Sanming Zhou

Quotient FCMs-a decomposition theory for fuzzy cognitive maps

In this paper, we introduce a decomposition theory for fuzzy cognitive maps (FCM). First, we partition the set of vertices of an FCM into blocks according to an equivalence relation, and by regarding these blocks as vertices we construct a quotient FCM. Second, each block induces a natural sectional FCM of the original FCM, which inherits the topological structure as well as the inference from the original FCM. In this way, we decompose the original FCM into a quotient FCM and some sectional FCM. As a result, the analysis of the original FCM is reduced to the analysis of the quotient and sectional FCM, which are often much smaller in size and complexity. Such a reduction is important in analyzing large-scale FCM. We also propose a causal algebra in the quotient FCM, which indicates that the effect that one vertex influences another in the quotient depends on the weights and states of the vertices along directed paths from the former to the latter. To illustrate the process involved, we apply our decomposition theory to university management networks. Finally, we discuss possible approaches to partitioning an FCM and major concerns in constructing quotient FCM. The results represented in this paper provide an effective framework for calculating and simplifying causal inference patterns in complicated real-world applications.

A Class of Arc-Transitive Cayley Graphs as Models for Interconnection Networks

We study a class of Cayley graphs as models for interconnection networks. With focus on efficient communication we prove that for any graph in the class there exists a gossiping protocol which exhibits attractive features, and, moreover, we give an algorithm for constructing such a protocol. In particular, these hold for two important subclasses of graphs, namely, Cayley graphs admitting a complete rotation and Frobenius graphs of a certain type. For such Frobenius graphs, we obtain the minimum gossip time and give an optimal gossiping protocol under which messages are transmitted along shortest paths and each arc is used exactly once at each time step. Moreover, for such Frobenius graphs we construct an all-to-all shortest path routing that is arc-transitive, edge- and arc-uniform, and optimal for the edge- and arc-forwarding indices simultaneously.

A class of finite symmetric graphs with 2-arc transitive quotients

Let Γ be a finite G-symmetric graph whose vertex set admits a non-trivial Ginvariant partition B with block size v. A framework for studying such graphs Γ was developed by Gardiner and Praeger which involved an analysis of the quotient graph ΓB relative to B, the bipartite subgraph Γ[B,C] of Γ induced by adjacent blocks B,C of ΓB and a certain 1-design D(B) induced by a block B ∈ B. The present paper studies the case where the size k of the blocks of D(B) satisfies k = v − 1. In the general case, where k = v − 1 > 2, the setwise stabilizer GB is doubly transitive on B and G is faithful on B. We prove that D(B) contains no repeated blocks if and only if ΓB is (G, 2)-arc transitive and give a method for constructing such a graph from a 2-arc transitive graph with a self-paired orbit on 3-arcs. We show further that each such graph may be constructed by this method. In particular every 3-arc transitive graph, and every 2-arc transitive graph of even valency, may occur as ΓB for some graph Γ with these properties. We prove further that Γ[B,C]%Kv−1,v−1 if and only if ΓB is (G, 3)-arc transitive.

Fuzzy causal networks: general model, inference, and convergence

In this paper, we first propose a general framework for fuzzy causal networks (FCNs). Then, we study the dynamics and convergence of such general FCNs. We prove that any general FCN with constant weight matrix converges to a limit cycle or a static state, or the trajectory of the FCN is not repetitive. We also prove that under certain conditions a discrete state general FCN converges to its limit cycle or static state in O(n) steps, where n is the number of vertices of the FCN. This is in striking contrast with the exponential running time 2/sup n/, which is accepted widely for classic FCNs.

Finite symmetric graphs with two-arc transitive quotients

This paper forms part of a study of 2-arc transitivity for finite imprimitive symmetric graphs, namely for graphs Γ admitting an automorphism group G that is transitive on ordered pairs of adjacent vertices, and leaves invariant a nontrivial vertex partition B. Such a group G is also transitive on the ordered pairs of adjacent vertices of the quotient graph ΓB corresponding to B. If in addition G is transitive on the 2-arcs of Γ (that is, on vertex triples (α, β γ) such that α, β and β, γ are adjacent and α ≠ γ), then G is not in general transitive on the 2-arcs of ΓB, although it does have this property in the special case where B is the orbit set of a vertex-intransitive normal subgroup of G. On the other hand, G is sometimes transitive on the 2-arcs of ΓB even if it is not transitive on the 2-arcs of Γ. We study conditions under which G is transitive on the 2-arcs of ΓB. Our conditions relate to the structure of the bipartite graph induced on B ∪ C for adjacent blocks B, C of B and a graph structure induced on B. We obtain necessary and sufficient conditions for G to be transitive on the 2-arcs of one or both of Γ, ΓB, for certain families of imprimitive symmetric graphs.

Networking for Big Data: A Survey

Complementary to the fancy big data applications, networking for big data is an indispensable supporting platform for these applications in practice. This emerging research branch has gained extensive attention from both academia and industry in recent years. In this new territory, researchers are facing many unprecedented theoretical and practical challenges. We are therefore motivated to solicit the latest works in this area, aiming to pave a comprehensive and solid starting ground for interested readers. We first clarify the definition of networking for big data based on the cross disciplinary nature and integrated needs of the domain. Second, we present the current understanding of big data from different levels, including its formation, networking features, mathematical representations, and the networking technologies. Third, we discuss the challenges and opportunities from various perspectives in this hopeful field. We further summarize the lessons we learned based on the survey. We humbly hope this paper will shed light for forthcoming researchers to further explore the uncharted part of this promising land.

Constructing a Class of Symmetric Graphs

We find a natural construction of a large class of symmetric graphs from point- and block-transitive 1-designs. The graphs in this class can be characterized as G -symmetric graphs whose vertex sets admit a G -invariant partition B of block size at least 3 such that, for any two blocks B, C of B, either there is no edge between B and C, or there exists only one vertex in B not adjacent to any vertex inC . The special case where the quotient graph ?Bof ? relative to B is a complete graph occurs if and only if the 1-design needed in the construction is a G -doubly transitive and G -block-transitive 2-design, and in this case we give an explicit classification of ? when G is a doubly transitive projective group or an affine group containing the affine general group. Examples of such graphs include cross ratio graphs studied recently by Gardiner, Praeger and Zhou and some other graphs with vertices the (point, line)-flags of the projective or affine geometry.

Almost covers of 2-arc transitive graphs

Let Γ be a G-symmetric graph whose vertex set admits a nontrivial G-invariant partition ℬ with block size v. Let Γ ℬ be the quotient graph of Γ relative to ℬ and Γ[B,C] the bipartite subgraph of Γ induced by adjacent blocks B,C of ℬ. In this paper we study such graphs for which Γ ℬ is connected, (G, 2)-arc transitive and is almost covered by Γ in the sense that Γ[B,C] is a matching of v-1 ≥ 2 edges. Such graphs arose as a natural extremal case in a previous study by the author with Li and Praeger. The case Γ ℬ ≅K v+1 is covered by results of Gardiner and Praeger. We consider here the general case where Γ ℬ ≇K v+1, and prove that, for some even integer n ≥ 4, Γ ℬ is a near n-gonal graph with respect to a certain G-orbit on n-cycles of Γ ℬ. Moreover, we prove that every (G, 2)-arc transitive near n-gonal graph with respect to a G-orbit on n-cycles arises as a quotient Γ ℬ of a graph with these properties. (A near n-gonal graph is a connected graph Σ of girth at least 4 together with a set ℰ of n-cycles of Σ such that each 2-arc of Σ is contained in a unique member of ℰ.)

Dynamic domination in fuzzy causal networks

This paper presents a dynamic domination theory for fuzzy causal networks (FCN). There are three major contributions. First, we propose a new inference procedure based on dominating sets. Second, we introduce the concepts of dynamic and minimal dynamic dominating sets (DDS and MDDS) in an FCN. To reflect changes of dominance with time, we also introduce the concept of a dynamic dominating process (DDP) that has significant implications in many real-world problems. We pay a special attention to the minimal dynamic dominating process (MDDP) and develop rules for generating DDP and MDDP. Third, we investigate dynamic dominating sets with extended feedback, which we call effective dynamic dominating sets (EDDS), and related effective dynamic dominating process (EDDP). This study unveils a very important phenomenon in FCN: At any time t, either an EDDS exists or there is a dramatic change of the states of vertices. In the latter case we also identify the special structure of the sub-FCN induced by active vertices.

Labelling Cayley Graphs on Abelian Groups

For given integers