One-alpha weighted network descriptors

Complex networks are often used to model objects and their relations. Network descriptors are graph-theoretical invariants assigned to graphs that correspond to complex networks. Transmission and betweenness centrality are well known network descriptors and networkness and network surplus have been recently analyzed. All these four descriptors are based on the unrealistic assumption about equal communication between all vertices. Here, we amend this by assuming that vertices on the distance larger then one communicate less than those that are neighbors. We analyze network descriptors for all possible values of the factor that measures reduction in the communication of the vertices that are not neighbors. We term these descriptors one-alpha descriptors and determine their extremal values

Vertex spans of multilayered cycles

In this paper we are defining a special class of graphs called multilayered graphs and its subclass, multilayered cycles. For that subclass of graphs we are giving the values of all vertex spans (strong, direct, or Cartesian span). Surprisingly, our results reveal that, irrespective of the chosen movement rules, the span values only depend on the length of the individual cycles, not the number of layers, which holds significant implications

Exponential Generalised Network Descriptors

In communication networks theory the concepts of networkness and network surplus have recently been defined. Together with transmission and betweenness centrality, they were based on the assumption of equal communication between vertices. Generalised versions of these four descriptors were presented, taking into account that communication between vertices $u$ and $v$ is decreasing as the distance between them is increasing. Therefore, we weight the quantity of communication by $\lambda^{d(u,v)}$ where $\lambda \in \left\langle0,1 \right\rangle$. Extremal values of these descriptors are analysed.Comment: 17 pages, 1 figur

Complex networks, network descriptors and safety in networks

U ovoj disertaciji izložena su istraživanja iz nekoliko područja teorije kompleksnih mreža. Definirane su poopćene verzije mrežnih deskriptora, kao što su transmisija, međupoloženost, vršna produktivnost i vršna profitabilnost koje uzimaju u obzir pretpostavku da u mreži vrhovi na manjim udaljenostima komuniciraju znatno više nego oni na većim udaljenostima. Proučavane su minimalne i maksimalne vrijednosti tih deskriptora i analizirane gornje i donje ograde tih vrijednosti. Nadalje, predložena je modificirana verzija Girvan-Newmanovog algoritma za detektiranje zajednica u mrežama, koja smanjuje broj operacija i dovodi do bržeg uočavanja strukture zajednica. U posljednjem dijelu su analizirane mreže s distribuiranim ključevima i proučavana njihova sigurnost na napad neprijateljskih agenata. Uz dvije različite pretpostavke o djelovanju agenata na mrežu određuju se minimalni brojevi vrhova u mreži i ključeva potrebnih da bi mreža bila sigurna.In this thesis several areas of theory of complex networks are explored. Generalized versions of network descriptors such as transmission, betweenness centrality, networkness and network surplus, which assume that the ammount of communication in the network is greater between vertices which are at smaller distances than that that are on greater distances, are defined. Minimal and maximal values of these descriptors are studied and lower and upper bounds are obtained. Further, a modified version of Girvan-Newman algorithm for community detection is proposed, which reduces the number of operations compared to the original and leads to faster community detection. In the last part, networks with distributed keys are analyzed and their safety under the attack of enemy agents is studied. Under two different assumptions on the behavior of agents in the network, minimal number of vertices in the network and minimal number of distributed keys needed to secure the network, are determined

Multicoloring of graphs to secure a secret

Vertex coloring and multicoloring of graphs are a well known subject in graph theory, as well as their applications. In vertex multicoloring, each vertex is assigned some subset of a given set of colors. Here we propose a new kind of vertex multicoloring, motivated by the situation of sharing a secret and securing it from the actions of some number of attackers. We name the multicoloring a highly a-resistant vertex k-multicoloring, where a is the number of the attackers, and k the number of colors. For small values a we determine what is the minimal number of vertices a graph must have in order to allow such a coloring, and what is the minimal number of colors needed

Multicoloring of Graphs to Secure a Secret

Vertex coloring and multicoloring of graphs are a well known subject in graph theory, as well as their applications. In vertex multicoloring, each vertex is assigned some subset of a given set of colors. Here we propose a new kind of vertex multicoloring, motivated by the situation of sharing a secret and securing it from the actions of some number of attackers. We name the multicoloring a highly $a$-resistant vertex $k$-multicoloring, where $a$ is the number of the attackers, and $k$ the number of colors. For small values $a$ we determine what is the minimal number of vertices a graph must have in order to allow such a coloring, and what is the minimal number of colors needed.Comment: 19 pages, 5 figure

Groups (S_{n}times S_{m}) in construction of flag-transitive block designs

In this paper, we observe the possibility that the group (S_{n}times S_{m}) acts as a flag-transitive automorphism group of a block design with point set ({1,ldots ,n}times {1,ldots ,m},4leq nleq mleq 70). We prove the equivalence of that problem to the existence of an appropriately defined smaller flag-transitive incidence structure. By developing and applying several algorithms for the construction of the latter structure, we manage to solve the existence problem for the desired designs with (nm) points in the given range. In the vast majority of the cases with confirmed existence, we obtain all possible structures up to isomorphism

Constructing flag-transitive incidence structures

The aim of this research is to develop efficient techniques to construct flag-transitive incidence structures. In this paper we describe those techniques, present the construction results and take a closer look at how some types of flag-transitive incidence structures relate to arctransitive graphs

Generalised network descriptors

Transmission and betweenness centrality are key concepts in communication networks theory. Based on this concept, new concepts of networkness and network surplus have recently been defined. However, all these four concepts include unrealistic assumption about equal communication between vertices. Here, we propose more realistic assumption that the amount of communication of vertices decreases as their distance increases. We assume that amount of communication between vertices u and v is proportional to d(u,v)Λ where Λ < 0. Taking this into account generalised versions of these four descriptors are defined. Extremal values of these descriptors are analysed