In this paper we introduce extensions of Kleinberg's Hubs & Authorities (HITS) algorithm to calculate the in uence of nodes in a network whose adjacency matrix takes values over dioids, zerosumfree semirings with a natural order. We relate these extensions to both the Singular Value Problem and the Eigen Problem of matrices in these semirings. We show the original HITS algorithm to be a particular instance of the generic construction, but also the advantages of working in idempotent semi elds. We also make some connections with extended K-Formal Concept Analysis, where the particular kind of dioid is an idempotent semi eld, and conclude that the type of knowledge extracted from a matrix by one procedure and the other are di erent.
The Singular Value Decomposition (SVD) [1] is a set of results about the decomposition of a rectangular matrix with applications in data processing, signal theory, machine learning and computer science at large.
In particular, it is a crucial tool in the dual-projection approach to the analysis of bipartite networks|also known as 2-mode, a liation or membership networks [2]. The dual projection approach postulates we can provide measures of centrality, core-vs.-periphery and structural equivalence for each of the projection networks with limited loss of global information, in terms of the left and right singular vectors.
On the other hand, one of the rst well-known approaches to link analysis on the Web, the Hubs and Authorities algorithm or Hyperlink-Induced Topic Search (HITS) [3, 4] also stems from the SVD. HITS was designed to solve the problem of ranking the nodes of a dynamic, directed 1-mode network of nodes obtained from a query against a search engine. It postulates the existence of two qualities in nodes: their \authoritativeness"|their quality of being authorities ? CPM has been supported by the Spanish Government-Comision Interministerial de
Ciencia y Tecnolog a project TEC2011-26807. with respect to a pervasive topic in the nodes|and their \hubness"|their quality of being good pointers to authorities|and solves this in terms of the left and right eigenvalues, again. 1.1
The original HITS formulation Consider a directed graph D = (V; E), where V = fvigin=1 is a set of nodes and E V V is a set of edges. It can alternatively be de ned by an adjacency matrix AD with 0 1 weights in its edges, (AD)ij = 1 if (vi; vj ) 2 E and (AD)ij = 0 otherwise. Note that this procedure can be reversed to obtain the digraph induced by a matrix A as DA = (n; EA) where n = f1; : : : ; ng and EA = f(i; j) j Aij = 1g .
Also note that the vertices and edges of the digraph model the nodes and links in any social network. Precisely based in this model, HITS nds an authority score a(vi) and a hub score h(vi) for each node aggregated as vectors, based in the following iterative procedure: (1) (2) (3) (4) Start with initial vector estimates h<0> and a<0> of the hub and authority scores.
Upgrade the scores with: h
Aa a
Ath so that in general, for k
1: h<k> = (M At)kh<0> h<k> = (AAt)k 1Aa<0> a<k> = (AtA)ka<0> a<k> = (AtA)k 1Ath<0> Since matrix A is non-negative, in general the sequences fh<k>gk and fa<k>gk would diverge, so the next step is to prove that the limits: lim k!1 h<k> ck = h< > lim k!1 a<k> dk = a< > exist, in which case they are eigenvectors of their respective matrices for seemingly arbitrary c and d, (AAt)h< > = ch< > (AtA)a< > = da< > : As long as the initial estimates do not inhabit the null space of these matrices| making them orthogonal to h< > and a< >, respectively|the iterative process will end up nding the principal eigenvectors. The proof of this fact entails that the initial estimates h<0> and a<0> should be non-negative. 1.2
The generalisation over complete dioids In this paper we generalise the setting above slightly to close the gap with the FCA problem: 1. First, as suggested by the form of (2), we consider directed weighted bipartite graphs between two sets G and M , because we want to study the quality of being a hub and being an authority separately. 2. Second, we consider edge weights R 2 DG M in a naturally ordered semiring or dioid D (cfr. x2.1) . This implies passing from directed graphs to networks, that is weighted directed graphs. Then D-formal contexts, denoted as (G; M; R), are a natural encoding for this type of bipartite graphs. Note that the original HITS setting can be recovered by using G = M = V and A = R and working in the S = R0+ semi eld with Rij = 1 if (vi; vj ) 2 E and Rij = 0 otherwise. Similarly, the original dual-projection approach deals only with the case where R is actually binary but is considered to be embedded in S = R0+ .
Let K = hK; ; ; ?i be a dioid. Then the space of hub scores is X = KG and that of authorities is Y = KM and they get mutually transformed by the actions of two linear functions:
But we want to emphasize their mutual dependence, so after [5] we write (1) in matrix form
Rt
: KG 7! KM x 7! Rt x
: KM 7! KG y 7! R y Equation (7) is the proof that HITS is actually trying to solve the singular valuesingular vector problem [1, 5], where h has the role of a left singular vector and a that of a right singular vector.
The previous Section suggests that the solution to the HITS problem entails the solution of the following eigenproblem in the variable zt = [xtyt]t, B z = z ,
Rt xy = xy where we have substituted zero matrices for dots, to lessen the visual clutter. The solutions to this problem are of the form wt = [htat]t that is, pairs of hub and authority vectors. To see this, we expand the system (6) into two equations called by Lanczos the \shifted eigenvalue problem" [5],
R a = h
Rt
h = a h a
Rt
R R
R h a (5) (6) (7)
To relate this problem back to the original one, we premultiply (7) by Rt, respectively R, to obtain (R
Rt) h = h 2 (Rt
R) a = a 2 (8) This proves that in order to solve the HITS problem in a dioid we have to solve the Singular Value Problem (7) which amounts to solving both decoupled Eigenvalue Problems (8).
In this paper we develop a similar tool for bipartite networks with edges with weights in a dioid, but we develop it as if it were an instance of the better understood, HITS problem.
To develop our program we rst introduce in Section 2.1 some de nitions and notation about semirings, in general, and about dioids and semi elds, in particular. In Section 2.2 we introduce the eigenproblem over dioids as a step to solving the singular value problem in dioids, and in Section 2.2 a very general technique to do so. Section 3 presents the weight of our results and we conclude with a short Example and a Discussion. 2 2.1
Theory and Methods
Semiring and semimodules over semirings A semiring is an algebra S = hS; ; ; ; ei whose additive structure, hS; ; i, is a commutative monoid and whose multiplicative structure, hSnf g; ; ei, is a monoid with multiplication distributing over addition from right and left and an additive neutral element absorbing for , i.e. 8a 2 S; a = . A semiring is commutative if its multiplication is commutative.
A semiring is zerosumfree if it does not have non-null additive factors of zero, a b = ) a = and b = ; 8a; b 2 S . Likewise, it is entire if a b = ) a = or b = ; 8a; b 2 S . Entire zerosumfree semirings are called sometimes information algebras, and have abundant applications [6]. Their importance stems from the fact that they model positive quantities.
Importantly, every commutative semiring accepts a canonical preorder, as a b if and only if there exists c 2 D with a c = b which is compatible with addition. A dioid is a commutative semiring D where this relation is actually an order. Dioids are zerosumfree. A dioid that is also entire is a positive dioid.
An idempotent semiring is a dioid whose addition is idempotent, and a selective semiring one where the arguments attaining the value of the additive operation can be identi ed.
A complete semiring S [7] is a semiring where for every (possibly in nite) family of elements faigi2I S we can de ne an element Pi2I ai 2 S such that 1. if I = ?, then Pi2I ai = , 2. if I = f1 : : : ng, then Pi2I ai = a1 : : : an , 3. if b 2 S, then b Pi2I ai = Pi2I b ai and Pi2I ai b = Pi2I ai b , and 4. if fIj gj2J is a partition of I, then Pi2I ai = Pj2J Pi2Ij ai . If I is countable in the de nitions above, then S is countably complete and already zerosumfree [8, Prop. 22.28]. The importance for us is that in complete semirings, the existence of the transitive closures is guaranteed (see below). Commutative complete dioids are already complete residuated lattices.
A semiring whose commutative multiplicative structure is a group will be called a semi eld and radicable if the equation ab = c can be solved for a. This term is not standard: for instance, [9] prefer to use \semiring with a multiplicative group structure", but we prefer semi eld to shorten out statements. Semi elds are all entire. We will use K to refer to them.
A semi eld which is also a dioid is both entire and naturally-ordered. These are sometimes called positive semi elds, examples of which are the positive rationals, the positive reals or the max-plus and min-plus semi elds. Semi elds are all incomplete except for the booleans, but they can be completed [10], and we will not di erentiate between complete or completed structures
A semimodule over a semiring is an analogue of a vector space over a eld. Semimodules inherit from their de ning semirings the qualities of being zerosumfree, complete or having a natural order. In fact, semimodules over complete commutative dioids are also complete lattices. Rectangular matrices over a semiring form a semimodule Mm n(S), and in particular, row- and columnspaces S1 n and Sn 1. The set of square matrices Mn(S) is also a semiring (but non-commutative unless n = 1). 2.2
The eigenvalue problem and transitive closures over dioids Given (4), understanding the HITS iteration is easier once understood the eigenvalue problem in a semiring. So let Mn(S) be the semiring of square matrices over a semiring S with the usual operations. Given A 2 Mn(S) the right (left) eigenproblem is the task of nding the right eigenvectors v 2 Sn 1 and right eigenvalues 2 S (respectively left eigenvectors u 2 S1 n and left eigenvalues 2 S) satisfying: u
A = u
A v = v The left and right eigenspaces and spectra are the sets of these solutions: Since (A) = P(At) and U (A) = V (At) , from now on we will omit references to left eigenvalues, eigenvectors and spectra, unless we want to emphasize di erences.
In order to solve (6) in dioids we have to use the following theorem [11, 9]: Theorem 1 (Gondran and Minoux, [11, Theorem 1]). Let A 2 Sn n. If A exists, the following two conditions are equivalent: 1. A:+i 2. A:+i = A:i (and A:i for some i 2 f1 : : : ng, and 2 S.
) is an eigenvector of A for e , A:+i 2 Ve(A) . where we de ne the transitive closure A+ = P1 k=1 and the transitive re exive closure A = Pk1=0 of A (also caleld Kleene's plus and star operators ).
In [12, 13, 14] this result was made more speci c in two directions: on the one hand, by focusing on particular types of completed idempotent semirings| semirings with a natural order where in nite additions of elements exist so transitive closures are guaranteed to exist and sets of generators can be found for the eigenspaces|and, on the other hand, by considering more easily visualizable subsemimodules than the whole eigenspace|a better choice for exploratory data analysis.
From Theorem 1 it is clear that we need e cient methods to obtain the closures. The following account is a summary of results in this respect, and we refer the reader to [12, 13] for proofs.
Lemma 1. Let A 2 Mn(A) be a square matrix over a commutative semiring S. A exists if and only if A+ exists and then:
A+ = A
A = A
A
A = I
A+ But since in incomplete semirings the existence of the closures is not warranted, our natural environment should be that of complete semirings. In dioids we have, Lemma 2. Let A 2 Mn(A) be a square matrix over a dioid S. For partition n = [ call Per (A) = A A A A . Then
A A
A A + =
A+
A A Per (A) A Per (A) A A
A
A
A Per (A) Per (A)+
Closures and simultaneous row and column permutations commute: Lemma 3. Let A; B 2 Mn(S) and let P be a permutation such that B = P tAP . Then B+ = P tA+P and B = P tA P .
Given the importance of the transitive closure of a matrix in the calculations of eigenvalues and eigenvectors highlighted by Theorem 1, we use the inductive structure of reducible matrices over dioids to calculate them. First a basic case: A square matrix is irreducible if it cannot be simultaneously permuted into a triangular upper (or lower) form. Otherwise we say it is reducible. Irreducibility expresses itself as a graph property on the induced digraph DA of Section 1.1. Lemma 4. If A 2 Mn(S) is irreducible, then:
The induced digraph DA has a single strongly connected component. All nodes in its induced digraph DA are connected by cycles. tu P t 3
A
2E
Lemma 5 (Recursive Upper Frobenius Normal Form, UFNF). Let A 2 Mn(S) be a matrix over a semiring and GA its condensation digraph. Then, 1. (UFNF3) If A has zero lines it can be transformed by a simultaneous row and column permutation of VA into the following form: (11) (12) (13) P t 2
A P t 1
A 2A1 where fAkgkK=1; K 1 are the matrices of connected components of GA. 3. (UFNF1) If A is reducible with no zero lines and a single connected component it can be simultaneously row- and column-permuted by P1 to where Arr are the matrices associated to each of its R strongly connected components (sorted in a topological ordering), and P1 = P (A11) : : : P (ARR). Note that irreducible blocks are the base case of UFNF1, so we sometimes refer to irreducible matrices as being in UFNF0. 2.3
Graphs, eigenvalues and eigenvectors of matrices over complete dioids Graph induced by a matrix. It is interesting to consider matrices under a di erent cryptomorphism: For a matrix A 2 Mn(S), the network or weighted digraph induced by A, NA = (VA; EA; wA), consists of a set of vertices VA, a set of arcs , EA = f(i; j) j Aij 6= S g, and a weight wA : VA VA ! S; (i; j) 7! wA(i; j) = aij .
This allows us to apply intuitively all notions from networks to matrices and vice versa, like the underlying graph GA = (VA; EA), the set of paths A+(i; j) between nodes i and j or the set of cycles CA+.
Orthogonality of eigenvectors. In spectral decomposition, orthogonality of the eigenvectors plays an important role. In zerosumfree semimodules orthogonality cannot be as prevalent as in standard vectors spaces. To see this, rst call the support of a vector, the set of indices of non-null coordinates supp(v) = fi 2 njvi 6= g, and consider a simple lemma: Lemma 6. In semimodules over entire, zerosumfree semirings, only vectors with empty intersection of supports are orthogonal.
Proof. Suppose v?u, then Pin=1 vi ui = . If any vi = or ui = then their product is null, so we need only consider a non-empty supp(v) \ supp(u) . In this case, vt u = Pi2supp(v)\supp(u) vi ui. But if S is zerosumfree, for the sum to be null every factor has to be null. And for a factor to be null, since S is entire, either vi is null, or ui is null, and then i would not belong in the common support.
Note that these are important in as much as they generate ? coordinates in the eigenvectors, that is, in the hubs and authorities vectors.
Eigenvalues and eigenvectors of matrices over positive semi elds. When S has more structure we can improve on the results in the previous section. The rst proposition advises us to concentrate on the irreducible blocks: Proposition 2. If K is a positive semi eld, A 2 Mn(K) is irreducible, and v 2 V (A) then > and 8i 2 n ; vi > .
The null eigenspaces. If any, the eigenvectors of the null eigenvalue are interesting in that they de ne the null eigenspace. Also, the particular eigenvalue ? can only appear in UFNF3. The following proposition describes the null eigenvalue and eigenspace: Proposition 1. Let S be a semiring and A 2 Mn(S). Then: 1. If the i-th column is zero then the characteristic vector ei is a fundamental eigenvector of for A and 2 Pp(A) . 2. Non-collinear eigenvectors of are orthogonal, so the order of multiplicity of ? 2 Pp(A) is the number of empty columns of A . 3. If S is entire, then GA has no cycles if and only if Pp(A) = f g . 4. If S is entire and zerosumfree, the null eigenspace if generated by the fundamental eigenvectors of for A, V (A) = hFEV ( ) Ai .
Proof. See [12, 3.6 & 3.7]. Claim 2 is a consequence of claim 1 and Lemma 6. tu tu
Note that these results apply to R0+, but not to R+; or to C+; , since the latter are Fnoort adioni ditse. 2 K in a semi eld, let (Ae )+ = ( 1 A)+ be the normalized transitive closure of A. The lemma below allows us to change the focus from the transitive closures to the circuit structure of GA and vice-versa. Lemma 7. If K is a semi eld and A 2 Mn(K), then if ( 1 A) exists and if either Pc2Ci w(c) ( 1)l(c) e = Pc2Ci w(c) ( 1)l(c) where Ci denotes the set of circuits in CA+ containing node vi , or ( 1
A) i = ( 1
A)+i then ( 1
A) i is an eigenvector of A for eigenvalue .
Proof. See [9, Chapter 6, Corollary 2.4].
When K is a radicable semi eld, the mean of cycle c is (c) = l(cp)w(c), If the semi eld is (additively) idempotent the aggregate cycle mean of A is (A) = Pf (c) j c 2 CA+g. If the semiring is idempotent and selective, the nodes in the circuits that attain this mean are called the critical nodes of A, VAc = fi 2 c j (c) = (A)g. +
Then the critical nodes are VAc = fi 2 VA j (Ae)ii = eg, and we de ne the set of (right) fundamental eigenvectors of A for as
+ + +
FEV (A) = f(Ae) i j i 2 VAcg = f(Ae) i j (Ae)ii = eg:
The basic building block is the spectrum of irreducible matrices: Theorem 2 ((Right) spectral theory for irreducible matrices, [12]). Let A 2 Mn(K) be an irreducible matrix over a complete commutative selective radicable semi eld. Then: 1. The right spectrum of the matrix includes the whole semiring but the zero: tu (14) tu 3. If an eigenvalue is improper 2 P(A) n Pp(A), then its eigenspace (and eigenlattice) is reduced to the two vectors:
V (A) = f?n; >ng = L (A) 4. The eigenspace for a nite proper eigenvalue = (A) < > is generated from its fundamental eigenvectors over the whole semi eld, while the eigenlattice is generated by the semi eld 3 = hf?; e; >g; ; ; ?; e; >i .
V (A) = hFEV (A)iK
L (A) = hFEV (A)i3 2. The right proper spectrum only comprises the aggregate cycle mean: P(A) = K n f?g Pp(A) = f (A)g Note how this theorem introduces the notion of eigenlattices to nitely represent an eigenspace over an idempotent semi eld. Refer to [12] for further details.
We will see in our results that the only other UFNF type we need be concerned about is UFNF2: Let the partition of VA generating the permutation that renders A in UFNF2, block diagonal form, be VA = fVkgkK=1, and write A = UK
k=1 Ak, Ak = A(Vk; Vk).
Lemma 8. Let A = UK
k=1 Ak 2 Mn(S) be a matrix in UFNF2, over a semiring, and V (Ak) (U (Ak)) a right (left) eigenspace of Ak for ( ). Then, K k=1
K k=1
Note that this procedure is constructive and how the combinatorial nature of the proof in [13] makes the claim hold in any semiring. Clearly, if 2 Pp(Ak) for any k, then 2 Pp(A). Since Pp(Ak) = p(Ak) for matrices admitting an UFNF2, Pp(A) = p(A) = SkK=1 Pp(Ak).
Corollary 1. Let A 2 Mn(S) be a matrix in UFNF2 over a semiring. Then the (left) right eigenspace of A for 2 P(A) is the product of the (left) right eigenspaces for the blocks, U (A) = kK=1 U (Ak) and V (A) = kK=1 V (Ak).
In complete semirings, looking for generators for the eigenspaces with k(k) = e and k(i) = ? for k 6= i, we de ne the right fundamental eigenvectors as
K " K FEV2(A) = [ k(i)
#
FEV1(Ai) : k=1 i=1 (16)
4. FEV2;>(A) = >
FEV2(A).
Lemma 9. Let A 2 Mn(D) be a matrix in UFNF2 over a complete idempotent semiring with 2 P(A). Then, 1. If 2. If 3. If 2 Pp(A), then FEV2,f(A) = Skj 2Pp(Ak) 2 P(A) n Pp(A) then FEV2(A) = FEV2;>(A). 2 Pp(A) then FEV (A) = FEV2,f(A) [ FEV2;>(A) n > K i=1 k(i) h
FEV1,f(Ai)i.
FEV2,f(A).
So call FEV2;>(A) the saturated fundamental eigenvectors of A, and de ne the (right) saturated eigenspace as V>(A) = hFEV2;>(A)iD.
Corollary 2. Let A 2 Mn(S) be a matrix in UFNF2 over a complete selective radicable idempotent semi eld. Then 1. For 2 Pp(A), V (A) V>(A). 2. For 2 P(A) n Pp(A), V (A) = V>(A).
Notice that the very general proposition below is for all complete dioids.
2 P(A) n Pp(A), Proposition 3. Let A 2 Mn(D) be a matrix in UFNF2 over a complete dioid. Then, 1. For
U >(A) = hFEV2;>(At)i3
V>(A) = hFEV2;>(A)i3 : 2. For involving the product of the component lattices, L (A) = K k=1 L (Ak). 3
The Hubs & Authorities algorithm over dioids Let (G; M; R) be an S-formal context over a complete dioid S, and A 2 Mn(S) as in (6). Equation (6) states the solution of the singular value-singular vector problem in dioids in terms of the eigenproblem of a symmetric matrix. 3.1
The eigenproblem in symmetric matrices Since A is symmetric, we no longer have to worry about the distinction between the left and right eigenproblem:
p(A) = Pp(A) and (U (A))t = V (A) .
We can also re ne the results in Proposition 1: Proposition 4. Let S be a semiring and a symmetric A 2 Mn(S). Then: 1. The multiplicity of ? 2 Pp(A) is the number of empty rows or columns of
A . 2. If S is entire, Pp(A) = f g if and only if A = En .
Proof. First, if A = At, then the number of empty rows and empty columns is the same, and Proposition 1.2 provides the result. Second, after 1.3 Pp(A) = f g means GA has no cycles. But if Aij 6= then c = i ! j ! i is a cycle with non-null weight w(c) = Aij Aji 6= , which is a contradiction. tu Note that empty rows of R generate left eigenvalues while empty columns generate right eigenvalues, so the multiplicity of the null singular value may change from left to right.
To use Lemma 7 and Theorem 2, we need the maximum cycle mean: Proposition 5. Let K be a complete idempotent semi eld and let A 2 Sn be symmetric. Then (A) = supi;j Aij , where the sup, taken in the natural order of the semi eld is attained.
Proof. Since A is symmetric, c = i ! j ! i is a cycle wheneverAij = Aji 6= ? . Then (c) = Aij . Consider one c0 such that (c0) = supi;j Aij = maxi;j Ai;j in the order of the semiring. This must exist since i; j are nite. If we can extend any of these critical cycles with another node k such that c00 = i ! j ! k ! i then w(c00) = Aij Ajk Aki Ai3j = (c0)l(c00), so in the aggregate mean (c0) (c00) = (c0) . So we induce on the length of any cycle that is an extension of c0 that Pc2CA+ = (c0) = supi;j Ai;j . tu To nd the cycle means easily, we use the UFNF form.
UFNF forms of symmetric matrices and their closures For symmetric reducible matrices, the feasible UFNF types are simpli ed: Proposition 6. Let S be a dioid, and a symmetric A 2 Mn(S). Then: (UFNF3) A admits a proper symmetric UFNF3 form if it has zero lines, and, in that case, the set of zero lines and zero rows are the same.
P t 3
A
P3 =
A
E (UFNF2) If a A has no zero lines it can be transformed by a simultaneous row and column permutation P2 = P (A1) : : : P (Ak) into symmetric block diagonal UFNF:
P t 2
A 2A1 where fAkgkK=1; K of GA. (no UFNF1) A cannot be permuted into a proper UFNF1 form.
1 are the symmetric matrices of connected components
We will see that this is almost the only structure we need worry about. Consider R of the form:
R =
R1 R12
R21 R2 If R12 and R21 are null, then we can nd P a permutation so that Now, if R2 = EG2M2 is null, then (20) is in UFN3 with E = E(G2+M2)(G2+M2) , while if R1 and R2 are both full, then (20) is in UFN2 with blocks A1 and A2 respectively. Other other blocked forms of R simply generate an irreducible A, since the UFNF1 form is not possible. So we can suppose that A can be simultaneously row and column permuted into a diagonal block form
E with the empty lines and rows permuted to the beginning E and irreducible blocks Ak. Recall that closures and permutations commute, whence the closures of the matrices in the forms above are really simple: the closure of (21) is straightforward in terms of the closures of the blocks: (22) (23) (24) where we are using the shorthand B = Re (A) to account for the normalization with the cycle means that we need to use to nd the eigenvectors. To nd the closures of such irreducible matrices apply Lemma 2 to (23):
A e (A) + = " (Re
R (A))t e (A)#+ = (B
Bt)+ Bt (B
Bt)
B
(Bt (Bt
B) B)+ where we have used that (Bt
B)
Bt = Bt (B
Bt) . 3.3
Pairs of singular vectors of symmetric matrices over idempotent semi elds.
To extract the eigenvectors corresponding to left singular vectors i 2 G (respectively, right singular vectors j 2 M ) we need to check Theorem 1 against each block in its diagonal: i 2 G; A
e j 2 M; A e (A) (A) i j = Ae
(Bt B)+ and (B Bt)+ as we could have expected from (8). Note that after (4), the hub and authority scores are those columns such that:
(A) + requires the existence of the transitive closures (B
Bt)+i = (B
Bt) i (Bt
B)+j = (Bt
B) j but, importantly, (24) gives us the form of the authority score related to a particular hub score and vice-versa, which is a kind of formal-concept property: hi = (B aj = (Bt
Bt) i , ai = Bt B) j , hj = B (B (Bt
Bt) i B) j (25) (26) This solves completely the description of the left and right singular vectors. To nd the singular values, we note from (8) that they are the square roots of the cycle means of the independent blocks or, equally, the proper eigenvalues of A, = fp j = (Ak) ; A = kAkg = fp j = Pp(A)g This would include the bottom if and only if one of the blocks is empty. 4
Example In this section we present a HITS analysis for a weighted two-mode network both using standard HITS and HITS over the max-min-plus idempotent semi eld. The data being analysed is the example in [16, p. 31], the worries data, which is a two-mode network of the type of worry declared as most prevalent by 1554 adult Israeli depending on their living countries|and sometimes those of their parents. The graph of the network is depicted in Fig. 1.(a).
The \authority" and \hub" scores are di erentiated for each of the modes: they return \type of worry" and \procedence" scores, respectively. We can see that HITS and i-HITS produce somehow di erent results: the actual meaning of these di erences is data-dependent and a matter for future, more specialized analyses. The idempotent primitives were developed in-house and are available from the authors upon request. 5
Discussion We have generalized the HITS algorithms for semirings, then instantiated it in dioids, semi elds (including the original semi eld where it was de ned), and nally in idempotent semi elds.
More importantly, solving these problems entails solving the Singular valueSingular vector problems in the semiring of choice. We have sketched the general solution for dioids and positive semi elds, and provided a complete solution in idempotent semi elds.
(a) Worries weighted bipartite network (b) Principal worry scores (c) Procedence scores Fig. 1: Weighted, directed graphof the worries data [16, p. 31] and its weighted idempotent and standard hub and authority scores. There are clear di erences in both approaches.
Note that the original HITS problem was set in a positive semi eld that is not idempotent, hence our method of solution does not apply yet to that case, but does apply to the max-min-plus and max-min-times semi elds, examples of which are given in terms of their normalized closures. In the case of the R0+ the Perron-Frobenius theorem is invoked to solve HITS iteratively by means of the power method.
A reviewer of this paper requested a consideration of the solution of the HITS problem for the rest of dioids which are not semi elds. The problem with further generalization of our scheme is the base case for the recursion of Frobenius normal forms. In the idempotent semi eld case, the cycle means provide the eigenvalues needed for the normalization of the matrices that allow the calculation of the closures in the irreducible case. But in the generic positive semi eld case the cycle means and the possibility of choosing the critical circuits to select the eigenvectors in the closures are not granted. Also, in inclines|and in the fuzzy semirings included in them|the base, irreducible case is completely di erent to that of semi elds [9]. But since the generic development on dioids for UFNF1 and UFNF2 is based in combinatorial considerations, we believe that a solution for the base case for other dioids could be plugged into this UFNF recursion to obtain analog results to those presented here. These extensions will be considered in future work.
On another note, the SVD in standard algebra makes a strong case about the orthogonality of the left and right singular vectors belonging to di erent singular values in order to guarantee certain properties of the bases of singular vectors in the reconstruction. But in entire zerosumfree semirings, and in entire dioids or positive semi elds a fortiori, orthogonality is a rare phenomenon, after Lemma 6.
Indeed, irreducible matrices do not have any orthogonal, but rather collinear left or right singular vectors. Regarding reducible matrices, note that (21) factors in all the possible orthogonality between eigenvectors. In fact we have, Corollary 3. Let (G; M; R) be an S-formal context over a dioid S, and A 2 Mn(S) as in (24). Then two of the eigenvectors for A or (left, right) singular vectors for R can only be orthogonal only if they arise from di erent blocks. and Proposition 3 proves that even in that case they might not be orthogonal.
Regarding the glimpse of a formal concept-like property of (25), a cursory analysis with the techniques in [10], shows that the analogue of the polars in FCA actually come from two di erent generalised Galois connections, so their order-properties are not the same. Yet, the isomorphism between the ranges of the operators in (5) de ning the duality between hubs and authorities is clear and attested in a more generic context than HITS was initially conceived.
Bibliography [1] Strang, G.: The fundamental theorem of linear algebra. The American
Mathematical Monthly 100 (1993) 848{855 [2] Everett, M.G., Borgatti, S.P.: The dual-projection approach for two-mode networks. Social networks 35 (2013) 204{210 [3] Kleinberg, J.M.: Authoritative sources in a hyperlinked environment. Journal of the ACM 46 (1999) 604{632 [4] Easley, D.A., Kleinberg, J.M.: Networks, Crowds, and Markets - Reasoning
About a Highly Connected World. Cambridge University Press 2010 (2010) [5] Lanczos, C.: Linear Di erential Operators. Dover Publications (1997) [6] Pouly, M., Kohlas, J.: Generic Inference. A Unifying Theory for Automated
Reasoning. John Wiley & Sons (2012) [7] Golan, J.S.: Power Algebras over Semirings. With Applications in Mathematics and Computer Science. Volume 488 of Mathematics and its applications. Kluwer Academic, Dordrecht, Boston, London (1999) [8] Golan, J.S.: Semirings and their Applications. Kluwer Academic (1999) [9] Gondran, M., Minoux, M.: Graphs, Dioids and Semirings. New Models and Algorithms. Operations research/Computer Science Interfaces. Springer (2008) [10] Valverde-Albacete, F.J., Pelaez-Moreno, C.: Extending conceptualisation modes for generalised Formal Concept Analysis. Information Sciences 181 (2011) 1888{1909 [11] Gondran, M., Minoux, M.: Valeurs propres et vecteurs propres dans les diodes et leur interpretation en theorie des graphes. EDF, Bulletin de la Direction des Etudes et Recherches, Serie C, Mathematiques Informatique 2 (1977) 25{41 [12] Valverde-Albacete, F.J., Pelaez-Moreno, C.: Spectral lattices of irreducible matrices over completed idempotent semi elds. Fuzzy Sets and Systems (2014) [13] Valverde Albacete, F.J., Pelaez-Moreno, C.: The spectra of reducible matrices over completed commutative idempotent semi elds and their spectral lattices. International Journal of General Systems (Accepted for publication.) [14] Valverde-Albacete, F.J., Pelaez-Moreno, C.: Spectral lattices of reducible matrices over completed idempotent semi elds. In Ojeda-Aciego, M., Outrata, J., eds.: Concept Lattices and Applications (CLA 2013), Universite de la Rochelle, Laboratory L31, La Rochelle (2013) 211{224 [15] Cohen, G., Gaubert, S., Quadrat, J.P.: Duality and separation theorems in idempotent semimodules. Linear Algebra and Its Applications 379 (2004) 395{422 [16] Mirkin, B.: Mathematical Classi cation and Clustering. Volume 11 of Nonconvex Optimization and Its Applications. Kluwer Academic Publishers (1996)