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Pavel Chebotarev

Publications and source records attributed to Pavel Chebotarev.

At least 37 records · Page 2Linked to original sources
arXiv2017

A graph theoretic interpretation of the mean first passage times

Let $m_{ij}$ be the mean first passage time from state $i$ to state $j$ in an $n$-state ergodic homogeneous Markov chain with transition matrix $T$. Let $G$ be the weighted digraph without loops whose vertex set coincides with the set of states of the Markov chain and arc weights are equal to the corresponding transition probabilities. We give a graph-theoretic interpretation to $m_{ij}$. Namely, We show that $m_{ij}=f_{ij}/q_j$ if $i\ne j$ and $m_{ij}=1/\tilde q_j$ if $i=j$, where $f_{ij}$ is the total weight of 2-tree spanning converging forests in $G$ that have one tree containing $i$ and the other tree converging to $j$, $q_j$ is the total weight of spanning trees converging to $j$ in $G$, and $\tilde q_j=q_j/\sum_{k=1}^nq_k$. The result is illustrated by an example. Keywords: Markov chain; Mean first passage time; Spanning rooted forest; Matrix forest theorem; Laplacian matrix

math.PR↗
arXiv2017

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.

cs.LG↗
arXiv2016

Models of latent consensus

The paper studies the problem of achieving consensus in multi-agent systems in the case where the dependency digraph $Γ$ has no spanning in-tree. We consider the regularization protocol that amounts to the addition of a dummy agent (hub) uniformly connected to the agents. The presence of such a hub guarantees the achievement of an asymptotic consensus. For the "evaporation" of the dummy agent, the strength of its influences on the other agents vanishes, which leads to the concept of latent consensus. We obtain a closed-form expression for the consensus when the connections of the hub are symmetric, in this case, the impact of the hub upon the consensus remains fixed. On the other hand, if the hub is essentially influenced by the agents, whereas its influence on them tends to zero, then the consensus is expressed by the scalar product of the vector of column means of the Laplacian eigenprojection of $Γ$ and the initial state vector of the system. Another protocol, which assumes the presence of vanishingly weak uniform background links between the agents, leads to the same latent consensus.

math.OC↗
arXiv2016

Protometrics

We introduce the concept of protometric and present some properties of protometrics.

math.MG↗
arXiv2015

Semi-supervised Learning with Regularized Laplacian

We study a semi-supervised learning method based on the similarity graph and RegularizedLaplacian. We give convenient optimization formulation of the Regularized Laplacian method and establishits various properties. In particular, we show that the kernel of the methodcan be interpreted in terms of discrete and continuous time random walks and possesses several importantproperties of proximity measures. Both optimization and linear algebra methods can be used for efficientcomputation of the classification functions. We demonstrate on numerical examples that theRegularized Laplacian method is competitive with respect to the other state of the art semi-supervisedlearning methods.

cs.LG↗
arXiv2015

The projection method for continuous-time consensus seeking

For the case where the dependency digraph has no spanning in-tree, we characterize the region of convergence of the basic continuous-time distributed consensus algorithm and show that consensus can be achieved by employing the method of orthogonal projection, which has been proposed for the discrete-time coordination problem.

eess.SY↗
arXiv2013

The effectiveness of altruistic lobbying: A model study

Altruistic lobbying is lobbying in the public interest or in the interest of the least protected part of the society. In fact, an altruist has a wide range of strategies, from behaving in the interest of the society as a whole to the support of the most disadvantaged ones. How can we compare the effectiveness of such strategies? Another question is: "Given a strategy, is it possible to estimate the optimal number of participants choosing it?" Finally, do the answers to these questions depend on the level of well-being in the society? Can we say that the poorer the society, the more important is to focus on the support of the poorest? We answer these questions within the framework of the model of social dynamics determined by voting in a stochastic environment.

cs.MA↗
arXiv2013

Studying new classes of graph metrics

In data analysis, there is a strong demand for graph metrics that differ from the classical shortest path and resistance distances. Recently, several new classes of graph metrics have been proposed. This paper presents some of them featuring the cutpoint additive distances. These include the path distances, the reliability distance, the walk distances, and the logarithmic forest distances among others. We discuss a number of connections between these and other distances.

math.MG↗
arXiv2013

Matrices of forests, analysis of networks, and ranking problems

The matrices of spanning rooted forests are studied as a tool for analysing the structure of networks and measuring their properties. The problems of revealing the basic bicomponents, measuring vertex proximity, and ranking from preference relations / sports competitions are considered. It is shown that the vertex accessibility measure based on spanning forests has a number of desirable properties. An interpretation for the stochastic matrix of out-forests in terms of information dissemination is given.

math.CO↗
arXiv2013

The forest consensus theorem

We show that the limiting state vector in the differential model of consensus seeking with an arbitrary communication digraph is obtained by multiplying the eigenprojection of the Laplacian matrix of the model by the vector of initial states. Furthermore, the eigenprojection coincides with the stochastic matrix of maximum out-forests of the weighted communication digraph. These statements make the forests consensus theorem. A similar result for DeGroot's iterative pooling model requires the Cesaro (time-average) limit in the general case. The forests consensus theorem is useful for the analysis of consensus protocols.

cs.MA↗
arXiv2012

Simple expressions for the long walk distance

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 connected weighted 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. In this paper, simple expressions for the long walk distance are obtained. They involve the generalized inverse, minors, and inverses of submatrices of the symmetric irreducible singular M-matrix ${\cal L}=ρI-A,$ where $ρ$ is the Perron root of $A.$

math.CO↗
arXiv2012

A topological interpretation of the walk distances

The walk distances in graphs have no direct interpretation in terms of walk weights, since they are introduced via the \emph{logarithms} of walk weights. Only in the limiting cases where the logarithms vanish such representations follow straightforwardly. The interpretation proposed in this paper rests on the identity $\ln\det B=\tr\ln B$ applied to the cofactors of the matrix $I-tA,$ where $A$ is the weighted adjacency matrix of a weighted multigraph and $t$ is a sufficiently small positive parameter. In addition, this interpretation is based on the power series expansion of the logarithm of a matrix. Kasteleyn (1967) was probably the first to apply the foregoing approach to expanding the determinant of $I-A$. We show that using a certain linear transformation the same approach can be extended to the cofactors of $I-tA,$ which provides a topological interpretation of the walk distances.

math.CO↗
arXiv2012

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.

math.CO↗
arXiv2012

A Cyclic Representation of Discrete Coordination Procedures

We show that any discrete opinion pooling procedure with positive weights can be asymptotically approximated by DeGroot's procedure whose communication digraph is a Hamiltonian cycle with loops. In this cycle, the weight of each arc (which is not a loop) is inversely proportional to the influence of the agent the arc leads to.

cs.MA↗
arXiv2011

The Forest Metrics for Graph Vertices

We propose a new graph metric and study its properties. In contrast to the standard distance in connected graphs, it takes into account all paths between vertices. Formally, it is defined as d(i,j)=q_{ii}+q_{jj}-q_{ij}-q_{ji}, where q_{ij} is the (i,j)-entry of the {\em relative forest accessibility matrix} Q(ε)=(I+εL)^{-1}, L is the Laplacian matrix of the (weighted) (multi)graph, and εis a positive parameter. By the matrix-forest theorem, the (i,j)-entry of the relative forest accessibility matrix of a graph provides the specific number of spanning rooted forests such that i and j belong to the same tree rooted at i. Extremely simple formulas express the modification of the proposed distance under the basic graph transformations. We give a topological interpretation of d(i,j) in terms of the probability of unsuccessful linking i and j in a model of random links. The properties of this metric are compared with those of some other graph metrics. An application of this metric is related to clustering procedures such as "centered partition." In another procedure, the relative forest accessibility and the corresponding distance serve to choose the centers of the clusters and to assign a cluster to each non-central vertex. The notion of cumulative weight of connections between two vertices is proposed. The reasoning involves a reciprocity principle for weighted multigraphs. Connections between the resistance distance and the forest distance are established.

math.CO↗
arXiv2011

The graph bottleneck identity

A matrix $S=(s_{ij})\in{\mathbb R}^{n\times n}$ is said to determine a \emph{transitional measure} for a digraph $G$ on $n$ vertices if for all $i,j,k\in\{1,\...,n\},$ the \emph{transition inequality} $s_{ij} s_{jk}\le s_{ik} s_{jj}$ holds and reduces to the equality (called the \emph{graph bottleneck identity}) if and only if every path in $G$ from $i$ to $k$ contains $j$. We show that every positive transitional measure produces a distance by means of a logarithmic transformation. Moreover, the resulting distance $d(\cdot,\cdot)$ is \emph{graph-geodetic}, that is, $d(i,j)+d(j,k)=d(i,k)$ holds if and only if every path in $G$ connecting $i$ and $k$ contains $j$. Five types of matrices that determine transitional measures for a digraph are considered, namely, the matrices of path weights, connection reliabilities, route weights, and the weights of in-forests and out-forests. The results obtained have undirected counterparts. In [P. Chebotarev, A class of graph-geodetic distances generalizing the shortest-path and the resistance distances, Discrete Appl. Math., URL http://dx.doi.org/10.1016/j.dam.2010.11.017] the present approach is used to fill the gap between the shortest path distance and the resistance distance.

math.CO↗
arXiv2011

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.

math.CO↗