Maximal Spreads Research Articles

A probability-driven structure-aware algorithm for influence maximization under independent cascade model

Influence maximization (IM) is the problem of finding a set of nodes that can achieve the maximal influence spreads into the network, which faces two significant but intractable issues in latest studies: (i) Curse of scales: with the increase of the network scale, traditional methods cost extensive times in guaranteeing accuracy, which re-evaluate influence spread of every node in network, leading to significant computational overhead; (ii) Generalization issue: with more and more studies on various networks and propagation parameters, it is difficult to find a universally appropriate algorithm that performs well in each topology. In this paper, we propose a novel probability-driven structure-aware (PDSA) algorithm, which begins by cutting/updating network according to the edge activation probability parameters of the IC model, and then uses a graph traversal algorithm (e.g., breadth first search algorithm) to evaluate the influence spread scores of each node. Meanwhile, we adopt a kind of centrality-based independent cascade (CIC) model to approximate a more realistic propagation scenario. Through extensive experiments with six real-world/synthetic networks and six CIC/IC models, we demonstrate that PDSA achieves great performance over state-of-the-art algorithms in terms of effect and efficiency. Even facing various complex topologies and propagation parameters, PDSA exhibits excellent robustness in solving IM problems.

On classifying objects with specified groups of automorphisms, friendly subgroups, and Sylow tower groups

We describe some group theory which is useful in the classification of combinatorial objects having given groups of automorphisms. In particular, we show the usefulness of the concept of a friendly subgroup: a subgroup H of a group K is a friendly subgroup of K if every subgroup of K isomorphic to H is conjugate in K to H . We explore easy-to-test sufficient conditions for a subgroup H to be a friendly subgroup of a finite group K , and for this, present an algorithm for determining whether a finite group H is a Sylow tower group. We also classify the maximal partial spreads invariant under a group of order 5 in both PG(3,7) and PG (3,8).

On maximal symplectic partial spreads

Abstract New types of maximal symplectic partial spreads are constructed.

Maximal Partial Spreads of Polar Spaces

Some constructions of maximal partial spreads of finite classical polar spaces are provided. In particular we show that, for $n \ge 1$, $\mathcal{H}(4n-1,q^2)$ has a maximal partial spread of size $q^{2n}+1$, $\mathcal{H}(4n+1,q^2)$ has a maximal partial spread of size $q^{2n+1}+1$ and, for $n \ge 2$, $\mathcal{Q}^+(4n-1,q)$, $\mathcal{Q}(4n-2,q)$, $\mathcal{W}(4n-1,q)$, $q$ even, $\mathcal{W}(4n-3,q)$, $q$ even, have a maximal partial spread of size $q^n+1$.

Tight sets in finite classical polar spaces

Abstract We show that every i-tight set in the Hermitian variety H(2r + 1, q) is a union of pairwise disjoint (2r + 1)-dimensional Baer subgeometries PG ( 2 r + 1 , q ) $\text{PG}(2r+1,\,\sqrt{q})$ and generators of H(2r + 1, q), if q ≥ 81 is an odd square and i < (q 2/3 − 1)/2. We also show that an i-tight set in the symplectic polar space W(2r + 1, q) is a union of pairwise disjoint generators of W(2r + 1, q), pairs of disjoint r-spaces {Δ, Δ ⊥}, and (2r + 1)-dimensional Baer subgeometries. For W(2r + 1, q) with r even, pairs of disjoint r-spaces {Δ, Δ ⊥} cannot occur. The (2r + 1)-dimensional Baer subgeometries in the i-tight set of W(2r + 1, q) are invariant under the symplectic polarity ⊥ of W(2r + 1, q) or they arise in pairs of disjoint Baer subgeometries corresponding to each other under ⊥. This improves previous results where i < q 5 / 8 / 2 + 1 $i \lt q^{5/8} / \sqrt{2} +1$ was assumed. Generalizing known techniques and using recent results on blocking sets and minihypers, we present an alternative proof of this result and consequently improve the upper bound on i to (q 2/3 − 1)/2. We also apply our results on tight sets to improve a known result on maximal partial spreads in W(2r + 1, q).

Constant Rank-Distance Sets of Hermitian Matrices and Partial Spreads in Hermitian Polar Spaces

In this paper we investigate partial spreads of $H(2n-1,q^2)$ through the related notion of partial spread sets of hermitian matrices, and the more general notion of constant rank-distance sets. We prove a tight upper bound on the maximum size of a linear constant rank-distance set of hermitian matrices over finite fields, and as a consequence prove the maximality of extensions of symplectic semifield spreads as partial spreads of $H(2n-1,q^2)$. We prove upper bounds for constant rank-distance sets for even rank, construct large examples of these, and construct maximal partial spreads of $H(3,q^2)$ for a range of sizes.

Maximal partial line spreads of non-singular quadrics

For $$n \ge 9$$ , we construct maximal partial line spreads for non-singular quadrics of $$PG(n,q)$$ for every size between approximately $$(cn+d)(q^{n-3}+q^{n-5})\log {2q}$$ and $$q^{n-2}$$ , for some small constants $$c$$ and $$d$$ . These results are similar to spectrum results on maximal partial line spreads in finite projective spaces by Heden, and by Gacs and Sz?nyi. These results also extend spectrum results on maximal partial line spreads in the finite generalized quadrangles $$W_3(q)$$ and $$Q(4,q)$$ by Pepe, Roβing and Storme.

Sets of generators blocking all generators in finite classical polar spaces

We introduce generator blocking sets of finite classical polar spaces. These sets are a generalisation of maximal partial spreads. We prove a characterization of these minimal sets of the polar spaces Q(2n,q), Q−(2n+1,q) and H(2n,q2), in terms of cones with vertex a subspace contained in the polar space and with base a generator blocking set in a polar space of rank 2.

A SURVEY OF THE DIFFERENT TYPES OF VECTOR SPACE PARTITIONS

A vector space partition is here a collection [Formula: see text] of subspaces of a finite vector space V(n, q), of dimension n over a finite field with q elements, with the property that every non-zero vector is contained in a unique member of [Formula: see text]. Vector space partitions relate to finite projective planes, design theory and error correcting codes. In the first part of the paper I will discuss some relations between vector space partitions and other branches of mathematics. The other part of the paper contains a survey of known results on the type of a vector space partition, more precisely: the theorem of Beutelspacher and Heden on T-partitions, rather recent results of El-Zanati et al. on the different types that appear in the spaces V(n, 2), for n ≤ 8, a result of Heden and Lehmann on vector space partitions and maximal partial spreads including their new necessary condition for the existence of a vector space partition, and furthermore, I will give a theorem of Heden on the length of the tail of a vector space partition. Finally, I will also give a few historical remarks.

The dimension of a subplane of a translation plane

It is shown that the commutative binary Knuth semifield planes of order $2^{n}$, for $n=5k~$ or $7k$ and $k$ odd, their transposes and transpose-duals admit subplanes of order $2^{2}$. In addition, many of the Kantor commutative semifield planes of order $2^{5k}$ or $2^{7k}$ also admit subplanes of order $4$. Furthermore, a large number of maximal partial spreads of order $p^{k}$ and deficiency at least $p^{k}-p^{k-1}$ or translation planes of order $p^{k}$ are constructed using direct sums of matrix spreads sets of different dimensions. Given any translation plane $\pi_{0}$ of order $p^{d}$, there is either a proper maximal partial spread of order $p^{c+d}$ whose associated translation net contains a subplane of order $p^{d}$ isomorphic to $\pi_{0}$ or there is a translation plane of order $p^{c+d}$ admitting a subplane of order $p^{d}$. Other than the semifield planes mentioned above and a few sporadic planes of even order, there are no other known translation planes of order $p^{c+d}$ admitting a subplane of order $p^{d}$, where $d$ does not divide $c$.

Small maximal partial spreads in classical finite polar spaces

Abstract We prove lower bounds on the size of small maximal partial spreads in Q +(4n + 1, q), W(2n + 1, q), and H(2n + 1, q 2). This research on the size of smallest maximal partial spreads in classical finite polar spaces is part of a detailed study on small and large maximal partial ovoids and spreads in classical finite polar spaces, performed in [De Beule, Klein, Metsch, Storme, Des. Codes Cryptogr 47: 21–34, 2008, De Beule, Klein, Metsch, Storme, European J. Combin 29: 1280–1297, 2008].

Random constructions and density results

In this paper we outline a construction method which has been used for minimal blocking sets in PG(2, q) and maximal partial line spreads in PG(n, q) and which must have a lot of more applications. We also give a survey on what is known about the spectrum of sizes of maximal partial line spreads in PG(n, q). At the end we list some more elaborate random techniques used in finite geometry.

Partial ovoids and partial spreads in symplectic and orthogonal polar spaces

We present improved lower bounds on the sizes of small maximal partial ovoids and small maximal partial spreads in the classical symplectic and orthogonal polar spaces, and improved upper bounds on the sizes of large maximal partial ovoids and large maximal partial spreads in the classical symplectic and orthogonal polar spaces. An overview of the status regarding these results is given in tables. The similar results for the hermitian classical polar spaces are presented in [J. De Beule, A. Klein, K. Metsch, L. Storme, Partial ovoids and partial spreads in hermitian polar spaces, Des. Codes Cryptogr. (in press)].

On maximal partial spreads of [formula omitted

A lower bound for the size of a maximal partial spread of H ( 2 n + 1 , q 2 ) is given. For H ( 2 n + 1 , q 2 ) in general, and for H ( 5 , q 2 ) in particular, new upper bounds for this size are also obtained. In [A. Aguglia, A. Cossidente, G.L. Ebert, Complete spans on Hermitian varieties, in: Proceedings of the Conference on Finite Geometries (Oberwolfach, 2001), vol. 29, 2003, pp. 7–15.], maximal partial spreads of H ( 3 , q 2 ) and H ( 5 , q 2 ) have been constructed from spreads of W 3 ( q ) and W 5 ( q ) , respectively; the construction for H ( 5 , q 2 ) will be generalized to H ( 4 n + 1 , q 2 ) , n ⩾ 1 , thus yielding examples of maximal partial spreads of H ( 4 n + 1 , q 2 ) for all n ⩾ 1 .

Partial ovoids and partial spreads in hermitian polar spaces

We present improved lower bounds on the sizes of small maximal partial ovoids in the classical hermitian polar spaces, and improved upper bounds on the sizes of large maximal partial spreads in the classical hermitian polar spaces. Of particular importance is the presented upper bound on the size of a maximal partial spread of H(3,q 2). For q = 2,3, the presented upper bound is sharp. For q = 3, our results confirm via theoretical arguments properties, deduced by computer searches performed by Ebert and Hirschfeld, for the largest partial spreads of H(3,9). An overview of the status regarding these results is given in two summarizing tables. The similar results for the classical symplectic and orthogonal polar spaces are presented in De Beule et al. [8].

Maximal partial ovoids and maximal partial spreads in hermitian generalized quadrangles

AbstractMaximal partial ovoids and maximal partial spreads of the hermitian generalized quadrangles H(3,q2) and H(4,q2) are studied in great detail. We present improved lower bounds on the size of maximal partial ovoids and maximal partial spreads in the hermitian quadrangle H(4,q2). We also construct in H(3,q2), q=22h+1, h≥ 1, maximal partial spreads of size smaller than the size q2+1 presently known. As a final result, we present a discrete spectrum result for the deficiencies of maximal partial spreads of H(4,q2) of small positive deficiency δ. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 101–116, 2008

On the smallest maximal partial ovoids and spreads of the generalized quadrangles W(q) and Q(4,q)

We present results on the size of the smallest maximal partial ovoids and on the size of the smallest maximal partial spreads of the generalized quadrangles W ( q ) and Q ( 4 , q ) .

The maximum size of a partial spread in [formula omitted] is [formula omitted

In this paper, we show that the largest maximal partial spreads of the hermitian variety H ( 5 , q 2 ) consist of q 3 + 1 generators. Previously, it was only known that q 4 is an upper bound for the size of these partial spreads. We also show for q ⩾ 7 that every maximal partial spread of H ( 5 , q 2 ) contains at least 2 q + 3 planes. Previously, only the lower bound q + 1 was known.

Maximal partial spreads in PG (3, q)

We transfer the whole geometry of PG(3, q) over a non-singular quadric Q4,q of PG(4, q) mapping suitably PG(3, q) over Q4,q. More precisely the points of PG(3, q) are the lines of Q4,q; the lines of PG(3, q) are the tangent cones of Q4,q and the reguli of the hyperbolic quadrics hyperplane section of Q4,q. A plane of PG(3, q) is the set of lines of Q4,q meeting a fixed line of Q4,q. We remark that this representation is valid also for a projective space \({\mathbb{P}}_{{3,{\mathbf{k}}}} \) over any field K and we apply the above representation to construct maximal partial spreads \({\mathfrak{F}}\) in PG(3, q). For q even we get new cardinalities for \({\mathfrak{F}}.\) For q odd the cardinalities are partially known.

Searching for maximal partial ovoids and spreads in generalized quadrangles

Searching for maximal partial ovoids and spreads in generalized quadrangles