation of how this Yet another internet network is the sys- measures like the scale-free exponent g. network structure evolves. tem of Internet domains and sub-domains. When scale-free networks are constructed, In the following section, the examples They may be seen as another structural le- as Barabasi and Albert suggested, the re- from table 1 will be discussed to get a bet- vel of the Internet (table 1, line 1). Domain

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which means the person is either immune Table 1 A Choice of Some Real World Networks within the IS Domain that Share the Scale-Free Pro- perty [WaCh03] or has died. The SIS-model, a variant of the SIR-model, changes the individual back to Network Size Clustering coefficient Degree exponent the susceptible status and another infection n C g may recur. How an infection spreads de- Internet, domain level [VaPV02] 32711 0.24 2.1 pends on l ¼ d=n and the structure of the network. The course of an infection can be Internet, router level [VaPV02] 228298 0.03 2.1 simulated using an iterative network mod- el. The intended number of time steps WWW [AlJB99] 153127 0.11 gin ¼ 2:1 gout ¼ 2:45 stands for the iterations that are used; d E-mail [EbMB02] 56969 0.03 1.81 and n are reviewed per time step for every Software [VaFS02] 1376 0.06 2.5 node within the network. The status of every node is changed over the course of the intended number of time steps. For and sub-domain names inter-linked ac- tion that a lower g indicates an improved every time step the number of infected cording to their hierarchy describe another software design process. The network ana- nodes is recorded. An infection path will Internet related scale-free network. The lysis may therefore provide a tool for veri- arise that is characteristic for l ¼ d=n and three levels of the Internet mentioned are fying the outcome of the efficient reuse of the network topology used. three distinct networks, each one created software modules. Within ER-networks there is a constant for a different purpose. From the physical lc > 0. If lc > d=n the epidemic dies out network of routers and cables, to the orga- quickly. With lc < d=n the epidemic nization of parts of the Internet in the form spreads over the whole network [AlBa02; of domains and sub-domains, to the struc- 3 Diffusion in Networks Leve02; LlMa01; Lope04]. Pastor-Satorras ture of web pages, which are linked accord- and Vespignani found that within scale-free ing to their content, all three networks The spread of epidemics within networks networks the constant lc is close to zero share the same scale-free exponent g of 2.1 is an interesting subject that everyone has [MaLo01; PaVe01]. This means that within (in the case of the WWW gin ¼ 2:1). occasionally been confronted with in real scale-free networks there is no resistance to An e-mail network (table 1, line 4) is less life. The computer worm Melissa and infections, and that an infection once ac- a technical network and more a social net- many other examples of epidemics have quired quickly develops into an epidemic. work, because it represents the social con- something in common. They are infections The diffusion behavior of epidemics there- tacts of users to other users. If an e-mail that disperse through complex networks by fore depends on the infection and recovery user is seen as a node, the links to other spreading from one individual to their di- rate on the one hand and the topology of nodes are created by tracing the e-mails rect neighbors. The prediction of epidemics the network on the other. In the following sent and received between the users. If a is especially important to prevent damage sections, the two network types will be user A sends an e-mail to user B, an edge to life as well as to material goods, and de- presented in the light of the SIS-model. between node A and B is drawn. Even pends on factors like how fast a neighbor though an e-mail network represents an becomes infected, how long they remain 3.2 Diffusion and ER-Networks area of human communication habits, it infectious and how the network the indivi- seems to have a similar structure, seen from dual is a part of is structured. The diffusion The degree distribution of ER-networks the standpoint of the degree distribution, of viruses or information through complex peak at an average value hji and decay expo- to a technical infrastructure like the Inter- networks may be studied with the help of nentially for j hji and j hji [MoPV02]. net. With a scale-free exponent g of 1.81 it simulation models, to produce predictions It has been shown that under these as- may be assumed that few people use e-mail of the course of epidemics [Keel99]. sumptions for the epidemic threshold of intensively with contacts to many other In the following three sections, a basic ER-networks there is a critical non-zero people, while most e-mail users only write model for diffusion alongside its applica- value lc: e-mails to a small group of receivers. tion on ER- and scale-free networks will l ¼ 1=hji : ð4Þ Valverde et al. [VaFS02] analyzed the be presented. c class diagram of the Java Development Fra- If l is below lc the infection dies, if l is mework (JDK 1.2) (table 1, line 5). In the 3.1 SIS / SIR-Model above lc the infection spreads and becomes case of software networks, we can analyze persistent [PaVe02]. If the infection and re- not only software structures that have The mathematical framework for the study covery rates of the network members are evolved over time, but also those which of epidemics goes back to Kermack and known, the number of nodes m that may were created intentionally for the goal of McKendrick 1927 [KeMc27]. They in- be infected within ER-networks is shown more efficient use of individual software vented the SIR-model, which stands for in figure 7. Until lc is reached the infection modules. The modular structure of modern susceptible-infected-removed. An indivi- is not able to remain in the network, it dies software may be represented by class dia- dual may be in one of the three states. If a out more quickly than it is able to infect grams. Software modules are taken as the susceptible person meets an infected per- new members of the network. If l is great- nodes of a with links ac- son, for example the direct neighbor within er than lc the infection can remain in the cording to their interaction. A lower de- a network, his or her status changes from network and infects broader regions of the gree coefficient in the scale-free structures susceptible to infected with probability d. system. The higher the l chosen, the more of the class diagrams, as found by Valverde With probability n a person may change regions of the network may be reached et al. [VaFS02], could lead to the assump- his or her status from infected to removed, during the course of an infection.

WIRTSCHAFTSINFORMATIK 48 (2006) 4, S. 267–275 272 Oliver Hein, Michael Schwind, Wolfgang Ko¨nig

spread of infections [DoMe03]. That find- booms and crashes, seems to play an espe- ing explains why the WWW is so vulner- cially important role [Bane93]. Herding is able to computer viruses. the result of the exchange of information between two or more investors and the willingness to deviate from rational valua- tion rules. To what extent does herding in- 4 A Scale-Free fluence the decision making process of an Communication Network individual security trader and what are the consequences for price building at a secur- of Security Traders ity exchange? We want to answer this question, by Network theory has made significant pro- means of a security market simulation (mi- gress during the last seven years, as a result cro-simulation) of individual security tra- of the publications of Watts, Strogatz ders connected by a well-defined commu- [WaSt98] and Barabasi, Albert [BaAl99]. nication network. In this way, the effects of The most important progress is that net- herding may be measured. The quality of work theory was taken out of the ivory the communication network plays a central Figure 7 Spreading of an Infection towers of purely mathematical interest and role when modeling the real-life exchange within ER-Networks Depending on l brought into everyday life by fitting theo- of information between investors. We retical expectations to empirical findings. choose a scale-free network as the commu- What is surprising is how powerful the nication model between agents. Simulation 3.3 Diffusion and Scale-Free analysis of real-world networks using to models always need validation, without the scale-free paradigm is, in terms of ex- which the results would be useless. When Networks planation and prediction. It seems that they speaking of stock market simulations, the For scale-free networks it has been shown are able to capture an important part of properties of the price development gener- natural mechanisms that lead to higher effi- ated by the simulation model are decisive. that there is no critical value lc > 0 below which an infection will not spread through ciency and robustness. Dorogovtsev, Real stock markets have important statisti- the system [DoMe03; MaLo01]. Figure 8 Mendes [DoMe03] is an appropriate source cal properties, called universal stylized shows the long-run prevalence of an infec- for a broader review of real-world applica- facts. If the daily percentage changes in tion in a scale-free network. The difference tions. stocks or currencies are reviewed, the dis- from ER-networks in figure 8 is evident. In the following an artificial stock mar- tribution of the changes shows many small There is no epidemic threshold, and the in- ket in which the traders communicate by a and some very large fluctuations (also fection starts to conquer distant parts of scale-free communications network is pre- called leptokurtic or “fat-tailed” distribu- the system from the outset. sented. It will be shown that the topology tion of changes). This statistical property If the scale-free exponent g in the degree of the communication network influences can only be achieved by complex systems distribution PðjÞ¼jg of the scale-free sys- the price building process. and is hard to generate. Another stylized tem is less or equal to 3, as is the case for What is the impact of general human in- fact is the correlation between the volatility many real life networks (table 1), there is teraction within specific network topolo- of prices and the volume of shares traded. no epidemic threshold, which means that gies with regard to the outcome of joint It means that volume rises with more the network is extremely sensitive to the goals? A broad empirical basis is necessary widely fluctuating stock prices. The clus- to secure sufficient evidence. A potential tering of volatility, as large fluctuations testing ground for a network theory of hu- tend to cluster together, is also experienced man interaction could be the international when examining real markets [Cont01]. A financial markets. They deliver a large valid stock market simulation must be able amount of second-by-second data and may to reproduce all the above-mentioned and be viewed as the aggregation of human in- other stylized facts of capital markets. This teraction through a price building process. is a difficult task, but is at the moment the Simulated model data could be tested only possible way to validate stock market against time series taken from real markets simulations with empirical findings. and could potentially lead to a better un- The latest models of capital market si- derstanding of joint interaction within a mulations [AHLP93; CoBo00; LeLS00; network environment. Lux00] are already able to reproduce the The decision making process of indivi- universal stylized facts of financial markets. dual security investors is a still undiscov- They differ in two key points from the ered field. The assumption of perfectly ra- model presented. First, they do not use an tional investors seems not to hold any auction method that is in use on real mar- more [Shle00]. The influence of factors kets, but mostly a function that maps sup- other than the fundamental value or the ply and demand to a price. The authors Figure 8 Spreading of an Infection technical analysis of a security seems to chose a realistic auction method that is in within Scale-Free Networks Depending guide the behavior of investors [Mand04; use at the Frankfurt Stock Exchange, on l. Shil01]. The herding effect of groups of in- namely the “Kassakurs”-Method [ScPr95]. vestors during some market situations, like Second, herding seems to be a major deter-

WIRTSCHAFTSINFORMATIK 48 (2006) 4, S. 267–275 Scale-Free Networks 273 minant for bubbles and crashes on the communication network has proved to be of the scale-free network, is the cause of stock exchange. This already well studied a driving force for the price building pro- rectified trading strategies within the agent phenomenon [Bane92] cannot be found in cess within security markets [HeSc05]. In- community. Their influence leads to homo- any capital market simulation models formation diffuses like an epidemic genous buy and sell behavior on the part of known to the authors. Therefore, a com- through a communication network that a majority of the agents and in conse- munication infrastructure was introduced has a scale-free structure. Agents adapt quence to an imbalance between buy and to enable agents to share information. The their beliefs as a result of the information sell orders. The majority of agents buys resulting diffusion of information through they get from their direct neighbors within and sells at the same time and causes prices a network with a well-defined topology is the communication network. If the major- to fall and rise with higher fluctuations. analyzed with regard to the properties of ity of the neighbors decide for a specific Only when extreme levels are reached does the simulated prices on the artificial stock trading strategy, the agent adapts its strat- the price revert, and the influence of the di- exchange. egy. If the simulation results are reviewed, rect neighbors weigh less than actual losses Multi-agent models with inter-agent it becomes evident that the scale-free struc- an agent experiences. In consequence, the communication are able to model the com- ture of the communication network is the agent returns to its original trading strat- plex interactions between the market parti- cause of extreme fluctuations within the egy. Figure 9 shows a typical simulation cipants within the controlled environment price building process [HeSS05]. The run with 200 agents who communicate of the computer lab. The structure of the power of opinion-making agents, the hubs through a scale-free network. The price

1.200 1.150 1.100 1.050

price 1.000 950 900 850 1 101 201 301 401 501 601 701 801 901 1001 iterations (trading days)

9%

4%

–1%

–6% daily changes in % –11% 1 101 201 301 401 501 601 701 801 901 1001 iterations (trading days)

250

200

150

100

50 volume in shares 0 1 101 201 301 401 501 601 701 801 901 1001 iterations (trading days)

Figure 9 Daily Prices, Daily Price Changes and the Daily Volume of Shares Traded of a Simulation of an Artificial Stock Market with a Scale- Free Communication Network [HeSS05]

WIRTSCHAFTSINFORMATIK 48 (2006) 4, S. 267–275 274 Oliver Hein, Michael Schwind, Wolfgang Ko¨nig

Micro-simulations of artificial stock Table 2 Statistical Moments of Simulation Runs with ER- and Scale-Free Communication Networks [HeSS05] markets with inter-agent communication networks seem to hint in the direction of “ER” “scale-free” difference in % the herding effect having a major influence Max. daily price change 7.06% 9.30% þ31.72% on the price building process. The effect of herding is made visible and reproducible Min. daily price change 5.15% 10.28% þ99.61% using scale-free networks in a multi-agent Standard deviation of daily price changes 0.87% 0.99% þ13.79% simulation.

Kurtosis 17.90 29.71 þ65.97% Hill-estimator 1.52 1.48 2.63% 5 Conclusion chart (top) exhibits the ups and downs of tion network yields higher volatility in Network topology does matter. As the ap- the price of an artificial stock, while the daily price changes than with an ER-net- plications of a scale-free network within a chart of the daily changes in percent (mid- work. In every volatility measure the re- communication structure of security tra- dle) indicates some clustering of daily sults are increased. The simulation results ders presented above show, the topology changes and the volume chart (below) seem to support the intuitive expectation of the network plays a crucial role for the shows some volume-volatility clustering that the diffusion process on scale-free net- dynamics of an artificial stock market. The when compared to the chart in the middle. works leads to a quicker distribution of network topology brings a new dimension Even visually, some typical phenomena of profitable market opinions. The buying into the simulation, as not only local deci- capital markets can be recognized. The sta- and selling within the model is synchro- sions, but also the spatial organization of tistical numbers are reviewed later in ta- nized and larger groups of agents buy and the underlying communication system in- ble 2. sell at the same time. The consequence is fluences the development of asset returns To make certain that the scale-free net- that there are not enough buy or sell orders in the network of traders investigated. The work causes the price fluctuations, another that can be matched, so that the price needs price of an asset in the system depends not communication network is needed as a to change in greater quantities which leads only on system time, but also on the rela- benchmark. In parallel to the scale-free net- to higher volatility. tive geographical position of the individual work, an ER-network was used for the si- mulation runs. The results show that the structure of the ER-network influences the prices on the artificial market in another way. There are no opinion leaders in an Abstract ER-network. Every trader has the same amount of influence; the diffusion of opi- nions through the network is slower and Scale-Free Networks – The Impact of Fat Tailed Degree Distribution on Diffusion and Com- sometimes an opinion diminishes before a munication Processes majority may be able to adopt a different The study of network topologies provides interesting insights into the way in which the princi- trading strategy. The result is less volatility ples on which interconnected systems are constructed influence the dynamics of diffusion and less extreme daily moves. Table 2 pre- and communication processes in many kinds of socio-technical systems. Empirical research sents the difference between an ER- and a has shown that there are principles of construction similar to those of the laws of nature for scale-free communication network in terms social networks and their technical derivatives, like E-mail networks, the internet, publication of market price volatility. Some statistical co-authoring, or business collaboration. For decades, the paradigm of a randomly con- measures are needed for the analysis of the nected network has been used as a model for real world networks, in ignorance of the fact different properties. A kurtosis > 3 signals that they are only a poor fit for such networks. Apparently, all the above-mentioned networks by definition a deviation from the normal share the same building blocks. They attach new members over time and the attachment distribution, which indicates a “fat-tailed” prefers existing members that are already well connected. This principle of “preferential at- distribution which means a higher prob- tachment” leads to interesting properties that have to be taken into consideration when ana- ability for large price changes, also called lyzing and designing systems with some kind of network background. returns, than in the normal distribution. What are called “scale-free” networks seems to be a better fit for the description of real The Hill estimator examines the tails; lower world networks. They use preferential attachment as a construction principle to resample real values stand for heavier tails in the return world networks. Their behavior in terms of diffusion and communication processes is funda- distribution and therefore a more extreme mentally different from that of random networks. distribution of returns between small re- To illustrate the potential value of the discovery of scale-free networks for applications in turns and large returns. A Hill estimator information systems related research, an example will be used in this article to illustrate their value around 1 and 2 indicates explosive usefulness for realistic network modeling. A scale-free communication network of security volatility after longer periods of quietness, traders will show what impact network topology has on the dynamics of complex socio-tech- which is commonly experienced on real nical systems. financial markets. The results are compar- able with the behavior of real stock mar- Keywords: Random Networks, Scale-Free Networks, Communication Networks, Socio- kets [HeSc05]. They indicate that the simu- Technical Networks, Diffusion lation runs with a scale-free communica-

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