The structure and function of complex networks(2003)

The structure and function of complex networks(2003) Author: M.E.J. Newman Presenter: Guoliang Liu Date:5/4/2012

Outline Networks in the real world Properties of networks Random graphs Exponential random graphs and Markov graphs The small-world model Models of network growth Processes happening or happened

Networks in the real world

Properties of networks The small-world effect Transitivity or clustering Degree distributions Network resilience Mixing patterns Degree correlations Community structure Network navigation Other network properties

The small-world effect • Started from 1960s by Stanley Milgram • 6 steps between two nodes. • Undirected networks, define l to be the mean geodesic distance between vertex pairs in a network: • In case that networks have more than one component • Small-world effect: the value l scales logarithmically or slower with network size or fixed mean degree.

Transitivity or clustering • In many networks it is found that if vertex A is connected to vertex B and vertex B to vertex C, then there is a heightened probability that A is connected to C. • Cluster coefficient: • Global value • Local value

Transitivity or clustering An example to calculate clustering coefficient C

Transitivity or clustering C = tend to a non-zero limit, as in real-world networks for a large n in random graph Clustering coefficient C measures the density of triangles in a network

Degree distributions • How to present degree data: • pk: the fraction of vertices in the network having degree k • Pk : cumulative distribution function with the probability that degree is greater than or equal to k. • For , • For ,

Degree distributions Scale-free networks, , = 2.5~3. Scale-free: functional form f(x) remains unchanged to within a multiplicative factor under a rescaling of the independent variable x. f(ax) = bf(x) Normally, scale-free is equal to power-law.

Degree distributions • Maximum degree • Early work by Aiello: • Assume that the maximum degree was approximately the value above which there is less than one vertex of that degree in the graph on average, i.e., . • , for . • However, not fit in real world networks

Degree distributions Maximum degree The probability of there being exactly m vertices of degree k and no vertices of higher degree is , the highest degree on the graph is k is : So,

Degree distributions Maximum degree For both small and large values of k, tends to zero.

Network resilience • Resilience to the removal of vertices(or edges.). Popular in Epidemiology, in CS, the attack in complex networks. • How to remove? • Remove vertices at random • Target some specific class of vertices, such as those with the highest degree. • How to measure resilience? • Distances l increasement on average. • Thorough study by Holme et al. • “attack vulnerability of complex networks”

Mixing patterns • In most kinds of networks, and the probabilities of connection between vertices often depends on types. • Food web • Social network of couples(In social also called assortative mixing or homophily)

Mixing patterns • How to measure mixing patterns? • Eij: the number of edges in a network that connect vertices of types i and j. • eij measure the fraction of edges that fall between vertices of types i and j. • Assortative mixing coefficient:

Degree correlations A special case of assortative mixing according to a scalar vertex property is mixing according to vertex degree, referred as degree correlation.

Community structure …

Network navigation • Ideas from Stanley Milgram’s experiment. • Not only small-world effect. • But also people are good at finding them. • Target: build efficient database structures or better peer-to-peer computer networks. E.g. ”local search in unstructured networks”

Other properties Giant component Betweenness centrality Self-similarity

Random graphs Poisson random graph Generalized random graphs

Poisson random graphs • Why we talk about Poisson random graphs? • Basic intuition about the networks behaviors from the study of random graph. • Poisson distribution is a classic one and later ideas all started from Poisson random graph

Poisson random graphs • Poisson random graphs • Early work, simple model of a network • Gn,p, each pair connects with probability p • Later, consider the mean degree z = p(n-1) • When we focus on low-density, low-p state and high-density, high-p state. • A single giant component • The remainder of vertices occupying smaller components with exponential size distribution and finite mean size,

Poisson random graphs Let u be the fraction of vertices on the graph that do not belong to the giant component. The fraction S of the graph occupied by the giant component is S = 1-u and

Poisson random graphs The mean size of the component:

Poisson random graphs • Talk about again about Small-world effect in Poisson random graphs. • The mean number of neighbors a distance l away from a vertex in a random graph is zd , and hence the value of d needed to encompass the entire network zl ~n. Thus l = logn/logz. • Shortcomings of Poisson • Low clustering coefficient C = p, when n is large, p~ n-1 ~0 • Unlike the real-world distribution.

Generalized random graphs • Configuration model • The vertex at the end of a randomly chosen edge is proportional to kpk. • Excess degree(how many edges there are leaving such a vertex other than the one we arrived along) qk . • Define two generating functions

Generalized random graphs • Consider the giant component: • Consider the clustering coefficient

Generalized random graphs • Power-law degree distribution • is the Riemann -function as a normalizing constant. • According the generating functions before, we get conclusions: • 0<<2, u=0, S=1. i.e., giant component occupies the entire graph • 7/3, C tends to zero as the graph becomes large. In the opposite, C becomes constant and actually increases with increasing system size.

Generalized random graphs Directed graphs Bipartite graphs Degree correlations

Exponential random graphs and Markov graphs Background: properties such as transitivity is not incorporated into random graph models.

Exponential random graphs and Markov graphs • Exponential random graphs (P* models) • is a set of measurable properties of a single graph. • is a set of inverse-temperature of field parameters • Each graph G appears with probability: • Partition function Z is

Exponential random graphs and Markov graphs The calculation of the ensemble average of a graph observable is then found by taking a suitable derivative of free energy f=-logZ. There are ways to express f in closed form. But carrying through the entire field-theoretic program is not easy The question of how to carry out calculations in exponential random graph ensembles is open

The small-world model Less sophisticated but more tractable model of a network with high transitivity. Built on lattices of any dimension or topology( Take one dimension with L vertices, each vertex has k neighbors with fewer spacing away) Each edge Rewires randomly with possibility p (create shortcuts)

The small-world model • Extreme 1, P=0, • clustering coefficient C=(3k-3)/(4k-2), • Mean geodesic distance tend to L/4k for large L • Extreme 2, P=1, • C~2k/L • Mean geodesic distance logL/logK • Numerical simulation by Watts and Strogatz showed there exists a sizable region in between there two extremes for which the model has both low path lengths and high transitivity.

The small-world model • Modification of the rewiring methods • Both ends of edges can be rewired • Allow double edges and self edges(maintain original edges)

The small-world model • A. Clustering coefficient • By Barrat and Weigt • By Newman • B. Degree distribution(Since it is not the goal, behaves badly compared with real-world networks)

The small-world model • C. Average path length • P=0, l~ L/4k , large-world • P=1, l~logL, small-world • 0<P<1, no exact solution for the value l • Attempts:

The small-world model • The scaling form shows that we can go from large-world regime to the small-world one through • Increasing p • Increasing the system size L

Models of network growth • Goal of Previous models: • Create networks that incorporate properties observed in real-world networks • Goal of Models with growth: • Explain network properties. • Mainly aimed at explaining the origin of the highly skewed degree distributions.

Models of network growth • Price’s Model • Basic work and later followed by many others • Explanation for power-law: “the rich get richer” , also called cumulative advantage or preferential attachment • Mean in-degree: • The probability of attachment to a vertex should be proportional to k+k0. k0 = 1. • The probability that a new edge attaches to any of the vertices with degree k is thus

Models of network growth Price’s Model Another classic model: model of Barabasi and Albert(using undirected network) Krapivsky and Redner consider the age of vertices and their degrees with older vertices having higher mean degree.

Outline Networks in the real world Properties of networks Random graphs Exponential random graphs and Markov graphs The small-world model Models of network growth Processes happening

Processes happening • A. percolation theory and network resilience • B. Epidemiological processes • The SIR model • The SIS model • C. Search on networks • Exhaustive network search(eigenvector centrality) • Guided network search • Network navigation • D. Other processes

Future work Study of graph models Other properties such as correlations, transitivity and community structure(degree distribution has been thoroughly done). Relations and differences of smaller components and largest component

The structure and function of complex networks(2003)