Semirings for Social Networks Analysis Vladimir Batagelj Department of Mathematics, University of Ljubljana Jadranska 19, 61 111 Ljubljana, Slovenia E–mail: VLADIMIR.BATAGELJ@UNI-LJ.SI Abstract In the paper four semirings for solving social networks problems are constructed. The closures of the matrix of a given signed graph over balance and cluster semirings can be used to decide whether the graph is balanced or clusterable. The closure of relational matrix over geodetic semiring contains for every pair of vertices ✁ and ✂ the length and the number of ✁ -✂ geodesics; and for geosetic semiring the length and the set of vertices on ✁ -✂ geodesics. The algorithms for computing the geodetic and the geosetic closure matrix are also given. Key words : balanced signed graphs, clusterable signed graphs, closed semirings, closure, geodesic, Freeman’s centrality indices, Boyle’s operation. Math. Subj. Class. (1991) : 05 C 50, 05 C 12, 05 C 75, 16 Y 60, 68 R 10, 92 H 30 There ✄✆are several applications of semirings in social networks analysis. For example, the ☎✞✝✠✟☛✡☞✟✍✌✎✟✑✏✓✒✔✟✖✕✘✗✚✙✛✟✖✕✢✜✤✣✠✥ can be applied to determine the connectedness matrix [19, p. semiring ✄✆☎✞✝✠✟✧✦★✟✧✩✪✒✔✟✖✕✘✗✚✙✫✟✖✕✢✜✤✣✠✥✬✟✭✝✯✮✰✦✱✮✰✩ ✦ ✩ 133]. Similar semiring ( 0 – no link, – uniplex links, – multiplex links) can be used to analyse the connectivity of social networks [12]. In this paper we shall construct four new semirings for solving social networks analysis problems: ✲ balance and cluster semirings which can be used to decide whether the graph is balanced or clusterable; ✦ ✲ ✴ ✲ geodetic semiring to determine for every pair of vertices and ✳ the length and the number ✦ of -✳ geodesics; geosetic semiring for computing Boyle’s operation – to determine for every pair of ver✦ ✦ tices and ✳ the length and the set of vertices on -✳ geodesics. Supported in part by the Ministry of Science and Technology of Slovenia. 1 1 Semirings An algebraic structure ✄✶✵✷✟✍✸✹✟✻✺✤✟✑✝✎✟✻✡✚✥ ✵ on the set ✲ ✄✶✵✷✟✍✸✹✟✑✝✼✥ ✲ ✄✶✵✷✟✻✺✤✟☛✡✚✥ ✲ multiplication distributes over addition ✝ is an abelian monoid with neutral element (zero); ✡ is a monoid with neutral element (unit); ✺ ✺✓✄✶✾❂✸❃✿❄✥❆❅ ✽ is a semiring [1, 5, 6, 22] iff: ✽ ✺✚✾❇✸ ✺✚✿ ✽ ✗❈✣❊❉ ✸ ✄ ✽ ; for all ✽ ✟✖✾✚✟✖✿❁❀✪✵ ✸❋✾❄✥❇✺✚✿●❅ ✺❍✿❆✸❋✾■✺✚✿ ✽ In the expressions we assume precedence of multiplication over addition. ✄❏✵❑✟✍✸✹✟✻✺▲✟✑✝✠✟✻✡❍✥ is complete iff the addition is well-defined also for countable sets The semiring of elements and the (generalized) commutativity, associativity (for addition) and distributivity hold also in this case. ❀▼✵ ✸ ❅ ✵ If the addition is idempotent, for every ✽ :✽ ✽ ✽ , the semiring over a finite set is complete. ✄✶✵❑✟✑✸✯✟☛✺✤✟❖◆P✟✑✝✠✟✻✡❍✥ ◆ The ❀✪ semiring is closed iff for the (unary) closure operation it holds for ✵ every ✽ ◆ ❅◗✡■✸ ◆ ✺ ✺ ◆ ❅◗✡■✸ ✽ ✽ ✽ ✽ ✽ There can exist different closures over the same semiring. The complete semiring is always closed for the closure defined by ◆ ❅ ✽ ❚ ❙❘ ✽ ❚✖❯❊❱ In the sequel we shall refer by the term closure to the operation defined by this expression. In a closed semiring we can define also a strict closure ✽ by Suppose that for a given graph ❲ value function ❅ ✺ ◆ ✽ ✽ ✽ ❅❳✄❩❨❆✟✑❬❭✥✍✟●❬❫❪❴❨❴❵❛❨ and a semiring ✄✶✵❑✟✑✸✯✟☛✺✤✟✑✝✠✟☛✡✚✥ a ❜✢❝ ❬❡❞❢✵ is given. ❅ ❱ ✟ ✳✼❤ ✟ ✳✞✐ ✟✻❥✻❥✻❥❄✟ ✳✬❦❍❧✛❤ ✟ ✳✬❦ is a walk of length ♠ on ❲ iff ❣ ✳ A finite sequence of vertices ❬ ✟✖♦P❅◗✡☞✟✻❥✻❥☛❥❄✟ ❬ ❬ ✟✖♦P❅s✡☞✟☛❥✻❥✻❥☛✟ on ❲ iff ✳♣♥✤❧✛❤ ✳✚♥✔qr✳✚♥ ✳✞♥▲❧✛❤ ✳✞♥✤❧✛❤ ✳✞♥ ♠ ; and is a semiwalk or chain ♠ . The ❅ ❱ (semi)walk is closed iff its ends coincide, ✳ ✳❄❦ . A walk in which no vertex appears twice is called a path; and if only its ends coincide it is called a cycle. ❜ We can extend the value function to walks and sets of walks on ❲ by ✲ let t✚✉ be a null walk in the vertex ✳ ✲ ✲ ❅ ✟ ✟ ✟✻❥✻❥✻❥❄✟ ❀✈❨ then ✟ ❜ ✄ t♣✉ ✥❆❅◗✡ ✳ ❤ ✳♣✐ ✳❄❦✻❧✛❤ ❄✳ ❦ be a walk of length ♠①✇ let ❣ ❜ ✳ ❱ ✔ ✄ ✭ ✥ ❅ ❜ ✄ ❱ ✟ ✥❇✺ ❜ ✄ ✟ ✥②❥☛❥✻❥ ❜ ✄ ✟ ✥ ❣ ✳ ✔ ✳ ❤ ✳✔❤ ♣ ✳ ✐ ✳✬❦✻❧✛❤ ✳✬❦ for the empty set of walks ③ we have ❜ ✄ ✭ ✥ ❅④✝ ③ 2 ✡ on ❲ then ✲ ❅s☎ let ⑤ ❜ ✄ ✟ ✻✟ ❥✻❥✻❥⑦✒ be a set of walks on ❲ ❣ ❤ ✫ ⑥ ❣ ✐ ✥✭❅ ❜ ✄ ⑧ ✥ ✸ ❜ ✄ ✥⑧✸⑨❥✻❥☛❥ ⑤ ❣⑥❤ ❣ ✐ ✫ then ✦ ❦ ◆ ⑩ We denote by ⑤ ⑩ ✉ the ✦ set of all walks of length ♠ from vertex to vertex ✳ ; by ⑤ ✉ the set of all ✳ ; and by ⑤ ⑩ ✉ the set of all nontrivial (different from t ⑩ ) walks walks from vertex to vertex ✦ from vertex to vertex ✳ . The value matrix of a graph is a matrix ❶ defined by ❶▼❷ ✦❸✟ ❜ ❼✄ ✄✶✦❸✟ ✥✧✥ ✳ ✝ ❅❻❺ ✳✼❹ ✄✶✦❸✟ ✥■❀✈❬ ❽❈❾✖❿❊✳ ➀❄➁❼➂●✜▲➃✧➀ ❀✪✵ In the following we shall assume that in the semiring for every ✽ ✺✚✝✷❅➄✝❭✺ ✽ ✽ ❅④✝ ❨s❅◗☎ ✟ ✟✻❥☛❥✻❥❄✟ ◆ ❅ ❚ ❙❘ ❶ ❚✖❯❊❱ ✒ holds and that the set of vertices is finite ✳✎❤ ✳♣✐ ✳✞➅ . Then the addition and multiplication can be extended in the usual way to square matrices of order ➆ which themselves form a semiring. The matrix semiring over complete semiring is also complete and therefore closed for ❶ There are two well known theorems [1, 6, 22] connecting values of walks in graphs and their matrices: THEOREM 1. Let ❶ THEOREM 2. Let ❶ closure matrix then ❦ be the ♠ -th power of value matrix ❶ ❜ ✄ ❦ ✥✭❅ ⑤ ⑩ ✉ ❶ ✦ ✟ ❦ ❸ ❷ ✳✼❹ be a value matrix over complete semiring, ❶ ❜ ✄ ◆ ✥➇❅ ⑤ ⑩ ✉ ❶ ◆ ❸ ✦ ✟ ❷ ✳✼❹ and ◆ ❜ ✄ To compute the closure matrix ❶ of a given matrix ❶ we can use the Fletcher’s algorithm [14]: ➊ ❱ then ❝❅ ⑤ ⑩ ✉ ✥❆❅ ❶➈❷ ◆ its closure and ❶ ✦❸✟ ✳✼❹ over a complete semiring ❝ ❅s ❶ ✡; for ➋ ♦ ❝ ❅s to✡ ➆ do begin ❝ s ❅ ✡ for ✿ ♦✧✟ to to♦❼➆ ✟ do✺✓✶✄ ✿ ❝ ❅④ ➆ ✿ do for ➌ ✟ ✥➎◆➏✺✚✿ ❚ ✟ ❚ ❷ ❝ ➌☞❹ ❚ ❧✛❤✬❷ ♦✧✟ ➌☞❹ ✸❋✿ ❚ ✛ ❚ ✛ ❧ ❤✍❷ ➋✓❹ ❧ ❤✍❷➍➋ ➋✓❹ ❧✛❤✬❷➍➋ ➌✼❹ ; ✿ ❚ ✟ ❅◗✡■✸❃✿ ❚ ✟ ❷➉➋ ➋✓❹ ❷➉➋ ➋✔❹ ; end; ◆ ❝❅ ➊ ❶ ➅ ; 3 its strict ✄❏✵❑✟✍✸✹✟✻✺▲✟✑✝✠✟✻✡❈✟➉◆②✥ ✿ ❝ ✟ ❅❫✡➐✸➄✿ ✟ ❚ ❷➉➋ ➋✔❹ we obtain the algorithm for computing the If we delete the statement ❚ ❷➍➋ ➋✓❹ strict closure ❶ . If the addition is idempotent we can by this algorithm compute the closure matrix in place ➊ – we omit the subscripts in matrices . There is a bijection between semiwalks on graph ❲ and walks in the graph ❲ ➑ which is the sum of the graph ❲ and its inverse – graph ❲ with reversed directions of its arcs. We have ✄ ❶ ✥✭❅ ❲➑ ✄ ❶ ❲ ✥⑧✸ ❶ ✄ ❲ ✥➓➒ ➒ where ❶ denotes the transpose of matrix ❶ . Let us define the symmetric closure of the value matrix by ❅→✄ ✸ ➒②✥ ◆ ❶①➔ ❶ ❶ 2 Balance and cluster semirings 2.1 Balanced and clusterable signed graphs ✄ ✟✖➣⑧✥ A signed graph is an ordered pair ❲ ❅ ✄↔❨✭✟✑❬❭✥ ❲ ❬◗❪⑨❨↕❵➙❨ ✲ ➣ ✲ ❝ ❬④❞ ☎ ♠ where is a directed graph (without loops) with set of vertices ❨ and set of arcs ✟ ; ➆ ✒ is a sign function. The arcs with the sign ♠ are positive and the arcs with ❬✷➛ the sign ➆ are negative. We denote the set of all positive arcs by and the set of all ❬ ❧ negative arcs by . The case when the graph is undirected can be reduced to the case of directed graph by replacing each edge ➜ by a pair of opposite arcs both signed with the sign of the edge ➜ . The signed graphs were introduced in [18] and later studied✄ by✟✖several authors [7, 8, 10, 11, ➣❸✥ 19, 20, 21]. Following Roberts [21, pp. 75–77] a signed graph ❲ is: ❨ ✲ ✲ balanced iff the set of vertices can be partitioned into two subsets so that every positive arc joins vertices of the same subset and every negative arc joins vertices of different subsets. ❨ clusterable iff the set of can be partitioned into subsets, called clusters, so that every positive arc joins vertices of the same subset and every negative arc joins vertices of different subsets. The (semi)walk on the signed graph is positive iff it contains an even number of negative arcs; otherwise it is negative. The balanced and clusterable signed graphs are characterised by the following theorems [9, 10, 19, 20, 21]. ✄ THEOREM 3. A signed graph ❲ ✄ ✟✖➣❸✥ THEOREM 4. A signed graph ❲ exactly one negative arc. ✟✧➣❸✥ is balanced iff every closed semiwalk is positive. is clusterable iff ❲ 4 contains no closed semiwalk with Table 1: Balance semiring ✸ ✺ 0 0 0 ➆ ➆ ♠ ➆ ♠ ✽ ➆ ♠ ✽ ➆ ✽ ✽ 0 ✽ ♠ ✽ ✽ ✽ ➆ ♠ ✽ ✽ ◆ 0 0 0 0 0 ♠ ✽ ✽ ➝ ➆ ♠ 0 0 ♠ 0 ➆ ➆ ➝ ✽ 0 ✽ ➆ ♠ ✽ ✽ ✽ ♠ ✽ ♠ ✽ ♠ ✽ ✽ Table 2: Cluster semiring ✸ ✺ 0 0 0 ➆ ➆ ♠ ➆ ♠ ✽ ➞ ➆ ♠ ✽ ➞ ➆ ✽ ♠ ✽ ✽ ✽ ➞ ✽ ➆ ♠ 0 ➆ ✽ ✽ ➞ ✽ ♠ ➆ ♠ ✽ ♠ ✽ ✽ ✽ ◆ 0 0 0 0 0 0 ➞ ➞ ➝ ➆ ♠ ✽ 0 ➞ 0 ➆ ➆ 0 0 ➞ ✽ ✽ ➞ 0 ➆ ♠ ➆ ➝ ➞ ➆ ➞ ✽ ➞ ✽ ♠ ♠ ✽ ➞ ➞ ♠ ➞ ✽ ➞ ♠ 2.2 Balance and cluster semirings ✵ To construct a semiring corresponding to the balance problem we take the set elements [11, 19]: with four ✝ no walk; all walks are negative; all walks are positive; at least one positive and at least one negative walk. ➆ ♠ ✽ Now it is easy to produce the Cayley tables for balance semiring (see Table 1). The balance semiring is idempotent closed semiring with zero and unit . For construction of the cluster semiring corresponding to the clusterability problem we need the set with five elements: ✝ ♠ ✵ ✝ ➆ ♠ ✽ ➞ no walk; at least one walk with exactly one negative arc; no walk with only positive arcs; at least one walk with only positive arcs; no walk with exactly one negative arc; at least one walk with only positive arcs; at least one walk with exactly one negative arc; each walk has at least two negative arcs. The Cayley tables for cluster semiring are given in Table 2. The cluster semiring is idempotent closed semiring with zero and unit . ✝ ♠ Combining the Theorem 2 with Theorems 3 and 4 we get. 5 ➟8 . . .. .... . . . . 9 ➟ .... ➟7 ... . . . .... ...... . . . . . . ... ..... .......➟ . . ... 6 ... ... ... . . ... ... ... ... . . . ... . . . ➟ 5 ... .. 1 ➟ .. . . . ... ...... .... . . . .... ... ... .... .... .... . .. ➟ ... ➟ ➟ ...... 2 3 4 1 2 3 4 5 6 7 8 9 1 0 2 ➆ 0 0 ♠ 0 ♠ 0 0 0 ♠ 0 ♠ ♠ 0 0 0 3 0 ➆ ➆ 0 0 4 0 0 5 0 0 ♠ 0 0 0 0 0 0 6 0 0 0 0 ♠ ➆ 0 ➆ 0 ➆ 0 0 7 0 ➆ 0 ➆ 0 ♠ 0 ♠ 0 0 0 8 0 0 0 0 0 0 ➆ 0 ➆ 0 0 0 0 0 ♠ 0 ♠ 0 ♠ 0 ♠ 9 ♠ ➆ 0 ➆ Figure 1: Chartrand’s example – graph ✄ ✟✖➣❸✥ THEOREM 3’. A signed graph ❲ is balanced iff the diagonal of its balance-closure matrix ❶ ➔➠ contains only elements with value ♠ . ✄ ✟✖➣⑧✥ THEOREM 4’. A signed graph ❲ is clusterable iff the diagonal of its cluster-closure matrix ❶ ➔➡ contains only elements with value ♠ . The balance-closure matrix of balanced signed graph contains no element with value ✽ , since in this case the corresponding diagonal elements should also have value ✽ . Similary the cluster-closure matrix of clusterable signed graph contains no element with value ✽ . A block is a maximal set of vertices with equal lines in matrix ❶ ➔ . In balance-closure of balanced signed graph and in cluster-closure of clusterable signed graph all the entries between vertices of two blocks have the same value. The value of entries between vertices of the same block is ♠ . In both cases different partitions of the set of vertices correspond to the (nonequivalent) colorings of the graph with blocks as vertices in which there is an edge between two vertices iff the entries between the corresponding blocks in matrix ❶ ➔ have value ➆ . There is another way to test the clusterability of a given signed graph: ✄ THEOREM 4”. A signed graph ❲ ✄✆☎✞✝✎✟✻✡❈✒✔✟ is computed in the semiring q ✟✖➣❸✥ is clusterable iff ✟✍➤➏✟✑✝✠✟☛✡✚✥ . ✄❩❬ ➛ ✥ ➔P➢ ❬ ❧ This form of the Theorem 4 is interesting because the intersection ✄ ✟✖➣❸✥ arcs which prevent the signed graph ❲ to be clusterable. ❅ ③ ✄❩❬ , where the closure ➔ ➛ ✥ ➔❂➢ ❬ ❧ consists of 2.3 Examples EXAMPLE 1. In Figure 1 the graph from [9, page 181] and its value matrix are given. The positive edges are drawn with solid lines, and the negative edges with dotted lines. 6 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ ✽ 1 p 1 2 3 4 5 6 7 8 9 ✽ ✽ ➆ ➆ ➆ 2 ➆ p p p p p p p p p ➆ ➆ ➆ ➆ ➆ ➆ ➆ ➆ ➆ ➆ ➆ ➆ 5 p 6 p ➆ ➆ 9 p ➆ ➆ ➆ ➆ ➆ ➆ ➆ ➆ ➆ ➆ ➆ 7 ➆ p 8 ➆ ➆ ➆ p p ➆ ➆ ➆ 4 ➆ p p p 3 ➆ ➆ ➆ ➆ ➆ ➆ p p ➆ ➆ p ➆ ➆ p p ➆ p p ➆ p p ➆ ➆ ➆ p Table 3: Chartrand’s example – closures u w y v x z Figure 2: Roberts’s example – graph On the left side of Table 3 the corresponding balance-closure is given – the graph is not balanced. From the cluster-closure on the right side of Table 3 we can see that the graph is clusterable and it has the clusters ❨ ❤ ❅➥☎✔✡☞✟✑➦✎✟✑➧✠✟✑➨✓✒✔✟ ❨ ✐ ❅s☎♣✌✎✟✑✏✠✟✧➩✠✒✔✟ ❨✫➫➭❅s☎♣➯✎✟✑➲✓✒ EXAMPLE 2. The signed graph from [21, page 77, exercise 16] is presented in Figure 2, and its value matrix on the left side of Table 4. In this case the balance-closure and the cluster-closure are equal (right side of Table 4). The corresponding partition is ❨ ❤ ❅s☎ ✟ ➝ ✟✧➳➵✒✔✟ ✳ ❨ 7 ✐ ❅s☎✚✦❸✟✖➸✷✟✑➺✎✒ u 0 u v w x y z v w 0 ♠ 0 0 0 0 0 ♠ 0 0 0 0 ➆ 0 0 0 0 x y 0 0 ➆ 0 ➆ z 0 0 0 ➆ 0 0 0 0 ♠ ➆ u v w x y z ♠ ➆ 0 u p w p y ➆ v ➆ x ➆ z p ➆ p ➆ p p ➆ p ➆ p ➆ ➆ p ➆ ➆ ➆ p p p p ➆ ➆ p p p ➆ p ➆ ➆ p ➆ Table 4: Roberts’s example – value matrix and its closure 3 Geodetic semiring 3.1 Construction of geodetic semiring Another example of a semiring we found reading the book [17, p. 34-38, 111-112]. In 1977 Freeman introduced the centrality index of a vertex based on betweenness [15, 16]: ➻ ➠ ❙ ✄➽➼❼✥✭❅ ❙ ⑩ ➆ ⑩✚➾ ✉ ✄➽➼❼✥ ➆ ⑩✚➾ ✉ ✉ ✦ ✄➽➼❼✥ where ➆ ⑩✚➾ ✉ is the✦ number of geodesics from vertex to vertex ✳ ; and ➆ ⑩❍➾ ✉ is the number of ➼ geodesics from to ✳ that contain vertex . ✄➽➼❼✥ ➻ For computing ➠ he proposed the methods given in Harary, Norman, Cartwright [19, p. 134-141]. ✄➽➼❼✥ ➻ Here we present an alternative approach to computing ➠ . Suppose that we know a matrix ❜ ✄ ✟ ✥ ❷ ❜ ⑩❍➾ ✉ ➆ ⑩❍➾ ✉ ✦ where ⑩✚➾ ✉ is the length of ✄➽➼❼✥ -✳ geodesics and also easy to determine ➆ ⑩❍➾ ✉ : ⑩ ➾✉ ➆ ❍ ✄ ❜ ✟ ➆ ⑩❍➾ ➚ ✝ ✄❏➼❼✥❆❅❻❺ ✺ ⑩ ➾✉ ➆ ❍ ❹ ✦ is the number of -✳ geodesics. Then it is ❜ ➆ ➚➪➾ ✉ ⑩❍➾ ➚ ❜ ✸ ➚➪➾ ✉ ❜ ❅ ⑩❍➾ ✉ otherwise ✥ Matrix ❷ ⑩❍➾ ✉ ➆ ⑩❍➾ ✉ ❹ can be obtained by computing the closure of relation matrix over the ❜ following geodetic semiring. ❬ ❅ ✄ ✟ ✥ ➆ ⑩✚➾ ✉✑❹ which has for entries pairs defined First we transform relation to a matrix R ❷ by ✄✆✡☞✟✻✡✚✥ ✄❏✦★✟ ✥➭❀✪❬ ✄ ❜ ✟ ➆ ✥ ⑩❍➾ ✉ ❅❻❺ ✄❩➶④✟✑✝✼✥ ✄❏✦★✟ ❜ ✳ ✳ ✥❭✪ ➹❀ ❬ where is the✵➄ length of path and ➆ is the number of shortest paths. ➛ a shortest ❅→✄ ☎♣➶➄✒♣✥➭❵➷✄ ☎♣➶➄✒♣✥ ➴ ❱ ➘ ➘ IR IN we define two operations: addition: In the set ✄ ♦ ✽ ✟✖♦➎✥⑧➬➮✄❩✾❍✟ ➌ ✥✭❅➱✄✶✕✢✜✤✣②✄ ✽ ✟✑✾✬✥✍✟➭❒ ✃❐ ❐❮ ♦➵✸ 8 ➌ ✮❰✾ ➌ ✽ ❅④✾ ✽ ✽✢Ï ✥ ✾ and multiplication: ✄ ✖✟ ♦➓✥❸Ð❡✄✶✾❍✟ ✥➇❅→✄ ✸❋✾❍✟✖♦⑧✺ ✥ ✽ ➌ ✽ ➌ ✄✶✵✷✟✍➬✯✟✍Ð✷✥ ✄❩➶④✟✑✝✼✥ ✄❩✝✠✟✻✡❍✥ It is easy to verify that is indeed a semiring with zero and identity . The verifications of semiring properties are straightforward. We present only the less trivial verifications of associativity of addition and of distributivity: Associativity: ✄✧✄ ✟✖♦➓✥⑧➬❡✄✶✾❍✟ ✥✧✥⑧➬➮✄❩✿✚✟ ✥✭❅→✄ ✟✖♦➓✥⑧➬➮✄❼✄❩✾❍✟ ✥⑧➬❡✄✶✿✞✟ ✥✧✥ Since the addition ➬ ✽ ➌ ➋ ✽ ➌ is commutative we can assume that ✽➴Ñ case Ó✷ß▼Ø❄ÔÜØ■ß➈Ú➇àáÓ❑ß➙Ú Ó✷ß▼Ø❄ÔÜØ✭âÞÚ➇àáÓ❑ß➙Ú Ó✷ß▼Ø❄ÔÜØ■å➈Ú Óæâ❛Ø❄ÔÜØ■ß➈Ú➇àáÓ❑ß➙Ú Óæâ❛Ø❄ÔÜØ✭âÞÚ Óæâ❛Ø❄ÔÜØ■å➈Ú Ò➓Ò➪Ó✎Ô➓Õ↔Ö➵×➷Ò➽Ø✬Ô✶Ù❈Ö➓Ö➵×❛Ò➪Ú☛ÔÜÛ✔Ö Ò➪Ó✓Ô➓Õ↔ÖÝ×ÞÒ➪Ú☛ÔÜÛ✔Ö★â❰Ò➪Ó✎Ô➓Õ↔Ö Ò➪Ó✓Ô➓Õ↔ÖÝ×ÞÒ➪Ú☛ÔÜÛ✔Ö★â❰Ò➪Ó✎Ô➓Õ↔Ö Ò➪Ó✓Ô➓Õ↔ÖÝ×ÞÒ➪Ú☛ÔÜÛ✔Ö Ò➪Ó✓Ô➓Õ✛ãçÙ♣Ö✫×➷Ò➪Ú❄ÔÜÛ✔ÖPâ❰Ò➪Ó✎Ô➓Õ❊ãçÙ❈Ö Ò➪Ó✓Ô➓Õ✛ãçÙ♣Ö✫×➷Ò➪Ú❄ÔÜÛ✔ÖPâ❰Ò➪Ó✎Ô➓Õ❊ãçÙ➏ã✈Û✔Ö Ò➪Ó✓Ô➓Õ✛ãçÙ♣Ö✫×➷Ò➪Ú❄ÔÜÛ✔ÖPâ❰Ò➪Ú☛ÔÜÛ✔Ö ✾ ➋ . Ò➪Ó✎Ô➓Õ↔Ö✫×➷Ò➓Ò➽Ø✬Ô✶Ù❈ÖÝ×ÞÒ➪Ú☛ÔÜÛ✔Ö➓Ö Ò➪Ó✓Ô➓Õ➎Ö✫×❛Ò➽Ø❄Ô✶Ù♣ÖPâ❰Ò➪Ó✓Ô➓Õ➎Ö Ò➪Ó✓Ô➓Õ➎Ö✫×❛Ò➽Ø❄Ô✶Ùäã✈Û✔Ö❸â✰Ò➪Ó✎Ô➓Õ↔Ö Ò➪Ó✓Ô➓Õ➎Ö✫×❛Ò➪Ú☛ÔÜÛ✔Ö Ò➪Ó✓Ô➓Õ➎Ö✫×❛Ò➽Ø❄Ô✶Ù♣ÖPâ❰Ò➪Ó✓Ô➓Õ✫ãrÙ♣Ö Ò➪Ó✓Ô➓Õ➎Ö✫×❛Ò➽Ø❄Ô✶Ùäã✈Û✔Ö❸â✰Ò➪Ó✎Ô➓Õ✛ãçÙ➏ã✪Û✔Ö Ò➪Ó✓Ô➓Õ➎Ö✫×❛Ò➪Ú☛ÔÜÛ✔Ö★â❰Ò➪Ú☛ÔÜÛ✔Ö From the table we can see that in all possible cases➬ the value of the left side term equals the value of the right side term. Therefore the addition is associative. Distributivity: ✄ ✖✟ ♦➎✥⑧Ð❡✄✧✄✶✾❍✟ ✥⑧➬➮✄❩✿✚✟ ✥✧✥✭❅→✄ ✖✟ ♦➓✥⑧Ð➮✄✶✾❍✟ ✥⑧➬➮✄ ✟✖♦➓✥⑧Ð④✄❩✿✚✟ ✥ ✽ ➌ ➋ ✽ ➌ ✽ ➋ ➬ ✾ ✿ Because of the commutativity of the addition , we can assume that Ñ . We obtain for the left side term ✄ ✖✟ ♦➓✥⑧Ð➮✄❼✄❩✾❍✟ ✥⑧➬➮✄❩✿✚✟ ✥❼✥✭❅→✄ ✟✧♦➓✥❸Ð❡✄❩✾❍✟ ❺ ✽ ➌ ➋ ✽ ➌ ✸ ➌ ✾è✮❰✿ ✾ä❅④✿ ➋ ✥❆❅→✄ ✽ ✸❃✾❍✟✖♦❸✺ ❺ ➌ ✸ ➌ ➋ ✾❁✮✰✿ ✾●❅➄✿ ✥ and for the right side term ✄ ✖✟ ♦➓✥⑧Ð➮✄✶✾✚✟ ✥⑧➬➮✄ ✟✖♦➓✥⑧Ð❡✄✶✿✞✟ ✥❆❅→✄ ✸❃✾❍✟✖♦❸✺ ✥⑧➬➮✄ ✸❃✿✚✟✖♦⑧✺ ✥✭❅➱✄ ✸❃✾❍✟❈❺ ✽ ➌ ✽ ➋ ✽ ➌ ✽ ➋ ✽ ♦⑧✺ ♦ ✺ ➌ ❋ ⑧ ✸ ♦❸✺ ➌ ➋ We get the same value sides. Therefore the distributivity holds. ✄✶✵❑✟✑➬éon ✟✍Ðêboth ✥ The semiring is also complete and closed, with a closure ✄ ✖✟ ♦➎✥ ◆ ❅ë❺ ✽ ✄✶✝✠✟✍➶❰✥ ✄✶✝✠✟✻✡✚✥ ✽ ❅➄✝✠✟✖♦➐❅⑨ ì ✝ otherwise This can be easily verified ✄❩✝✎✟✻✡✚✥⑥➬❡✄ ✖✟ ♦➓✥❸Ð❡✄ ✟✧♦➓✥ ◆ ✽ ✽ ❅ ❅ ✄❩✝✎✟✻✡✚✥⑥➬➮✄ ❅ ✸ ✟✧♦★✺☛♦ ✥ ◆ ✽ ◆ ✄✶✝✠✟✍➶❰✥ ✄❩✝✎✟✻✡✚✥⑥➬ ❒ ❐✃ ✄✶✝✠✟✑✝✼✥ ✄ ✸ ✟✖♦⑧✺❍♦ ✥ ◆ ❮❐ ✽ ✽ ◆ ✄✶✝✠✟✍➶❰✥ ❅➄✝✠✟✖♦➐❅⑨ ì ✝ ❺ ✽ ✄✶✝✠✟✻✡✚✥ ✽ otherwise 9 ❅ ✸ ✽ ✸ ✽ ✸ ✽ ❅➄✝✠✟✧♦í❅⑨ ì ✝ ✽ ◆ ➄ ❅ ✝✠✟✧♦❇❅⑨✝ ✽ ◆ ✝ ✽ ◆ Ï ✄ ✟✧♦➓✥ ◆ ✽ ✾è✮✰✿ ✾ä❅➄✿ ✥ 2ø î û 4 ï ïð ï î ï ñ ö óô ö î ö÷ ô ô ô ô 6 ñ ñ ô î ñ ïð ô ï ô 3 ö î ö ö ñ ñ❼ò û❴ü ó î ö ❴ ù 5ú ö ô ö ö ô ö ô ö ô ñ ö ö ö óô ö ö î ö÷ ô ù 1 ô ô➎õ ï î ï ó 7 ô ô➎õ 8 ñ❼ò Figure 3: Example graph ✡ ✡ ✌ ✏ ➩ ➦ ➧ ➯ ➲ ✌ ✏ ➩ ✄✆✡☞✟✻✡✚✥ ✄❩✌✎✟✍✌☞✥ ✄↔✌✓✟✍✌☞✥❢✄❩✏✎✟✖➩✔✥❢✄❩✏✎✟✖➩✔✥❢✄✶➩✠✟✑➲✼✥ ✄↔➶④✟✑✝✼✥❢✄❏➩❊✟✖➩✔✥❢✄❩➶④✟✑✝✼✥❢✄✆✡☞✟✻✡✚✥ ✄Ü✡❈✟✻✡✚✥❢✄↔✌✓✟✍✌☞✥❢✄↔✌✓✟✍✌☞✥❢✄❩✏✎✟✖➩✔✥ ✄↔➶④✟✑✝✼✥❢✄❏➩❊✟✖➩✔✥❢✄❩➶④✟✑✝✼✥❢✄✆✡☞✟✻✡✚✥ ✄Ü✡❈✟✻✡✚✥❢✄↔✌✓✟✍✌☞✥❢✄↔✌✓✟✍✌☞✥❢✄❩✏✎✟✖➩✔✥ ✄↔➶④✟✑✝✼✥❢✄✶✏✠✟✍✌☞✥❢✄❩➶④✟✑✝✼✥❢✄❏➩❊✟✍✌☞✥ ✄✶➩✠✟✍✌☞✥❢✄Ü✡❈✟✻✡✚✥❢✄Ü✡❈✟✻✡✚✥❢✄↔✌✓✟✍✌☞✥ ✄↔➶④✟✑✝✼✥❢✄✶✏✠✟✍✌☞✥❢✄❩➶④✟✑✝✼✥❢✄❏➩❊✟✍✌☞✥ ✄Ü✡❈✟✻✡✚✥❢✄Ü✡❈✟✻✡✚✥❢✄Ü✡❈✟✻✡✚✥❢✄↔✌✓✟✍✌☞✥ ✄↔➶④✟✑✝✼✥❢✄❩✌✎✟✻✡✚✥❢✄❩➶④✟✑✝✼✥❢✄✶✏✠✟✻✡✚✥ ✄❩✏✎✟✻✡✚✥❢✄✶➩✠✟✍✌☞✥❢✄✶➩✠✟✍✌☞✥❢✄Ü✡❈✟✻✡✚✥ ✄↔➶④✟✑✝✼✥❢✄❩✌✎✟✻✡✚✥❢✄❩➶④✟✑✝✼✥❢✄✶✏✠✟✻✡✚✥ ✄❩✏✎✟✻✡✚✥❢✄✶➩✠✟✍✌☞✥❢✄✶➩✠✟✍✌☞✥❢✄Ü✡❈✟✻✡✚✥ ✄↔➶④✟✑✝✼✥❢✄✆✡☞✟✻✡✚✥❢✄❩➶④✟✑✝✼✥❢✄❩✌✎✟✻✡✚✥ ✄↔✌✓✟✻✡✚✥❢✄❩✏✎✟✍✌☞✥❢✄❩✏✎✟✍✌☞✥❢✄✶➩✠✟✖➩✔✥ ✄↔➶④✟✑✝✼✥❢✄✆✡☞✟✻✡✚✥ ➦ ➧ ➯ ➲ Table 5: Geodetic closure ✄ ✟✖♦➓✥ ✝ ✵ ➛ A semiring element ✽ is positive iff ✽✘Ï . The set of all positive elements is closed for addition and multiplication. Note also that for positive elements the absorbtion property ✄❩✝✠✟☛✡✚✥⑥➬➮✄ ✽ ✟✖♦➎✥✭❅➱✄❩✝✎✟✻✡✚✥ holds. 3.2 Algorithm and example ✄ ❜ ✟✑✿❄✥ ⑩❍➾ ✉ be the entry of the strict closure ý Let ❜ of✦ relation matrix R over geodetic semiring. Then equals to the length of a shortest nontrivial -✳ path (a geodesic or a shortest cycle), and ✿ ✦ equals to the number of different -✳ geodesics. Using this algorithm we obtained for a graph represented in Figure 3. the strict geodetic closure presented in Table 5. To adapt the Fletcher’s algorithm for computing the strict geodetic closure we have to consider some properties of geodetic semiring. By the construction all entries of the relation ma- 10 trix follows ✿ ❚ R ✟ are positive. ✄✶✿ ❚ ✟ From ✥↔◆ the description of Fletcher’s ✄❩algorithm ✿❚ ✟ ✥↔it◆➐❅→ ✄❩✝✠✟✻✡❍✥ that also the entry and we can omit it ❧✛❤✍❷➍➋ ➋✓❹ in ❧✛❤✬❷➍➋ ➋✓❹ is always positive. Therefore ❧✛❤✬❷➍➋ ➋✓❹ from the expression. A detailed analysis of the algorithm shows that we can compute the strict ➊ geodetic closure matrix in place – we omit the subscripts in matrices . Representing the relation matrix R (and its geodetic closure) by two matrices: the shortest ➊ paths length matrix ❶ and the geodesics count matrix , we obtain the following adapted version of Fletcher’s algorithm for computing the strict geodetic closure: ❝ ❅s✡ to✡ ➆ do begin ❝ ❅s✡ for ➋ ♦ ❝ ❅s for ❜✓þ ➼ ❝ ❅➄to ❜ ➆ ✟ do✥ begin ✕✢✜✤➆ ✣②do ✄❩✾✑♦❩ÿ➵for ✟ ❜ ➌ ♦✧✟ ✸ to ❜ ♦❼✟ ❜✓þ ➼ ❷ ➋✓❹ ❷➍➋ ➌☞❹ ; if ❷ ✿ ➌☞➼ ❹❸❝ ❅④ ✇ ✿ ♦✧✟ then✿ begin ✟ ➆ ❜ ♦❼✟ ❷❅ ➋✓❜ ❹✁þ ➼ ❷➍➋ ➌☞✿ ❹ ; ♦✧✟ ❝ ❅④✿ ♦❼✟ ✸❃✿ ➼ ❷ ➌☞✿ ❹ ➼✄✂ ❜ ♦❼❷ ✟ ➌☞❹ ❝ ❅ ❜✓➆ þ ➼ if ❷ ➌☞❹ ✿ then ♦❼✟ ❝ ❅④ ➆ ❷ ➌☞❹ else begin ❷ ➌☞❹ end; end; end; end; The constant ✾✑♦❩ÿ in the algorithm is a number representing the infinity ➶ . 4 Semiring for Boyle’s operation 4.1 Construction of geosetic semiring In [4, 13] Boyle introduced the operation ✦ ➭✳ ❅ ❨ set of vertices of a graph ❲ that belong to the shortest paths from ❅→✄↔❨✭✟✑❬❭✥✍✟ ❬s❪⑨❨↕❵➙❨ ✦ to ✳ over the set of vertices of a graph ❲ . This operation has several interesting properties. For example: ✦ ✦ ➭✳✆❅ ☎ ➸ ➭✳ ✦ ➼✞✝ ➼✠✝ ➼ ✦ ➼ ✳✟✟✖✦❸✟✖➸✷ ☎ ✟✧➼ , ✳ belong to the same cycle. But, it is not associative. Nevertheless it is possible to embed it into a semiring in the following way. We start with quadruples of the form ✄✶✦❸✟☛ä ✡ ✟ ✟ ✥ ✳ ♠ ✦❛❀ ❨ ❀❃❨ ✡❴❪➱❨ with the interpretation: – initial vertex, ✳ – terminal vertex, vertices ✦ ❀ ☎✚✦❸✟ – ✒➙ ❪☞✡ on ✳ the✮✍shortest –✳ paths, ♠ IN – length of these paths.✄✶✦❸We also and ✌✻✗❈➁❼❉✎✡ ✟☛☎✚✦⑧✒✔require ✟✖✦❸✟✑✝✼✥✍✟✖✦ç ❀➙❨ that ♠ . Quadruples with length 0 are of the form . ❀ ✏ We build the geosetic semiring over the set ✏ of finite sets of quadruples ✑ ❅s☎ ✒✔✟ ❅➱✄❏✦ ✟☛✡ ✟ ✟ ✥ ✑ ✽☞♥ ✽✼♥ ♥ ♥ ✳✚♥ ♠❊♥ 11 which also satisfy the requirement that no pair of quadruples in ✑ has the same initial and terminal vertices ✦ ❅➄✦✓❂ ✒ ➤ ❅ ✒ ✝❢♦❇❅ ✔ ♥ ✳✞♥ ✳ ➌ Let us first consider one element sets ❅s☎ ✑ ✒✔✟ ✽ ❅➱✄✶✦❸☛ ✟ ✡ä✟ ✽ We can define addition as follows: ☎✎✄✶✦❸☛ ✟ ✡ ❒ ✃❐❐❐ ➬✗✕➙❅ ✑ ✕✑ ❐❐ ✖é✟ ➘ ✥ ♠ ✟ ✳ ✕✈❅➥☎✞✾❍✒✔✟❇✾ä❅➱✄❏➸❑✟☛é ✖ ✟✧➼✍✟ ✗♣✣❊❉ ✥✑✒ ♠ ➘ ➞ ✮ ➞ ✕ ✘ ❅ ♠ ♠ ✑ ❐❮ ✟ ✳ ✮ ➞ ❐✙ ✟❇✦r❅➄➸❑✟ ✚❐ ♠ ➞ ✥ ❅⑨➼ ✳ otherwise ✕➷❅ ☎ ✟✑✾✻✒ ✽ Note that in the last case ✑ ➘ which is not a one element set so that the addition is not closed over one element sets. To extend it to ✏ we first introduce the operation of reduction Red ✑ , which from an arbitrary finite set of quadruples, ✑ , eliminates multiple occurrences of quadruples with the same initial and terminal vertices. It is defined by ❅ Red ✑ ❷ ✄ ✑✜✛ ☎ ✟ ✽☞♥ ✒✻✒♣✥ ✽ ➘ ✄✆☎ ✽☞♥ ✒ä➬⑨☎ ✒❍✒♣✥✬✟✖♦✭✮ ✽ ➌✼❹ or in a procedural form ❝ ✕ ❅ ✑ ✂ ❝ ♦✧✟ while ✢ ❝ ❅✥➌ ✕ Red ✑ ✄✶♦➐ì❅ ➌ ➤ ✦ ✦✁✒❇➤ ♥ ❅ ✳✚♥ ❅ ✳ ✒❄✥ do ❝ ✕ ❅➱✄✣✕ ☎ ✛ ✽☞♥ ✟ ✽ ✒✻✒♣✥ ➘ ✄➓☎ ✽✼♥ ✒ä➬ ☎ ✒❍✒♣✥✤✂ ✽ ❀ ✏ . Now it is easy to extend the addition to ✏ . Evidently✟✦Red ✕▼❀ ✑ ✏ then Let ✑ ➬✍✪ ✕ ❅ ✄ ✕P✥ ✑ Red ✑ ➘ ✄ ✏ ✟✍➬é★ ✟ ✧➭✥ is an idempotent Abelian monoid (semilattice) [3]. A neutral element for the addition is the empty set ③ which can also be represented by a special element ✧❛❅➥☎✎✄✪✩✬✟ ✟☛✩✬✟✍➶❰✥✑✒ ③ ✩ where represents a ”joker”-vertex, which matchs any vertex. We could base the introduction ✧ ✄✪✫❇✟✄✬✼✥ of also on the theorem that every semigroup without a✭ neutral element can be extended ✄✪✫ ☎ ✒✔✟☛✬✠✟ ✥ ➹❀ ✫ ➝ ❀ ➘ ❝ ➜ ➜ ➜ ✮ to a monoid by addition of a new element satisfying the relation ✫ ☎ ✒ ✬ ➝ ❅ ➝ ✬ ❅ ➝ ➘ ➜ ➜ ➜ . ❅ ☎✎✄✶✦❸☛ ✟ ✡ä✟ ✟ ✥✑✒ ✕❛❅ and Now we can also define the multiplication of one element sets ✑ ✳ ♠ ☎✎✄✶➸❑✟✯✢ ✖ ✟✧➼✍✟ ✥✑✒ : ➞ ✑ Ð✰✕✈❅ ❒ ❐❐❐ ❐ ✃❐❐ ✕✑ ☎✎✄✶✦❸✟☛✡ ❐❐❮ ❐❐ ❐❐ ✧ ➘ ✖é✟✧➼✍✟ ♠ ♠ ✸ ➞ ✥✍✒ ✘ ❅④✝ ➞ ✦ ❅➄✝ ⑨ ì❅ ➼✍✟✱✡ ✦r❅⑨➼✍✟✱✡ otherwise 12 ➢ ➢ ✖➥❅➥☎ ✖➥❅➥☎ ✳ ✳ ✒ ❐❐ ❐✙ ✟✖✦⑧✒✳✲ ✳ ➹❀ ☎✚✦❸✟✧➼✑✒ ✚ ❐❐❐ ✳ ❅➄➸ v=w v=w z Q P t u Q P t u a b Figure 4: Geosetic multiplication ✕ Ð ✕ ✑ ✴ ✧ This definition needs some explanation. The is different from only if ➸ product coincide (see Figure 4a). Assume that the❀ ➹ terminal vertex ☎✚✦❸✟✧➼✑✒ ✦Þ❅⑨ ì ➼ ✳ of and the intial ➺é❀ vertex ☎ of ✒ ✦ ➼ . If and there exists ✦ ➼ ➢ ✳ ✳ then there also exists a shorter - path on ➘ (see ➺✷❅⑨Figure ✦ ➺❑4b). ❅ ➼ Therefore the - paths through ✳ are not geodesics. This also covers the case or . For example ✡ ✑ ✖ ✞✡ ✖ ✛ ☎✎✄✶✦❸✟☛☎✚✦❸✟ ➝ ✟ ✔ ✒ ✟ ✟✑✌☞✥✍✒➇Ð ✳ ✳ ✵✧ ☎✎✄ ☛✟ ☎ ✟✖➳➵✟✖✦❸✟✧➼✑✒✔✟✧➼✍✟✑✏✼✥✑✒➐❅ ✳ ✳ Other cases are dealt with similarly. We extend the multiplication to by ✏ Ð ✕✈❅ ✶ ☎✽ ✑ ✍ ✷☛✸✺❈✹ ➾ ✻✼✸✺✽ Since for ✄✏ ✏ ✄ t ✒➏Ð➄☎✞✾✻✒ ❅➥☎✎✄ ❄✟ ☎ ✔ ✒ ✟ ✟✖✝✼✥ ❝ ❀✈❨é✒ ✳ ✳ ✳ ✳ Ð ✟✍Ðé✟ ✥ ➬ is a monoid and the multiplication distributes over the addition , the structure t ✟✍➬é✟✑Ðé✟ ➐✟ ✥ t is an idempotent semiring; and since is finite it is also a complete semiring. ✾✧ ✏ ✧ The neutral element for addition is an absorptive element for multiplication – a zero: for ✑ ❀ ✏ we have ✧✈Ð ✑ ❅ ✑ Ð✰❛ ✧ ❅✵✧ ❅s☎✎✄❏✦★✟✯● ✡ ✟ ✳ ✟ ♠ ✥✍✒✔✟ ♠①Ï ✝ that From the definition of multiplication it follows for ✑ ✑ ✐ ❅ ✑ Ð ✑ ❅✿✧ ❚ ❅✵➐ ✧ ✟ ➋ ✇ ✌ and Therefore, in this case, ✑ ✑ ◆ ❅❀❚ ✶ ✸ ✑ ❚ ❅ t ➬ ✑ ❅á☎✎✄ ✟☛ä ✡ ✟ ✳ ✟ ♠ ✥✑✒ (used in Fletcher’s algorithm) we have ✳ In the case element” ✑ ◆●❅ of a➬ “diagonal ❅ t ✑ t. further ✑ IN 13 û❴ü ❑ ❁ ❁ ❁ ó ❁❂ ❁ î ❃❅ ❃ ❇❈ ❇ ❅ ù❴ú ✽ ❁ î ❋■ ❅ ❅ ❇ ❃ ❇ ❅ ❅✪❆ î ❇ ❋ ÿ ❋ î●❋ ó î ❃ ❃✪❄ ❃ ❋ ❏ ❋ ✾ ó ➜ ❉ ❇ ó ❉❊ ❉ î❜ ❉ ❉ ❋ ❉ ❉ ❋✪❍ ❉ø î ❉ ✿ Figure 5: Example graph ✄ ✽ ✾ ✿ ❜ ➜ ❑ ÿ ✽ ✄ ✄ ③ ✽ ✾ ✟✻✡❍✥ ✟✑➶❰✥ ❜ ✿ ✄ ✾❍✟✻✡✚✥ ✽ ✄✶✾✬✿ ❑ ✟✖✏✼✥ ✄ ✟✑➶❰✥ ✄✶✾✬✿ ❑ ✟✑✌☞✥ ③ ❜ ✄ ✟✑➶❰✥ ✄✶✾ ❑ ✟✍✌☞✥ ③ ❜ ✄ ✟✑➶❰✥❢✄❩✾ ❑ ✟✑✏✼✥ ③ ➜ ✄ ✟✑➶❰✥ ✄❩✾ ❑ ✟✻✡✚✥ ③ ✄ ÿ➵✟✻✡❍✥ ✄ ✾✖ÿ➵✟✍✌☞✥ ✽ ✽ ✾✍✿ ❑ ✟✍✌☞✥ ✽ ✄ ✄✶✾✍✿✞✟✻✡❍✥ ✽ ✄❩✿ ❑ ❜ ✄❩✿ ❑ ❜ ✄✶✿ ➜ ✄❩✿ ❑ ✾✍✿ ✄❩✿ ❜ ✄✶✿ ❑ ✟✖✏✼✥ ✟✻✡✚✥ ✄❩✿ ❜➜ ✟✍✌☞✥ ✟☛✡✚✥ ❑ ✟✑✏✼✥ ✟✻✡✚✥ ❜ ➜ ✄✶✿ ✄✶✿✍ÿ➵✟✻✡✚✥ ❜ ➜ ❜ ➜ ✄ ❑ ✟✖✏✼✥ ❜ ➜ ✄✶✾✬✿ ✟✍✌☞✥ ✟✑✌☞✥ ❜ ❑ ✟✍✌☞✥ ÿ➵✟✍✌❈✥ ✄ ✾ ✽ ➜ ➜ ÿ ❑ ❑ ✟✍✌❈✥ ✄✶✾ ✻ ✟ ✡✚✥ ➜ ✄❩✿ ❑ ✍ ✟ ✌❈✥ ❜ ➜ ✄ ❑ ✟✍✌❈✥ ❜ ➜ ✄ ❑ ✟✑✏☞✥ ➜ ✄ ❑ ✟☛✡✚✥ ➜ ✄ ✾✍✿ ❑ ÿ➵✟✑✏☞✥❢✄ ✽ ➜ ✄ ❑ ✟☛✡✚✥ ✽ ✄❩✾✍✿ ❑ ✍ ✟ ✌❈✥ ✄ ✄ ✄❩✿ ❑ ✻ ✟ ✡❍✥ ❜ ✄ ❑ ✟✻✚ ✡ ✥ ❜ ✄ ✍ ✟ ✌☞✥ ❑ ➜ ✄❩✿ ❑ ✟✍✌❈✥ ✽ ✿ ❑ ÿÝ✟✑✌☞✥❢✄ ✟✍➶❰✥ ③ ✟✍➶❰✥ ③ ✄ ✟✍➶❰✥ ③ ✄ ✟✍➶❰✥ ③ ✄ ✟✍➶❰✥ ✄ ③ ✟✍➶❰✥ ③ ③ ✟✍➶❰✥ Table 6: Geosetic closure 4.2 Algorithm and example This semiring can be used to compute Cayley’s table of Boyle’s operation for a given network by computing a strict closure of ❬ a relation matrix over it. ❅ ✄✣✡●✟ ✥ ❷ ♠ ⑩❍➾ ✉✖❹ which has for the entries pairs First we transform relation to a matrix R defined by ✄✆☎✚✦❸✟ ✒✔✟✻✡✚✥ ✄✶✦❸✟ ✥➭❀✈❬ ✄✪✡ä✟ ✡ ♠ ✥ ⑩❍➾ ✉ ❅ë❺ ✄ ③ ✳ ✟✍➶❰✥ ✦ ✄✶✦❸✟ ✳ ✳ ✥❭✈ ➹❀ ❬ where is the set of vertices on the shortest -✳ paths and ♠ is their length. Since the✟ addition is idempotent we can apply the Fletcher’s algorithm [14] in place. The ✄✶✿ ✥↔◆➐❅ ❚ ❧✛❤✬❷➍➋ ➋✓❹ t and therefore it can be omitted. term Representing the relation matrix R (and its strict closure) by two matrices: the matrix ▲ of sets of vertices on the shortest paths, and the shortest paths lengths matrix ❶ , we obtain the following adapted version of Fletcher’s algorithm: 14 ❝ ❅s✡ for ➋ ♦ ❝ ❅s to✡ ➆ do begin ❝ ❅s✡ to ➆ do begin for ♦ ❝ ❅ to þ ♦✧➆ ✟ do for þ ➌✟ ♠ ➞✄ ♦ ❷ ♦ ➋✓✥❆❹▼❅ ❷➉➋ ➌☞❹ ; if ♠ ✦ ➞❖◆⑨❝ ❅ ❷ ❹ þ ♦✧✟ ❷➍➋✔❹✸ then þ ✟ begin ♠ ❜✓þ ➼ ➞ ❝ ❅ ❜ ❷ ♦✧✟ ➋✓❹ ✸ ❜ ❷➍➋ ✟ ➌☞❹ ; ➋ þ❹ ➼ ❷➍➋ þ➌☞❹ ♦✧;✟ ❝ ❅ þ ♦✧✟ ✸ ✦ ❜ ♦❼✟ ❅ ❷ ❜✓ ❜✓þ ➼ ❷ ➌☞❹ ❷ ➌☞❹ ♠ ➞ if ❷ ➌☞❜ ❹ ♦✧✟ then else if ❷ þ ➌☞❹❸♦❼✟ Ï ❝ ❅ then ✦ ✂ ❜ ♦✧✟ ❝ ❅ ❜✎þ ➼ ♠ ➞ ❷ ➌✼❹ begin ❷ ➌☞❹ end; end; end; end; ✸ Note, for sets in pascal denotes the union ➘ , and denotes the intersection ➢ . Using this algorithm we obtained for a graph represented in Figure 5 a strict geosetic closure presented in Table 6. 5 Conclusion In the paper four semirings for solving social networks analysis problems were constructed. For balance and cluster semirings Fletcher’s algorithm for computing the closure of a network matrix can be applied directly; and for geodetic and geosetic semirings we have adapted versions of Fletcher’s algorithm. The algorithm for geodetic closure is more efficient than the usual Harary, Norman, Cartwright algorithm [19]. We do not know any other algorithm for computing Cayley’s table of Boyle’s operation (geosetic closure). The embedding technique used in the construction of a geosetic semiring can be used also as a general framework for path problems in valued graphs. References [1] A. V. Aho, J. E. Hopcroft, J. D. Ullman, The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, Massachusetts (1976). 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