A continuous time model for network evolution. Gail Gilboa-Freedman, Refael Hassin. January 2, 2011

Transcription

1 A continuous time model for network evolution Gail Gilboa-Freedman, Refael Hassin January 2, 2011 Abstract The study of networks is in great focus of many branches of science. We suggest a novel approach to modeling network evolution, based on the dynamics of independent Markov chains. Evolution is measured in continuous time units, as opposed to other models, where it is measured by a discrete counter of iterations. We derive a closed solution for the expected time until a node has any specific degree. The model produces a highly skewed degree distribution and a small world s criteria, in agreement with real world networks. Our study demonstrates that a network of complex topology can be composed of identical elements, that have independent behavior. 1 Introduction Networks are found everywhere: human bodies [1, 2, 3], technological systems [4, 5, 6], nature [7, 8, 9], social relationships [10, 11], etc. It is a challenge to understand the evolution of real world networks. For example, a common modeling goal is to explain how a given network comes to have its particular degree distribution or clustering at time t. Network evolution models have been proposed over the past few decades, such as the Erdös and Renyi s random graph [12, 13, 14], the small-world [15] and the scale free network model [16]. We briefly describe these models, providing the basis and the inspiration for our model. The following evolution process is a series of random graphs: starting with a graph of n nodes and no edges, iteratively add a random edge according to a uniform distribution on the missing edges. The evolution ends when the proportion of chosen edges is p. This model demonstrates a formation Department of Statistics and Operations Research, Tel Aviv University, Israel. s: gailgf,hassin@post.tau.ac.il 1

2 of a giant component [12, 13], in agreement with real networks like scientific collaboration [17] and neural connection [18]. It also demonstrate a small average shortest path [19] like the six degree separation concept [20, 21]. In 1998 Watts and Strogats reported the small world phenomena: a coexistence of small diameter and high clustering [15]. They measure the tendency to cluster by the clustering coefficient, which is the fraction of triples that have their third edge filled in to complete the triangle [10, 15, 22]. Examples for small world are the World Wide Web [23] and the co-occurrence of words [24]. Watts and Strogats introduce a network evolution model that captures the small world s criteria: starting with a ring lattice with n nodes in which every node is connected to its first k neighbors (k/2 on either side), reassign each edge to a distant nodes with probability p, such that self-connections and duplicate edges are excluded. This model has been further investigated in [25]. In 1999 Barabàsi and Albert [16] explored the observed static degree distribution of networks. The degree k i of a node i is the number of its neighbors. In many real networks, such as scientific papers citations [26], the degree distribution is typically right-skewed with a heavy tail. Moreover, the degree distribution follows a power-law distribution, which means it is scale-free. Barabàsi and Albert introduced a network evolution model, producing a power-law degree distribution: starting with a small number (m 0 ) of nodes, iteratively add a new node and link it to m m o different nodes already present in the system. The new node is connected to node i with probability d i j d j (preferential attachment process). One can identify two approaches for modeling network evolution. The traditional approach, which is inspired by the theory of random graphs, emphasizes the advantage of having a simple model, which is exactly solvable for many of its properties. The later modeling approach, known as the new science of networks [27, 28], emphasizes the topological structure of the network. Motivated by these approaches, we raise the following questions: Can a simple model, which is based on identical and consistent behavior of elements, produce a network with complex topology? What is the right way to describe a network evolution over continuous time? To explore these questions, we suggest a novel approach to model network evolution. The evolution is represented by the dynamics of independent Markov chains, each represents an element of the network. Chains move among a common space of states. Sometimes chains intersect, being in the same state at the same time. These intersections relate the chains with each 2

3 other and imply many interesting processes, including network evolution. As a first step, we consider a simple version of the model: Markov chains are identical; the time a chain spends at a given state is exponential; there are only two states, one of them is a meeting-state. Starting with a network of no edges, we add an edge between two nodes when their representative chains are both in the meeting-state for the first time. The paper is structured as follows: In Section 2, we describe the model. In Section 3, we achieve a closed formula for the degree evolution. In Section 4, we present the structural features of the evolving network, which are in agrement with the real world: the diameter is small, the clustering is high and the degree distribution is highly skewed. 2 The model We list the assumptions underlying the model: N Markov chains are identical and independent. Each chain has two states: M and L. The duration times in M and L are independent, identically and exponentially distributed with parameters µ and λ, respectively. The evolution is a series of changing graphs, based on the dynamics of the chains. Evolution starts with G = (V, E), where V = 1, 2,..., N and E = φ. Each node is associated with a chain. Chains move between state M and state L. M is a meeting state: if at any given time two chains are both in M, we say that the chains meet. When two chains, say chain i and chain j, meet for the first time, we add an undirected edge (i, j) to G. The model parameters can be normalized, so there are only two relevant parameters: N which is the number of chains, and ρ = λ, which is the ratio µ of the expected times spent in M and L. The model is an extension of the random graph model: when ρ goes to zero, edges are randomly added one by one. On the other hand, when ρ is high, there is a high probability that the corresponding evolving network is a series of cliques with growing sizes. 3 Analysis of degree distribution We derive a closed formula for the expected time until the degree of an arbitrary node in the evolving network is h. One distinct chain plays the role of a leader. All other N chains are non-leaders. The meetings of the leader 3

4 represent the edges incident to a specific node in the corresponding evolving network. We analyze these meetings. Specifically, we derive the expected time until the leader has met h non-leaders. 3.1 Recursion Let S i,m,l denote a state of the system, where i equals 1 or 0 to indicate whether the leader is in state M or L, respectively; m 0,..., N 1 is the number of non-leaders in state M which are not yet acquainted with the leader; l 0,..., N m is the number of non-leaders in state L which are not yet acquainted with the leader. Let M l denote the expected time until the system goes from state S 1,0,l to state S 1,0,0. Let L m,l denote the the expected time until the system goes from state S 0,m,l to state S 1,0,0. M l and L m,l are the expected times until the leader has been in state M with all of the non-leaders (at least once). We compute M N and L 0,N as a function of µ, λ and N. When l = 0, M l = 0. For any l > 0, the first transition takes the system from S 1,0,l to S 1,0,l 1 or S 0,0,l, depending on whether it is a transition of the leader or one of the non-leaders, respectively. Hence, for l = 1,..., N: M l = 1 µ + lλ + lλ µ + lλ M l 1 + µ µ + lλ L 0,l. (1) When l = 0, L m,l = 0, for all m values. For any l > 0, the first transition takes the system from S 0,m,l to S 0,m+1,l 1, S 0,m 1,l+1 or S 1,l, depending on the type of the transiting chain and the direction of this transition. Hence, for m = 0,..., N l and l = 1,..., N m : L m,l = + 1 mµ + (l + 1)λ + lλ mµ mµ + (l + 1)λ L m 1,l+1 mµ + (l + 1)λ L m+1,l 1 + λ mµ + (l + 1)λ M l. (2) L 1,l+1 is not defined. However, its coefficient is Recursion for an embedded process Solving the recursion system 1 and 2 is not straightforward, since L m,l is not induced by lower values of m and l. To derive a closed solution, we embed the system in states S 1,0,l and S 0,0,l, where the non-leaders who are not acquainted with the leader are in state L. Let L l = L 0,l. When the system is in state S 0,0,l and l > 0, the first transition of the leader takes 4

5 the system to S 1,0,r, for some 0 r l, depending on the number of nonleaders, that have not yet met the leader and are in state M at the time of this transition. Let T denote the time of the first transition of the leader. For l = 1,... N, (2) is replaced by: [ ] l L l = T + [P l,r (T )M r ] λe λt dt T =0 = 1 λ + T =0 r=0 [ l ] [P l,r (T )M r ] λe λt dt, r=1 where P l,r (T ) denotes the probability that X = r, where X is the number of chains in state L at time T, if initially there are l chains and all of them are in state L. X has been studied by Enns [29]. Enns model is equivalent to the generalization of Uppuluri [30] for the Ehrenfest model [31]. As a conclusion from Enns study, X is distributed according to binomial distribution B(l, u), where u, the probability that a chain is in state L at time T, given that it was in state L at time 0, satisfies: Therefore, L l = 1 λ + T =0 [ l P r=1 = 1 l [M λ + r r=1 u = µ + λe (λ+µ)t λ + µ ( B (l, µ + λe (λ+µ)t λ + µ P T =0 ( B. ) (l, µ + ) λe (λ+µ)t λ + µ ) ] = r M r λe λt dt ) ] = r λe λt dt = 1 l ( ) ( ) l l 1 λ + M r f(r, l), (3) r 1 + ρ r=1 5

6 where f(r, l) = 1 µ l T =0 [ = ρ l r G r, [ λ G λ + µ = ρ l r G G ( µ + λe (λ+µ)t ) r ( λ(1 e (λ+µ)t ) ) l r λe λt dt λ µ + λ ; 1 r + l + λ 2λ + µ, r l; λ + µ ; 1 ρ 1 + ρ ; 1 r + l + ρ [ r, [ ρ 2ρ + 1, r l; ρ + 1 ρ + 1 ; 1 where G is the Gauss s hypergeometric function: (a) l (b) l G[a, b; c, z] = (c) l l=0 ] λ + µ ; λ µ ], ] ρ + 1 ; ρ ], (4) z l l!, and (a) k is the Pochhammer symbol of the rising factorial: { a(a + 1) (a + k 1) k 1 (a) k = 1 k = 0. Substituting (3) in (1), M l can be computed recursively, for l = 1,... N as a function of all M a, a = 0,..., l 1. Substituting M 0 = 0, we get: where l 1 M l = χ(l) + ξ(r, l)m a, (5) λχ(l) = a=1 1 + ρ ( ) l, 1 + lρ f(a, l) 1 1+ρ ξ(a, l) = ( l ) ( l 1 a 1+ρ) f(a, l) + δ(a, l)lρ ( 1 + lρ 1 1+ρ ) l f(l, l), and δ(a, l) is the Kronecker function: { 1 a = l 1 δ(a, l) = 0 otherwise. 6

7 3.3 Expected time until a node is connected with all others Consider a directed acyclic weighted graph H = (V, E), with V = {1,..., N} and E = {(i, j) i < j}. Denote the weight of edge (i, j) by w(i, j), and the weight of vertex i by v(i). For 1 k n N, let P k,n denote the set of directed pathes from k to n in H. Define { p P Ω(k, n) = k,n (i,j) p w(i, j) 1 k < n, (6) 1 k = n. Notice that the last edge in a path from k to n is (q, n) for some k q < n. Therefore, for any k and n, Ω(k, n) = p P k,n (i,j) p n 1 w(i, j) = w(q, n)ω(k, q). (7) Consider the function q(n), defined recursively by q(1) = v(1) and for n = 2,..., N q=k n 1 q(n) = v(n) + w(s, n)q(s). (8) Lemma 3.1. q(n) = n k=1 v(k)ω(k, n), for n = 1, 2,..., N. s=1 Proof. We verify that the claim is true for n = 1. By the definitions of q(n) and Ω(k, n), q(1) = v(1) and Ω(1, 1) = 1. Therefore, q(1) = v(1) = v(1)ω(1, 1) = 1 v(k)ω(k, 1). k=1 We assume that the claim is true for all n < n 1, for some n 1 N. We prove 7

8 it for n 1. From (7) and (8): q(n 1 ) = v(n 1 ) + = v(n 1 ) + n 1 1 s=1 n 1 1 s=1 w(s, n 1 ) n 1 1 = v(n 1 ) + v(k) k=1 n 1 1 w(s, n 1 )q(s) k=1 [ n1 1 s v(k)ω(k, s) ] w(s, n 1 )Ω(k, s) s=k = v(n 1 ) + v(k)ω(k, n 1 ) k=1 = v(n 1 )Ω(n 1, n 1 ) + n 1 = v(k)ω(k, n 1 ). k=1 n 1 1 k=1 v(k)ω(k, n 1 ) We return to our problem and calculate M N and λl N as functions of ρ and N. We consider an acyclic weighted graph H = (V, E), in which the weights of the edges are ξ(i, j) and the weights of the vertices are χ(i). Let Ω be defined by substituting w(i, j) = ξ(i, j) in (6): { p P Ω(k, n) = k,n (i,j) p ξ(i, j) 1 k < n, 1 k = n. Theorem 3.2. M N = N k=1 χ(k) Ω(k, N). Proof. By (5), (7) and Lemma 3.1. Theorem 3.3. λl N = 1 + N r=0 [ (N r ) ( ) N [ 1 ρ+1 f(r, N) r k=1 χ(k) Ω(k, r)] ]. Proof. By (3): λl N = 1 + N r=0 ( ) ( ) N N 1 f(r, N)M r, r ρ + 1 and using Theorem 3.2, [ N (N ) ( ) N 1 λl N = 1 + f(r, N) r ρ + 1 r=0 8 r k=1 [ χ(k) Ω(k, r)] ],

9 where f(r, N) is given in (4). Figure 1 shows λl N as a function of N for various µ values. Functions are monotonously increasing as it takes more time to become acquainted with a larger population. The functions are also concave. It is a result of the fact that, adding a fixed number of non-leaders N has a stronger effect when the original number of chains is small. Intuitively, L N increases as a result of adding chains, only because the leader may meet some of the new N chains after he meets all the original N chains. When N is higher, the period to infect N chains becomes longer and less of the new N chains are left to the end of the process. The inset shows the same plots using a horizontal log scale. The affine relation of L N and log(n), graphically shows that L N can be well-approximated by a linear function of log(l) ρ= ρ=5 ρ= λ L N 20 ρ= N Figure 1: Expected time to meet all chains. The solution for λl N is shown as a function of the size of the network N, for different ρ values. The inset shows the same plot, using a horizontal log scale. 9

10 3.4 Expected time until a node has a certain degree We extend the formula for M N and L N, to compute the expected time until the leader meets a subset of non-leaders. These subsets represent the neighbors of the node corresponding to the leader. The growing size of these subsets represents the degree evolution of the same node. Let Ml h denote the expected time until the system goes from state S 1,0,l to any state S 1,0,z where z N h. Let L h m,l denote the expected time until the system goes to the same states from S 0,m,l. Ml h and L h m,l are the expected time until the leader is acquainted with at least h out of the N non-leaders. We compute Ml h and L h m,l as a function of λ, µ, N and h. By definition, Ml h equals 0 for all l N h. The development of Ml h is similar to the development M l. The only difference lies at the range of index r in (3), which now starts at N h + 1 to compute L h. Substituting L h to (1), we get the following modification of (5): M h l = χ(l) + l 1 a=n h+1 ξ(r, l)m h a. A corresponding modification of index a in Theorem 3.2 gives: r=0 M h l = l k=n h+1 [χ(k)ω(k, l)]. (9) The solution for Ml h is a generalization of the solution for M l, as Ml N = M l. Let L h l denote L N 0,l. LN l = 0 for m + l N h, otherwise it is achieved by substituting (4) and (9) in (3): [ L h l = 1 l (N ) ( ) ] N 1 l λ + f(r, N) [χ(k)ω(k, l)]. r λ + µ k=n h+1 4 Network features during the evolution We use numerical simulations to investigate the structural features of the evolving network. We start a simulation from a state where all the Markov chains are in L. We continue with an iterative process of acquiring the next transition of any chain. We update the states of the system and at the same time we update the link propagation in the evolving network. Specifically, in every epoch when chain i transits into state M, we examine the chains in M. Some of the them, say chains j 1, j 2,..., represent nodes which are not 10

11 yet connected to i in the evolving network. We connect these nodes to i by edges (i, j 1 ), (i, j 2 )... The process terminates when the network becomes a clique with N nodes. We investigate the network evolution in dependency with the parameters of the model. For each choice of ρ and N, the results are averaged over ten runs of simulations. We recognize interesting features in our model: small average shortest path (section 4.1), high clustering (section 4.2) and highly skewed degree distribution (section 4.3). Following the format of previous studies, we demonstrate the evolution of the main structures versus p - the proportion of edges. Furthermore, we demonstrate the same structures versus λt - the time, normalized by the expected time a chain spends in state L. Each feature is demonstrated in a graph, for a specific case when N = 200, for different values of ρ. We find special interest when ρ goes to zero, or when ρ is high enough to imply a series of growing cliques. 4.1 Small average shortest path Two nodes are connected is the network contains a path between them. For any couple of connected nodes, the shortest path is the minimum number of edges to traverse from one node to another. Let l denote the average of all the shortest pathes in a graph. For a connected graph, l characterizes the spread of the graph. For a disconnected graph, l is defined as the average of the average lengthes of shortest pathes of its components. Figure 2 demonstrates the evolution of l in our model. 11

12 A ρ=10 1 ρ=10 2 ρ=10 6 RANDOM B l p λ t x 10 4 Figure 2: Average shortest path during the evolution. (A) The volution of l versus p, for various ρ. (B) The evolution of l versus continuous normalized time, for the same ρ values. Like in random graphs [19, 32, 33, 34, 35, 36, 37], the value of l in our model is low, provided p is not too small. It is similar to Real networks, like the World Wide Web [23] and the co-occurrences of words [24]. Moreover, when p is small, the value of l in our model is smaller than ln N, which is the the average shortest path of a random graph with the same size (shown as a dashed line). The reason for this is that when p is small, l decreases with ρ as a chain meets smaller sets of chains when it moves to M. 4.2 High clustering Inspired by a concept in sociology, named fraction of transitive triples [10], Watts and Strogats quantified the tendency to cluster by the clustering coefficient [15]. The clustering coefficient of a node quantifies how close the 12

13 node and its neighbors are to being a clique. For each selected node i, the nodes which are connected to it are examined. These are called the neighbors of node i. Having k i neighbors, let e i be the number of edges that exist between pairs of neighbors of node i. The clustering coefficient of node i is defined as Cl i = 2e i. In words, Cl k i (k i 1) i gives the proportion of triangles that go through node i, whereas k i(k i 1) is the total number of triangles that could 2 pass through node i, should all of its neighbors be connected to each other. The clustering coefficient of a network is the average of its local coefficients: Cl = n i=1 Cl i n. Figure 3 demonstrates Cl in our model. 1 A B Cl 0.5 ρ=1 ρ=10 2 ρ=10 3 RANDOM p λ t Figure 3: Clustering coefficient during the evolution. (A) The evolution of Cl versus p, for different ρ values. (B) The evolution of Cl versus continuous normalized time, for the same ρ values. In general, Cl increases from 0 to 1 as the network evolves to a clique. Let as look at Figure 3(A). When ρ 0, Cl goes to a limit function which is Cl = p as for a random graph with the same size (shown as a dashed line). 13

14 Otherwise, Cl is higher than p. Figure 3(B) shows the evolution Cl versus time. The clustering in most, if not all, real networks is much higher than the clustering coefficient of a random graph. The clustering coefficient of the internet, for example, ranges between 0.18 and 0.3 [38], in comparison with for random networks with similar parameters. Together with the result in the previous section, our model gives a satisfactory result: exhibition of a short average shortest path along with a clustering coefficient which is higher than that of a random graph. Our model represents the small world criteria better than a random graph. 4.3 Highly skewed degree distribution A degree of a node is the number of its neighbors. Many real networks, the degree distribution is typically right-skewed with a heavy tail, meaning that a small fraction of the nodes are many times better connected than the average. These nodes are called hubs. Figure 6 demonstrates some samples of degree distributions in our model for different ρ value for the same p. 14

15 ρ= A p= ρ= B 0 ρ= C DEGREE Figure 4: Histograms of degree distribution, for p = 0.12, for different ρ values, in comparison with the distribution of a random graph (red dashed line). (A) when ρ is low, the distribution is well approximated by a Poisson distribution like in random graphs. (B) for intermediate value of ρ, the distribution is highly skewed. (C) when ρ is high, the evolution is a series of cliques and there is a single degree value. When ρ is low (upper graph), the distribution is well approximated by a Poisson distribution with the same mean. When ρ is high (lower graph), the network is a clique and the connected nodes have the same degree. In random graphs, the number of nodes with degree k follows a Poisson distribution, of which the variance is small (dashed lines in all graph). The high skewness is not captured by random graphs. In contrast, it is captured in our model, for intermediate values of ρ (middle graph). 15

16 5 Summery and conclusions We suggest modeling network evolution according to the dynamics of Markov chains. We examined a simple version of such modeling. The model assumes a finite number of independent Markov chains with two states, one of which is a meeting state and a meeting there imply an addition of corresponding edge to the evolving network. Some properties of the model can be analyzed by the theory of stochastic process. For example, we formulated a closed formula for the expected time until a node has a certain degree. Recall the questions in the introduction section: Can identical and consistent behavior of elements produce a network with complex topology? Yes. Our model demonstrates that complex features in real world networks are not necessarily produced by diverse behavior of the elements. Instead, interactions of identical elements, whose behavioral rules are not influenced by the dynamic structure of the network, may also imply familiar features like: high clustering, small diameter and a skewed degree distribution. What is the right way to describe a network evolution over continuous time? We believe that time is an important parameter in a model for network evolution. Our model gives an example for mechanism that relates the the duration of processes in a network with the underlying processes of its elements. The underlying processes in our model are the transitions of the Markov chains. These processes can be measured in real time units because they are associated with the changes of an element in any real network. For example, people move among feasible states: home, work, pub, etc, making new friends in these places. For another example, neurons may be spiking or non spiking, while getting connected through synapses, following a principle which is often summarized as fire together, wire together [39]. The spiking state of the neuron resembles the meeting state in our model. As for further study, the model could be examined under more general assumptions: having any number of states for each Markov chain (instead of two), assuming any Poisson rate of meetings in each state (instead of either 0 or ) and considering a diversity of transition rates (instead of identical rates for all the chains). Also, there is room for natural extension of the model: to allow an elimination of the edges; to distinguish between the chain 16

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