Clusterization, Frustration and Collectivity in Random Networks

Abstract

We consider the random Erdős-Rényi network with enhanced clusterization and Ising spins s = ±1 at the network nodes. Mutually linked spins interact with energy J. Magnetic properties of the system as dependent on the clustering coefficient C are investigated with the Monte Carlo heat bath algorithm. For J > 0 the Curie temperature Tc increases from 3.9 to 5.5 when C increases from almost zero to 0.18. These results deviate only slightly from the mean field theory. For J < 0 the spin-glass phase appears below TSG; this temperature decreases with C, on the contrary to the mean field calculations. The results are interpreted in terms of social systems.

Clustering in random line graphs

“…This limit value can be lower, if G → L(G) is applied to selected local subnetworks and not to the whole system. The density of the transformed nodes can be used to tune C, similarly to [11,12,13,14,15,16]. …”

“…The idea is to enhance C gradually by linking nodes which are neighbours of the same node. When a node has three neighbours, it is convenient to replace it by a triangle [16]; the trick is similar to the star-triangle or star-delta transformation. The latter has been used also to construct Apollonian networks with a high values of the clustering coefficient [17], and to prove some theorems in the theory of percolation [18].…”

“…First, we share the point of view expressed by Mark Newman, as reported in our first paragraph. Our former simulations [16,23] can actually be seen as the same transformation limited to local subgraphs and to nodes of degree three. This technique allowed us to enhance the clustering coefficient in networks.…”

Competing Contact Processes on Homogeneous Networks With Tunable Clusterization

“…The latter networks are prepared from the usual Erdös-R'enyi network by adding some links between neighbors of the same nodes. The procedure, applied by Holme and Kim to the scale-free networks [35], was adapted later [36] to the Erdös-R'enyi networks. As a rule, the modified networks contained 10 3 nodes, with mean degree < k >= 10.00 ± 0.5.…”

Frustration and Collectivity in Spatial Networks

“…In our former texts, we calculated the transition temperature T SG of the Erdös-Rényi networks [10] and of the regular network [11]. The results indicated that on the contrary to the anticipations of the Bethe theory T SG decreases with the clustering coefficient C. However, in both cases we dealt with the networks endowed with the small-world property.…”