@article{
author = "Maletić, Slobodan and Stamenić, Ljubisav and Rajković, Milan",
year = "2011",
abstract = "The interest in real world systems such as particular types of social (the network of acquaintances between individuals, collaboration of scientists, collaboration between actors in who acted in same movies, ...), biological (the network of metabolic pathways, genetic regulatory network, food web, protein interaction network, ...), technological (Internet, Power Grid, the telephone network, ...), informational (the network of citations between academic papers, World Wide Web, ...), etc. systems was in recent years remarkable [1], [2]. The above mentioned systems are characterized by two sets - a set of items and a set of connections or relationships between them. This kind of system we call network. The overall connectivity of the whole system is not simple, regardless of pairwise connections of items, and far from regularity. Hence, we call these physical systems complex networks, and they induced the development of complex networks theory. In the essence of this theory was the attempt to make classification of such systems with respect to the underlying structure, as well as to understand organizational principles which lead to formation of characteristic structures. Different kinds of dynamical processes on real world systems (such as deletion/addition of elements, and/or connections; information flow, etc.) takes place, and properties of these dynamical processes are influenced by the underlying structure. Hence, the notion of structure is certainly one of central concepts of complex networks theory. The most usual property for characterizing a structure is the distribution of number of connections which items have. Since the beginning of its development, the complex networks theory relied on the concepts of graph theory. This reliance is understandable since the treatment of real world networks as sets of entities and their pairwise connections was natural and easiest to represent mathematically as a graph. On the other hand, the branch of mathematics called combinatorial algebraic topology [3], [4] introduced us the concepts of simplices and simplicial complexes, as objects which have local connectivity properties not so simple as those in graph theory. Let us in short make some parallels between graphs and simplicial complexes. While the main entity in graph is a node, in simplicial complex main entity is simplex, which is defined as a set of vertices. A pair of nodes in graph are connected by link, while in simplicial complex two simplices are connected if sets which define them have some vertices in common (these shared common vertices we call face). From this short comparison we can anticipate that simplices form connectivity structure which is more complex than graph. Furthermore, connectivity, as well as structural properties, can be considered from three aspects: combinatorial, algebraic, and topological. Hence, at this point we came to the main idea of this paper, which is the following: can we represent physical complex network as simplicial complex, and if we can, what are the statistical mechanics properties of measures which emerge from combinatorial, algebraic, and topological aspect, analogously to the statistical mechanics approach to graph representation of complex network? Furthermore, if we can do all this, another problem arises: can we make some relationship between properties of complex networks which emerges from two representation - graph and simplicial complex representation?",
journal = "Atti del Seminario Matematico E Fisico Dell' Universita di Modena",
title = "Statistical Mechanics of simplicial complexes",
volume = "58",
number = "1",
pages = "245-262",
url = "https://hdl.handle.net/21.15107/rcub_vinar_12846"
}
