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

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

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

Network descriptors and curriculum networks

U disertaciji su izloženi rezultati istraživanja iz više područja teorije mreža. U prvom dijelu proučavane su ekstremalne vrijednosti poopćenih mrežnih deskriptora transmisije, međupoloženosti, vršne produktivnosti i vršne profitabilnosti. U dva promatrana slučaja uzeta je u obzir pretpostavka da vrhovi u mreži koji se nalaze na većim udaljenostima komuniciraju manje od onih na manjim udaljenostima. U prvom slučaju količina komunikacije medu vrhovima utežena je s $d(u, v)^\lambda$ za $\lambda < 0$, a u drugom slučaju s $\lambda^{d(u,v)}$ za $\lambda \in (0, 1)$, pri čemu je $d(u, v)$ udaljenost između vrhova $u$ i $v$. Analizirane su gornje i donje ograde vrijednosti deskriptora. Nadalje, definirane su kurikulne mreže i analizirana neka njihova svojstva. Riječ je o jednostavnim usmjerenim grafovima u kojima vrhovi predstavljaju edukacijske jedinice, a usmjereni brid između dva vrha označava da je poznavanje jedne jedinice potrebno za učenje druge. Definirane su mjere pomoću kojih je moguće napraviti evaluaciju valjanosti redoslijeda edukacijskih jedinica. Mjere su analizirane s više različitih gledišta i određeni su grafovi koji odgovaraju najmanje i najviše složenim nastavnim planovima. Predložen je algoritam za određivanje optimalne ekspozicije edukacijskih jedinica u odnosu na odabranu mjeru složenosti. Konačno, detaljno je analiziran problem detekcije zajednica u kurikulnim mrežama i predloženi su algoritmi za rješavanje tog problema.In this thesis results of research from several different areas of complex networks theory have been introduced. First, generalized versions of network descriptors such as transmission, betweeness centrality, networkness and network surplus have been defined and analyzed. Two different approaches that were studied both take into account the assumption that vertices in networks, which are closer to each other, communicate more than the vertices on greater distances. In the first case the amount of communication between vertices has been amended by $d(u, v)^\lambda$where $\lambda < 0$ and in the second case by $\lambda^{d(u,v)}$, where $d(u, v)$ represents the distance between vertices $u$ and $v$, and $\lambda \in (0, 1)$. Upper and lower bounds for minimal and maximal values of these descriptors have been analyzed. Furthermore, special kind of networks has been defined. Curriculum network is a simple, directed graph in which vertices represent educational units, and directed arc from $u$ to $v$ indicates that understanding of a unit $u$ is necessary for learning and understanding of unit $v$. Measures for evaluating the quality of curriculum content sequencing have been proposed and analyzed from sever different points of view. Graphs corresponding to most and least complex curricula have been found. Some real-world curriculum networks and their properties have been analyzed. Finally, the problem of community detection in curriculum networks has been analyzed and two algorithms for solving this problem have been suggested

Network descriptors and curriculum networks

U disertaciji su izloženi rezultati istraživanja iz više područja teorije mreža. U prvom dijelu proučavane su ekstremalne vrijednosti poopćenih mrežnih deskriptora transmisije, međupoloženosti, vršne produktivnosti i vršne profitabilnosti. U dva promatrana slučaja uzeta je u obzir pretpostavka da vrhovi u mreži koji se nalaze na većim udaljenostima komuniciraju manje od onih na manjim udaljenostima. U prvom slučaju količina komunikacije medu vrhovima utežena je s $d(u, v)^\lambda$ za $\lambda < 0$, a u drugom slučaju s $\lambda^{d(u,v)}$ za $\lambda \in (0, 1)$, pri čemu je $d(u, v)$ udaljenost između vrhova $u$ i $v$. Analizirane su gornje i donje ograde vrijednosti deskriptora. Nadalje, definirane su kurikulne mreže i analizirana neka njihova svojstva. Riječ je o jednostavnim usmjerenim grafovima u kojima vrhovi predstavljaju edukacijske jedinice, a usmjereni brid između dva vrha označava da je poznavanje jedne jedinice potrebno za učenje druge. Definirane su mjere pomoću kojih je moguće napraviti evaluaciju valjanosti redoslijeda edukacijskih jedinica. Mjere su analizirane s više različitih gledišta i određeni su grafovi koji odgovaraju najmanje i najviše složenim nastavnim planovima. Predložen je algoritam za određivanje optimalne ekspozicije edukacijskih jedinica u odnosu na odabranu mjeru složenosti. Konačno, detaljno je analiziran problem detekcije zajednica u kurikulnim mrežama i predloženi su algoritmi za rješavanje tog problema.In this thesis results of research from several different areas of complex networks theory have been introduced. First, generalized versions of network descriptors such as transmission, betweeness centrality, networkness and network surplus have been defined and analyzed. Two different approaches that were studied both take into account the assumption that vertices in networks, which are closer to each other, communicate more than the vertices on greater distances. In the first case the amount of communication between vertices has been amended by $d(u, v)^\lambda$where $\lambda < 0$ and in the second case by $\lambda^{d(u,v)}$, where $d(u, v)$ represents the distance between vertices $u$ and $v$, and $\lambda \in (0, 1)$. Upper and lower bounds for minimal and maximal values of these descriptors have been analyzed. Furthermore, special kind of networks has been defined. Curriculum network is a simple, directed graph in which vertices represent educational units, and directed arc from $u$ to $v$ indicates that understanding of a unit $u$ is necessary for learning and understanding of unit $v$. Measures for evaluating the quality of curriculum content sequencing have been proposed and analyzed from sever different points of view. Graphs corresponding to most and least complex curricula have been found. Some real-world curriculum networks and their properties have been analyzed. Finally, the problem of community detection in curriculum networks has been analyzed and two algorithms for solving this problem have been suggested

Network descriptors and curriculum networks

U disertaciji su izloženi rezultati istraživanja iz više područja teorije mreža. U prvom dijelu proučavane su ekstremalne vrijednosti poopćenih mrežnih deskriptora transmisije, međupoloženosti, vršne produktivnosti i vršne profitabilnosti. U dva promatrana slučaja uzeta je u obzir pretpostavka da vrhovi u mreži koji se nalaze na većim udaljenostima komuniciraju manje od onih na manjim udaljenostima. U prvom slučaju količina komunikacije medu vrhovima utežena je s $d(u, v)^\lambda$ za $\lambda < 0$, a u drugom slučaju s $\lambda^{d(u,v)}$ za $\lambda \in (0, 1)$, pri čemu je $d(u, v)$ udaljenost između vrhova $u$ i $v$. Analizirane su gornje i donje ograde vrijednosti deskriptora. Nadalje, definirane su kurikulne mreže i analizirana neka njihova svojstva. Riječ je o jednostavnim usmjerenim grafovima u kojima vrhovi predstavljaju edukacijske jedinice, a usmjereni brid između dva vrha označava da je poznavanje jedne jedinice potrebno za učenje druge. Definirane su mjere pomoću kojih je moguće napraviti evaluaciju valjanosti redoslijeda edukacijskih jedinica. Mjere su analizirane s više različitih gledišta i određeni su grafovi koji odgovaraju najmanje i najviše složenim nastavnim planovima. Predložen je algoritam za određivanje optimalne ekspozicije edukacijskih jedinica u odnosu na odabranu mjeru složenosti. Konačno, detaljno je analiziran problem detekcije zajednica u kurikulnim mrežama i predloženi su algoritmi za rješavanje tog problema.In this thesis results of research from several different areas of complex networks theory have been introduced. First, generalized versions of network descriptors such as transmission, betweeness centrality, networkness and network surplus have been defined and analyzed. Two different approaches that were studied both take into account the assumption that vertices in networks, which are closer to each other, communicate more than the vertices on greater distances. In the first case the amount of communication between vertices has been amended by $d(u, v)^\lambda$where $\lambda < 0$ and in the second case by $\lambda^{d(u,v)}$, where $d(u, v)$ represents the distance between vertices $u$ and $v$, and $\lambda \in (0, 1)$. Upper and lower bounds for minimal and maximal values of these descriptors have been analyzed. Furthermore, special kind of networks has been defined. Curriculum network is a simple, directed graph in which vertices represent educational units, and directed arc from $u$ to $v$ indicates that understanding of a unit $u$ is necessary for learning and understanding of unit $v$. Measures for evaluating the quality of curriculum content sequencing have been proposed and analyzed from sever different points of view. Graphs corresponding to most and least complex curricula have been found. Some real-world curriculum networks and their properties have been analyzed. Finally, the problem of community detection in curriculum networks has been analyzed and two algorithms for solving this problem have been suggested