Statistical Properties of Massive Graphs (Networks) Networks

Statistical Properties of Massive Graphs (Networks) Networks and Measurements

What is an information network? Network: a collection of entities that are interconnected A link (edge) between two entities (nodes) denotes an interaction between two entities We view this interaction as information exchange, hence, Information Networks The term encompasses more general networks

Why do we care? Networks are everywhere more and more systems can be modeled as networks, and more data is collected traditional graph models no longer work Large scale networks require new tools to study them A fascinating “new” field (“new science”?) involves multiple disciplines: computer science, mathematics, physics, biology, sociology. economics

Types of networks Social networks Knowledge (Information) networks Technology networks Biological networks

Social Networks Links denote a social interaction Networks of acquaintances actor networks co-authorship networks director networks phone-call networks e-mail networks IM networks Microsoft buddy network Bluetooth networks sexual networks home page networks

Knowledge (Information) Networks Nodes store information, links associate information Citation network (directed acyclic) The Web (directed) Peer-to-Peer networks Word networks Networks of Trust Bluetooth networks

Technological networks Networks built for distribution of commodity The Internet router level, AS level Power Grids Airline networks Telephone networks Transportation Networks roads, railways, pedestrian traffic Software graphs

Biological networks Biological systems represented as networks Protein-Protein Interaction Networks Gene regulation networks Metabolic pathways The Food Web Neural Networks

Now what? The world is full with networks. What do we do with them? understand their topology and measure their properties study their evolution and dynamics create realistic models create algorithms that make use of the network structure

Erdös-Renyi Random graphs Paul Erdös (1913-1996)

Erdös-Renyi Random Graphs The Gn,p model n : the number of vertices 0 p 1 for each pair (i,j), generate the edge (i,j) independently with probability p Related, but not identical: The Gn,m model

Graph properties A property P holds almost surely (or for almost every graph), if limP G hasP 1 n Evolution of the graph: which properties hold as the probability p increases? Threshold phenomena: Many properties appear suddenly. That is, there exist a probability pc such that for p pc the property does not hold a.s. and for p pc the property holds a.s.

The giant component Let z np be the average degree If z 1, then almost surely, the largest component has size at most O(ln n) if z 1, then almost surely, the largest component has size Θ(n). The second largest component has size O(ln n) if z ω(ln n), then the graph is almost surely connected.

The phase transition When z 1, there is a phase transition The largest component is O(n2/3) The sizes of the components follow a powerlaw distribution.

Random graphs degree distributions The degree distribution follows a binomial n n k p(k) B(n;k;p) pk 1 p k Assuming z np is fixed, as n B(n,k,p) is approximated by a Poisson distribution zk z p(k) P(k;z) e k! Highly concentrated around the mean, with a tail that drops exponentially

Random graphs and real life A beautiful and elegant theory studied exhaustively Random graphs had been used as idealized generative models Unfortunately, they don’t capture reality

Measuring Networks Degree distributions Small world phenomena Clustering Coefficient Mixing patterns Degree correlations Communities and clusters

Degree distributions frequency fk fraction of nodes with degree k probability of a randomly selected node to have degree k fk k degree Problem: find the probability distribution that best fits the observed data

Power-law distributions The degree distributions of most real-life networks follow a power law p(k) Ck-α Right-skewed/Heavy-tail distribution there is a non-negligible fraction of nodes that has very high degree (hubs) scale-free: no characteristic scale, average is not informative In stark contrast with the random graph model! highly concentrated around the mean the probability of very high degree nodes is exponentially small

Power-law signature Power-law distribution gives a line in the log-log plot log p(k) -α logk logC log frequency frequency degree α log degree α : power-law exponent (typically 2 α 3)

Examples Taken from [Newman 2003]

A random graph example

Maximum degree For random graphs, the maximum degree is highly concentrated around the average degree z For power law graphs 1/(α 1) kmax n Rough argument: solve nP[X k] 1

Exponential distribution Observed in some technological or collaboration networks p(k) λe-λk Identified by a line in the log-linear plot log p(k) - λk log λ log frequency λ degree

Collective Statistics (M. Newman 2003)

Clustering (Transitivity) coefficient Measures the density of triangles (local clusters) in the graph Two different ways to measure it: C(1) trianglescenteredat nodei triplescenteredat nodei i i The ratio of the means

Example 1 4 3 2 5 3 3 C(1) 1 1 6 8

Clustering (Transitivity) coefficient Clustering coefficient for node i trianglescenteredat nodei Ci triplescenteredat nodei (2) C 1 Ci n The mean of the ratios

Example 1 4 (2) C 1 13 1 1 1 6 5 30 3 2 5 C(1) 3 8 The two clustering coefficients give different measures C(2) increases with nodes with low degree

Collective Statistics (M. Newman 2003)

Clustering coefficient for random graphs The probability of two of your neighbors also being neighbors is p, independent of local structure clustering coefficient C p when z is fixed C z/n O(1/n)

Small world phenomena Small worlds: networks with short paths Stanley Milgram (1933-1984): “The man who shocked the world” Obedience to authority (1963) Small world experiment (1967)

Small world experiment Letters were handed out to people in Nebraska to be sent to a target in Boston People were instructed to pass on the letters to someone they knew on first-name basis The letters that reached the destination followed paths of length around 6 Six degrees of separation: (play of John Guare) Also: The Kevin Bacon game The Erdös number Small world project: http://smallworld.columbia.edu/index.html

Measuring the small world phenomenon dij shortest path between i and j Diameter: d maxdij i,j Characteristic path length: 1 dij n(n- 1)/2 i j Harmonic mean 1 -1 dij n(n- 1)/2 i j 1

Collective Statistics (M. Newman 2003)

Mixing patterns Assume that we have various types of nodes. What is the probability that two nodes of different type are linked? assortative mixing (homophily) E : mixing matrix p(i,j) mixing probability p(i,j) E(i, j) E(i,j) i,j p(j i) conditional mixing probability p(j i) E(i, j) E(i,j) j

Mixing coefficient Gupta, Anderson, May 1989 p(i i) - 1 Q i N 1 Advantages: Q 1 if the matrix is diagonal Q 0 if the matrix is uniform Disadvantages sensitive to transposition does not weight the entries

Mixing coefficient Newman 2003 a(i) p(i,j) (row marginal) b(i) p(j,i) (column marginal) j j p(i i) - a(i)b(i) r 1 a(i)b(i) i i i r 0.621 Q 0.528 Advantages: r 1 for diagonal matrix , r 0 for uniform matrix not sensitive to transposition, accounts for weighting

Degree correlations Do high degree nodes tend to link to high degree nodes? Pastor Satoras et al. plot the mean degree of the neighbors as a function of the degree Newman compute the correlation coefficient of the degrees of the two endpoints of an edge assortative/disassortative

Collective Statistics (M. Newman 2003)

Communities and Clusters Use the graph structure to discover communities of nodes essentially clustering and classification on graphs

Other measures Frequent (or interesting) motifs bipartite cliques in the web graph patterns in biological and software graphs Use graphlets to compare models [Przulj,Corneil,Jurisica 2004]

Other measures Network resilience against random or targeted node deletions Graph eigenvalues

Other measures The giant component Other?

References M. E. J. Newman, The structure and function of complex networks, SIAM Reviews, 45(2): 167256, 2003 M. E. J. Newman, Random graphs as models of networks in Handbook of Graphs and Networks, S. Bornholdt and H. G. Schuster (eds.), WileyVCH, Berlin (2003). N. Alon J. Spencer, The Probabilistic Method