School of Information University of Michigan SI 614 Random graphs & power law networks preferential attachment Lecture 7 Instructor: Lada Adamic

Outline Erdos-Renyi random graphs BA model scale free networks in Pajek modifications of preferential attachment other processes that lead to power law networks randomizing networks but preserving network properties assortative mixing

Simplest random network Erdos-Renyi model: randomly draw E edges between N nodes Conserves only the average number of neighbors (connectivity) of a node =2E/N = p N No hubs! Narrow distribution of connectivities Poisson distribution

Real world networks are often power law though... Sexual networks Great variation in contact numbers

Yule model

Basic BA-model Very simple algorithm to implement start with an initial set of m 0 fully connected nodes e.g. m 0 = 3 now add new vertices one by one, each one with exactly m edges each new edge connects to an existing vertex in proportion to the number of edges that vertex already has → preferential attachment easiest if you keep track of edge endpoints in one large array and select an element from this array at random the probability of selecting any one vertex will be proportional to the number of times it appears in the array – which corresponds to its degree ….

generating BA graphs – cont’d To start, each vertex has an equal number of edges (2) the probability of choosing any vertex is 1/3 We add a new vertex, and it will have m edges, here take m=2 draw 2 random elements from the array – suppose they are 2 and 3 Now the probabilities of selecting 1,2,3,or 4 are 1/5, 3/10, 3/10, 1/5 Add a new vertex, draw a vertex for it to connect from the array etc

Properties of the BA graph The distribution is scale free with exponent  = 3 P(k) = 2 m 2 /k 3 The graph is connected Every new vertex is born with a link or several links (depending on whether m = 1 or m > 1) It then connects to an ‘older’ vertex, which itself connected to another vertex when it was introduced And we started from a connected core The older are richer Nodes accumulate links as time goes on, which gives older nodes an advantage since newer nodes are going to attach preferentially – and older nodes have a higher degree to tempt them with than some new kid on the block

vertex introduced at time t=5 vertex introduced at time t=95 Time evolution of the connectivity of a vertex in the BA model Younger vertex does not stand a chance: at t=95 older vertex has ~ 20 edges, and younger vertex is starting out with 5 at t ~ 10,000 older vertex has 200 edges and younger vertex has 50

Generating scale free networks with Pajek Two general options Scale free D.M. Pennock et al. (2002) Winners don’t take all, PNAS, 99/8, Pajek command: Net > Random Network > Scale Free Differs from the BA model primarily in that: new vertices are not automatically assigned edges probability of attaching is partially independent of degree Extended model Albert R., Barabasi A.L.: Topology of evolving networks: local events and universality Pajek command: Net > Random Network > Extended Model Differs from the simple BA model in that: edges are added between existing nodes, not only the newcomer edges are rewired between existing nodes

‘Scale free’ network option in Pajek Network starts with m 0 vertices, which link to each other with probability p 0 (as in an Erdos-Renyi random graph) At each time step t, one vertex and m edges are added to the network Instead of attaching one end point of each edge to the newly introduced vertex, choose each end point according to the probability: fraction of edges in the graph that start at v fraction of edges in the graph that end at v the credit v gets just for being one of the vertices

‘Scale free’ network generation in Pajek-cont’d Observations:  = 1, so can vary the relative importance of indegree, outdegree, and independent probability in an undirected network  since indegree and outdegree are the same Not all vertices will be connected, since they are not ‘born’ with an edge The larger g is, the less scale-free the degree distribution edges are added at without regard to degree Original BA paper showed that in that case the degree distribution P(k) ~ exp(-  k) so an exponential distribution

Pennock model Example: It is reasonable to assume that some webpages will be linked to in part because of what they are rather than the number of links they already have

fits to various subsets of web data, and web pages in general

‘Scale free’ in Pajek For the network you can specify ‘undirected’, ‘directed’, or ‘acyclic’ an ‘adding > free’ option? # of vertices # of lines average degree of vertices Initial Erdos-Renyi Graph (these are the first few vertices present) # of vertices (use something small, a couple of vertices) probability p of connecting – type to have them fully connected, or anything between 0 and 1 doesn’t matter much  – this is between 0 and 0.5 for an undirected graph the higher  the more scale-free your distribution will be but watch out, if you set  = 0.5, then  =0.5 and  = 0, and your new, edgeless vertices will never get new connections – you will only have the original Erdos-Renyi component connected in theory you can leave either the # of vertices or # of lines unconstrained, but leaving the # of lines unconstrained (enter in ‘0’) works for me

Extended BA model (undirected network) start with m 0 isolated nodes at each timestep perform one of the following operations: w/ prob. p add m (m≤ m 0 ) new links for each link select ‘from’ vertex at random select ‘to’ vertex in proportion to its degree (+1 so that isolated vertices have a chance of getting links) w/ prob. q where 0 < q < 1 – p rewire m links select node i at random and one of i’s links rewire the endpoint of i’s link to another node j randomly chosen with probability  (k j )

Extended BA model – cont’d w/ prob. 1 – p - q add a new node with m links connect endpoints of the m links to vertices in proportion to their degree (  (k j ) In the p=q=0 limit, reduces to the simple BA model rewire m links select node i at random and one of i’s links rewire the endpoint of i’s link to another node j randomly chosen with probability  (k j ) In the high q (q -> 1) limit, extended model produces a network with an exponential tail because growth is very slow (only rewiring is occurring)

parameter space of the extended BA model In the high p (p > 0.5) limit, have a scale free distribution, because adding new edges preferentially saturation effect for small k (degree) because edges keep being added, but vertices are not being added that quickly, eventually even the low degree vertices get a few more edges power-law exponent varies between 2 and , depending on parameters

Extended BA model in Pajek Net > Random Network > Extended Model Specify n = # of vertices m 0 = # of initial, disconnected nodes m ≤ m 0, number of edges to add/rewire at a time p = probability to add new lines q = probability to rewire edges, 0 ≤ q ≤ 1-p can ask for network without multiple lines

How can we randomize a network while preserving the degree distribution? Stub reconnection algorithm (M. E. Newman, et al, 2001, also known in mathematical literature since 1960s) Break every edge in two “edge stubs” A  B to A   B Randomly reconnect stubs Problems: Leads to multiple edges Cannot be modified to preserve additional topological properties

Local rewiring algorithm Randomly select and rewire two edges (Maslov, Sneppen, 2002, also known in mathematical literature since 1960s) Repeat many times Preserves both the number of upstream and downstream neighbors of each node

Conserving additional low-level topological properties In addition to k i one may also conserve: The exact numbers of loops or other motifs The size and numbers of components: Internet – all nodes have to be connected to each other Metropolis algorithm: two edges are rewired based on E=(N actual -N desired ) 2 /N desired If  E  0 rewiring step is always accepted If  E>0 rewiring step is accepted with p=exp(-  E/T)

Assortativity Social networks are assortative: the gregarious people associate with other gregarious people the loners associate with other loners The Internet is disassorative: Assortative: hubs connect to hubs RandomDisassortative: hubs are in the periphery

Correlation profile of a network Detects preferences in linking of nodes to each other based on their connectivity Measure N(k 0,k 1 ) – the number of edges between nodes with connectivities k 0 and k 1 Compare it to N r (k 0,k 1 ) – the same property in a properly randomized network Very noise-tolerant with respect to both false positives and negatives

Correlation profiles give complex networks unique identities Internet Protein interactions slide by Sergei Maslov 2D picture

Correlation profiles give complex networks unique identities Internet Protein interactions Sergei Maslov: 2D histogram

Correlation profiles -cont’d Pastor-Satorras and Vespignani: 2D plot average degree of the node’s neighbors degree of node

Correlation profiles -cont’d Newman: single number internet degree correlation coefficient The Pearson correlation coefficient of nodes on each side on an edge

Other examples of assortative mixing Assortativity is not limited to degree-degree correlations other attributes social networks: race, income, gender, age food webs: herbivores, carnivores internet: high level connectivity providers, ISPs, consumers Tendency of like individuals to associate: ‘homophily’ more about this later…