Frontiers of Network Science Fall 2019 Class 13: Barabasi-Albert Model II Evolving Networks (Chapter 6 in Textbook) Degree Correlations I (Chapter 7 in Textbook) Boleslaw Szymanski based on slides by Albert-László Barabási www.BarabasiLab.com and Roberta Sinatr a
Section 7 Measuring preferential attachment
Section 7 Measuring preferential attachment ∂ ∆ k k ∝ Π i i ( k ) ~ ∂ ∆ i t t Plot the change in the degree Δ k during a fixed time Δ t for nodes with degree k . To reduce noise, plot the integral of Π(k) over k : ∑ κ = Π ( k ) ( K ) < K k No pref. attach: κ~ k Linear pref. attach: κ ~k 2 (Jeong, Neda, A.-L. B, Europhys Letter 2003; cond-mat/0104131) N t k S i E l i N t k M d l
Section 7 Measuring preferential attachment citation Plots shows the integral of Internet Π(k) over k: network ∑ κ = Π ( k ) ( K ) < K k No pref. attach: κ ~k actor neurosci collab. collab Linear pref. attach: κ ~k 2 α Π ≈ + α ≤ k A k ( ) , 1 Network Science: Evolving Network Models
Section 8 Nonlinear preferential attachment
Section 8 Nonlinear preferential attachment α =0: Reduces to Model A discussed in Section 5.4. The degree distribution follows the simple exponential function. α =1: Barabási-Albert model, a scale-free network with degree exponent 3. α >1: Superlinear preferential attachment. The tendency to link to highly connected nodes is enhanced, accelerating the “rich-gets-richer” process. The consequence of this is most obvious for , when the model predicts a winner-takes-all phenomenon: almost all nodes connect to a single or a few super-hubs.
Section 8 Nonlinear preferential attachment α =0: Reduces to Model A discussed in Section 5.4. The degree distribution follows the simple exponential function. α =1: Barabási-Albert model, a scale-free network with degree exponent 3. 0<α<1: Sublinear preferential attachment. New nodes favor the more connected nodes over the less connected nodes. Yet, for the bias is not sufficient to generate a scale-free degree distribution. Instead, in this regime the degrees follow the stretched exponential distribution:
Section 8 Nonlinear preferential attachment The growth of the hubs. The nature of preferential attachment affects the degree of the largest node. While in a scale-free network the biggest hub grows as (green curve), for sublinear preferential attachment this dependence becomes logarithmic (red curve). For superlinear preferential attachment the biggest hub grows linearly with time, always grabbing a finite fraction of all links (blue curve)). The symbols are provided by a numerical simulation; the dotted lines represent the analytical predictions.
Section 9 The origins of preferential attachment
Section 9 Link selection model Link selection model -- perhaps the simplest example of a local or random mechanism capable of generating preferential attachment. Growth : at each time step we add a new node to the network. Link selection : we select a link at random and connect the new node to one of nodes at the two ends of the selected link. To show that this simple mechanism generates linear preferential attachment, we write the probability that the node at the end of a randomly chosen link has degree k as
Section 9 Originators of preferential attachments
MECHANISMS RESPONSIBLE FOR PREFERENTIAL ATTACHMENT 1. Copying mechanism directed network select a node and an edge of this node attach to the endpoint of this edge 2. Walking on a network directed network the new node connects to a node, then to every first, second, … neighbor of this node 3. Attaching to edges select an edge attach to both endpoints of this edge 4. Node duplication duplicate a node with all its edges randomly prune edges of new node Network Science: Evolving Network Models
Section 9 Copying model (a) Random Connection: with probability p the new node links to u. (b) Copying : with probability we randomly choose an outgoing link of node u and connect the new node to the selected link's target. Hence the new node “copies” one of the links of an earlier node (a) the probability of selecting a node is 1/N. (b) is equivalent with selecting a node linked to a randomly selected link. The probability of selecting a degree-k node through the copying process of step (b) is k/2L for undirected networks. The likelihood that the new node will connect to a degree-k node follows preferential attachment Social networks: Copy your friend’s friends . Citation Networks : Copy references from papers we read. Protein interaction networks : gene duplication,
Preferential Attachment in Cellular Networks: GENOME protein-gene interactions PROTEOME protein-protein interactions METABOLISM Bio-chemical reactions Citrate Cycle
Protein interactions: Yeast two-hybrid method
Comparison of proteins through evolution Use Protein-Protein BLAST (Basic Local Alignment Search Tool) -check each yeast protein against whole organism dataset -identify significant matches (if any) Eisenberg E, Levanon EY, Phys. Rev. Lett. 2003.
Preferential Attachment! ∂ ∆ k k ∝ Π For given ∆ t : ∆ k ∝ Π ( k ) i i ( k ) ~ ∂ ∆ i t t k vs. ∆ k : linear increase in the # of links S. Cerevisiae PIN: proteins classified into 4 age groups Eisenberg E, Levanon EY, Phys. Rev. Lett. 2003.
SUMMARY: PROPERTIES OF THE BA MODEL • Nr. of nodes: • Nr. of links: • Average degree: β: dynamical exponent • Degree dynamics γ: degree exponent • Degree distribution: ln N ≈ • Average Path Length: l ln ln N • Clustering Coefficient: The network grows, but the degree distribution is stationary. Network Science: Evolving Network Models
DEGREE EXPONENTS γ collab γ metab out γ intern γ synonyms γ w γ w in γ actor γ sex γ cita γ=1 γ=2 γ=3 <k 2 > diverges <k 2 > finite BA model Can we change the degree exponent? Network Science: Evolving Network Models
Section 9 Optimization model
Section 9 Optimization model Star Network
Section 9 Optimization model Scale-Free Network
Section 9 Optimization model Exponential Networks
Section 10 Diameter and clustering coefficient
Section 10 Diameter Bollobas, Riordan, 2002
Section 10 Clustering coefficient Reminder: for a random graph we have: What is the functional form of C(N)? Konstantin Klemm, Victor M. Eguiluz, Growing scale-free networks with small-world behavior, Phys. Rev. E 65, 057102 (2002), cond-mat/0107607
CLUSTERING COEFFICIENT OF THE BA MODEL Nr( ∆ ) 1 2 Denote the probability to have a link between node i and j with P(i,j) The probability that three nodes i,j,l form a triangle is P(i,j)P(i,l)P(j,l) The expected number of triangles in which a node l with degree k l participates is thus: ( ∆ ) We need to calculate P(i,j). Network Science: Evolving Network Models
CLUSTERING COEFFICIENT OF THE BA MODEL Calculate P(i,j). Node j arrives at time t j =j and the probability that it will link to node i with degree k i already in the network is determined by preferential attachment: Where we used that the arrival time of node j is t j =j and the arrival time of node is t i =i ( ∆ ) = Let us approximate: Which is the degree of node l at current time, at time t=N There is a factor of two difference... Where does it come from? Network Science: Evolving Network Models
Evolving network models Network Science: Evolving Network Models
EVOLVING NETWORK MODELS The BA model is only a minimal model. Makes the simplest assumptions: k = • linear growth 2 m Π ∝ • linear preferential attachment ( k ) k i i Does not capture variations in the shape of the degree distribution variations in the degree exponent the size-independent clustering coefficient Hypothesis : The BA model can be adapted to describe most features of real networks. We need to incorporate mechanisms that are known to take place in real networks: addition of links without new nodes, link rewiring, link removal; node removal, constraints or optimization Network Science: Evolving Network Models
BA ALGORITHM WITH DIRECTED EDGES (the simplest way to change the degree exponent) Undirected BA network: Directed BA network: − 2 P ( k ) ~ k in β=1: dynamical exponent γ in =2: degree exponent; P(k out )=δ(k out -m) Undirected BA: β=1/2; γ=3 Network Science: Evolving Network Models
EXTENDED MODEL: Other ways to change the exponent Extended Model • prob. p : internal links • prob. q : link deletion • prob. 1-p-q : add node P(k) ~ (k+ κ (p,q,m)) - γ (p,q,m) γ ∈ [1, ∞ ) Network Science: Evolving Network Models
EXTENDED MODEL: Small-k cutoff P(k) ~ (k+ κ (p,q,m)) - γ (p,q,m) γ ∈ [1, ∞ ) Extended Model • prob. p : internal links Predicts a small-k cutoff • prob. q : link deletion a correct model should predict all aspects of the • prob. 1-p-q : add node degree distribution, not only the degree exponent. Degree exponent is a continuous function of p,q , m p=0.937 m=1 κ = 31.68 γ = 3.07 Actor network Network Science: Evolving Network Models
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