Network Topologies: Analysis And Simulations

Transcription

1 Networ Topologes: Analyss And Smulatons MARJAN STERJEV and LJUPCO KOCAREV Insttute for Nonlnear Scence Unversty of Calforna San Dego, 95 Glman Drve, La Jolla, CA USA Abstract:-In ths paper we present some of the networ models and topologes that have been defned n the past few years as a result of the communty efforts to model real world networs. For each presented model we provde computer calculated networ measures of nterest and restate some conclusons that separate one networ model from another. Key-Words:- networ, topology 1 Introducton Graphs and ther propertes have been studed for a long tme n the classc graph theory. In the past few years there was a lot of wor done n order to provde models for real world networs. At the current technology level an expermental data s also avalable mang the model verfcaton easy and accurate. In the networ research communty there are already accepted and well establshed networ models. The rest of the paper s outlned as follows. Secton presents networ measures of nterest. Sectons 3, 4 and 5 present the Random, Small- World and Scale-Free networ models. For each model we provde computer smulaton results. Secton 6 concludes. Networ Representaton and Measures Networ s usually represented as a graph G ( N, E) where N s a set of nodes and E s a set of edges. Let n represents the number of nodes n n n( n 1) N. E contans from up to edges. A node degree s the number of edges connected to that node. In the case of drected edges someone can dfferentate between out degree and n degree. The Laplacan matrx L(G) of a graph G, where G s an undrected, unweghted graph wthout graph loops or multple edges from one node to another, s nxn symmetrc matrx wth one row and column for each node. The matrx elements are defned as follows: node degree f j L j ( G) 1 f j edge (, j) (1) otherwse For example, a rng lattce of 4 elements can be represented by the followng Laplacan matrx: L ( G) () The egenvalues of L(G) are nonnegatve and ordered: λ1 λ... λn. Connected component s a subset of mutually reachable nodes. The number of connected components of G s equal to the number of λ equal to. In partcular, λ! f and only f G s connected,.e. there s only one connected component contanng all networ nodes. Networ measures of nterest n our observatons are: 1. Average dstance between two nodes;. Clusterng coeffcent; 3. Degree dstrbuton. The dstance between two nodes s defned as the number of edges along the shortest path connectng them. In our computer smulatons we use the teratve Bellman-Ford algorthm for the shortest

2 dstance computaton. If the value of λ s, the graph s dsconnected and we tae -1 as the average dstance between nodes. We use the followng defnton of the clusterng coeffcent. Let the node has edges that connect t to the neghbors. The maxmum number of edges that can exst among these nodes s ( 1). If E represents the number of edges that actually exst among these nodes, clusterng coeffcent for the node s defned as the followng quotent: E C (3) ( 1) The clusterng coeffcent C for the whole graph s the average of all C 's. The degree dstrbuton P() s defned as the probablty that a randomly chosen node has exactly edges. The followng text presents 3 networ models that have been dfferentated n the contemporary networ topology research: Random, Small-World and Scale-Free networ model. For each model we provde networ measures of nterest collected through the computer calculatons and smulatons. 3 Random Model The Random model s the oldest networ model among the models presented n ths paper. It s based on the random graph theory ntroduced by Paul Erdős and Alfréd Rény ([1],[]). Ther wor was followed by Bélla Bollobás [3], today's one of the leadng scentsts on ths feld. Many complex networs wth unnown organzng prncples appear random and are nvestgated usng random graph theory. Erdős and Rény have defned two models. The frst one s defned as a graph wth N nodes and n edges, where the edges are randomly chosen from the N( N 1) possble ones. In the second model two nodes are connected wth probablty p. These two models become equvalent n the lmt N. Random graph theory s most concerned by the queston at what probablty p a specfc graph property appears. For example at what connecton probablty p a graph wth N nodes becomes fully connected? Are there trangles of connected nodes? The greatest dscovery n the Random model research s that many propertes of these graphs appear very suddenly,.e. bellow some probablty threshold the property doesn't exst, but above that threshold almost every random graph wth the same number of nodes has that property. A sngle node has from up to N-1 edges. In a Random networ model havng connecton probablty p, the node has degree wth probablty followng bnomal dstrbuton: N 1 p 1 N 1 ( p) P( ) C (4) We can assume that the equaton (4) holds for the whole graph. In the lmt N equaton (4) becomes Posson dstrbuton wth the mean value pn: ( pn ) pn P( ) e (5)! Another mportant result from the random graph theory s that f E( ) pn ln ( N ) almost every graph s connected. In that case the average dstance between nodes has the followng approxmaton: l random ( N ) ( E( )) ( N ) ( pn ) ln ln ~ (6) ln ln We can thn of the equaton (6) ths way. Every node has n average pn outgong edges. After l steps, the spannng tree contanng ( pn ) l nodes has reached every node n the graph,.e. ( pn ) l N. The average clusterng coeffcent n the Random networ model s gven as: C random p (7) The equaton (7) has also ntutve bacground because the quotent between the number of exstng edges and number of all possble edges s always p for every possble node's neghborhood. Fg.1 shows the smulaton results for the Random networ model wth N14 nodes. The networ contans undrected edges and does not contan loops or multple edges between nodes. We vary the connecton probablty n the nterval [,1]. In

3 Fg.1-a we plot the average dstance between nodes. It can be seen that bellow p.1 the average dstance s -1,.e. graph s dsconnected. In Fg.1-b we plot the average clusterng coeffcent that s a straght lne wth the slope value equal to 1. In Fg.1-c we plot the node's degree dstrbuton. It can be seen that the curve resembles the Posson dstrbuton. a) 4. Average dstance b) c) Average clusterrng Fracton of nodes p p Node degree Fg.1. Random networ model wth N14 nodes a) Average dstance b) Average clusterng c) Degree dstrbuton We also created networ vsualzaton tool. Fg. s a vsualzaton of the Random networ model wth the connecton probablty equal to.1. The nodes wth a hgher degree are postoned closer to the center. Fg.. Vsualzaton of the Random networ model wth N14 nodes and p.1 4 Small-World Model The dea behnd the Small-World networ model comes from the socal systems and the relatonshps theren. Most of the people n the socal networs are frends wth ther mmedate neghbors mposng that way very large short dstance clusterng. However, some people have frends who are a long way away (old relatonshp or acquantance). Ths relatonshp "shortcuts" mae the average dstance between people relatvely small. In a fol wsdom ths s nown as "sx degrees of separaton". The Small-World networ model was frst ntroduced by Watts and Strogats [4]. The model was extended and mathematcally analyzed by Watts and Newman [5],[6]. They model the networ as an ordered lattce wth clusterng coeffcent whch s networ sze (N) ndependent. Clusterng coeffcent n a lattce depends only on the coordnaton number K (the number of neghbors the node s connected to). A rng lattce wth N nodes and coordnaton NK number K has edges. In such lattce Watts and Strogats randomly rewre each edge wth a small NK probablty p leadng to a small number of p rewred edges. Instead of edge rewrng, Watts and NK Newman n [5] add p new edges wthout removng the old ones. Ths small change maes the model more realstc (there s no loops or solated

4 components) and t s easer to analyze. Fg.3 shows a rng lattce wth N nodes, coordnaton number K4 and 3 shortcuts added. Fg.3. Rng lattce wth 3 shortcuts A rng lattce wthout shortcuts has clasterng coeffcent gven as [7]: 3( K ) C lattce. (8) 4( K 1) and average dstance between nodes gven as: N l lattce. (9) K The Small-World networ model nterpolates between an ordered lattce (p) and Random networ model (p1). Watts and Strogats has frst notced that there s a regon of small p wth large and almost unchanged clusterng coeffcent (property of an ordered lattce) and small average dstance that scales as logarthm of N (property of the Random networ model). Watts and Newman n [5], usng recombnaton group prncples, have derved the followng equaton for the average dstance between nodes n one dmensonal lattce: ordered lattce. In Fg.4-b we plot the average clusterng whch s almost unchanged. a) Average dstance b) Average clusterng Number of shortcuts Number of shortcuts Fg.4. Small-World networ model wth N14 nodes a) Average dstance b) Average clusterng l SW f N K f ( pkn ) constant x for x ( x) ~ log( x). for x << >> 1 1 (1) Fg.4 shows the smulaton results for the Small- World model wth N14 nodes. The networ contans undrected edges and does not contan loops or multple edges between nodes. We vary the number of shortcuts n the nterval [,1]. In Fg.4-a we plot the average dstance between nodes. It can be seen that the average dstance n the case of 1 shortcuts s greatly reduced compared wth an Fg.5. Vsualzaton of the Small-World networ model wth N14 nodes, K4 and 1 shortcuts 5 Scale-Free Model The Scale-Free networ model s an attempt to descrbe the results obtaned by nvestgatng data avalable from some real networs. Many systems

5 (networs) have the property that the degree dstrbuton P() follows a power low,.e. γ P ( ) ~ and t s ndependent of the networ sze N (scale-free). For example, the analyzes of the move actors collaboraton graph [8] have shown that P() follows a power low wth γ actor.3 ± 1. World Wde Web s also consdered as a huge networ where every HTML page represents a node and hyperlns between pages represent edges. It has been shown [8] that WWW's degree dstrbuton also follows a power n low, havng γ. 1 for ncomng lns and www γ.45 for outgong lns. out www The frst and most exploted Scale-Free networ model s suggested by Barabás and Albert [8],[9]. They obtan the scale-free networ propertes usng the followng two aspects: growth and preferental attachment. The growth aspect states that the number of vertces ncreases durng the tme.e. t s not fxed (the growth of the WWW). The preferental attachment aspect states that a new node s more probably attached to the nodes havng hgher degrees (rch-becomes-rcher prncple). For example, a newly ntroduced WWW page ponts more lely to some well nown and establshed WWW stes. The above two aspects are ncorporated nto the Barabás -Albert model (BA) as follows: Growth: Startng wth a small number of nodes ( m ), at every tme step a new node s added wth m m edges connectng t to the nodes already present n the system. Preferental attachment: A new ntroduced node connects to the node wth probablty proportonal to the node's degree,.e.: P ( ) m j j. (11) Usng the mean-feld approach, Barabás and Albert have derved the followng approxmatve degree dstrbuton formula: m t 1 P( ). (1) 3 m + t In the above equaton t s the number of executed tme steps that correspond to networ wth m + t nodes and mt edges (plus the edges among the ntal nodes). The equaton (1) shows that the degree dstrbuton follows a power low wth 3 γ 3 ( ). We tae m m n the computer smulaton. In order to avod the possblty of havng dsconnected nodes we start wth a networ of m connected nodes. At every tme step we add a new node wth m new edges. The ntal state of two connected nodes doesn't matter n the lmt of large N. It s worth mentonng that the BA model allows multple edges between nodes, but doesn't allow loops. When we calculate the average clusterng or average dstance we count only one edge. The LCD model, ntroduced n [1], allows both multple edges and loops. Fg.6 plots the degree dstrbuton of the BA networ ( m m ) at two growng levels: N14 nodes and N48 nodes. The red colored dotted lnes plot a degree dstrbuton accordng equaton (1). It can be seen that n both cases the degree dstrbuton n a large range follows a power low wth γ ~ 3. a) b) Fracton of nodes Fracton of nodes 1.E+ 1.E E- 1.E-3 1.E-4 1.E-5 1.E-6 1.E- 1.E-3 1.E-4 1.E-5 1.E-6 1.E-7 Node degree 1.E+ 1.E Node degree Fg.6. Degree dstrbuton n the BA networ model wth m m a) N14 nodes b) N48 nodes

6 The calculated average dstance n the BA networ model wth N14 nodes s 4.. So, the BA networ s hghly connected graph. The calculated average clusterng n the same networ vares n the nterval [.,.5] and t s greater than the clusterng n a random networ wth the same number of nodes and edges. [3] B. Bollobás, Random Graphs, Cambrdge Unversty Press, 1 [4] D. Watts, S. Strogatz, Collectve Dynamcs of 'small-world' networs, Nature, Vol.4, 1998, pp [5] M.Newman, D.Watts, Renormalzaton group analyss of the small-world networ model, Physcs Letters, A 63, 1999, pp [6] M.Newman, D.Watts, Scalng and percolaton n the small-world networ model, Physcs revews, E 6, 1999, pp [7] R.Albert, A. Barabás, Statstcal mechancs of complex networs, Revews of modern physcs, Vol. 74,, pp [8] R.Albert, A. Barabás, Emergence of Scalng n Random Networs, Scence, Vol. 86, 1999, pp [9] R.Albert, H. Jeong, A. Barabás, The dameter of the World-Wde Web, Nature, Vol. 41, 1999, pp. 13 [1] B. Bollobás, O.Rordan, The dameter of a scale-free random graph, Combnatorca, 3 [11] B. Bollobás, Mathematcal results on scale-free random graphs, 3 Fg.7. Vsualzaton of the BA networ model wth N14 nodes and m m 6. Conclusons We presented three networ models that mae the basc networ classfcaton n the contemporary networ topology research. However, varants of the models exst. Each of them s tryng to ncorporate addtonal networ topology concepts. For example, an overvew of the current scale-free networ models s gven n [11]. Networ models and analyss theren are very mportant for predctng future networ behavour. Analyzng average dstance and clusterng may help n desgnng more effectve and robust networ servces. References [1] P.Erdős, A. Rény, On random graphs I., Publcatones Mathematcae Debrecen, Vol. 5, 1959, pp [] P.Erdős, A. Rény, On the evoluton of random graphs, Magyar Tud. Aad.Mat.Kutató Int.Közl., Vol. 5, 196, pp