1. Author: M.E.J. Newman Presenter: Guoliang Liu Date:5/4/2012

2.  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

3. Social networksInformation networksTechnological networksBiological networks

4.  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

5.  The small-world effect  Transitivity or clustering  Degree distributions  Network resilience  Mixing patterns  Degree correlations  Community structure  Network navigation  Other network properties

6.  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.

7.  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

8.  An example to calculate clustering coefficient C

15.  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”

16.  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)

17.  How to measure mixing patterns? E ij : the number of edges in a network that connect vertices of types i and j. e ij measure the fraction of edges that fall between vertices of types i and j. Assortative mixing coefficient:

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

19. ……

20.  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”

21.  Giant component  Betweenness centrality  Self-similarity

22.  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

23.  Poisson random graph  Generalized random graphs

24.  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

25.  Poisson random graphs Early work, simple model of a network  G n,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,

26.  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

28.  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 z d, and hence the value of d needed to encompass the entire network z l ~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.

29.  Configuration model The vertex at the end of a randomly chosen edge is proportional to kp k. Excess degree(how many edges there are leaving such a vertex other than the one we arrived along) q k. Define two generating functions

30.  Consider the giant component:  Consider the clustering coefficient

32.  Directed graphs  Bipartite graphs  Degree correlations

33.  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

34.  Background: properties such as transitivity is not incorporated into random graph models.

35.  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

36.  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

37.  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

38.  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)

39.  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.

40.  Modification of the rewiring methods Both ends of edges can be rewired Allow double edges and self edges(maintain original edges)

41.  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)

42.  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:

43.  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

44.  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

45.  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.

46.  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+k 0. k 0 = 1. The probability that a new edge attaches to any of the vertices with degree k is thus

47.  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.

48.  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

49.  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

50.  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