Network Questions: Structural 1. How many connections does the average node have? 2. Are some nodes more connected than others? 3. Is the entire network connected? 4. On average, how many links are there between nodes? 5. Are there clusters or groupings within which the connections are particularly strong? 6. What is the best way to characterize a complex network? 7. How can we tell if two networks are “different”? 8. Are there useful ways of classifying or categorizing networks? slides from David P. Feldman 1

Network Questions: Communities 1. Are there clusters or groupings within which the connections are particularly strong? 2. What is the best way to discover communities, especially in large networks? 3. How can we tell if these communities are statistically significant? 4. What do these clusters tell us in specific applications? 2

Network Questions: Dynamics of 1. How can we model the growth of networks? 2. What are the important features of networks that our models should capture? 3. Are there “universal” models of network growth? What details matter and what details don’t? 4. To what extent are these models appropriate null models for statistical inference? 5. What’s the deal with power laws, anyway? 3

Network Questions: Dynamics on 1. How do diseases/computer viruses/innovations/ rumors/revolutions propagate on networks? 2. What properties of networks are relevant to the answer of the above question? 3. If you wanted to prevent (or encourage) spread of something on a network, what should you do? 4. What types of networks are robust to random attack or failure? 5. What types of networks are robust to directed attack? 6. How are dynamics of and dynamics on coupled? 4

Network Questions: Algorithms 1. What types of networks are searchable or navigable? 2. What are good ways to visualize complex networks? 3. How does google page rank work? 4. If the internet were to double in size, would it still work? 5

Network Questions: Algorithms There are also many domain-specific questions: 1. Are networks a sensible way to think about gene regulation or protein interactions or food webs? 2. What can social networks tell us about how people interact and form communities and make friends and enemies? 3. Lots and lots of other theoretical and methodological questions What else can be viewed as a network? Many applications await. 6

Network Questions: Outlook Advances in available data, computing speed, and algorithms have made it possible to apply network analysis to a vast and growing number of phenomena. This means that there is lots of exciting, novel work being done. This work is a mixture of awesome, exploratory, misleading, irrelevant, relevant, fascinating, ground-breaking, important, and just plain wrong. It is relatively easy to fool oneself into seeing thing that aren’t there when analyzing networks. This is the case with almost anything, not just networks. For networks, how can we be more careful and scientific, and not just descriptive and empirical? 7

Lecture 3: Mathematics of Networks CS 765: 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 9

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 10

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 11

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 12

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 = 13

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 15

HyperGraphs Edges join more than two nodes at a time (hyperEdge) Affliation networks Examples Families Subnetworks Can be transformed to a bipartite network 16 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’ =