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This paper investigates the problem of resilient vector consensus for a group of dynamic agents against mobile malicious attacks. As real networks often operate under random environment and noises, we approach this problem by considering general random dynamic networks with weighted directed topologies. We propose three types of mobile attack models, which differ in the timing of moving of attackers and the capability of detecting such moving. By employing distributed discrete-time algorithms, the Lyapunov theory and martingale convergence theorem, resilient vector consensus is shown to be reached for all three models when the underlying network satisfies certain stochastic robustness conditions. [ABSTRACT FROM AUTHOR]

A fundamental organizing principle of real-world complex networked systems is modularity, where networks have interactions at different levels. In this paper we consider a modular graph G having modules with arbitrary intraconnections and random interconnections between activated vertices in different modules. The vertices in different modules are activated with probability r and linked by an interconnecting edge with probability p independently. We present results regarding the Cheeger constant, robustness, algebraic connectivity as well as the smallest eigenvalue for the Dirichlet Laplacian matrix of G with high probability. Our results suggest that r = (ln n) / n is a potential scaling for the recently observed external field-like phenomena of modular networks in statistical mechanics. [ABSTRACT FROM AUTHOR]

This paper initiates the study of the mathematical aspects of the ad-hoc Lanzhou index. If G is a graph with the vertex set { x 1 , ... , x n } , then the ad-hoc Lanzhou index of G is defined by L z ˜ (G) = ∑ i = 1 n d i (n − 1 − d i) 2 , where d i represents the degree of the vertex x i . Several identities for the ad-hoc Lanzhou index, involving some existing topological indices, are established. The problems of finding graphs with the extremum values of the ad-hoc Lanzhou index from the following sets of graphs are also attacked: (i) the set of all connected ξ -cyclic graphs of a fixed order, (ii) the set of all connected molecular ξ -cyclic graphs of a fixed order, (iii) the set of all graphs of a fixed order, and (iv) the set of all connected molecular graphs of a fixed order. [ABSTRACT FROM AUTHOR]

Let D be a digraph with n vertices and a arcs. The Laplacian and the signless Laplacian matrices of D are, respectively, defined as L (D) = D e g + (D) − A (D) and Q (D) = D e g + (D) + A (D) , where A (D) represents the adjacency matrix and D e g + (D) represents the diagonal matrix whose diagonal elements are the out-degrees of the vertices in D. We derive a combinatorial representation regarding the first few coefficients of the (signless) Laplacian characteristic polynomial of D. We provide concrete directed motifs to highlight some applications and implications of our results. The paper is concluded with digraph examples demonstrating detailed calculations. [ABSTRACT FROM AUTHOR]

We investigate the rainbow k -connectivity rc k and (t , k) -rainbow index rx t , k of the inhomogeneous random graph G (n , p) , where any two vertices i and j are joined by an edge e i j with probability p (e i j) independently of all other edges, and p = { p (e i j) }. We show that the known threshold functions for the monotone properties rc k (G (n , p)) ≤ r and rx t , k (G (n , p)) ≤ t for integers k , r and t in the Erdős–Rényi random graph G (n , p) can be extended to 'threshold landscapes' in terms of G (n , p). In contrast to the traditional plain thresholds characterized as a watershed, our threshold landscapes have two surfaces that are inherently interwoven with each other. This sheds some light on the network connectivity as appropriate trade-offs are allowed and is potentially applicable in network science where connections are not always equal. [ABSTRACT FROM AUTHOR]

This paper studies resilient coordinated control over networks with hybrid dynamics and malicious agents. In a hybrid multi‐agent system, continuous‐time and discrete‐time agents concurrently exist and communicate through local interaction. We introduce the notion of heterogeneous robustness to capture the topological structure and facilitate convergence analysis of hybrid agents over multiple subnetworks, where the exact number and identities of malicious agents are not known. A hybrid resilient strategy is first designed to ensure group consensus of the heterogeneously robust network admitting completely distributed implementation. We then develop a scaled consensus protocol which allows different clusters within each subnetwork, providing further flexibility over the resilient control tasks. Finally, some numerical examples are worked out to illustrate the effectiveness of theoretical results. [ABSTRACT FROM AUTHOR]

Summary: This paper considers the resilient interval consensus problems for continuous‐time time‐varying multi‐agent systems when a normal agent is surrounded by no more than r misbehaving agents. Each normal agent individually proposes a constraint interval which specifies their acceptable consensus range and misbehaving agents are anonymous and exert different arbitrary rules posing threats to the global performance of the systems. On the basis of our purely distributed resilient interval consensus strategy, we showed that if the network is (2r + 1)‐robust and the interval intersection is nonempty, the normal agents are able to reach an agreement with the final consensus equilibrium in the interval intersection. A numerical example is presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this letter, we propose a continuous-time dynamics for social network that represents patterns of both amity and enmity through directed signed graphs. The introduction of discrepancies between true and perceived sentiments gives rise to a non-autonomous system and distinguishes itself from the prior models. We show that for almost all initial configurations, the system will evolve into at most four factions. Under some mild assumptions on the initial conditions, structural balance with at most two factions can be achieved, which extends the previous results for symmetric or normal initial configurations without considering the effect of perceived sentiment. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a novel finite-time median-related group consensus problem, where the finial consensus value can be identified as a desired function of the median of initial states instead of the much studied average value. The underlying communication topology is modelled by a weighted dynamical directed network. A distributed control protocol is firstly introduced to ensure that the agents can reach a median-related consensus in finite time in a collaboration network, meaning that all edge-weights of the communication network are non-negative. We then generalise the results to cooperation–competition networks, where the communication network is divided into predetermined collaboration subnetworks allowing possibly negative weights. Effective group control protocols are designed to guarantee the median-related group consensus in finite time. Finally, numerical simulations are presented to illustrate the availability of our theoretical results. [ABSTRACT FROM AUTHOR]

Recent theoretical studies on network robustness have focused primarily on attacks by random selection and global vision, but numerous real-life networks suffer from proximity-based breakdown. Here we introduce the multi-hop generalized core percolation on complex networks, where nodes with degree less than k and their neighbors within L -hop distance are removed progressively from the network. The resulting subgraph is referred to as G (k , L) -core, extending the recently proposed G k -core and classical core of a network. We develop analytical frameworks based upon generating function formalism and rate equation method, showing for instance continuous phase transition for G (2 , 1) -core and discontinuous phase transition for G (k , L) -core with any other combination of k and L. We test our theoretical results on synthetic homogeneous and heterogeneous networks, as well as on a selection of large-scale real-world networks. This unravels, e.g., a unique crossover phenomenon rooted in heterogeneous networks, which raises a caution that endeavor to promote network-level robustness could backfire when multi-hop tracing is involved. [ABSTRACT FROM AUTHOR]

Network measures derived from empirical observations are often poor estimators of the true structure of system as it is impossible to observe all components and all interactions in many real world complex systems. Here, we study attack robustness of complex networks with data missing caused by: 1) a uniform random sampling and 2) a nonuniform random sampling. By introducing the subgraph robustness problem, we develop analytically a framework to investigate robustness properties of the two types of subgraphs under random attacks, localized attacks, and targeted attacks. Interestingly, we find that the benchmark models, such as Erdős-Rényi graphs, random regular networks, and scale-free networks possess distinct characteristic subgraph robustness features. We show that the network robustness depends on several factors including network topology, attack mode, sampling method and the amount of data missing, generalizing some well-known robustness principles of complex networks. Our results offer insight into the structural effect of missing data in networks and highlight the significance of understanding different sampling processes and their consequences on attack robustness, which may be instrumental in designing robust systems. [ABSTRACT FROM AUTHOR]

Abstract This letter considers the resilient consensus problem for switched multi-agent systems composed of continuous-time and discrete-time subsystems. We propose a switched filtering strategy for cooperative nodes based upon available local information, withstanding the threat of non-cooperative nodes. We provide conditions that guarantee resilient consensus in the presence of locally bounded Byzantine nodes in directed networks under arbitrary switching. Resilient scaled consensus and resilient scaled formation generation problems for switched multi-agent systems are solved as generalizations. Simulations are also provided to illustrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper studies the problem of resilient consensus in continuous-time multiagent systems against potential malicious agents. Network topology conditions have been developed to ensure asymptotic consensus of cooperative agents in the network under multi-hop communication and path-dependent heterogeneous delays. We develop two new distributed protocols based on the idea of weighted mean subsequence reduced algorithm to address scalar resilient consensus and vector resilient consensus problems in path-weighted directed networks leveraging on the capability of multi-hop communication. Our frameworks are flexible in that both linear and general nonlinear network coupling functions are featured. Resilient consensus problem for dynamical agents with higher-order integrators is solved as a byproduct. We illustrate the error introduced by the resilience mechanisms when the network is free of malicious agents through numerical simulations. [ABSTRACT FROM AUTHOR]

TheHindex, also known as Hirsch index, quantifies and compares the citation impact of scientific researchers. In the general context of networks, we define a node as a leader if itsHindex is not less than the average of theHindices of its neighbors. We show that in a randomly connected network, the proportion of leaders is almost always close to a half. [ABSTRACT FROM PUBLISHER]

Let G be a simple connected graph on n vertices. The distance Estrada index DEE(G) of G is defined as the sum of eλi(D) over 1≤i≤n, where λ1(D),λ2(D),…,λn(D) are the eigenvalues of its distance matrix D. In this paper, we establish lower and upper bounds to DEE(G) for almost all bipartite graphs G. [ABSTRACT FROM AUTHOR]

In this study, the author studies fixed‐time group consensus problem in networks of dynamic agents with intrinsic non‐linear dynamics and bounded uncertainties. Three types of distributed control protocols are proposed to achieve fixed‐time group consensus when the subgroups are connected and have inter‐group common influence. By using Lyapunov theory, algebraic graph theory, and fixed‐time stability, some conditions are derived to select the controller gains to ensure the convergence in a prescribed time regardless of the initial conditions. Numerical examples are worked out to illustrate the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

Robustness against attacks serves as evidence for complex network structures and failure mechanisms that lie behind them. Most often, due to detection capability limitation or good disguises, attacks on networks are subject to false positives and false negatives, meaning that functional nodes may be falsely regarded as compromised by the attacker and vice versa. In this work, we initiate a study of false positive/negative effects on network robustness against three fundamental types of attack strategies, namely, random attacks (RA), localized attacks (LA), and targeted attack (TA). By developing a general mathematical framework based upon the percolation model, we investigate analytically and by numerical simulations of attack robustness with false positive/negative rate (FPR/FNR) on three benchmark models including Erdős-Rényi (ER) networks, random regular (RR) networks, and scale-free (SF) networks. We show that ER networks are equivalently robust against RA and LA only when FPR equals zero or the initial network is intact. We find several interesting crossovers in RR and SF networks when FPR is taken into consideration. By defining the cost of attack, we observe diminishing marginal attack efficiency for RA, LA, and TA. Our finding highlights the potential risk of underestimating or ignoring FPR in understanding attack robustness. The results may provide insights into ways of enhancing robustness of network architecture and improve the level of protection of critical infrastructures. [ABSTRACT FROM AUTHOR]

This paper deals with finite-time scaled consensus problems over undirected and directed topologies, wherein agents’ states reach prescribed ratios in a finite time. We develop distributed linear iterations as a function of a linear operator on the underlying network and present necessary and sufficient conditions guaranteeing scaled consensus in a fixed number of steps equal to the number of distinct eigenvalues of a related linear operator. We identify the dependence of the final consensus states on the initial state condition, which can be conveniently and freely tuned by designing suitable parameters. Our results extend the recently developed approach on successive nulling of eigenvalues from complete consensus to scaled consensus, and from undirected topologies to directed topologies. Numerical examples and comparison studies are provided to illustrate the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

We consider a multiplex random graph G (n , m , p) with m independent color layers over a common vertex set V of order n. In each layer, the edges are independent following the Erdős-Rényi model with edge probability p = c (ln ⁡ n + (k − 1) ln ⁡ ln ⁡ n) / m n for some constant c > 1. A rainbow path in this context means a path with all edges from distinct layers. For a graph G ∈ G (n , m , p) , let r c k (G) be its rainbow k -connectivity, namely the smallest required number of layers so that any pair of vertices in G can be connected by k internally vertex-disjoint rainbow paths. We show that with high probability, r c k (G (n , m , p)) is concentrated on three consecutive numbers. These numbers are at distance Θ (ln ⁡ n / (ln ⁡ (ln ⁡ n + (k − 1) ln ⁡ ln ⁡ n)) 2) to the diameter of the graph. [ABSTRACT FROM AUTHOR]

The study of the maximum and minimal characteristics of graphs is the focus of the significant field of mathematics known as extreme graph theory. Finding the biggest or smallest graphs that meet specified criteria is the main goal of this discipline. There are several applications of extremal graph theory in various fields, including computer science, physics, and chemistry. Some of the important applications include: Computer networking, social networking, chemistry and physics as well. Recently, in 2021 exponential multiplicative Zagreb indices were introduced. In generalization, we introduce the generalized form of exponential multiplicative Zagreb indices for α ∈ R + \ { 1 }. Furthermore, to see the behaviour of generalized first and second exponential Zagreb indices for α ∈ R + \ { 1 } , we used a transformation method. In term of the two newly developed generalized exponential multiplicative Zagreb indices, we will investigate the extremal bicyclic, uni-cyclic and trees graphs. Four graph transformations are used and some bounds are presented in terms of generalized exponential multiplicative Zagreb indices. [ABSTRACT FROM AUTHOR]

Here, we deal with a model of multitype network with nonpreferential attachment growth. The connection between two nodes depends asymmetrically on their types, reflecting the implication of time order in temporal networks. Based upon graph limit theory, we analytically determined the limit of the network model characterized by a kernel, in the sense that the number of copies of any fixed subgraph converges when network size tends to infinity. The results are confirmed by extensive simulations. Our work thus provides a theoretical framework for quantitatively understanding grown temporal complex networks as a whole. [ABSTRACT FROM AUTHOR]

In this letter, cluster consensus of discrete-time linear multi-agent systems is investigated. A set of stochastic matrices P is said to be a cluster consensus set if the system achieves cluster consensus for any initial state and any sequence of matrices taken from P . By introducing a cluster ergodicity coefficient, we present an equivalence relation between a range of characterization of cluster consensus set under some mild conditions including the widely adopted inter-cluster common influence. We obtain a combinatorial necessary and sufficient condition for a compact set P to be a cluster consensus set. This combinatorial condition is an extension of the avoiding set condition for global consensus, and can be easily checked by an elementary routine. As a byproduct, our result unveils that the cluster-spanning tree condition is not only sufficient but necessary in some sense for cluster consensus problems. [ABSTRACT FROM AUTHOR]

How complex a network is crucially impacts its function and performance. In many modern applications, the networks involved have a growth property and sparse structures, which pose challenges to physicists and applied mathematicians. In this paper, we introduce the forest likelihood as a plausible measure to gauge how difficult it is to construct a forest in a non-preferential attachment way. Based on the notions of admittable labeling and path construction, we propose algorithms for computing the forest likelihood of a given forest. Concrete examples as well as the distributions of forest likelihoods for all forests with some fixed numbers of nodes are presented. Moreover, we illustrate the ideas on real-life networks, including a benzenoid tree, a mathematical family tree, and a peer-to-peer network. [ABSTRACT FROM AUTHOR]

A groupie in a graph is a vertex whose degree is not less than the average degree of its neighbors. Under some mild conditions, we show that the proportion of groupies is very close to 1/2 in multitype random graphs (such as stochastic block models), which include Erdős-Rényi random graphs, random bipartite, and multipartite graphs as special examples. Numerical examples are provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Stochastic consensus problems for linear time-invariant multi-agent systems over Markovian switching networks with time-varying delays and topology uncertainties are dealt with. By using the linear matrix inequality method and the stability theory of Markovian jump linear system, we show that consensus can be achieved for appropriate time delays and topology uncertainties which are not caused by the Markov process, provided the union of topologies associated with the positive recurrent states of the Markov process admits a spanning tree and the agent dynamics is stabilizable. Feasible linear matrix inequalities are established to determine the maximal allowable upper bound of time-varying delays. Numerical examples are given to show the feasibility of theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, group consensus problems in fixed directed networks of dynamic agents are investigated. Group consensus means that the agents in each group share a consistent value while there is no agreement between any two groups. Based on algebraic graph theory, sufficient conditions guaranteeing group consensus under the proposed control protocol in the presence of random noises and communication delays are derived. The analysis uses a stability result of Mao for stochastic differential delay equations, which ensures the consensus can be achieved almost surely and exponentially fast. Numerical examples are provided to demonstrate the availability of the obtained results as well as the effect of time delay/noise intensity. [ABSTRACT FROM PUBLISHER]

Let N be a prime near-ring. We show two main results on the commutativity of N: (1) If there exist k, l ∈ ℕ such that N admits a generalized derivation D satisfying either D([x,y])=xk [x,y]xl for all x, y ∈ N or D([x,y])=-xk [x,y]xl for all x, y ∈ N, then N is a commutative ring. (2) If there exist k, l ∈ ℕ such that N admits a generalized derivation D satisfying either D(x ◦ y)= xk (x ◦ y) xl for all x, y ∈ N or D(x ◦ y)= -xk (x ◦ y) xl for all x, y ∈ N, then N is a commutative ring. Moreover, some interesting relations between the prime graph and zero-divisor graph of N are studied. [ABSTRACT FROM AUTHOR]

We study a model for inhomogeneous long-range percolation on the hierarchical lattice Ω N of order N with an ultrametric d . Each vertex x ∈ Ω N is assigned a nonnegative weight W x , where ( W x ) x∈Ω N are i.i.d. random variables. Conditionally on the weights, and given two parameters α ≥ 0, β > 0, the edges are independent and the probability that there is an edge between two vertices x and y is of the form 1 - exp{-α W x W y /β d ( x, y ) }. Conditions on the weight distribution and the parameter β are formulated for the existence of a critical percolation value α c ∈ (0, ∞) such that the resulting graph contains an infinite component when α > α c and no infinite component when α < α c . Numerical simulations are also provided to validate the theoretical results. [ABSTRACT FROM AUTHOR]

• A growing simplicial network model is proposed to study higher-order interactions. • Degree and clique distributions are analytically derived. • Sombor index and its computation are compared with known bounds. • Power-law and small-word effect of the network are observed. • The scaling constant for Sombor index is determined for the model. Many dynamical effects in biology, social and technological complex systems have recently revealed their relevance to group interactions beyond traditional dyadic relationships between individual units. In this paper, we propose a growing simplicial network to model the higher-order interactions represented by clique structures. We analytically study the degree distribution and clique distribution of the network model. As an important degree-based topological index, Sombor index of the model has been derived in an iterative manner and an approximation method with closed expression is proposed. Moreover, we observe power-law and small-word effect for the simplicial networks and examine the effectiveness of the approximation method for Sombor index through computational experiments. We discover the scaling constant for Sombor index with the evolution of the network when the initial seed network is modeled as an Erdős-Rényi random graph. Our findings suggest the relevance and potential applicability of simplicial networks in modelling higher-order interactions in complex networked systems. [ABSTRACT FROM AUTHOR]

Large-scale time-evolving networks have been generated by many natural and technological applications, posing challenges for computation and modeling. Thus, it is of theoretical and practical significance to probe mathematical tools tailored for evolving networks. In this paper, on top of the dynamic Estrada index, we study the dynamic Laplacian Estrada index and the dynamic normalized Laplacian Estrada index of evolving graphs. Using linear algebra techniques, we established general upper and lower bounds for these graph-spectrum-based invariants through a couple of intuitive graph-theoretic measures, including the number of vertices or edges. Synthetic random evolving small-world networks are employed to show the relevance of the proposed dynamic Estrada indices. It is found that neither the static snapshot graphs nor the aggregated graph can approximate the evolving graph itself, indicating the fundamental difference between the static and dynamic Estrada indices. [ABSTRACT FROM AUTHOR]

Supposeis a simple graph andare the eigenvalues of its distance matrix. The distance Estrada indexofis defined as the sum of,. In this paper, we establish better lower and upper bounds tofor almost all graphs. [ABSTRACT FROM AUTHOR]

In this paper, we investigate average consensus problem in continuous-time multi-agent systems with uncertain topologies as well as multiple time-varying communication delays. The network of dynamic agents is directed and both fixed and switching topologies are considered. By using the linear matrix inequality method, we prove that all the nodes in the network can achieve average consensus asymptotically for appropriate delays and uncertainties if the network topology is weakly connected and balanced. Feasible linear matrix inequalities are established to determine the maximal allowable upper bounds of multiple delays. Numerical examples are presented to illustrate the availability of the theoretical results. [ABSTRACT FROM AUTHOR]

The average consensus in continuous-time multi-agent systems with dynamically changing topologies and multiple time-varying delays is studied in this paper. The network topology is captured by weighted digraphs which are weakly connected and balanced. Some feasible linear matrix inequalities are established to determine the allowable upper bounds of multiple delays that guarantee the average consensus of the system. Numerical examples are provided to show the usefulness of the theoretical results. [ABSTRACT FROM AUTHOR]

The act of accessing the exact location, or position, of a node in a network is known as the localization of a network. In this methodology, the precise location of each node within a network can be made in the terms of certain chosen nodes in a subset. This subset is known as the locating set and its minimum cardinality is called the locating number of a network. The generalized hexagonal cellular network is a novel structure for the planning and analysis of a network. In this work, we considered conducting the localization of a generalized hexagonal cellular network. Moreover, we determined and proved the exact locating number for this network. Furthermore, in this technique, each node of a generalized hexagonal cellular network can be accessed uniquely. Lastly, we also discussed the generalized version of the locating set and locating number. [ABSTRACT FROM AUTHOR]

Recent empirical and theoretical works on collective behaviors based on a topological interaction are beginning to offer some explanations as for the physical reasons behind the selection of a particular number of nearest neighbors locally affecting each individual's dynamics. Recently, flocking starlings have been shown to topologically interact with a very specific number of neighbors, between six to eight, while metric-free interactions were found to govern human crowd dynamics. Here, we use network- and graph-theoretic approaches combined with a dynamical model of locally interacting self-propelled particles to study how the consensus reaching process and its dynamics are influenced by the number k of topological neighbors. Specifically, we prove exactly that, in the absence of noise, consensus is always attained with a speed to consensus strictly increasing with k. The analysis of both speed and time to consensus reveals that, irrespective of the swarm size, a value of k ∼10 speeds up the rate of convergence to consensus to levels close to the one of the optimal all-to-all interaction signaling. Furthermore, this effect is found to be more pronounced in the presence of environmental noise. [ABSTRACT FROM AUTHOR]

We consider the random graph model G(w) for a given expected degree sequence w = (w1, w2, . . ., wn). Warmth, introduced by Brightwell and Winkler in the context of combinatorial statistical mechanics, is a graph parameter related to lower bounds of chromatic number. We present new upper and lower bounds on warmth of G(w). In particular, the minimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degree m = O(nα) with 0 < α < 1/2. [ABSTRACT FROM AUTHOR]

This paper studies the group consensus problem for generic linear multi-agent systems under directed information flow. External adapted inputs are introduced to realize the intra-group synchronization as well as the inter-group separation. Without imposing complicated algebraic criteria or restrictive graphic conditions on the interaction topology, we show that the group consensus can be achieved by designing appropriate gains given any magnitude of the coupling strengths among the agents. Numerical examples are presented to illustrate the availability of our results. [ABSTRACT FROM AUTHOR]

Resilience or recovery of multi-agent systems against attacks is believed to be one of the most important topics in control and management of complex systems. Previous studies have mainly focused on the random failure and intentional attack targeted on node degrees, whereas consensus recovery is much less investigated. In this paper, two kinds of targeted attacks are introduced and corresponding recovery strategies are proposed to address the consensus recovery of continuous-time nonlinear multi-agent systems under directed topologies against subsequent attacks. The cut-nodes and cut-edges are chosen as the targets of attack, and the multi-agent system becomes disconnected when they are deleted from the system. The neighbor nodes of the cut-node and end nodes of the cut-edge are used to recover the connectivity of the network. It is shown that our proposed strategies can significantly improve the convergence speed of reaching consensus of the resulting multi-agent systems. A cost function is also proposed to compare different recovery strategies. [ABSTRACT FROM AUTHOR]

Abstract: Understanding how interacting subsystems of an overall system lead to cluster/group consensus is a key issue in the investigation of multi-agent systems. In this Letter, we study the group consensus problem of discrete-time multi-agent systems with external stochastic inputs. Based on ergodicity theory and matrix analysis, group consensus criteria are obtained for multi-agent systems with switching topologies. Some numerical examples are provided to illustrate the effectiveness and feasibility of the theoretical results. [Copyright &y& Elsevier]

In this paper, we consider a long-range percolation model on the hierarchical group ΩNTwo nodes in ΩNseparated by distancekbecome connected with probability min {αβ−k, 1}, whereα≥ 0 andβ> 0. The percolation functionθ(α,β) is defined as the probability of having a infinite component contain the origin 0 ∊ ΩN. We show thatθ(α,β) is continuous with respect to bothαandβ. [ABSTRACT FROM PUBLISHER]

The geometry of complex networks has a close relationship with their structure and function. In this article, we investigate Gromov-hyperbolicity of inhomogeneous random networks modeled by the Chung-Lu modelG(w). When the maximum expected degreewmaxand minimum expected degreewminsatisfywmax ≤ 21/3wmin, we prove that for any positive δ,G(w) has a positive probability of containing δ-fat triangles asn → ∞. Our numerical simulations illustrate this non-hyperbolicity ofG(w) for power law degree distributions among others. [ABSTRACT FROM AUTHOR]

Let $G$ be a general weighted graph (with possible self-loops) on $n$ vertices and $\lambda _1,\lambda _2,\ldots ,\lambda _n$ be its eigenvalues. The Estrada index of $G$ is a graph invariant defined as $EE=\sum _{i=1}^ne^{\lambda _i}$. We present a generic expression for $EE$ based on weights of short closed walks in $G$. We establish lower and upper bounds for $EE$in terms of low-order spectral moments involving the weights of closed walks. A concrete example of calculation is provided. [ABSTRACT FROM AUTHOR]

During an epidemic outbreak in a human population, susceptibility to infection can be reduced by raising awareness of the disease. In this paper, we investigate the effects of three forms of awareness (i.e., contact, local, and global) on the spread of a disease in a random network. Connectivity-correlated transmission rates are assumed. By using the mean-field theory and numerical simulation, we show that both local and contact awareness can raise the epidemic thresholds while the global awareness cannot, which mirrors the recent results of Wu et al. The obtained results point out that individual behaviors in the presence of an infectious disease has a great influence on the epidemic dynamics. Our method enriches mean-field analysis in epidemic models. [ABSTRACT FROM AUTHOR]

Our main interest in this paper is the large dicliques in a directed inhomogeneous random graph modelG(n,α, Φ) onnvertices, which has power-law out/in-degree distributions with scaling exponent α>0 and community structures involved in the homophyly matrix Φ. We show that there is a major difference in the size of the largest diclique ωd(G(n,α, Φ)) between the case α<2 and α>2 with an intermediate result for α=2. In addition, we show that a simple algorithm with high probability finds a large diclique of size ωd(G(n,α, Φ)) in a polynomial time. Our simulation results reveal that the connections between different subcommunities are essential for the formation of large clusters in the networks. [ABSTRACT FROM PUBLISHER]

Abstract: Analytical description of disease propagation on random networks has become one of the most productive fields in recent years, yet more complex contact patterns and dynamics have been resorted to numerical study. In this paper, an epidemic model is defined in which each individual, once infected, has chances to recover from infection at certain rate. The chance is represented by a parameter . This model can be viewed as an interpolation between classical SI model () and SIR model (). We develop a low-dimensional system of non-linear ordinary differential equations to model the mixed susceptible-infected (-recovered) SI (R) epidemics on random network with general degree distributions. Both the final size of epidemics and the time-dependent behaviors are derived within this simple framework. In addition, we present the exact transmissibility and the epidemic threshold for this model. [Copyright &y& Elsevier]