Memory effect in growing trees

We show that the structure of a growing tree preserves an information on the shape of an initial graph. For the exponential trees, evidence of this kind of memory is provided by means of the iterative equations, derived for the moments of the node-node distance distribution. Numerical calculations confirm the result and allow to extend the conclusion to the Barabasi--Albert scale-free trees. The memory effect almost disappears, if subsequent nodes are connected to the network with more than one link.Comment: 9 pages, 9 figure

The Sznajd dynamics on a directed clustered network

The Sznajd model is investigated in the directed Erdos--Renyi network with the clusterization coefficient enhanced to 0.3 by the method of Holme and Kim (Phys. Rev. E65 (2002) 026107). Within additional triangles, all six links are present. In this network, some nodes preserve the minority opinion. The time tau of getting equilibrium is found to follow the log-normal distribution and it increases linearly with the system size. Its dependence on the initial opinion distribution is different from the analytical results for fully connected networks.Comment: dedicated to Dietrich Stauffer for his 65-th birthda

Dependence of the average to-node distance on the node degree for random graphs and growing networks

In a graph, nodes can be characterized locally (with their degree $k$) or globally (e.g. with their average length path $\xi$ to other nodes). Here we investigate how $\xi$ depends on $k$. Our earlier algorithm of the construction of the distance matrix is applied to the random graphs. Numerical calculations are performed for the random graphs and the growing networks: the scale-free ones and the exponential ones. The results are relevant for search strategies in different networks.Comment: 7 pages, 2 figure

Heider Balance in Human Networks

Recently, a continuous dynamics was proposed to simulate dynamics of interpersonal relations in a society represented by a fully connected graph. Final state of such a society was found to be identical with the so-called Heider balance (HB), where the society is divided into two mutually hostile groups. In the continuous model, a polarization of opinions was found in HB. Here we demonstrate that the polarization occurs also in Barabasi-Albert networks, where the Heider balance is not necessarily present. In the second part of this work we demonstrate the results of our formalism, when applied to reference examples: the Southern women and the Zachary club.Comment: 9 pages, 5 figures. Presented on 8th Granada Seminar on Computational and Statistical Physics, Modeling Cooperative Behavior in the Social Sciences, Granada, Spain, 7-11 February 200