Categories, norms and weights

The well-known Lawvere category R of extended real positive numbers comes with a monoidal closed structure where the tensor product is the sum. But R has another such structure, given by multiplication, which is *-autonomous. Normed sets, with a norm in R, inherit thus two symmetric monoidal closed structures, and categories enriched on one of them have a 'subadditive' or 'submultiplicative' norm, respectively. Typically, the first case occurs when the norm expresses a cost, the second with Lipschitz norms. This paper is a preparation for a sequel, devoted to 'weighted algebraic topology', an enrichment of directed algebraic topology. The structure of R, and its extension to the complex projective line, might be a first step in abstracting a notion of algebra of weights, linked with physical measures.Comment: Revised version, 16 pages. Some minor correction

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

A Class of Graph-Geodetic Distances Generalizing the Shortest-Path and the Resistance Distances

A new class of distances for graph vertices is proposed. This class contains parametric families of distances which reduce to the shortest-path, weighted shortest-path, and the resistance distances at the limiting values of the family parameters. The main property of the class is that all distances it comprises are graph-geodetic: $d(i,j)+d(j,k)=d(i,k)$ if and only if every path from $i$ to $k$ passes through $j$. The construction of the class is based on the matrix forest theorem and the transition inequality.Comment: 14 pages. Discrete Applied Mathematic

Do logarithmic proximity measures outperform plain ones in graph clustering?

We consider a number of graph kernels and proximity measures including commute time kernel, regularized Laplacian kernel, heat kernel, exponential diffusion kernel (also called "communicability"), etc., and the corresponding distances as applied to clustering nodes in random graphs and several well-known datasets. The model of generating random graphs involves edge probabilities for the pairs of nodes that belong to the same class or different predefined classes of nodes. It turns out that in most cases, logarithmic measures (i.e., measures resulting after taking logarithm of the proximities) perform better while distinguishing underlying classes than the "plain" measures. A comparison in terms of reject curves of inter-class and intra-class distances confirms this conclusion. A similar conclusion can be made for several well-known datasets. A possible origin of this effect is that most kernels have a multiplicative nature, while the nature of distances used in cluster algorithms is an additive one (cf. the triangle inequality). The logarithmic transformation is a tool to transform the first nature to the second one. Moreover, some distances corresponding to the logarithmic measures possess a meaningful cutpoint additivity property. In our experiments, the leader is usually the logarithmic Communicability measure. However, we indicate some more complicated cases in which other measures, typically, Communicability and plain Walk, can be the winners.Comment: 11 pages, 5 tables, 9 figures. Accepted for publication in the Proceedings of 6th International Conference on Network Analysis, May 26-28, 2016, Nizhny Novgorod, Russi