Lecture 3: Mathematics of Networks

1. Lecture 3: Mathematics of Networks CS 765: Complex Networks Slides are modified from Networks: Theory and Application by Lada Adamic

2. What are networks? • Networks are collections of points joined by lines. “Network” ≡ “Graph” node edge

3. Network elements: edges • Directed (also called arcs) • A -> B (EBA) • 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

4. 2 2 2 Louise Ada Lena 1 1 Adele 1 1 Marion 2 1 2 1 2 Jane 2 2 1 2 2 Frances Cora 1 2 1 2 1 Eva Maxine Mary 1 1 1 2 2 1 Anna Ruth 1 1 Edna 1 Robin Betty Martha 2 1 1 2 2 2 2 2 2 2 2 1 Jean 1 1 1 Laura Alice 1 2 Hazel Helen Hilda 1 2 1 Ellen 2 Ella Irene Directed networks • girls’ school dormitory dining-table partners (Moreno, The sociometry reader, 1960) • first and second choices shown

5. 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

6. i i i Adjacency matrices • Representing edges (who is adjacent to whom) as a matrix • Aij = 1 if node i has an edge to node j = 0 if node i does not have an edge to j • Aii = 0 unless the network has self-loops • If self-loop, Aii=1 • Aij = Aji if the network is undirected,or if i and j share a reciprocated edge j j Example: 2 3 A = 1 4 5

7. Edge / Adjacency lists • Edge list • 2 3 • 2 4 • 3 2 • 3 4 • 4 5 • 5 2 • 5 1 • 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: 1 2 2 3 1 4 5

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

9. HyperGraphs • Edges join more than two nodes at a time (hyperEdge) • Affliation networks • Examples • Families • Coauthors, project members, etc • Subnetworks Can be transformed to a bipartite network A B C D A B C D

10. 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

11. in matrix notation • Bij • = 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 =

12. going from a bipartite to a one-mode graph group 1 • 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 2

13. Collapsing to a one-mode network i j • i and j are linked if they both link to k • Pij = k Bik Bjk • P’ = B BT • the transpose of a matrix swaps Bxy and Byx • if B is an nxm matrix, BT is an mxn matrix k=1 k=2 B = BT =

14. Matrix multiplication • general formula for matrix multiplication Zij= k Xik Ykj • let Z = P’, X = B, Y = BT = P’ = 1 1 = 1*1+1*1 + 1*0 + 1*0= 2 1 1 2

15. 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 • P’ is symmetric • 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 1 1 P’ = 1 1 2

16. Human Disease Network 

17. Multi-partite Networks

18. Multi-partite Networks

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

20. 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

21. Cliques and complete graphs • Kn is the complete graph (clique) with K vertices • each vertex is connected to every other vertex • there are n*(n-1)/2 undirected edges K5 K3 K8

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

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

24. Kuratowski’s theorem • Every non-planar network contains at least one subgraph that is an expansion of K5 or K3,3. K5 K3,3 Expansion: Addition of new node in the middle of edges. • Research challenge: Degree of planarity?

25. 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 (K5) or the complete bipartite graph K3, 3. (Kuratowski's Theorem)

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

27. Bi-cliques (cliques in bipartite graphs) • Km,n is the complete bipartite graph with m and n vertices of the two different types • K3,3 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? Utility graph K3,3

28. Lecture 4

29. B e1 e6 e2 C A D e5 e4 A D e3 e7 B B Eulerian Path • Euler’s Seven Bridges of Königsberg • the first problem in graph theory • 1735 • Is there a route that crosses each bridge only once and returns to the starting point? http://barabasi.com/networksciencebook/chapter/2

30. A Characterization for Eulerian Graphs Theorem: G is Eulerian  G is even and connected Idea: • In an Eulerian circuit C, each time C goes through a vertex exactly two incident edges are used • The first edge of C is paired with the last one • Hence every vertex has even degree Start (The 1st) In Out End (The last)

31. Eulerian Circuits and Trails • A graph is Eulerian if it has a circuit containing all edges (may repeat vertices but not edges) • Eulerian circuit: same start and end • Eulerian trail: different start and end Each contains all edges without repetition

32. Eulerian and Hamiltonian paths • Hamiltonian path is self avoiding If starting point and end point are the same: only possible if no nodes have an odd degree as each path must visit and leave each shore If don’t need to return to starting point can have 0 or 2 nodes with an odd degree Eulerian path: traverse each edge exactly once Hamiltonian path: visit each vertex exactly once

33. Structural metrics: Average path length 1 ≤ L ≤ D ≤ N-1

34. Characterizing networks:How far apart are things?

35. Network metrics: paths • A path is any sequence of vertices such that every consecutive pair of vertices in the sequence is connected by an edge in the network. • For directed: traversed in the correct direction for the edges. • path can visit itself (vertex or edge) more than once • Self-avoiding paths do not intersect themselves. • Path length r is the number of edges on the path • Called hops

36. Network metrics: paths

37. Network metrics: shortest paths B 3 C A 2 1 3 D E 2 3

38. Paths • A path between nodes i0 and in is an ordered list of n links P = {(i0, i1), (i1, i2), (i2, i3), ... ,(in-1, in)}. • The length of the path is n. • The path shown in orange in (a) follows the route 1→2→5→7→4→6, hence its length is n = 5 • The shortest paths between nodes 1 and 7, or the distance d17, correspond to the path with the fewest number of links that connect nodes 1 to 7. • There can be multiple paths of the same length, as illustrated by the two paths shown in orange and grey. • The network diameter is the largest distance in the network • being dmax = 3 here.

39. Paths path Shortest path Diameter Hamiltonian path Average path length Cycle Eulerian path

40. 2 Node degree 3 1 • Outdegree = 4 5 A = example: outdegree for node 3 is 2, which we obtain by summing the number of non-zero entries in the 3rdrow • Indegree = A = example: the indegree for node 3 is 1, which we obtain by summing the number of non-zero entries in the 3rdcolumn

41. Node degrees

42. Degree sequence and Degree frequency • 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 frequency: A frequency count of the occurrence of each degree In-degree frequency: [(2,3) (1,4) (0,1)] Out-degree frequency : [(2,4) (1,3) (0,1)] (undirected) frequency : [(3,3) (2,2) (1,3)]

43. Degree distribution • The degree distribution is a function P(k), which gives the probability of a randomly chosen node from the graph having degree k

44. Structural Metrics: Degree distribution

45. Degree distributions • Imagine I have a graph with 1000 nodes, but no links. Now I start adding links randomly, one by one. • After 10 random additions, what do you expect the degree distribution to be? • What will the average node degree be after 1000 additions? • The standard situation in a network where links are added completely at random. • If there are n nodes, and m edges randomly added, then the peak of this is at 2m/n, the average degree. • For a randomly picked node, the most likely degree is the average one. • The probabilities then drop quickly either side.

46. Degree Distributions Protein interactions of yeast

47. Characterizing networks:How dense are they?

48. network metrics: graph density • Of the connections that may exist between n nodes • directed graph Lmax= n*(n-1) • undirected graph Lmax= n*(n-1)/2 • What fraction are present? • density = L / Lmax • In real networks L << Lmax • For example, out of 12 possible connections, this graph has 7, giving it a density of 7/12 = 0.583

49. Graph density • Would this measure be useful for comparing networks of different sizes (different numbers of nodes)? • As n → ∞, a graph whose density reaches • 0 is a sparse graph • a constant is a dense graph

50. Characterizing networks:Is everything connected?