Lecture 4: Mathematics of Networks CS 790g: Complex Networks Slides are modified from Networks: Theory and Application by Lada Adamic

What are networks? Networks are collections of points joined by lines. “Network” ≡ “Graph” pointslinesDomain verticesedges, arcsmath nodeslinkscomputer science sitesbondsphysics actorsties, relationssociology node edge 2

Network elements: edges Directed (also called arcs) A -> B (E BA ) A likes B, A gave a gift to B, A is B’s child Undirected A B or A – B A and B like each other A and B are siblings A and B are co-authors Edge attributes weight (e.g. frequency of communication) ranking (best friend, second best friend…) type (friend, relative, co-worker) properties depending on the structure of the rest of the graph: e.g. betweenness Multiedge: multiple edges between two pair of nodes Self-edge: from a node to itself 3

Directed networks Ada Cora Louise Jean Helen Martha Alice Robin Marion Maxine Lena Hazel Hilda Frances Eva Ruth Edna Adele Jane Anna Mary Betty Ella Ellen Laura Irene girls’ school dormitory dining-table partners (Moreno, The sociometry reader, 1960) first and second choices shown 4

Edge weights can have positive or negative values One gene activates/ inhibits another One person trusting/ distrusting another Research challenge: How does one ‘propagate’ negative feelings in a social network? Is my enemy’s enemy my friend? Transcription regulatory network in baker’s yeast 5

Adjacency matrices Representing edges (who is adjacent to whom) as a matrix A ij = 1 if node i has an edge to node j = 0 if node i does not have an edge to j A ii = 0 unless the network has self-loops If self-loop, A ii =? A ij = A ji if the network is undirected, or if i and j share a reciprocated edge i j i i j Example: A = 6

Adjacency lists Edge list Adjacency list is easier to work with if network is large sparse quickly retrieve all neighbors for a node 1: 2: 3 4 3: 2 4 4: 5 5:

Nodes Node network properties from immediate connections indegree how many directed edges (arcs) are incident on a node outdegree how many directed edges (arcs) originate at a node degree (in or out) number of edges incident on a node outdegree=2 indegree=3 degree=5 8

HyperGraphs Edges join more than two nodes at a time (hyperEdge) Affliation networks Examples Families Subnetworks Can be transformed to a bipartite network 9 CD AB CD AB

Bipartite (two-mode) networks edges occur only between two groups of nodes, not within those groups for example, we may have individuals and events directors and boards of directors customers and the items they purchase metabolites and the reactions they participate in

in matrix notation B ij = 1 if node i from the first group links to node j from the second group = 0 otherwise B is usually not a square matrix! for example: we have n customers and m products i j B =

going from a bipartite to a one-mode graph One mode projection two nodes from the first group are connected if they link to the same node in the second group naturally high occurrence of cliques some loss of information Can use weighted edges to preserve group occurrences Two-mode network group 1 group 2

Collapsing to a one-mode network i and k are linked if they both link to j P ij =  k B ki B kj P’ = B B T the transpose of a matrix swaps B xy and B yx if B is an nxm matrix, B T is an mxn matrix i j=1 k j=2 B =B T =

Matrix multiplication general formula for matrix multiplication Z ij =  k X ik Y kj let Z = P’, X = B, Y = B T P’ = = = 1*1+1*1 + 1*0 + 1*0 = 2

Collapsing a two-mode network to a one mode-network Assume the nodes in group 1 are people and the nodes in group 2 are movies The diagonal entries of P’ give the number of movies each person has seen The off-diagonal elements of P’ give the number of movies that both people have seen P’ is symmetric P’ =

Trees Trees are undirected graphs that contain no cycles For n nodes, number of edges m = n-1 A node can be dedicated as the root

examples of trees In nature trees river networks arteries (or veins, but not both) Man made sewer system Computer science binary search trees decision trees (AI) Network analysis minimum spanning trees from one node – how to reach all other nodes most quickly may not be unique, because shortest paths are not always unique depends on weight of edges

Planar graphs A graph is planar if it can be drawn on a plane without any edges crossing

Kuratowski’s theorem Every non-planar network contains at least one subgraph that is an expansion of K 5 or K 3,3. K 5 K 3,3 Expansion: Addition of new node in the middle of edges. Research challenge: Degree of planarity? 19

#s of planar graphs of different sizes 1:1 2:2 3:4 4:11 Every planar graph has a straight line embedding

Edge contractions defined A finite graph G is planar if and only if it has no subgraph that is homeomorphic or edge-contractible to the complete graph in five vertices (K 5 ) or the complete bipartite graph K 3, 3. (Kuratowski's Theorem)

Peterson graph Example of using edge contractions to show a graph is not planar

Cliques and complete graphs K n is the complete graph (clique) with K vertices each vertex is connected to every other vertex there are n*(n-1)/2 undirected edges K5K5 K8K8 K3K3

Bi-cliques (cliques in bipartite graphs) K m,n is the complete bipartite graph with m and n vertices of the two different types K 3,3 maps to the utility graph Is there a way to connect three utilities, e.g. gas, water, electricity to three houses without having any of the pipes cross? K 3,3 Utility graph

Node degree Outdegree = A = example: outdegree for node 3 is 2, which we obtain by summing the number of non- zero entries in the 3 rd row Indegree = A = example: the indegree for node 3 is 1, which we obtain by summing the number of non-zero entries in the 3 rd column

Degree sequence and Degree distribution Degree sequence: An ordered list of the (in,out) degree of each node In-degree sequence: [2, 2, 2, 1, 1, 1, 1, 0] Out-degree sequence: [2, 2, 2, 2, 1, 1, 1, 0] (undirected) degree sequence: [3, 3, 3, 2, 2, 1, 1, 1] Degree distribution: A frequency count of the occurrence of each degree In-degree distribution: [(2,3) (1,4) (0,1)] Out-degree distribution: [(2,4) (1,3) (0,1)] (undirected) distribution: [(3,3) (2,2) (1,3)] 26

Structural Metrics: Degree distribution 27 What if it is directed ?

Graph density 28

Characterizing networks: How dense are they?

network metrics: graph density Of the connections that may exist between n nodes directed graph e max = n*(n-1) undirected graph e max = n*(n-1)/2 What fraction are present? density = e/ e max For example, out of 12 possible connections, this graph has 7, giving it a density of 7/12 = Would this measure be useful for comparing networks of different sizes (different numbers of nodes)? 30