Communicability Angles Reveal Critical Edges for Network Consensus Dynamics

We consider the question of determining how the topological structure influences a consensus dynamical process taking place on a network. By considering a large dataset of real-world networks we first determine that the removal of edges according to their communicability angle -an angle between position vectors of the nodes in an Euclidean communicability space- increases the average time of consensus by a factor of 5.68 in real-world networks. The edge betweenness centrality also identifies -in a smaller proportion- those critical edges for the consensus dynamics, i.e., its removal increases the time of consensus by a factor of 3.70. We justify theoretically these findings on the basis of the role played by the algebraic connectivity and the isoperimetric number of networks on the dynamical process studied, and their connections with the properties mentioned before. Finally, we study the role played by global topological parameters of networks on the consensus dynamics. We determine that the network density and the average distance-sum -an analogous of the node degree for shortest-path distances, account for more than 80% of the variance of the average time of consensus in the real-world networks studied.Comment: 15 pages, 2 figure

Node-node distance distribution for growing networks

We present the simulation of the time evolution of the distance matrix. The result is the node-node distance distribution for various kinds of networks. For the exponential trees, analytical formulas are derived for the moments of the distance distribution.Comment: presented during the 37-th Polish Physicists' Meeting, Gdansk, Poland, 15-19 Sep. 2003, 6 pages, 3 figure

Leader-follower consensus under peer-pressure in complex networks

Synchronisation is an important process for different kinds of systems, such as biological, chemical, physical and social. Among the related synchronisation problems, consensus has received high attention because of the distributed properties shown by its models and the possibility they offer for controlling complex systems. When dealing with consensus processes in social networks, we known from empirical evidence that the formation of opinions is not free from being influenced by people around every actor, and more, it is well known that some of the actors may play a leading role and guide a social system to a final state different from the pure average consensus. A main paradigm while modelling interactions among actors in social networks is that every actor receives and transmits information from and to her nearest neighbours, thus implicitly assuming that the decisions of a given actor only are influenced by their directly connected peers, and not tking into account indirect influences coming from not directly connnected peers in the same social network, for example, the influence coming from the friend's friend of a friend. Our work studies consensus processes in the presence of influence coming from not only those directly connected actors, but from other ones in the same network. We call this influence peer pressure (PP). We propose a consensus model that takes into account direct and indirect PP modelled as a function of the social distance among actors. We apply this consensus model to different real social networks assuming three different decay laws for the strength of PP, and in the presence of leaders and without them. We choose those nodes acting as leaders according to different centrality criteria, as well as randomly, and compare thier performance for driving the system. Since it is natural that different leaders may diverge in their positions, we introduce a divergence parameter among the initial states of the leaders with respect to the avreage consensus of the system, to take the feature into account in our model. We then analyse the effects of PP on two different real cases of diffusion of innovation processes. We show that as the strength of indirect PP increases, the centrality criteria used to select the leaders has a decaying effect on the effectiveness of such leaders to better drive a consensus process, allowing random leaders to be as good as those with better centrality. Our work also shows that, despite divergence among leaders induces higher times for reaching consensus, this effect is reduced for stronger levels of PP present in the system. For the case of diffusion innovations our model reproduces the behaviour of the empirical data, and we demonstrate that certainlevels of PP are necessary to match the results coming from two different studies, supporting our hypothesis that indirect PP is an important factor to be taken into account when modelling opinion formations in social networks. Leaders emerging by global centrality criteria in networks with tightly connected groups can be counterproductive. This can be tackled by selecting node-leaders in a local basis. This effect is also reduced when indirect PP is allowed to be higher. This finding points to the fact that distance among nodes is an important characteristic for consenus processes. For the purpose of studying this structural feature, we propose a distance-sum heterogeneity index based on a fictional consensus process. We conjecture that an special type of graph, that we call complete split graph, is related with the maximization of the index, and based on this conjecture we study the relative distance-sum heterogeneity of random graphs and different real-world networks, which allows us to characterise them. We propose a spectral representation of the distance-sum heterogeneity index for networks that we call S-plots. We also study the relation between the time for consensus and the distance-sum heterogeneities in complex networks from different nature.Synchronisation is an important process for different kinds of systems, such as biological, chemical, physical and social. Among the related synchronisation problems, consensus has received high attention because of the distributed properties shown by its models and the possibility they offer for controlling complex systems. When dealing with consensus processes in social networks, we known from empirical evidence that the formation of opinions is not free from being influenced by people around every actor, and more, it is well known that some of the actors may play a leading role and guide a social system to a final state different from the pure average consensus. A main paradigm while modelling interactions among actors in social networks is that every actor receives and transmits information from and to her nearest neighbours, thus implicitly assuming that the decisions of a given actor only are influenced by their directly connected peers, and not tking into account indirect influences coming from not directly connnected peers in the same social network, for example, the influence coming from the friend's friend of a friend. Our work studies consensus processes in the presence of influence coming from not only those directly connected actors, but from other ones in the same network. We call this influence peer pressure (PP). We propose a consensus model that takes into account direct and indirect PP modelled as a function of the social distance among actors. We apply this consensus model to different real social networks assuming three different decay laws for the strength of PP, and in the presence of leaders and without them. We choose those nodes acting as leaders according to different centrality criteria, as well as randomly, and compare thier performance for driving the system. Since it is natural that different leaders may diverge in their positions, we introduce a divergence parameter among the initial states of the leaders with respect to the avreage consensus of the system, to take the feature into account in our model. We then analyse the effects of PP on two different real cases of diffusion of innovation processes. We show that as the strength of indirect PP increases, the centrality criteria used to select the leaders has a decaying effect on the effectiveness of such leaders to better drive a consensus process, allowing random leaders to be as good as those with better centrality. Our work also shows that, despite divergence among leaders induces higher times for reaching consensus, this effect is reduced for stronger levels of PP present in the system. For the case of diffusion innovations our model reproduces the behaviour of the empirical data, and we demonstrate that certainlevels of PP are necessary to match the results coming from two different studies, supporting our hypothesis that indirect PP is an important factor to be taken into account when modelling opinion formations in social networks. Leaders emerging by global centrality criteria in networks with tightly connected groups can be counterproductive. This can be tackled by selecting node-leaders in a local basis. This effect is also reduced when indirect PP is allowed to be higher. This finding points to the fact that distance among nodes is an important characteristic for consenus processes. For the purpose of studying this structural feature, we propose a distance-sum heterogeneity index based on a fictional consensus process. We conjecture that an special type of graph, that we call complete split graph, is related with the maximization of the index, and based on this conjecture we study the relative distance-sum heterogeneity of random graphs and different real-world networks, which allows us to characterise them. We propose a spectral representation of the distance-sum heterogeneity index for networks that we call S-plots. We also study the relation between the time for consensus and the distance-sum heterogeneities in complex networks from different nature

Funciones distancia y arcos de mínima longitud : marco teórico, modelado y aplicaciones a problemas de optimización

Funciones distancia y arcos de mínima longitud : Marco teórico, modelado y aplicaciones a problemas de optimizació

The impact of bus punctuality on users’ decisions and welfare

This paper proposes a public transport users’ scheduling model that considers crowding inside vehicles, waiting time, and punctuality as a reliability measure. Commuters simultaneously make two choices: the preferred bus and the timing to arrive at the bus stop (on time or late). Public transport punctuality is the probability of being on time or late, generating a parameter of public transport reliability. We compute users’ equilibrium, social optimum, and first-best pricing and analytically devise a methodology to obtain the second-best pricing. Using numerical analysis, we show that (i) punctuality plays an essential role in the commuter strategy modifying according to its level, (ii) commuters’ strategy changes according to how reliable the system is, and (iii) second-best pricing is efficient only for limited cases