arXiv subjects

Pavel Chebotarev

Publications and source records attributed to Pavel Chebotarev.

At least 19 recordsLinked to original sources

arXiv2025

Modeling society with a responsible elite

Within the framework of the ViSE (Voting in a Stochastic Environment) model, we examine the dynamics in a society, part of which can be considered an elite. The model allows us to analyze the influence of social attitudes, such as collectivism, individualism, altruism on the well-being of agents. The dynamics is determined by collective decisions and changes in the structure of society, in particular, by the formation of groups of cooperating agents. It is found that the presence of a "responsible elite", combining the support of other agents with limited concern for their own benefit, stabilizes society and eliminates the "pit of losses" paradox. The benefit to society from having a responsible elite is comparable to that from having a prosocial group of the same size. If the elite radically increases the weight of the group component in its combined voting strategy, then its incomes rise sharply, while society's incomes decline. If, in response to the selfish transformation of the elite, a new responsible elite emerges, proportionally larger than the previous one, then society will stabilize again, and the old elite will lose its dominant position. This process can be repeated as long as the size of society allows the formation of new responsible elites of the required size.

physics.soc-ph↗

arXiv2025

Evolution of Society Caused by Collective and Individual Decisions

Decision-making societies may vary in their level of cooperation and degree of conservatism, both of which influence their overall performance. Moreover, these factors are not fixed -- they can change based on the decisions agents in the society make in their interests. But can these changes lead to cyclical patterns in societal evolution? To explore this question, we use the ViSE (Voting in Stochastic Environment) model. In this framework, the level of cooperation can be measured by group size, while the degree of conservatism is determined by the voting threshold. Agents can adopt either individualistic or group-oriented strategies when voting on stochastically generated external proposals. For Gaussian proposal generators, the expected capital gain (ECG) -- a measure of agents' performance -- can be expressed in standard mathematical functions. Our findings show that in neutral environments, societal evolution with open or democratic groups can follow cyclic patterns. We also find that highly conservative societies or conservative societies with low levels of cooperation can evolve into liberal (less conservative than majoritarian) societies and that mafia groups never let their members go when they want to.

physics.soc-ph↗

arXiv2024

How to choose the most appropriate centrality measure? A decision tree approach

Centrality metrics play a crucial role in network analysis, while the choice of specific measures significantly influences the accuracy of conclusions as each measure represents a unique concept of node importance. Among over 400 proposed indices, selecting the most suitable ones for specific applications remains a challenge. Existing approaches -- model-based, data-driven, and axiomatic -- have limitations, requiring association with models, training datasets, or restrictive axioms for each specific application. To address this, we introduce the culling method, which relies on the expert concept of centrality behavior on simple graphs. The culling method involves forming a set of candidate measures, generating a list of as small graphs as possible needed to distinguish the measures from each other, constructing a decision-tree survey, and identifying the measure consistent with the expert's concept. We apply this approach to a diverse set of 40 centralities, including novel kernel-based indices, and combine it with the axiomatic approach. Remarkably, only 13 small 1-trees are sufficient to separate all 40 measures, even for pairs of closely related ones. By adopting simple ordinal axioms like Self-consistency or Bridge axiom, the set of measures can be drastically reduced making the culling survey short. Applying the culling method provides insightful findings on some centrality indices, such as PageRank, Bridging, and dissimilarity-based Eigencentrality measures, among others. The proposed approach offers a cost-effective solution in terms of labor and time, complementing existing methods for measure selection, and providing deeper insights into the underlying mechanisms of centrality measures.

physics.soc-ph↗

arXiv2023

Majority voting is not good for heaven or hell, with mirrored performance

The ViSE (Voting in Stochastic Environment) model studies voting strategies and social decision rules under a stochastic agenda of proposals with random gains. The pit-of-losses paradox established earlier shows that, in hostile ("hell") environments, majority voting can be worse than rejecting all proposals. We show a counterpart in favorable ("heaven") environments, where majority voting can be worse than accepting all proposals. A one-line identity underlies this symmetry: for an antisymmetric voting body, the expected capital gain of each agent under a gain generator of mean $\mu>0$ exceeds that under the opposite generator by exactly $\mu$. Complementary voting strategies together with a complementary social decision rule provide a broad sufficient, but not necessary, condition for antisymmetry. This includes symmetrized majority with individualistic, utilitarian-group, and majoritarian-group strategies. For symmetric location families, the result yields mirror symmetry of performance relative to the baseline that rejects all proposals in unfavorable environments and accepts all in favorable ones. We also strengthen the Gaussian pit-of-losses result: simple majority has a pit in every society of $n>2$ individualists, and every rule requiring at least $r$ of $n$ votes, $r<n$, has one as well, while unanimity avoids the pit. Finally, under complementary strategies, optimal voting thresholds inherit the mirror structure. For stochastic vote-count rules under the same condition, a complementary optimal rule can always be selected, and optimal total performance is even in $\mu$ in symmetric location families. In the two-plus-trio paradox, the optimal vote-count rule outperforms all threshold rules even in a neutral symmetric environment.

physics.soc-ph↗

arXiv2023

Matrix-Forest Theorems

The Laplacian matrix of a graph $G$ is $L(G)=D(G)-A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the diagonal matrix of vertex degrees. According to the Matrix-Tree Theorem, the number of spanning trees in $G$ is equal to any cofactor of an entry of $L(G)$. A rooted forest is a union of disjoint rooted trees. We consider the matrix $W(G)=I+L(G)$ and prove that the $(i,j)$-cofactor of $W(G)$ is equal to the number of spanning rooted forests of $G$, in which the vertices $i$ and $j$ belong to the same tree rooted at $i$. The determinant of $W(G)$ equals the total number of spanning rooted forests, therefore the $(i,j)$-entry of the matrix $W^{-1}(G)$ can be considered as a measure of relative ''forest-accessibility'' of vertex $i$ from $j$ (or $j$ from $i$). These results follow from somewhat more general theorems we prove, which concern weighted multigraphs. The analogous theorems for (multi)digraphs are also established. These results provide a graph-theoretic interpretation for the adjugate to the Laplacian characteristic matrix.

math.CO↗

arXiv2023

Extending Utility Functions on Arbitrary Sets

We consider the problem of extending a function $f^{}_P$ defined on a subset $P$ of an arbitrary set $X$ to $X$ strictly monotonically with respect to a preorder $\succcurlyeq$ defined on $X$, without imposing continuity constraints. We show that whenever $\succcurlyeq$ has a utility representation, $f^{}_P$ is extendable if and only if it is gap-safe increasing. A class of extensions involving an arbitrary utility representation of $\succcurlyeq$ is proposed and investigated. Connections to related topological results are discussed. The condition of extendability and the form of the extension are simplified when $P$ is a Pareto set.

math.OC↗

arXiv2023

Selection of Centrality Measures Using Self-Consistency and Bridge Axioms

We consider several families of network centrality measures induced by graph kernels, which include some well-known measures and many new ones. The Self-consistency and Bridge axioms, which appeared earlier in the literature, are closely related to certain kernels and one of the families. We obtain a necessary and sufficient condition for Self-consistency, a sufficient condition for the Bridge axiom, indicate specific measures that satisfy these axioms, and show that under some additional conditions they are incompatible. PageRank centrality applied to undirected networks violates most conditions under study and has a property that according to some authors is ``hard to imagine'' for a centrality measure. We explain this phenomenon. Adopting the Self-consistency or Bridge axiom leads to a drastic reduction in survey time in the culling method designed to select the most appropriate centrality measures.

physics.soc-ph↗

arXiv2023

The Power of Small Coalitions under Two-Tier Majority on Regular Graphs

In this paper, we study the following problem. Consider a setting where a proposal is offered to the vertices of a given network $G$, and the vertices must conduct a vote and decide whether to accept the proposal or reject it. Each vertex $v$ has its own valuation of the proposal; we say that $v$ is ``happy'' if its valuation is positive (i.e., it expects to gain from adopting the proposal) and ``sad'' if its valuation is negative. However, vertices do not base their vote merely on their own valuation. Rather, a vertex $v$ is a \emph{proponent} of the proposal if the majority of its neighbors are happy with it and an \emph{opponent} in the opposite case. At the end of the vote, the network collectively accepts the proposal whenever the majority of its vertices are proponents. We study this problem for regular graphs with loops. Specifically, we consider the class $\mathcal{G}_{n|d|h}$ of $d$-regular graphs of odd order $n$ with all $n$ loops and $h$ happy vertices. We are interested in establishing necessary and sufficient conditions for the class $\mathcal{G}_{n|d|h}$ to contain a labeled graph accepting the proposal, as well as conditions to contain a graph rejecting the proposal. We also discuss connections to the existing literature, including that on majority domination, and investigate the properties of the obtained conditions.

math.CO↗

arXiv2022

Dissecting graph measure performance for node clustering in LFR parameter space

Graph measures that express closeness or distance between nodes can be employed for graph nodes clustering using metric clustering algorithms. There are numerous measures applicable to this task, and which one performs better is an open question. We study the performance of 25 graph measures on generated graphs with different parameters. While usually measure comparisons are limited to general measure ranking on a particular dataset, we aim to explore the performance of various measures depending on graph features. Using an LFR graph generator, we create a dataset of 11780 graphs covering the whole LFR parameter space. For each graph, we assess the quality of clustering with k-means algorithm for each considered measure. Based on this, we determine the best measure for each area of the parameter space. We find that the parameter space consists of distinct zones where one particular measure is the best. We analyze the geometry of the resulting zones and describe it with simple criteria. Given particular graph parameters, this allows us to recommend a particular measure to use for clustering.

cs.SI↗

arXiv2021

Measuring Proximity in Attributed Networks for Community Detection

Proximity measures on graphs have a variety of applications in network analysis, including community detection. Previously they have been mainly studied in the context of networks without attributes. If node attributes are taken into account, however, this can provide more insight into the network structure. In this paper, we extend the definition of some well-studied proximity measures to attributed networks. To account for attributes, several attribute similarity measures are used. Finally, the obtained proximity measures are applied to detect the community structure in some real-world networks using the spectral clustering algorithm.

cs.SI↗

arXiv2019

Clustering as a means of leader selection in consensus networks

In the leader-follower approach, one or more agents are selected as leaders who do not change their states or have autonomous dynamics and can influence other agents, while the other agents, called followers, perform a simple protocol based on the states of their neighbors. This approach provides a natural link between control theory and networked agents with their input data. Despite the fact that the leader-follower approach is widely used, the fundamental question still remains: how to choose leaders from a set of agent. This question is called the problem of choosing leaders. There is still no selection algorithm that is both optimal under a natural criterion and fast. In this paper, for agents that obey a linear consensus protocol, we propose to choose leaders using graph nodes' clustering algorithms and show that this method is the most accurate among the fast existing algorithms of choosing leaders.

math.OC↗

arXiv2019

Modeling Responsible Elite

Within the ViSE (Voting in Stochastic Environment) model, we study social dynamics determined by collective decisions in a society with an elite. The model allows the analysis of the influence of participants' social attitudes, such as the effects of selfishness, collectivism, lobbying, altruism, etc., on the welfare of the society and its strata. Social dynamics is determined by the change in capital over time, as well as the formation and dissociation of groups. We show that the presence of a responsible elite, which partially cares for its own benefit, stabilizes society and removes the `pit of damage' paradox. Society's gain from having a responsible elite is comparable to that from an altruistic group of the same size. If the responsible elite succumbs to the temptation to dramatically increase the weight of the group component in its combined voting strategy, then its income rises sharply, while the income of the society decreases. The rest of the participants benefit from joining the elite, while for the elite it is beneficial to maintain a moderate size. If, in response to insufficient responsibility of the elite, an altruistic group emerges that outnumbers the elite and becomes a new responsible elite, then society again stabilizes, and monopoly of the previous elite (a `clique') ends. If the responsible elite competing with the clique becomes a second clique, then tough competition between the cliques is still preferable for society over having a unique clique.

physics.soc-ph↗

arXiv2018

Two Models of Latent Consensus in Multi-Agent Systems

In this paper, we propose several consensus protocols of the first and second order for networked multi-agent systems and provide explicit representations for their asymptotic states. These representations involve the eigenprojection of the Laplacian matrix of the dependency digraph. In particular, we study regularization models for the problem of coordination when the dependency digraph does not contain a converging tree. In such models of the first kind, the system is supplemented by a dummy agent, a "hub" that uniformly, but very weakly influences the agents and, in turn, depends on them. In the models of the second kind, we assume the presence of very weak background links between the agents. Besides that, we present a description of the asymptotics of the classical second-order consensus protocol.

math.OC↗

arXiv2018

Second-Order Agents on Ring Digraphs

The paper addresses the problem of consensus seeking among second-order linear agents interconnected in a specific ring topology. Unlike the existing results in the field dealing with one-directional digraphs arising in various cyclic pursuit algorithms or two-directional graphs, we focus on the case where some arcs in a two-directional ring graph are dropped in a regular fashion. The derived condition for achieving consensus turns out to be independent of the number of agents in a network.

cs.MA↗

arXiv2018

Comparative Efficiency of Altruism and Egoism as Voting Strategies in Stochastic Environment

In this paper, we study the efficiency of egoistic and altruistic strategies within the model of social dynamics determined by voting in a stochastic environment (the ViSE model) using two criteria: maximizing the average capital increment and minimizing the number of bankrupt participants. The proposals are generated stochastically; three families of the corresponding distributions are considered: normal distributions, symmetrized Pareto distributions, and Student's $t$-distributions. It is found that the "pit of losses" paradox described earlier does not occur in the case of heavy-tailed distributions. The egoistic strategy better protects agents from extinction in aggressive environments than the altruistic ones, however, the efficiency of altruism is higher in more favorable environments. A comparison of altruistic strategies with each other shows that in aggressive environments, everyone should be supported to minimize extinction, while under more favorable conditions, it is more efficient to support the weakest participants. Studying the dynamics of participants' capitals we identify situations where the two considered criteria contradict each other. At the next stage of the study, combined voting strategies and societies involving participants with selfish and altruistic strategies will be explored.

math.OC↗

arXiv2018

Hitting Time Quasi-metric and Its Forest Representation

Let $\hat m_{ij}$ be the hitting (mean first passage) time from state $i$ to state $j$ in an $n$-state ergodic homogeneous Markov chain with transition matrix $T$. Let $Γ$ be the weighted digraph whose vertex set coincides with the set of states of the Markov chain and arc weights are equal to the corresponding transition probabilities. It holds that $$ \hat m_{ij}= q_j^{-1}\cdot \begin{cases} f_{ij},&\text{if }\;\; i\ne j,\\ q, &\text{if }\;\; i=j, \end{cases} $$ where $f_{ij}$ is the total weight of 2-tree spanning converging forests in $Γ$ 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 $Γ,$ and $q=\sum_{j=1}^nq_j$ is the total weight of all spanning trees in $Γ.$ Moreover, $f_{ij}$ and $q_j$ can be calculated by an algebraic recurrent procedure. A forest expression for Kemeny's constant is an immediate consequence of this result. Further, we discuss the properties of the hitting time quasi-metric $m$ on the set of vertices of $Γ$: $m(i,j)=\hat m_{ij}$, $i\neq j$, and $m(i,i)=0$. We also consider a number of other metric structures on the set of graph vertices related to the hitting time quasi-metric $m$---along with various connections between them. The notions and relationships under study are illustrated by two examples.

math.CO↗

arXiv2018

Similarities on Graphs: Kernels versus Proximity Measures

We analytically study proximity and distance properties of various kernels and similarity measures on graphs. This helps to understand the mathematical nature of such measures and can potentially be useful for recommending the adoption of specific similarity measures in data analysis.

math.CO↗

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↗