More Complex Graph Structures The vertexlabelled edgelabelled graph

More Complex Graph Structures? The vertex-labelled, edge-labelled graph We can interpret this structure many ways, 1. as a relationship with entity tables; 2. as a AN[lysist] Table with attributes, the AN attributes (SA, Ct, C, Sal) plus each Ticker. Symbol p. Tree as an additional attribute (the TS attributes (Dow? , Ct, BHS, SA) are not captured in this interpretation); 3. as a T[icker] S[ymbol] or Stock Table with attributes, the TS attributes (Dow? , Ct, TS AN p. Tree SA Ct C 3 8 2 3 7 3 2 8 0 1 7 1 3 4 0 2 4 1 3 4 0 1 3 0 3 4 0 2 4 0 1 6 0 1 7 2 1 8 0 3 5 0 1 2 0 2 2 0 3 2 3 ANalyst Ticker. Symbol Relationship with labels 1 0 0 1 1 0 3 3 6 4 8 8 a e c 5 4 6 3 3 Dow? AN p. Tree Ct H B B SS S S H H B B B SB Buy-Hold-Sell 1 2 2 3 3 TS SA 1 3 3 1 2 3 1 Sal AN 1 2 3 4 5 6 7 8 9 10 11 12 13 14 7 5 7 6 2 1 2 3 4 5 1 1 0 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 0 1 0 0 0 0 0 0 0 1 6 4 5 7 6 7 8 9 10 0 0 1 0 0 0 0 0 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 4 5 6 3 4 11 12 13 14 15 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 1 6 1 16 17 18 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 1 2 11 11 10 01 00 0 0 1 1 0 0 1 3 10 11 11 10 10 1 0 0 0 1 1 1 4 11 01 11 1 1 0 0 0 1 0 1 1 5 10 10 10 1 1 0 0 0 0 0 1 6 01 01 00 1 1 0 0 0 1 0 7 01 11 10 0 1 1 1 0 0 0 1 0 8 10 10 10 01 00 1 1 1 0 0 0 1 9 10 01 11 01 00 0 0 1 1 0 0 0 0 0 1 1 0 10 11 10 00 01 00 10 00 1 1 0 0 1 0 12 00 00 00 1 0 0 1 1 0 0 0 1 0 13 00 00 00 0 0 0 1 1 1 14 00 00 00 TS BHS, SA) plus each Analyst p. Tree as an additional attribute (the AN attributes (SA, Ct, F, Sal) are not captured in this interpretation); We can include this relationship with other relationships sharing entities by using the Rolo. Dex Model (next slide). The graph could be 3 D, 4 D (i. e. , edges are triples, quadruples), etc. The graph could also be edge labelled. A convenient way to capture edge labels is by making the cell content of each matrix cell into the label structure rather than just a yes/no bit. As a simple but pertinent example, suppose we have a 0 -3 rating of each Analyst-Stock pair which measure how much that Analysts know about that stock. We just change each bit to a decimal number in [0, 3] (or bitslice those using two bits instead of on, so that the matrix columns are 2 -bit p. Tree. Sets rather than just one p. Tree). 10 11 00 11 10 10 0 0 1 01 1 10 0 01 1 01 0 0 0 111 101 110 010 1 2 3 4 5 1 0 0 0 1 1 10 01 00 10 10 10 00 01 0 0 0 00 1 0 0 0 01 0 001 110 101 111 6 7 8 9 10 01 01 00 00 11 00 00 01 00 00 10 10 00 01 01 01 00 00 01 10 10 10 01 10 00 00 10 10 10 00 01 10 00 00 01 01 01 10 11 00 00 00 01 01 10 00 10 11 01 0 01 1 0 0 01 1 1 0 10 0 01 0 100 101 110 010 100 11 12 13 14 15 00 00 00 00 00 00 00 10 00 01 01 01 10 10 01 00 10 01 10 00 11 11 11 00 00 00 10 01 11 11 11 00 10 10 10 00 01 11 11 11 01 In full p. Tree form: SS AA CCCC F S S S AN 1 0 32 1 0 1 0 0 0 0 1 16 0 0 1 1 0 17 1 0 0 0 0 1 18 Dow? C 3 C 2 C 1 C 0 SS S H B SB SA 1 SA 0 TS 00 00 00 01 10 01 01 00 00 00 01 00 11 10 10 01 00 00 00 01 01 00 11 01 00 01 If C measures the “Correctness Level” of the Analyst over recent days or weeks over all stock (e. g. , based on backward analysis of previous sentiment analysis and the actual performance of the stock) and the cell numbers measure the correctness of that Analyst on that Stock, then a signal might be to mask C>=2 and for those Analysts find the average Correctness for each stock, then mask out those Stock for which the number of Analysts is between two thresholds (want a high average but also more than one analyst but not too many).

Every Entity (Gene, Term, Experiment, Person, Document, Item, Stock, Course, Movie) has an Entity. Table of many descriptive attributes (columns). They aren’t shown here. For example, on the previous slide we show the descriptive columns of Stocks(Dow? , Count, BHS, SA) and Analysts(SA, Count, Female? , Salary. In. Billions), not shown here. Tweets are Documents, so the Tweet-Tweeter relationship is a Document-Author relationship (Tweetee, hashtag, etc. are Edge Labels). Stock In looking for signals that no one else uses: What if an Investor BUYS an island in the Mediterranean? What if an Investor’s best friend buys lots of stock in an Online University? 1 2 Stock. Investor relationship 3 4 Conf(A B) Supp(A) = Cus. Freq(Item. Set) =Supp(A B)/Supp(A) 5 6 7 16 Item Friends relationship 6 itemset 5 BUYS 4 3 2 Author People Customer Term. Document 1 1 1 PI 2 32 43 45 56 7 Item. Set 1 2 3 4 0 0 antecedent 1 1 1 Auth 1 1 1 Doc gene rel (ppi) 3 2 1 1 1 2 1 3 1 docdoc 1 1 1 Gene 1 movie 4 Do c 1 Item. Set 1 Enroll 4 Course 2 People term G exp. PI 1 1 2 3 4 5 6 7 3 4 3 0 0 Doc 3 0 0 2 1 0 0 0 Expg ene t Ex p 3 1 1 The Multi-Relationship Model 3 1 0 0 0 5 6 7 0 0 0 4 5 0 0 0 customer rates movie 0 2 Share. Stem term rel Cell. Label=stem 0 0 0 1 0 0 4 0 0 1 1 0 0 3 5 0 0 0 0 customer rates movie as 5 relationships 0 0 0 1 0 5 6 16

More Complex Graph Structures? Hyper. Graphs, clique. Trees (c. Trees), GRAPH (linear edges, 2 vertices) k. PARTITE Graph or just k. PART Graph (V=! Vi i=1. . k (x, y) E x, y same Vi ) Motifs k. Hyper. Graph (edges=k vertices) k. PART Hyper. Graph (V=! Vi i=1. . k (x 1. . xk) E xj, xj same Vi ) Bi. PART Clique Mining finds Max. Cliques at cost of pairwise &s. Each LETp. Tree MCLQ unless pairwise & with same count. A&B, B w Ct(A&B)=Ct(A) is a MCLQ. potential for a k-plex [k-core] mining alg here. Instead of Ct(A&B)=Ct(A), consider. E. g. , Ct(A&B)=Ct(A)-1. Each such p. Tree, C, would be missing just 1 vertex (1 edge). Taking any MCLQ as above, ANDing in Cp. Tree would produce a 1 -plex. ANDing in k such C’s would produce a k-plex. In fact, suppose we have produced a k-plex in such a manner, then ANDing in any C with Ct(C)=Ct(A)-h would produce a (K+h)-plex. &i=1. . n. Ai is a [ i=1. . n. Ct(Ai)]-Core Tri. PART Clique Mining Algorithm? In a Tripartite Graph edges must start and end in different vertex parts. E. g. , PART 1=tweeters; PART 2=hashtags; PART 3=tweets. Tweeters-to-hashtags is many-to-many? Tweeters-to-tweets is many-to-many (incl. retweets)? ; hashtags-to-tweets is many-to-many? Multi. PART Graphs Bi. PART, Tri. PART (have 2, 3 PARTs respectively but still an edge is a linear (between two vertices) … No edge can start and end in the same PART. D Hyper. Clique Mining: A 3 hyper. Graph has 3 vertex PARTS and each edge is a planar triangle (defined by a vertex triple, one from each PART). Stock recommender is 3 hyper. Graph (Investors, Stocks, Days). A triangular edge connects Investor k, Stock X, and Day n if k recommends X on day n. A 3 hyper. Clique is I a community s. t. all investors in clique recommend all stocks in the clique on each day in clique. Tweet ex: PART 1=tweeters; PART 2=hashtags; PART 3=tweets. Conjecture: Kmulti. Cliques and Khyper. Cliques are in 1 -1 correspondence (both are defined by a K PART vertex set)? So, only one mining process needed? S We will represent these common objects with clique. Trees (c. Trees). A c. Tree bitmaps each PART of the clique. E. g. , the c. Tree for Inv={2, 3}; Stock={A, B} Day={ , }: Cts Cliques, Kplexes and Kcores are subgraphs (communities) defined using an internal edge count. A Motif is a subgraph defined using external “isomorphism into the graph” count. A motif must occur (isomorphically) in the graph more times than “expected”. Criticism: Some authors argue[62] that a motif structure does not necessarily determine function. Recent research [64] shows the connections of a motif to the network, is too important to draw function inferences just from local structure. [65] Research shows certain topological features of biological networks naturally give rise to canonical motifs, . [66] Most find induced Motifs. A graph, G′, is a subgraph of G (G′⊆G) if V′⊆V and E′⊆E∩(V′×V′). If G′⊆G and G′ contains all ‹u, v›∈E with u, v∈V′, G′ is induced subgraph. G′ and G are isomorphic (G′↔G), if a bijection f: V′→V with ‹u, v›∈E′⇔‹f(u), f(v)›∈E u, v∈V′. G″⊂G and an isomorphism between G″ and G′, G′ appears in G). The number of appearances G′ in G is the frequency FG of G′ in G, FG(G’). G is recurrent or frequent in G, when FG(G’)>threshold (pattern=frequent subgraph). Motif discovery includes exact counting, sampling, pattern growth. Motif discovery has 2 steps: calculate the # of occurrences; evaluating the significance. Are Stock-Inv or Stock-Inv-Day Motifs useful? Some questions/theorems/thoughts: 1. 2. 3. 4. 5. All K-Paths are isomorphic (thus, there’s alway a Kpath motif). A Shortest. KPath is an Induced subgraph. What does sequence Frequency(1 Path. Motif)=|V|, Frequency(2 Path. Motif), …tell? Sequence of Frequency(Shortest 1 Path), Frequency(Shortest 2 Path), …? Sequence Frequency(Max. Shortest 1 Path), Frequency(Max. Shortest 2 Path)… tell us? where a Max. S 2 P is not part of a S 3 P. Extend to Hyper. Edges? What is a path in, e. g. , a 3 Hyper. Graph? Both? 2 HGInterface 3 Hyper. Graph. Path. 1 HGI 3 HGP. (In general, h. HGIk. HGP, where 0<h<k) At the other extreme (all SPs are length=1: Or? 6. 7. I’ll bet most important motifs, M(V’, E’) in G are “Shortest Path Motifs”: x, y V’, a G-Shortest. Path in M running from x to y. I. e. , M is made up of G-SPs. 8. A Clique is a SPMotif (made up entirely of Shortest 1 Paths) A 4 PARThyper. Graph or just 4 Hyper. Graph has 4 vertex PARTS and each edge is a solid tetrahedron (defined by a vertex quadruple, one from each PART). Stock Recommender 4 hyper. Graph (Investors, Stocks, Strengh(Stron. Buy, …), Days). A tetrahedral "edge" connects Investor k, Stock X, Strength B and Day n iff k recommends X as a Buy on day n. A 4 hyper. Clique is a community s. t. all the investors recommend all the stocks as strength=B on each day in the clique. some degeneracy since the Strength will always be singleton? One might argue that this is just a series of 3 Hyper. Graphs, one for each strength level. ) A Tweet 4 Hyper. Graph: PART 1=tweeters; PART 2=hashtags; PART 3=tweets, PART 4=day. A 4 hyper. Clique: all tweeters send all tweets on all hashtags each day of the clique. A MBR 4 Hyper. Graph: PART 1=customers; PART 2=items; PART 3=days, PART 4=store. A 4 hyper. Clique: all customers buy all items at all stores on each day of the clique. 0 1 1 1 1 0 2 2 2

Introduction to 2 PART Graph Community Search: For a multipartite graph the concept of community is still related to a large density of edges between members of the same group. A clique in a 2 PART (bipartite) graph to be a bipartite subset of vertices with all possible edges. 2 PART Induction thm: In a bipartite graph, a Kclique and 3 clique that share an edge form a (K+1)clique iff all edges that can exist, from the nonshared Kclique vertices to the non-shared 3 clique vertex, do exist. 2 PART 3 Clique thm: a pair of vertices from part 1, a, b and a vertex from the part 2, 1, form a 3 Clique iff both possible edges a 1, b 1 exist. CLQ 3 is constructed by listing each vertex pair in each p. Tree along with the naming vertex of the p. Tree. a a The 2 3 cliques ab 1 and b 12 sharing b 1 form a 4 clique iff the non-shared vertex pair a 2 is an edge The 2 3 cliques ab 1 and bc 1 sharing b 1 form a 4 clique. b 1 b 2 c 1 a a 1 b 2 The 4 clique ab 12 and 3 clique bc 2 sharing b 2 form a 5 clique iff the non-shared vertex pair c 1 is an edge. The 4 clique abc 1 and 3 clique cd 1 sharing c 1 form a 5 clique c c d a a a 1 b b 2 5 clique abc 12 and 3 clique c 23 sharing c 2 form a 6 clique iff the non-shared vertex pairs a 3 and b 3 are edges. 5 clique abc 12 and 3 clique d 12 sharing vertices 1 and 2 form a 6 clique. 5 clique abcd 1 and 3 clique de 1 sharing edge e 1 form a 6 clique. c 3 b 1 2 c 3 d b b 2 c d d e 1 a a 1 b 2 c 3 d c d 1 c a a 6 clique abc 123 and 3 clique cd 3 sharing c 3 form a 7 clique iff the non-shared vertex pairs d 1 and d 2 are edges. 6 clique abc 123 and 3 clique d 23 sharing vertices 2 and 3 form a 7 clique iff vertex pair d 1 is an edge. 6 clique abcd 12 and 3 clique de 2 sharing edge d 2 form a 7 clique iff vertex pair e 1 is an edge 6 clique abcde 1 and 3 clique ef 1 sharing edge e 1 form a 7 clique. 1 b e 1 1 b 2 c d e f Although the pattern seems complex, the 2 PART Clique Algorithm can be stated: A Kclique and 3 clique sharing 2 vertices form a K+1 clique iff all edges from the non-shared 3 clique vertex to each non-shared Kclique vertex (from the other PART) exist. That is, check edge existence between all non-shared vertices.

clique. Trees 2 PART G 11: Inv(12345) rec Stock(ABCDE) DI Stock. Base. Clique. Trees NPZp. T st=5 I 23 1 00 0 10 0 01 1 0 0 0 1 I 3 S A 1 10 B 0 10 C 1 11 1 0 1 0 1 1 1 0 0 1 1 S 1 11 2 31 1 1 1 2 2 1 1 1 1 3 2 1 11 0 00 0 11 1 1 0 0 0 1 1 00 0 10 0 01 1 0 0 0 1 A 1 1 0 B 0 1 0 C 1 1 0 1 22 1 11 2 31 0 0 0 1 1 1 5 1 0 0 1 1 3 0 1 0 1 1 4 Stock EBCTs I 12 11 01 01 01 3 4 5 S A B C D E 0 0 0 1 1 0 2 4 0 0 0 1 1 1 0 0 0 0 0 1 1 00 0 10 0 01 1 0 0 0 1 A 1 10 B 0 10 C 1 11 1 0 1 0 1 1 1 0 0 1 11 2 31 1 1 1 2 2 1 1 1 1 3 2 1 11 0 00 0 0 0 1 1 1 0 0 0 0 0 1 1 01 1 11 0 01 1 0 0 0 1 0 1 1 0 1 1 0 1 A 1 1 0 B 0 1 0 C 1 1 0 1 0 1 1 1 0 0 1 1 3 3 1 1 1 2 2 1 11 2 13 2 31 1 2 1 1 1 2 2 1 1 1 3 1 2 1 3 2 1 11 0 00 0 11 1 1 0 0 0 1 1 11 0 00 0 10 0 1 1 1 0 0 1 1 00 1 11 0 01 1 0 0 0 1 0 1 1 1 01 1 11 0 01 1 0 0 0 1 0 1 1 0 0 1 1 2 1 0 1 3 A 1 10 B 0 10 C 1 11 1 0 1 0 1 0 1 A 1 10 B 0 10 C 1 11 1 0 1 0 1 22 2 12 2 31 3 3 1 1 1 2 2 1 21 2 13 2 31 2 3 1 2 1 1 1 2 2 1 2 3 oaa aoa 1 11 0 00 D aoa oaa 1 2 3 A B C D E 0 0 0 1 1 0 1 4 A 1 1 A 1 2 A 0 3 A 1 4 A 0 5 B 1 1 B 1 2 B 0 3 B 0 4 B 0 5 C 1 1 C 1 2 C 1 3 C 1 4 C 0 5 D 1 1 D 1 2 D 1 3 D 1 4 D 0 5 E 0 1 E 1 2 E 1 3 E 1 4 E 0 5 0 0 0 1 1 1 5 1 1 0 0 0 1 1 1 3 3 0 1 0 1 1 1 2 4 NPZp. Tr st=5 L=2 1 1 L=1 1 1 111 L=0 001 101 000 1 2 3 H 1 Stock EBCTs 1 1 0 0 1 1 1 0 324 43 Investor BCTs S ABC 10 01 00 00 00 G 11 Stock EBCTs 1 0 0 0 1 I 12 1 1 1 S I 234 5 1 11 0 00 0 10 0 1 1 1 0 0 1 11 01 11 1 01 1 11 0 01 1 0 0 0 1 0 1 1 0 1 01 10 10 1 1 1 0 0 1 1 A 1 10 B 0 10 C 1 11 1 0 1 0 1 1 1 0 0 1 1 01 00 10 1 2 1 3 2 1 21 2 13 2 31 2 3 1 2 1 1 1 2 2 1 3 1 2 1 3 2 23 21 11 0 0 D 0 0 E 0 0 1 1 0 0 0 1 1 3 2 1 0 0 1 1 0 1 4 0 1 1 1 1 0 1 4 0 0 1 1 1 0 1 3 3 4 5 A B C D E 0 0 0 1 1 0 2 4 0 0 0 1 1 1 5 1 1 0 0 0 1 1 1 3 3 0 1 0 1 1 1 2 4 0 1 0 1 1 0 3 3 1 1 0 0 0 1 1 0 4 2 Inv EBGTs =C a Max. Clique. Then 1 of 1 1 0 0 A B C D E 1 2 3 4 5 DI Stock. Base. Clique. Trees 0 0 0 1 1 2 3 S 4534 0 0 0 1 1 1 L=1 1 0 1101 L=0 1100 1111 0111 1 11 0 00 D 3 4 5 New DSs: L=2 Stock BCTs I 12 10 01 00 00 oa 2 E 3 C 3 D 3 E 4 A 4 C 4 D 4 E A B C D E A B C D E oa 12 3 4 5 11 0 0 0 11 1 1 0 01 1 1 0 Edge. Map A B C D E Adj Matrix Traditional data structures 1 A Edge. Tbl 1 A 2 B 1 B 3 C 1 C 1 D 4 D 2 A 2 B 5 Graph E 2 C G 11 2 D 11 11 10 21 21 21 30 30 31 31 31 41 40 41 41 41 50 50 50 0 1 1 0 1 0 3 3 1 1 1 0 0 0 4 2 0 1 1 1 1 0 2 4 0 1 1 1 0 3 3

Baseclique. Trees (Bc. Ts), Expanded. Base c. Trees (EBc. Ts)=Max. Cliques for: 1 2 IDS Base c. Trees 1 1 0 0 0 1 A 0 B 1 1 1 0 0 A B 0 1 1 1 1 1 1 2 1 1 0 0 1 1 0 A B 0 1 1 1 1 2 1 0 1 A B 0 1 1 1 1 1 1 2 1 0 0 0 1 1 A B 0 1 1 1 0 0 1 0 1 0 0 1 1 1 2 aao 1 2 1 1 A 1 1 B 0 0 0 1 0 0 1 1 1 0 1 2 1 0 0 1 1 1 A 1 1 B 0 0 0 1 0 0 1 1 1 0 0 1 1 1 2 0 1 1 0 0 1 1 0 1 1 1 2 1 1 1 0 1 A 0 B 1 1 2 1 0 0 1 1 2 0 1 1 0 A 1 0 B 0 1 1 0 0 0 1 1 1 0 0 1 1 0 0 1 1 1 1 1 A 1 B 0 0 1 1 1 ISD Bc. Trees 1 2 1 1 0 0 0 1 1 0 A 1 0 B 0 1 1 0 0 1 1 0 0 1 1 2 1 1 1 1 SID Base c. Trees 1 2 1 1 0 0 SID Base c. Trees SDI Base c. Trees A 1 B 0 1 2 1 1 1 SDI Base c. Trees DSI Base c. Trees The c. Tree. Set is closed under aao as well and applying aao to DSI and SDI gives the same 2 Max. Clique c. Trees The non-leaf c. Tree PARTs are a lexico ordering of singletons so construct the EBc. Ts using only the p. Tree Leaves? 1 1 1 1 2 SDI Base c. Trees DSI Base c. Trees 1 1 0 ISD Bc. Trees A 1 B 0 0 1 1 0 A 1 0 B 0 1 0 1 1 0 0 1 1 0 1 2 1 1 1 2 1 1 1 0 A B 1 2 1 0 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 A B 1 2 A B oaa A B 1 2 1 0 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 DIS Bc. Ts 1 2 1 0 0 0 1 1 1 0 0 1 A 0 B 1 1 1 0 0 SDI Bc. Ts 1 2 DSI Base c. Trees 1 2 1 0 0 0 1 1 A 0 B 1 1 1 0 0 1 1 1 1 1 aoa 1 2 0 1 1 0 A 1 0 B 0 1 1 0 0 1 1 0 0 1 0 0 1 1 1 0 0 1 DSI Base c. Trees 1 2 1 0 0 0 1 1 A 0 B 1 1 1 0 0 1 1 1 1 1 1 2 DIS NPZ p. Tree Stride=2 Max. Cliques(H 2)= 1 Bc. Ts + 1 EBc. T 1 0 0 1 1 1 0 1 1 2 A B 0 1 1 0 1 2 1 1 10 11 20 20 11 10 21 20 1 1 A L=2 B 1 A B L=1 0 A B 0 A L=0 1 B 1 1 1 0 IDS NPZ p. Tree Stride=2 1 1 2 2 1 1 0 A 1 B 1 A 0 B 0 A 0 B 1 1 0 DSI NPZ p. Tree Stride=2 A A B B 1 1 01 02 11 12 01 02 0 1 1 SDI NPZ p. Tree Stride=2 A A B B ISD Bc. Ts 1 1 0 0 A 1 1 B 0 0 2 Seems to be no need to apply the 3 ops to 6 p. Tree. Sets (apply aoa, oaa, aao to any 1 p. Tree. Set only). And if the 3 ops commute use any order). Can we concatenate p. Tree. Sets into 1 and use only 1 op on it to get all EBc. Ts? If so, 1 op on 1 p. TS gives EBc. Ts=MCLQs. Take oaa on one Concatenated Bc. T (CBc. T), generate all EBc. Ts=Max. Cliques! SID CBc. Ts 1 ISD Bc. Trees B 1 1 0 0 1 DSI Base c. Trees 1 0 0 1 oaa 1 0 1 1 1 1 2 aao A B 0 1 1 1 0 oaa 1 2 1 1 0 0 oaa 1 0 1 2 1 0 0 1 oaa A B 0 1 1 0 1 2 0 0 1 1 A 1 1 B 0 0 A 1 2 1 0 1 1 0 0 oaa 0 1 SID Base c. Trees A S A B 0 1 Investors(12) recommend Stocks(AB) B 0 1 A 1 1 B 0 0 aoa 1 0 0 1 1 3 PART Hyper. Graph H 2: On Days( ) aoa 1 0 0 1 aoa 1 0 aoa 1 2 I SDI Base c. Trees aoa DIS Base c. Trees 0 0 D 10 1 1 DSI Base c. Trees 0 0 1 1 1 0 0 0 1 1 1 2 1 2 0 1 1 1 1 0 SID NPZ p. Tree Stride=2 DSI Base c. Trees 1 2 1 0 0 0 1 1 A 0 B 1 1 1 0 0 1 1 1 1 1 DSI Base c. Trees 1 2 aao 1 0 0 1 A 0 B 1 1 0 1 1 1 1 2 A A B B 10 11 20 21 11 10 20 20 1 1 0 ISD NPZ p. Tree Stride=2 1 1 2 2 A 0 A 1 B 0 1 1 0 0 1

Maximal Base Clique. Trees for H 3 Stock Day Investor Base c. Trees A B C D E 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 1 0 0 0 0 1 1 1 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 1 2 3 4 5 1 1 0 1 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 0 1 1 1 1 0 1 0 0 0 0 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 0 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 3 5 2 1 3 2 3 3 Ct. I 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 4 3 3 2 2 5 3 2 4 2 4 3 1 3 2 4 3 3 1 1 1 1 3 2 1 2 3 2 2 5 3 2 4 2 1 1 3 4 2 3 3 2 4 2 1 1 1 3 5 2 3 1 3 2 4 3 1 1 5 4 2 1 3 3 4 4 5 1 5 3 2 3 3 2 1 2 3 2 2 5 3 2 4 2 3 2 2 3 4 2 3 2 4 oaa (all of these will be Max Cliques) aoa We can count the S=1 D=1 I=4 motifs? 6 + COMBO(5, 4)=5 = 11 113? 10+6 C(4, 3)+C(5, 3) = 54 112? 7+10 C 3, 2+6 C 4, 2+C 5, 2 = 83 Day Stock Investor Base c. Trees 1 1 1 4 2 1 3 3 4 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 1 1 1 A B C D E 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 1 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 1 0 0 1 1 0 0 1 0 0 0 1 1 5 1 4 5 1 5 3 1 1 2 3 3 4 3 2 2 1 1 1 5 1 4 5 2 5 3 3 4 2 3 3 4 3 2 2 1 1 2 3 4 5 Ct. I 1 1 1 1 1 1 1 1 1 1 1 1 1 3 2 4 4 3 3 2 2 4 5 4 4 4 3 3 3 2 2 1 3 Stock Investor Day Base c. Trees 1 1 1 3 4 2 3 2 4 1 1 1 3 1 5 3 3 2 aoa 3 3 2 3 4 2 3 2 4 3 2 2 3 1 5 3 3 2 oaa (all of these Max Cliques, only 3 new ones) A B C D E 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 1 1 0 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 2 3 4 5 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 1 0 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0 0 1 0 1 1 1 0 0 1 0 1 1 0 1 1 1 1 0 0 1 0 1 1 0 0 0 1 1 1 1 1 1 0 0 1 0 1 1 1 1 0 1 1 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 0 1 0 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 1 3 5 3 4 1 2 5 1 1 3 3 2 4 1 1 3 1 2 2 3 5 3 4 2 2 5 2 3 3 3 2 4 4 5 5 4 1 2 1 1 2 2 5 2 4 4 2 3 1 4 1 3 3 4 3 1 2 5 3 2 3 3 2 2 4 Ct. D 1 1 1 1 5 5 2 4 1 1 1 1 1 4 4 2 3 1 4 5 3 3 1 1 1 1 3 5 3 3 aoa 1 1 1 1 2 2 5 2 4 4 2 3 1 4 1 1 3 3 1 1 3 1 2 3 3 5 3 3 oaa (all of these Max Cliques, only 3 new ones)

Base Clique. Trees for H 3 last 3. Investor Stock Day c. Trees 1 2 3 4 5 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 1 A B C D E 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 1 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 1 1 1 0 0 1 1 1 0 1 1 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 0 0 1 0 1 1 1 0 0 1 1 0 1 0 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 5 4 5 3 3 5 4 5 5 5 2 2 3 3 3 4 2 3 3 4 1 1 1 2 3 4 5 4 4 1 1 1 4 4 5 3 5 4 1 5 2 1 3 3 1 1 1 2 5 4 4 2 3 1 1 1 2 3 4 5 4 4 1 1 1 4 4 5 3 5 4 2 5 2 2 3 3 2 2 5 4 4 2 3 5 1 1 aoa Day Investor Stock c. Trees 1 2 3 4 5 A B C D E oaa (all of these will be Max Cliques) 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 1 1 0 1 0 0 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0 1 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 1 0 0 0 1 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 5 5 1 1 1 1 4 4 1 1 3 1 1 2 1 3 2 4 5 3 4 1 1 3 2 1 5 1 4 5 1 1 5 3 1 5 1 1 4 4 1 2 3 3 4 2 3 2 1 3 2 4 5 3 4 2 2 3 2 1 5 1 4 5 1 2 5 3 1 5 3 5 1 1 4 4 3 2 3 1 1 1 1 4 5 3 4 1 1 1 1 5 5 oaa (all of these will be Max Cliques) aoa Investor Day Stock c. Trees 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 1 0 0 A B C D E 1 1 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 4 1 5 5 4 3 5 5 1 1 4 1 1 1 5 5 3 1 1 1 3 5 2 4 1 5 1 1 1 5 3 2 4 3 5 1 3 4 1 2 5 1 3 3 1 1 1 2 2 3 3 5 2 4 1 5 1 2 2 5 3 2 4 3 5 2 3 4 3 2 5 3 3 3 5 1 1 1 2 3 4 5 aoa 1 1 3 4 4 5 oaa (all of these will be Max Cliques)

Maximal Base Clique. Trees for H 3 A B C D E 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 2 3 4 5 1 1 5 4 2 1 3 3 4 4 5 1 5 3 2 3 3 2 1 2 3 2 2 5 3 2 4 2 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 1 0 1 1 1 1 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0 1 1 0 1 0 0 1 0 1 1 1 0 0 0 0 1 0 aoa then oaa on the 6 c. Trees (removing duplicates - no covers since aoa then oaa gives Maximal Cliques only). We get 34 MCs below. Theorem: These 34 MCs are the only Maxmal Cliques. General thm: {a. . ao(a. . oa(…oa. . a(B)|B=Base. Clique} is the Max. Clique. Set Thus, for a bipartite graph, ao(B) is MCS. 1 1 5 4 5 1 5 3 2 3 3 2 2 1 3 2 3 3 4 2 2 3 4 5 4 1 4 2 1 3 2 1 2 3 3 4 2 5 2 3 3 3 4 5 2 4 4 4 3 3 5 4 3 3 4 2 2 5 3 2 4 2 3 2 4 1 3 3 3 2 1 2 3 3 1 1 2 3 1 2 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 1 1 1 A B C D E 1 0 1 1 1 1 0 0 0 1 1 1 1 1 0 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 0 1 0 0 1 0 0 0 1 1 5 1 4 5 2 5 3 3 4 2 3 3 4 3 2 2 1 3 2 2 3 1 5 3 3 2 1 0 0 3 2 2 1 3 2 3 4 2 5 2 3 3 2 4 1 3 3 1 0 1 1 0 3 3 2 3 4 2 3 2 4 1 0 0 1 2 3 4 5 A B C D E 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 1 A B C D E 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 1 0 1 1 1 1 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 A B C D E 1 0 0 1 1 0 1 1 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 0 0 1 1 0 0 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 1 1 0 0 1 1 1 0 0 1 1 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0 1 1 0 1 0 0 1 0 1 1 1 0 0 0 0 1 0 1 0 1 1 1 4 4 5 3 5 4 2 5 2 2 3 3 2 2 5 4 4 2 3 5 1 1 1 2 3 4 5 2 1 1 2 3 4 5 4 4 1 2 3 4 5 1 1 5 4 5 1 5 3 2 3 3 2 2 1 3 2 3 3 4 2 2 3 4 5 4 1 4 2 3 4 5 4 4 2 1 3 2 1 2 3 3 4 2 5 2 3 3 3 4 5 2 4 4 4 3 3 5 4 4 3 2 3 3 3 4 2 2 5 3 2 4 2 3 2 4 1 3 3 3 2 1 2 3 3 1 1 2 3 1 2 1 1 2 2 A B C D E 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 A B C D E 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 1 0 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 1 0 1 1 1 0 0 0 0 1 0 1 0 1 2 3 4 5 1 1 5 4 5 1 5 3 2 3 3 2 2 1 3 2 3 3 4 4 2 1 3 2 1 2 3 3 4 2 5 2 3 3 3 4 2 2 5 3 2 4 2 3 2 4 1 3 3 3 2 1 1 1 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 0 0 1 1 1 0 0 A B C D E 1 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 0 1 0 0 0 1 2 2 5 3 2 4 3 5 2 3 4 3 2 5 3 3 3 5 1 1 1 2 3 4 5 1 1 5 4 5 1 5 3 2 3 3 2 2 1 3 2 3 3 4 2 2 3 4 5 4 1 4 2 3 4 5 4 4 2 1 3 2 1 2 3 3 4 2 5 2 3 3 3 4 5 2 4 4 4 3 3 5 4 4 3 2 3 3 3 4 2 2 5 3 2 4 2 3 2 4 1 3 3 3 2 1 2 3 3 1 1 2 3 1 2 1 1 2 2 A B C D E 1 0 1 0 1 1 1 1 0 1 1 0 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 2 3 4 5 1 1 0 0 0 1 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 2 2 5 2 3 3 3 2 4 4 5 5 4 1 2 1 1 2 2 5 2 4 4 2 3 1 4 1 3 3 4 3 1 2 5 3 2 2 3 4 1 2 3 4 5 A B C D E 2 2 3 3 5 2 4 1 5 1 1 3 4 4 5 1 0 1 1 1 1 0 1 0 0 1 1 0 0 0 0 1 1 1 1 1 0 0 1 1 1 A B C D E 1 0 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 1 0 1 1 1 1 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 1 1 0 0 0 1 1 0 0 0 0 1 1 1 0 0 1 0 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 0 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 1 0 1 0 0 0 1 1 1 1 1 0 0 1 1 1 0 0 1 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 1 1 1 0 0 1 1 0 1 0 0 1 0 1 1 1 0 0 0 0 1 0 1 0 1 0 0 1 2 2 3 2 1 5 1 4 5 1 2 5 3 1 5 3 5 1 1 4 4 3 2 3 3 4 2 3 2 1 3 2 4 5 3 4 1 2 3 4 5 1 1 5 4 5 1 5 3 2 3 3 2 2 1 3 2 3 3 4 2 2 3 4 5 4 1 4 2 3 4 5 4 23 4 2 1 3 2 1 2 3 3 4 2 5 2 3 3 3 4 5 2 4 4 4 3 3 5 4 4 3 2 3 53 3 3 4 2 2 5 3 2 4 2 3 2 4 1 3 3 3 2 1 2 3 3 1 1 2 3 1 2 1 1 2 2 13

DOSPPE: Measure between-ness using One Shortest Path per Edge. Delete edges with highest betweenness until a Connected Component peals off, recalculate SP Counts, delete edges with highest, etc 2 3 4 5 6 7 8 9 a b c d e f g h i j k l m n o p q r s t u v w x y 13 SPPCs 24 14 12 6 7 31 0 0 0 0 31 0 0 0 3 2 17 9 15 2 0 0 16 0 9 0 0 0 21 0 0 0 31 0 0 0 2 3 34 0 0 0 33 0 0 2 0 0 16 5 15 10 0 0 0 0 32 31 3 0 0 0 0 0 18 17 0 0 0 0 0 0 0 0 0 29 0 0 0 21 0 0 0 0 0 16 0 0 21 0 0 0 0 26 0 0 0 0 0 1 2 3 4 5 6 0 0 0 0 0 3 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 11 0 0 16 14 8 9 a SPPC after deleting 5 6 7 11 17 10 12 13 1 2 3 4 8 9 a c d 1 2 13 3 24 14 4 12 6 7 8 17 9 15 2 9 16 0 9 0 0 a 0 0 21 0 0 0 c 34 0 0 0 d 31 0 0 2 0 0 e 16 5 15 10 0 0 f 0 0 0 0 0 g 0 0 0 0 0 i 31 3 0 0 0 0 j 0 0 0 0 0 k 18 3 0 0 0 0 l 0 0 0 0 0 m 18 17 0 0 0 0 n 0 0 0 0 0 o 0 0 0 0 0 p 0 0 0 0 0 q 0 0 0 0 0 r 0 0 0 0 0 s 0 0 29 0 0 0 t 0 0 21 0 0 0 u 0 0 0 0 0 v 0 16 0 0 0 2 0 0 0 w 21 0 0 0 0 x 0 0 26 0 0 11 0 0 0 y 0 0 0 16 14 0 0 0 0 0 0 0 b 0 0 0 0 0 0 c 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 26 26 3 9 9 e f g 0 0 0 0 0 h Max SPPC=34 is at edge {1, 12}DNI Next Max SPPC=33 is at edge {1, 13} Next Max SPPC=32 is at edge {6, 17} Next Max SPPC=31 is at edges {1, 5} {5, 6, 7, 11, 17}=CC. Delete. Recalc. Start over 0 0 0 0 0 0 0 0 0 0 0 0 0 26 0 25 0 9 15 9 i j k l 0 0 0 0 16 4 0 0 0 0 19 6 0 0 0 0 0 5 0 0 0 27 21 0 0 26 10 0 0 9 7 0 0 29 m n o p q r 0 0 0 0 4 0 0 25 10 4 12 7 8 s t u v 8 4 w 2 x 14 15 16 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 e f g i j k l m n o p q r s t u v w x 0 0 0 0 0 0 0 0 0 0 0 0 0 26 26 3 9 9 0 0 0 0 0 0 0 0 0 0 0 0 0 26 0 25 0 9 15 9 0 0 0 0 16 4 0 0 0 0 19 6 0 0 0 0 0 5 0 0 0 27 21 0 0 26 10 0 0 9 7 0 0 29 0 0 0 0 4 0 0 25 10 4 12 7 8 1 2 3 8 4 4 2 5 6 7 Next Max SPPC=13 is at edges {1, 2} {1, 4} {29, 34} Max SPPC=34 is at edge {1, 12}DNI 8 9 Next Max SPPC=31 is edges {1, 18} {1, 13}DNI 10 Next Max SPPC=11 is at edges {9, 33} Next Max SPPC=29 is edges {3, 29} {27, 34} Next Max SPPC=10 is at edges {4, 14} {24, 33} {31, 33} 11 12 13 Next Max SPPC=27 is at edge {25, 32} Next Max SPPC=9 edges {2, 8} {3, 9} 14 15 Next Max SPPC=26 is at edges {3, 33} {15, 33} DNI{15, 34} {16, 34} {19, 34} {23, 34} 16 {16, 33} {19, 33} {23, 33} 17 Next Max SPPC=8 edges {31, 34}DNI {32, 33} 18 19 Next Max SPPC=25 is at edges {21, 33} {30, 33} Next Max SPPC=7 edges {3, 4}{30, 34}DNI {24, 34} 20 Next Max SPPC=6 edges {25, 28}DNI {2, 14} Next Max SPPC=24 is at edges {1, 3} {2, 3} 21 22 Next Max SPPC=21 is at edges {3, 10} {1, 32} {1, 2, 3, 4, 8, 12, 13, 18, 20, 22}=CC. 23 24 {26, 32} {24, 28} {3, 28} Delete. Recalc. Start over 25 26 Next Max SPPC=18 is at edges {1, 20} {1, 22} 27 28 Next Max SPPC=17 is at edges {1, 8} {2, 22} 29 30 Next Max SPPC=16 is at edges {1, 9} {1, 14} {24, 26} {2, 31} {9, 34} 31 32 33 Next Max SPPC=15 is at edges {3, 8} {3, 14} {10, 34}DNI {20, 34} 34 a b c d e f g h i j k l m n o p q r s t u v w x y 1 0 1 1 1 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 2 1 0 1 1 0 0 0 0 0 1 0 0 0 3 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 0 4 1 1 1 0 0 0 0 0 0 0 56 11 00 00 00 00 00 01 00 00 00 00 00 7 1 0 0 0 1 1 0 0 0 0 0 0 0 8 1 1 0 0 0 0 0 0 0 0 9 1 0 0 0 0 0 0 0 1 0 1 1 a 0 0 1 0 0 0 0 0 0 0 0 b 1 0 0 0 1 1 0 0 0 0 0 0 0 c 1 0 0 0 0 0 0 0 0 d 1 0 0 0 0 0 0 0 0 e 1 1 0 0 0 0 0 0 0 0 1 f 0 0 0 0 0 0 0 0 1 g 0 0 0 0 0 0 0 0 1 h 0 0 0 1 1 0 0 0 0 0 0 0 i 1 1 0 0 0 0 0 0 0 0 j 0 0 0 0 0 0 0 0 1 k 1 1 0 0 0 0 0 0 0 0 l 0 0 0 0 0 0 0 0 1 m 1 1 0 0 0 0 0 0 0 0 n 0 0 0 0 0 0 0 0 1 o 0 0 0 0 0 0 0 1 0 1 0 0 1 p 0 0 0 0 0 0 0 1 0 0 0 1 0 q 0 0 0 0 0 0 1 1 0 0 0 1 0 r 0 0 0 0 0 0 0 0 1 0 0 0 s 0 0 1 0 0 0 0 0 1 1 0 0 0 0 t 0 0 1 0 0 0 0 0 0 0 1 0 u 0 0 0 0 0 0 1 0 0 0 1 v 0 1 0 0 0 0 0 0 0 1 w 1 0 0 0 0 0 0 1 1 0 0 0 1 x 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 1 0 y 0 0 0 0 1 1 1 0 1 1 0 0 1 1 1 0

DOSPPE: Measure between-ness using One Shortest Path per Edge. Delete edges with highest betweenness until a Connected Component peals off, recalc, del next highest, etc SPPC after deleting 5 6 7 11 17 1 2 3 4 8 9 a c d e f g i j k l m n o p q r s t u v w x y 1 2 3 4 13 24 14 12 6 7 17 9 15 2 16 0 9 0 0 0 21 0 34 0 0 0 31 0 0 2 16 5 15 10 0 0 0 0 31 3 0 0 0 18 17 0 0 0 0 0 0 29 0 0 0 21 0 0 0 16 0 0 21 0 0 0 26 0 0 0 8 9 10 12 13 14 15 16 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 a c d e f g i j k l m n o p q r s t u v w x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 11 0 0 16 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 26 26 3 9 9 0 0 0 0 0 0 0 0 0 0 0 0 0 26 0 25 0 9 15 9 0 0 0 0 16 4 0 0 0 0 19 6 0 0 0 0 0 5 0 0 0 27 21 0 0 26 10 0 0 9 7 0 0 29 Max SPPC=29 is at edge {27, 34} Max SPPC=27 is at edge {15, 33} {16, 33} {19, 33} {23, 33} Max SPPC=25 is at edge {21, 33} {30, 33} Max SPPC=21 is at edge {21, 33} {30, 33} Max SPPC=19 is at edge {24, 28} Max SPPC=16 is at edge {24, 26} {9, 34} Max SPPC=14 is at edge {10, 34}DNI (but del would remove only green) Max SPPC=12 is at edge {29, 34} Max SPPC=11 is at edge {9, 33} Max SPPC=10 is at edge {24, 33} Max SPPC=9 is at edge {15, 33} {16, 33} {19, 33} {21, 33} {23, 33} all DNI Max SPPC=8 is at edge {32, 33} {31, 34}DNI Max SPPC=7 is at edge {24, 34} {30, 34}DNI Max SPPC=6 is at edge {25, 26}DNI Max SPPC=5 is at edge {24, 30}DNI Max SPPC=4 is at edge {28, 34} {29, 32}DNI 0 0 0 0 4 0 0 25 10 4 12 7 8 8 4 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 a b c d e f g h i j k l m n o p q r s t u v w x y 1 0 1 1 1 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 2 1 0 1 1 0 0 0 0 0 1 0 0 0 3 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 0 4 1 1 1 0 0 0 0 0 0 0 56 11 00 00 00 00 00 01 00 00 00 00 00 7 1 0 0 0 1 1 0 0 0 0 0 0 0 8 1 1 0 0 0 0 0 0 0 0 9 1 0 0 0 0 0 0 0 1 0 1 1 a 0 0 1 0 0 0 0 0 0 0 0 b 1 0 0 0 1 1 0 0 0 0 0 0 0 c 1 0 0 0 0 0 0 0 0 d 1 0 0 0 0 0 0 0 0 e 1 1 0 0 0 0 0 0 0 0 1 f 0 0 0 0 0 0 0 0 1 g 0 0 0 0 0 0 0 0 1 h 0 0 0 1 1 0 0 0 0 0 0 0 i 1 1 0 0 0 0 0 0 0 0 j 0 0 0 0 0 0 0 0 1 k 1 1 0 0 0 0 0 0 0 0 l 0 0 0 0 0 0 0 0 1 m 1 1 0 0 0 0 0 0 0 0 n 0 0 0 0 0 0 0 0 1 o 0 0 0 0 0 0 0 1 0 1 0 0 1 p 0 0 0 0 0 0 0 1 0 0 0 1 0 q 0 0 0 0 0 0 1 1 0 0 0 1 0 r 0 0 0 0 0 0 0 0 1 0 0 0 s 0 0 1 0 0 0 0 0 1 1 0 0 0 0 t 0 0 1 0 0 0 0 0 0 0 1 0 u 0 0 0 0 0 0 1 0 0 0 1 v 0 1 0 0 0 0 0 0 0 1 w 1 0 0 0 0 0 0 1 1 0 0 0 1 x 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 1 0 y 0 0 0 0 1 1 1 0 1 1 0 0 1 1 1 0

DOSPPE: Measure between-ness using One Shortest Path per Edge. Delete edges with highest betweenness until a Connected Component peals off, recalc, del next highest, etc 2 3 4 5 6 7 8 9 a b c d e f g h i j k l m n o p q r s t u v w x y 13 SPPCs 24 14 12 6 7 31 0 0 0 0 31 0 0 0 3 2 17 9 15 2 0 0 16 0 9 0 0 0 21 0 0 0 31 0 0 0 2 3 34 0 0 0 33 0 0 2 0 0 16 5 15 10 0 0 0 0 32 31 3 0 0 0 0 0 18 17 0 0 0 0 0 0 0 0 0 29 0 0 0 21 0 0 0 0 0 16 0 0 21 0 0 0 0 26 0 0 0 0 0 1 2 3 4 5 6 0 0 0 0 0 3 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 11 0 0 16 14 8 9 a SPPC after deleting 5 6 7 11 17 10 12 13 1 2 3 4 8 9 a c d 1 2 13 3 24 14 4 12 6 7 8 17 9 15 2 9 16 0 9 0 0 a 0 0 21 0 0 0 c 34 0 0 0 d 31 0 0 2 0 0 e 16 5 15 10 0 0 f 0 0 0 0 0 g 0 0 0 0 0 i 31 3 0 0 0 0 j 0 0 0 0 0 k 18 3 0 0 0 0 l 0 0 0 0 0 m 18 17 0 0 0 0 n 0 0 0 0 0 o 0 0 0 0 0 p 0 0 0 0 0 q 0 0 0 0 0 r 0 0 0 0 0 s 0 0 29 0 0 0 t 0 0 21 0 0 0 u 0 0 0 0 0 v 0 16 0 0 0 2 0 0 0 w 21 0 0 0 0 x 0 0 26 0 0 11 0 0 0 y 0 0 0 16 14 0 0 0 0 0 0 0 b 0 0 0 0 0 0 c 0 0 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 26 26 3 9 9 e f g 0 0 0 0 0 h Max SPPC=34 is at edge {1, 12}DNI Next Max SPPC=33 is at edge {1, 13} Next Max SPPC=32 is at edge {6, 17} Next Max SPPC=31 is at edges {1, 5} {5, 6, 7, 11, 17}=CC. Delete. Recalc. Start over 0 0 0 0 0 0 0 0 0 0 0 0 0 26 0 25 0 9 15 9 i j k l 0 0 0 0 16 4 0 0 0 0 19 6 0 0 0 0 0 5 0 0 0 27 21 0 0 26 10 0 0 9 7 0 0 29 m n o p q r 0 0 0 0 4 0 0 25 10 4 12 7 8 s t u v 8 4 w 2 x 14 15 16 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 e f g i j k l m n o p q r s t u v w x 0 0 0 0 0 0 0 0 0 0 0 0 0 26 26 3 9 9 0 0 0 0 0 0 0 0 0 0 0 0 0 26 0 25 0 9 15 9 0 0 0 0 16 4 0 0 0 0 19 6 0 0 0 0 0 5 0 0 0 27 21 0 0 26 10 0 0 9 7 0 0 29 0 0 0 0 4 0 0 25 10 4 12 7 8 1 2 3 8 4 4 2 5 6 7 Next Max SPPC=13 is at edges {1, 2} {1, 4} {29, 34} Max SPPC=34 is at edge {1, 12}DNI 8 9 Next Max SPPC=31 is edges {1, 18} {1, 13}DNI 10 Next Max SPPC=11 is at edges {9, 33} Next Max SPPC=29 is edges {3, 29} {27, 34} Next Max SPPC=10 is at edges {4, 14} {24, 33} {31, 33} 11 12 13 Next Max SPPC=27 is at edge {25, 32} Next Max SPPC=9 edges {2, 8} {3, 9} 14 15 Next Max SPPC=26 is at edges {3, 33} {15, 33} DNI{15, 34} {16, 34} {19, 34} {23, 34} 16 {16, 33} {19, 33} {23, 33} 17 Next Max SPPC=8 edges {31, 34}DNI {32, 33} 18 19 Next Max SPPC=25 is at edges {21, 33} {30, 33} Next Max SPPC=7 edges {3, 4}{30, 34}DNI {24, 34} 20 Next Max SPPC=6 edges {25, 28}DNI {2, 14} Next Max SPPC=24 is at edges {1, 3} {2, 3} 21 22 Next Max SPPC=21 is at edges {3, 10} {1, 32} {1, 2, 3, 4, 8, 12, 13, 18, 20, 22}=CC. 23 24 {26, 32} {24, 28} {3, 28} Delete. Recalc. Start over 25 26 Next Max SPPC=18 is at edges {1, 20} {1, 22} 27 28 Next Max SPPC=17 is at edges {1, 8} {2, 22} 29 30 Next Max SPPC=16 is at edges {1, 9} {1, 14} {24, 26} {2, 31} {9, 34} 31 32 33 Next Max SPPC=15 is at edges {3, 8} {3, 14} {10, 34}DNI {20, 34} 34 a b c d e f g h i j k l m n o p q r s t u v w x y 1 0 1 1 1 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 2 1 0 1 1 0 0 0 0 0 1 0 0 0 3 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 0 4 1 1 1 0 0 0 0 0 0 0 56 11 00 00 00 00 00 01 00 00 00 00 00 7 1 0 0 0 1 1 0 0 0 0 0 0 0 8 1 1 0 0 0 0 0 0 0 0 9 1 0 0 0 0 0 0 0 1 0 1 1 a 0 0 1 0 0 0 0 0 0 0 0 b 1 0 0 0 1 1 0 0 0 0 0 0 0 c 1 0 0 0 0 0 0 0 0 d 1 0 0 0 0 0 0 0 0 e 1 1 0 0 0 0 0 0 0 0 1 f 0 0 0 0 0 0 0 0 1 g 0 0 0 0 0 0 0 0 1 h 0 0 0 1 1 0 0 0 0 0 0 0 i 1 1 0 0 0 0 0 0 0 0 j 0 0 0 0 0 0 0 0 1 k 1 1 0 0 0 0 0 0 0 0 l 0 0 0 0 0 0 0 0 1 m 1 1 0 0 0 0 0 0 0 0 n 0 0 0 0 0 0 0 0 1 o 0 0 0 0 0 0 0 1 0 1 0 0 1 p 0 0 0 0 0 0 0 1 0 0 0 1 0 q 0 0 0 0 0 0 1 1 0 0 0 1 0 r 0 0 0 0 0 0 0 0 1 0 0 0 s 0 0 1 0 0 0 0 0 1 1 0 0 0 0 t 0 0 1 0 0 0 0 0 0 0 1 0 u 0 0 0 0 0 0 1 0 0 0 1 v 0 1 0 0 0 0 0 0 0 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 a b c d e f g h i j k l m n o p q r s t u v w x y w 1 0 0 0 0 0 0 1 1 0 0 0 1 y 0 0 0 0 1 1 1 0 1 1 0 0 1 1 1 0 x 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 1 0