65 research outputs found
On dynamic monopolies of graphs: the average and strict majority thresholds
Let be a graph and
be an assignment of thresholds to the vertices of . A subset of vertices
is said to be a dynamic monopoly corresponding to if the vertices
of can be partitioned into subsets such that
and for any , each vertex in has at least
neighbors in . Dynamic monopolies are in fact
modeling the irreversible spread of influence in social networks. In this paper
we first obtain a lower bound for the smallest size of any dynamic monopoly in
terms of the average threshold and the order of graph. Also we obtain an upper
bound in terms of the minimum vertex cover of graphs. Then we derive the upper
bound for the smallest size of any dynamic monopoly when the graph
contains at least one odd vertex, where the threshold of any vertex is set
as (i.e. strict majority threshold). This bound
improves the best known bound for strict majority threshold. We show that the
latter bound can be achieved by a polynomial time algorithm. We also show that
is an upper bound for the size of strict majority dynamic
monopoly, where stands for the matching number of . Finally, we
obtain a basic upper bound for the smallest size of any dynamic monopoly, in
terms of the average threshold and vertex degrees. Using this bound we derive
some other upper bounds
NÚMERO ENVOLTÓRIO NA CONVEXIDADE P3: RESULTADOS E APLICAÇÕES
Este artigo apresenta uma revisão sistemática da literatura sobre os resultados e aplicações do número envoltório na convexidade P3 em grafos. A determinação deste parâmetro é equivalente ao problema de se encontrar o menor número de vértices de um grafo que permitam disseminar uma informação, influência, ou contaminação, para todos os vértices do grafo. Em particular, esta revisão descreve um panorama sobre estudos teóricos e aplicados acerca do número envoltório P3 considerando a modelagem de fenômenos sociais. Os resultados mostram que o parâmetro é pouco explorado em sociologia computacional para a modelagem de fenômenos sociais. Por outro lado, com o surgimento das redes sociais, pesquisas teóricas têm sido impulsionadas nas últimas décadas. Pesquisadores têm direcionado esforços com o objetivo de contribuir para a solução de problemas relacionados à influência social e disseminação de informação. Entretanto, ainda há espaço para estudos envolvendo o número envoltório na convexidade P3
Irreversible k-threshold conversion number of some graphs
Purpose – This paper aims to study Irreversible conversion processes, which examine the spread of a one way change of state (from state 0 to state 1) through a specified society (the spread of disease through populations, the spread of opinion through social networks, etc.) where the conversion rule is determined at the beginning of the study. These processes can be modeled into graph theoretical models where the vertex set V(G) represents the set of individuals on which the conversion is spreading. Design/methodology/approach – The irreversible k-threshold conversion process on a graph G=(V,E) is an iterative process which starts by choosing a set S_0?V, and for each step t (t = 1, 2,…,), S_t is obtained from S_(t−1) by adjoining all vertices that have at least k neighbors in S_(t−1). S_0 is called the seed set of the k-threshold conversion process and is called an irreversible k-threshold conversion set (IkCS) of G if S_t = V(G) for some t = 0. The minimum cardinality of all the IkCSs of G is referred to as the irreversible k-threshold conversion number of G and is denoted by C_k (G). Findings – In this paper the authors determine C_k (G) for generalized Jahangir graph J_(s,m) for 1 < k = m and s, m are arbitraries. The authors also determine C_k (G) for strong grids P_2? P_n when k = 4, 5. Finally, the authors determine C_2 (G) for P_n? P_n when n is arbitrary. Originality/value – This work is 100% original and has important use in real life problems like Anti-Bioterrorism
Dissipative, Entropy-Production Systems across Condensed Matter and Interdisciplinary Classical VS. Quantum Physics
The thematic range of this book is wide and can loosely be described as polydispersive. Figuratively, it resembles a polynuclear path of yielding (poly)crystals. Such path can be taken when looking at it from the first side. However, a closer inspection of the book’s contents gives rise to a much more monodispersive/single-crystal and compacted (than crudely expected) picture of the book’s contents presented to a potential reader. Namely, all contributions collected can be united under the common denominator of maximum-entropy and entropy production principles experienced by both classical and quantum systems in (non)equilibrium conditions. The proposed order of presenting the material commences with properly subordinated classical systems (seven contributions) and ends up with three remaining quantum systems, presented by the chapters’ authors. The overarching editorial makes the presentation of the wide-range material self-contained and compact, irrespective of whether comprehending it from classical or quantum physical viewpoints
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