Optimization Based Modeling of Social Network Yong-Yeol Ahn, Hawoong Jeong

Outline About real networks and models Motivation Simulation method Result Conclusion

Real Complex Networks Social networks Acquaintance, scientific collaboration, actor, bbs, etc. Internet, WWW, , other communication networks

Real Complex Networks Biological networks Metabolic network Genetic network Protein interaction network Neuronal network

Basic Concepts of Network Nodes Links Degree = 3 A shortest path with path length=3 (Equivalent with 3 clicks in WWW)

Clustering Coefficient A B C A knows B A knows C The probability that B knows C is large Clustering coefficient for a node represent how many links are there between neighbors Clustering coefficient for a network is the average of all nodes ’ s clustering coefficient C= # of links between neighbor # of possible neighbor pair

Clustering Coefficient A clique or a community C=1 C=0

Clustering Coefficient ‘ Triangle ’  the building block. Alternative definition of clustering coefficient 3 x # of triangle # of connected triples C=

Real Network ’ s Universal Characters Short path length High clustering Large inhomogeneity (power-law degree distribution)

Modeling Real Networks Static network model Erdös-Rényi model(random network) Connect All pairs of nodes with probability p

Erdös-Rényi Model Randomness  short path length Homogeneous model

Modeling Real Networks Static network model Watts-Strogatz model (small world)

Modeling Real Networks Watts-Strogatz model C: clustering coefficient L: average path length

Small World Network Model Randomness  short path length Regularity  high clustering Balance between regularity and randomness

Modeling Real Networks Growing network model BA model From the power law degree distribution of real networks Many models after BA model adopted the growing scheme

Network Models: BA Model Growing New nodes and links are added continuously Linear preferential attachment New nodes make links with preferential attachment rule Rule : “Riches get richer”

Scale-free Network Model Scale-free network model ‘Hub’ and power-law degree distribution  ‘inhomogeneity’ “Network is growing and inhomogeneous” P(k) ~k -3

New Scheme: Optimization BA model says “ A network is growing ” New models say “ The evolution is more important than growth. Let ’ s ignore the growth ” (Mathias et al.)

Growth and Evolution WWW is growing exponentially Rewiring in WWW is faster than growth Bacteria  Human (Growth of biological networks) Origin of species (Numerous rewiring in biological networks) Growth : Addition of nodes Evolution : Rewiring of links

Evolutionary Pressure So, the rewiring occur randomly?  No. Biological networks Natural selection Artificial networks(electrical circuit,…) Cost, High performances

New Design : Optimization Models Origin of biological networks and man- made networks Timescale of link dynamics vs. Timescale of node dynamics  Take a snapshot ‘Growth’  ‘rewiring, evolution’

Examples of Optimization In biosystems Metabolic network’s path length conservation Allometric scaling In artificial systems JAVA class network(A structure of computer program) Electric circuit

Optimization Scheme How to model the natural selection and optimization?  Nature want to enlarge network’s ‘efficiency’ while want to cut down ‘cost’ So, High ‘efficiency’  short path length (Information flow) Low ‘cost’  fewer links  Energy = p L + (1-p)E (p:parameter, L:path length, E: expense, cost)

Star Network Trivial case: optimizing only average path length ‘ Star network ’ To shorten path lengthmakes a hub

Result of Optimization Model Power law degree distribution in some range of p (parameter) k (Degree) P(k) (Cancho and Sole)

Our Motivation Real networks have large clustering coefficient and community structures Then, What kind of network will we get, if we maximize a network’s clustering cofficient?

Method Greedy algorithm Choose a link and rewire it randomly If energy decreases, keep it If energy increases, discard it We calculate with or without ‘connection constraint’ A triangle is formed, we ’ ll keep this rewiring

Method: Supplement This link is weak under our method Strong link

Energy Optimization Maximizing clustering coefficient Energy= 1 - C (C: Clustering coefficient) We try to maximize clustering coefficient Generalized form Energy= p(1-C) + (1-p)d P balances contributions from C and d We try to maximize clustering and to minimize normalized vertex-vertex distance

Results:Clustering Only (NotConnected) Scale free network with exponent – 2.2 (N=10000,L~20000) Clustering coeff. : 0.83 P(k) Degree

Results:Clustering Only (NotConnected) Structure of the network. N=300, L~600, Clustering coeff. ~0.9

Results: Clustering Only(connected) P(k) k Exponent ~ – 2.9 (N=10000,L~20000) Clustering coeff. : 0.79

Results: Clustering Only(connected) Structure of the network

Results: Clustering and Distance p=0 p=0.1 p=1 We can observe large differences in topology Only by path length Only by clustering coefficient

Discussion Let’s see social networks Can we define ‘cost’ in social networks? Can we define ‘efficiency’ in social networks?  Social networks are different from biological and artificial networks.

Discussion Functional networks : Metabolic network, Electrical circuit network,..  ‘global’ Non-functional network : Social networks, network,..  ‘Local’

Discussion Creation and deletion of a link in non- functional network. Creation of link  through friends Deletion of link  through ‘out of sight, out of mind’  Simplified to ‘rewiring’

Discussion Two forces Make triangles! Make hubs!

Discussion The two forces make power-law degree distribution If we add average path length in energy function, large hubs result.

Conclusion We categorize networks into two groups We explain the meaning of clustering- driving scheme With clustering optimization, we get highly clustered scale-free network