The Walk Distances in Graphs

The walk distances in graphs are defined as the result of appropriate transformations of the $\sum_{k=0}^\infty(tA)^k$ proximity measures, where $A$ is the weighted adjacency matrix of a graph and $t$ is a sufficiently small positive parameter. The walk distances are graph-geodetic; moreover, they converge to the shortest path distance and to the so-called long walk distance as the parameter $t$ approaches its limiting values. We also show that the logarithmic forest distances which are known to generalize the resistance distance and the shortest path distance are a subclass of walk distances. On the other hand, the long walk distance is equal to the resistance distance in a transformed graph.Comment: Accepted for publication in Discrete Applied Mathematics. 26 pages, 3 figure

Social Balance on Networks: The Dynamics of Friendship and Enmity

How do social networks evolve when both friendly and unfriendly relations exist? Here we propose a simple dynamics for social networks in which the sense of a relationship can change so as to eliminate imbalanced triads--relationship triangles that contains 1 or 3 unfriendly links. In this dynamics, a friendly link changes to unfriendly or vice versa in an imbalanced triad to make the triad balanced. Such networks undergo a dynamic phase transition from a steady state to "utopia"--all friendly links--as the amount of network friendliness is changed. Basic features of the long-time dynamics and the phase transition are discussed.Comment: 16 pages, 11 figures, paper based on an invited talk at Dyonet06, Dynamics on Complex Networks and Applications, Dresden, Germany, Feburary 200

The t-improper chromatic number of random graphs

We consider the $t$-improper chromatic number of the Erd{\H o}s-R{\'e}nyi random graph $G(n,p)$. The t-improper chromatic number $\chi^t(G)$ of $G$ is the smallest number of colours needed in a colouring of the vertices in which each colour class induces a subgraph of maximum degree at most $t$. If $t = 0$, then this is the usual notion of proper colouring. When the edge probability $p$ is constant, we provide a detailed description of the asymptotic behaviour of $\chi^t(G(n,p))$ over the range of choices for the growth of $t = t(n)$.Comment: 12 page

Multi-hop Byzantine reliable broadcast with honest dealer made practical

We revisit Byzantine tolerant reliable broadcast with honest dealer algorithms in multi-hop networks. To tolerate Byzantine faulty nodes arbitrarily spread over the network, previous solutions require a factorial number of messages to be sent over the network if the messages are not authenticated (e.g., digital signatures are not available). We propose modifications that preserve the safety and liveness properties of the original unauthenticated protocols, while highly decreasing their observed message complexity when simulated on several classes of graph topologies, potentially opening to their employment